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ABSTRACT

THE MATHEMATICAL BEHAVIORS DERIVABLE FROM THE
PROGRAM OF UNIFIED SCIENCE AND MATHEMATICS
FOR ELEMENTARY SCHOOLS

By
Sompop Krairojananan

This Case-Study is a comprehensive account of inten-
sive continual observation.of four 'USHES' classes at
Lansing, Michigan, during the nine weeks from late September
to early December, 1972. USMES (Unified Science and
Mathematics for Elementary Schools) is an activity-oriented
and integrated course in science, mathematics and social
science. Many units, designed to promote the problem-
solving skills of students in their attempt to answer some
major 'challenges', have been developed, and four such units
(Soft Drink Design; Dice Design; Designing for Human Pro-
portion; and Burglar Alarm Design) were selected for
observation in.this Case-Study with emphasis on.mathe-
matical behaviors which arose naturally from these courses
of activities. The mathematical topics observed were then
categorized into seven areas: Arithmetic, Algebra, Graph
and Tabulation, Geometry, Application.and Practical Mathe-
matics, Statistics or Physics, and Foundation of Mathematics.
The mathematical behaviors arising from these four USMES

:8 In
seven a:
units 52

Six
to a lo:
arcse we
data. 1
niques a
clildre:
scientif
was that
results
by recon

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oriented
could be
rather t:

Despite ‘

Sompop Krairoj ananan

units were found to be distributed fairly evenly in all
seven areas, and thus it could be deduced that these USMES
units gave rise to a well-balanced mathematics program.

Since each unit aimed at providing a partial solution
to a long-range challenge, the mathematical problems that
arose were thus highly relevant and did not contain artificial
data. These problems were systematically tackled by tech-
niques as near to the 'scientific method' as possible.
Children were observed to have learned Just as much about
scientific process as mathematics. One outstanding feature
was that students had to verify the correctness of their
results by checking with the practical outcomes, and 923
by recourse to teachers.

Uniformity was not to be expected in an activity-
oriented almost-realistic program like USMES. In fact, it
could be observed that diversity of students' achievements,
rather than uniformity, was encouraged in these classes.
Despite divergence in.activities, all students in.these
four unitseventually learned at least the following
mathematical topics, which the l963-Cambridge Conference
had termed ”the bed-rock foundation of elementary school
mathematics," namely, counting and fractions, m
and invariant properties of geometric figgaes, real
experience in collecti_n_g 9.21:3, 222 9; Eaphs and other
visual displays of data, and the vocabulary of elementarz

logic.

THE MATHEMATI CAL BEHAVIORS DERIVABLE FROM
THE PROGRAM OF UNIFIED SCIENCE AND
MATHEMATICS FOR ELEMENTARY SCHOOLS

By

Sompop KrairoJananan

ATHESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

College of Education

1973

__ 3'
6? ACKNOWLEDGEMENTS

It is a pleasure to express my heartfelt thanks to
Professor William M. Fitzgerald, the thesis director as well
as academic adviser, for giving me such an excellent guid-
ance in every respect from the beginning to the end of my
graduate study at Michigan State University. I also
appreciate the invaluable advice given by the members of
the Ph.D. Committee at various critical stages of my
graduate program. Thanks are also due to Professor Earle
Lemon, Department of Theoretical Physics, Massachusetts
Institute of Technology, for giving me permission to do a
Case-Study on USMES, and to the Faculty of Uexford and
Pleasant View Schools in Lansing, Michigan, for giving me
such a friendly welcomewhile I observed the USMES classes
there. Mrs. Cosette Rodefeld, who typed this thesis, also
deserves a word of thanks for her cheerfulness and efficiency
throughout this arduous work.

Finally, I would like to thank the Mathematics Depart-
ment, Michigan State University, for giving me partial
financial support, and the Thai Government for giving me
the 3-year study leave.

ii

”
uiapt

TABLE OF CONTENTS

Chapter
1 mTRODUCTIONOOOOOOOO...OOOOOOOOOOOOOOOO0.0...

What is USES?O0.0.000000000000000000000000
Statement of the Research Problem..........
Justification of the Method Used...........
Limitation Of the StudyOOOOOOOOOOOOOOOOOOOO

2 mm OF WTED LITMTUREOOOOOOCOCOOOOOOOO

The literature related to activity-
oriented curricula.........................
The literature on the integration of
mathematics with science, social science
and Other cou89800000OOOOOOOOOOOOOOOOOOOO.
The literature on the philosophical

and psychological aspects of an activity-
oriented and integrated curriculum.........
The literature on the Case-Study type

of research in mathematics education.......

3 CATEGORIZING THE MATHEMATICAL BEHAVIORS
DERIVABLE FROM FOUR 'USMES' UNITS............
4 DETAILS OF THE FOUR CASE-STUDIES.............

The .SOft Drink D6819. unitOOOOOOOOOOOOOO.
The 'Dice D3819. unit.....................
The 'Designing for Human.Proportion' Unit..
The 'Burglar Alarm Design' Unit............

5 DISCUSSIm AND CONCLUSIW...OOOOOOOOOOOOOOOOO
BIBLIOGRAPHYOOOOOO00.000.000.000...0.0.0.0...

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CHAPTER 1

INTRODUCTION

This research is a case-study type of work which
involves intensive continual observations of four USMES
classes in Lansing, Michigan, from late September to early
December 1972. The variable singled out for this observation
was the mathematical behavior, pure or applied, exhibited by
students while participating in USMES activities.

What is USMES?

USMES (Unified Science and Mathematics for Elementary
Schools) is a working project of Education.Development
Center, funded by the National Science Foundation, to
develop interdisciplinary units focusing on problems whose
partial solution implies some realistically useful achieve-
ment and a substantial amount of learning in science and
mathematics. The USMES project grew out of the 1967
Cambridge Conference on the Correlation of Science and
Mathematics in the Schools. This Conference Report1 stated
the primary aim of drawing maximum.benefits in scientific
education from an integrated program in.elementary schools.

 

1Cambridge Conference, "Goals for the Correlation of
Elementary Science and Mathematics," Boston: Houghton
Mifflin, 1968.
l

2

In the implementing phase of the project, this integrated
program makes use of both activity learning and expository
instructions (frequently in the form of ”How to" cards).
Its underlying philosophical principle has been 'macro-
openness and micro-structuring.‘ USMES has selected special
types of practical or 'real' problems which meet the
following criteria: attainable results within the time
and resources available (in elementary schools), $2222?
disciplinary nature with emphasis on problem-solving, and
definite academic achievement in elementary science and
mathematics appropriate to the age level.

The USMES challenges are hypothesized to have four
important advantages as follows:

(a) Motivation, based on appeal and direct application
to the students' own environment,

(b) Accuracy of learning, to be checked against the
practical outcomes,

(c) Opportunities for problem defining,

(d) Experience in total problem-solving, in contrast
to isolated problemssolving in.most programs. (Total
problempsolving resembles the working of a real scientist.
The students will be actively involved in observation,
quantification, formulation, trial of successive models,
and, above all, critical thinking.)

Statement of the Research Problem:

A case-study is to be undertaken to ascertain the
mathematical'behaviors, or potential mathematical topics,

derivable from.the USMES Program.

 

 

'4
5‘.
m

"h.”
I

.3.
observat.
other me‘
two reas.
Hill per:
CCnSider;
mush t2
Design U1
(and hem
31” Desi
dice‘roll
PrObabilj
into desj
for an 8]
graphic E

bone-st m

3

Justification of the Method Used:

Why should the case-study method involving intensive
observation of a few units have the preference over the
other methods involving a larger sample size? There are
two reasons. First, the nature of a program like USMES
will permit a high degree of flexibility. There will be
considerable divergence between any two USMES units even
though they may bear the same title. For example, one Dice
Design Unit may concentrate on the construction of polyhedra
(and hence learn a great deal of Geometry), while another
Dice Design Unit may collect data on coin-tossing and
dice-rolling (and hence learn many topics in Elementary
Probability). The 'Human Proportion' Unit may branch out
into designing garments, or designing furniture and fixtures
for an elementary school, or’measuring a human body to study
graphic art, or even going to the extent of studying the
bone-structure, muscles, and other anatomical topics. Secondly,

J. Piaget,1 K. Lovell,2 z. P. Dienes,3 the Soviet Academy

 

lSee e.g., Piaget, Jean, "The Child's Conception of
Number,” London: Routledge and Paul, 1952.

2Lovell, Kenneth, "The Growth of Basic Mathematical and
Scientific Concepts in Children," London: London.University
Press, 1961.

3Dienes, Zoltan 2., "An Experimental Study of Mathematics
Learning," London: Hutchinson, 1963.

Ti

 

of Pedago!
have sire.-
about mat]

intensive

’4
H-
A

I

”4 conch
Of all US)
3 Farms,
can b0 dez

like that
°f teachin

”Gaent .

a
of Pedagogical Science and other clinical psychologists,
have already demonstrated the immense gain in the knowledge
about mathematics learning when case-studies based on

1

intensive continuous observations were carried out.

Limitation of the Study:

This study is not intended for generalizing the results
and conclusions about the observed populations to those
of all USMES populations. Rather, it is meant to provide
a paradigm case showing what amount of. mathematical learning
can be derived from part of the USMES program.

In a way, the content of this Case-Study is very much
like that of the teachers ' Handbooksa im the new approach
of teaching mathematics in British primary schobls at

present .

 

J'Kilpatriek, Jeremy and Wirsup, Izaak (ed.), "Soviet
Studies in the Psychology of Learning and Teaching Mathe-
matics,“, Chicago: The University Press, 1969.

2Schools Council, "Mathematics in PrimarySchools,”
Curriculum Bulletin No. 1, London: E. M. Stationery Office,

1965 .

 

 

 

1.
curricula

The
Pestalozz
use of re
lid-met
Stressed
&(“Tested
in class,
natural.
'Deflocrac

(1925), ..

6

1. The literature related to activity-oriented
curricula:

The idea of an activity-oriented curriculum is not new:
ZE’estalozzi1 and Herbert2 emphasized real experience and the
use of real objects for their pupils' learning ever since
mid-nineteenth century. Also in the last century, Froebel3
stressed the value of play in learning. Later, Montessori4
advocated the idea of allowing children freedom of movements
in class, and other reasonable liberties to promote 'more
natural' learning. Finally, with John Dewey's publications:5
"Democracy and Education" (1916), "hperience and Nature”
(1925), “Experience and Education" (1938), the child-
centered activity-oriented classroom did begin to take
shape and lingered on even after the zenith of the 'progres-
sive education' movement . More recently, the Plowden

Report6

recommended a completely Open classroom in British

 

1'Eeafford, Michael, ”Pestalozzi, His Thought and Its
Relevance Today, " London: Methuen, 1967.

2Dunkel, Harold B., "Herbert and Education," New York:
Random House , 1969.

aPriestman, Barbara, "Froebel Education Today," London
University Press, 1960.

. ~ 4Montessori, Maria, "The Montessori Method,” translated
by Anne E. George, New York: Schoecher Books, Inc., 1964.

5Dewe , John, "Democracy and Education”; ”kperience
and Nature ; 'kperience and Education," New York: Macmillan,
1916, 1925 and 1958 respectively.

6Central Advisory Council for Education (hgland) ,
”Children and Their Primary Schools," London: E. M.
Stationery Office, 1967.

Ti

 

prism so]
oriented c1
Perry's ad:
Ldvanoenem
Hooro'a pr:
Society in
mthenatica
quotations

ideas here:

I‘ 8T8
tione

and pa.
“V0 8
IGDCB 1
in gee:
calla ;
labora‘
t0 dev«
Spirit
88 well
Of acic

the above q.
or at least

The act
dur1118 the 2
bum, Bob
bend or the

Show. 81v.

0k Place 11

7
primary schools. For activity-oriented or experience-
oriented curricula in mathematics, it began with John

1 to the British Association For the

Perry's address
Advancement of Science in 1901, followed closely by E. H.
Moore's presidential address"2 to the American Mathematical
Society in 1902. Both men strongly urged reforms in the
mathematics curriculum and mathematics teaching. A few
quotations from Moore's speech might help to sum up the
ideas here:
"A grade teacher must make wiser use of the founda-
tions furnished by the kindergarten. The drawing
and paper folding must lead on directly to...intui-
tive geometry, the construction of models, the ele-
ments of mechanical drawing, (and) simple exercises
in geometrical reasoning....‘1‘his program of reforms
calls for the development of a thorough-going
laboratory system of instruction in mathematics...
to develop on the part of every student the true
spirit of research, and an appreciation practical
as well as theoretic, of the fundamental methods
of science.“
The above quotation is most relevant to the proper conduct
of at least two USMES classes observed in this Case-Study.
The activity curricula seemed to be quite popular
during the 20's and the 30's, maybe due to Dewey's influence.
Franklin Bobbitt (1934. 1936) wrote two articles, ”The
Trend of the Activity Curriculum,“ and ”What is the Activity
School?" giving indication of the kind of learning that

took place in some elementary schools. G. W. Diemer (1925)

 

lBidwell, James K. and Clason, Robert G. (ed.),
"Readings in the History of Mathematics Education,"
Washington: E.C.T.M., 1970, pp. 221-245.

ZIbid. , pp. 246-255.

T

earlie
School
to act:

study <

 

school:
common
schools
that oh
reading
Practic
his Stu
in the ;

'S<

I‘m

me, 31
Class 1:
'urban 1
”flea
3,, cm

8

earlier described a half-way progressive school--the Platoon
School, whose schedule in the afternoon was devoted entirely
to activities. C. M. Reinoehl (1934) did a comparative
study on the relative efficiency of platoon and non-platoon
schools with regard to pupils' progress and achievement in
common school subjects, and his finding was that platoon
schools, on the average, did somewhat better. The concept
that children could learn 'homeroom' subjects (arithmetic,
reading, spelling, etc.) from activities was put into
practice even earlier. R. Beatley (1921) reported that
his students at Horace Mann School for Boys learned 'math'
in the following manner:

”Some students trail a tape measure around the

room,...,squint along a rod out of the window,
...,learn to use a pantograph,...or a slide rule,

Alice Stewart (1939) of Lincoln School described how her
class learned from a 'community lab' and afterwards an
'urban lab' by actually living in those settings over a
period-ma variation from the usual 'camping' activities.
Her children's logs revealed a great deal of mathematical
behaviors learned from such experiences. Frank Freeman
(1935) summed up the advantages of an activity curriculum
as follows:

"Activities furnish the acquaintance with the world

of things and persons, which is the necessary basis

for ideas and thinking; the traditional school has

done violence to the children's nature by attempting

to curb their natural disposition to be active."

The 'activity' trend was not confined to pupils alone;

good teachers, too, initiated and conducted 'activity-

T

 

 

 

omientc
teachiz
grader:
she rec
Out of
Period,
6,051 a
0.1%) a
feels 1
train:
or lath

'1“:
that Q
5.83qu
develol
“My-1
to the
Hons].

Gene 0]

(repla‘

9
oriented' experiments which would finally result in better

teaching. Claire Zyre (1927) experimented with 31 third-
graders by giving them free 'conversation periods, ' and
she recorded the subjects of the children' s conversation.
Out of the 47,575 items of conversation recorded over a
period, 12,983 were about play at home; 7,0#8 about animals;
6,051 about auto-trips and picnics and only 49 (i.e.
0.1%) about arithmetic. The writer of this Case-Study
feels that, if this experiment were replicated today in a
traditional school, the percentage pertaining to arithmetic
or mathematics might not have altered very much.

The Activity Movement then captured so much attention
that the Uational Society for the Study of Education (1934)
issued a yearbook devoted entirely to the historical
development, pros and cons of the activity curriculum.
Forty-two leading personalities in education contributed
to the writing of this yearbook. They defined the func-
tions1 of Activity in the Learning Process as encompassing
some or all of the following: to be the basis of learning
(replacing the former basis of passive reception); to be a
natural mode of living (not Just studying); to explore,
experiment and interact with the environment , to develop
body, mind and spirit, and, as a result, to make school
life a happier phase of child life; to promote further
growth and critical thinking.

 

1’1!..ES.S.E., "The Activity Hovement," 33rd Yearbook,
Pm II, ppe “.500

r

 

is ex;
«valuation
wide questi
personnel,
palatal we:
considerati
“3 Sained,
whats (:
determine t
t1111183. how
independenc
habits, m

he 'a
had inPlica
We! (192
'Dart of B
Widermng :
“In“ to 1
Douay, the
“quiz." 3h]

10

As expected, there was heated arguement over the
evaluation in an activity program. By means of a nation-
wide questionnaire sent to a cross-section of school
personnel, the yearbook committee found that the following
points1 were dominant in the responses: "The important
consideration is not how much factual information the child
has gained, but how able he is to use it in new situations.
...Tests (for the Activity programs) should be designed to
determine how learning takes place, whether a child can $3
things, how much progress he has made in the growth of
independence, purposefulness, tolerant understanding, social
habits, and social attitudes in general."

The 'activity' concept of teaching children has always
had implication for the teaching of mathematics. U. 1'.
Downey (1928) published an article: "New Mathematics as
a part of New Education" in the Mathematics Teacher. The
underlying principle of his l928-essey was strikingly
similar to the l970-guideline of USERS. According to
Downey, the process of education through self-activity
requires three conditions:

(a) The pupils should be given opportunity to be
problem-finders as well as problem-solvers,

(b) Whatever activity is undertaken: academic studies
or mechanical, practical, and fine arts, or athletics, the
pupils should feel that it is worth mastering,

(c) Before considering any problem as completed, the
pupils should feel sure, through checks and other means

11bide’ Pp. 153‘1560

flux his w,
success' 1:.
undertakin

H. Ru
revising n
Principles
enough. 5.

'Pire‘

Pupil:
they 1

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eaterj
not ':
Charla
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between COu
“W! and .
commas or
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11

that his work is correct. Then they would reach 'assured
success' which would bring encouragement for the next

undertaking.

H. Rugg (1924) who sat in the National Committee for
revising mathematics curriculum suggested three guiding
principles for the future, most of which sounded progressive
enough. He wrote:

”First what are the abilities and interests of the

pupils‘} Secondly, what mathematics materials do

they need, both as growing youths in schools and as

prospective citizens in a highly industrialised

society? Thirdly, in what grades should these
materials be utilised?" (notice the word 'utilised',
not 'verbally‘t'i'fi'gfi't").

Charles B. Judd (1925). who was a prominent educational
psychologist at the time, designed a laboratory study
investigating the early stages of M consciousness.

By means of elaborate apparatuses, he noted the difference
between counting in the abstract and counting obJ ects,
sounds and flashes of light. He concluded that concrete
counting of objects, sounds, light-flashes, etc. aroused
number consciousness more rapidly. Judd (1925) also
advocated more emphasis on the informational aspect of
arithmetic as opposed to the cut-and—dry computational
aspect. In fact, the mathematical behaviors observed in
the Soft Drink Design Unit (See Chapter 4) utilised more
informational aspect than computational aspect of cardinal
and ordinal numbers, as well as fractions.

Although the 29th Yearbook of the National Society for
the Study of Education (1930) came out at a time when

progressive education was popular, the Yearbook Committee

_T7

(i.e. the

would not .
degenerate
thereby ne.‘
child's cu:

 

become fun:
”Sued the
merry, but
398131:» the
th‘ IGarboo
gradual aov
“may.

~The j
less

“Lug'

 

l2

(i.e. the Committee on Arithmetic) made it plain that they
would not allow children's interest and 'felt needs' to
degenerate into whims (often requiring less effort) and
thereby neglect all the other dynamics of learning. A
child's curriculum had to include materials which would
become fundamentals for competent and useful adult living,
argued the Committee. "Let the child eat, drink and be
merry, but (forget not) that tomorrow he will be an adult.
Despite these harsh warnings, the curriculum proposed in

.1

the Yearbook was relatively liberal , even suggesting the

gradual movement from 'mental discipline" to 'activity

learning'. Examples of the suggestions2 were:

“The place of object-teaching: more concrete work,
less abstract drill;

-—Larger use of the pupils ' environment and increased
provision for self—activity: counting toys, marbles,
objects in a picture, candles for birthday cakes,
scores in games, etc.;

«Motivating abstract calculations through games and
other devices.

Some people questioned the actual mathematical learning
obtained from an activity curriculum. Many research studies
have been, and are being carried out to measure the mathe-
matical outcomes (if any) of activity-oriented programs.
These included studies done by MacLatchy (1930), Woody
(1931), Riser and Harap (1932), Thiele (1938), Tompkins and

J'13.S.S.l?.., "Report of the Society's Committee on
Arithmetic,” 29th Yearbook, Bloomington: Public School
Pub. Co., 1930, pp. 4-5.

21bid.,.pp. 85, 89, 9s.

 

Stokes (Q
Host mat}
activity-
evaluatix
findings
were car:
PaoLatclu
Children
Possessed

W

W

13
Stokes (1940), Lowry and Inez.(l944) and Villiams (1949).
Most mathematics and science programs in the 50's and 60's,
activity-oriented or otherwise, included research studies
evaluating their own outcomes. It is relevant to note the
findings of the first two studies cited above, since they
were carried out during the 'progressive' era. Both
MacLatchy and woody investigated the number ideas of young
children and the types of arithmetical information they
possessed. Both studies concluded that zo_u_r_1_g children
acguired incidentally through their informal activities

outside schools a much larger arithmetical backggound than

has been believed possible.
During the Second world Var, the traditional method

of instruction seemed to take priority over activity classes.
But many teachers, for example Catherine Stern,1 still
conducted their mathematics classes in a healthy manner by
allowing the students to manipulate concrete materials
before verbalising abstract rules. In a way, Stern's
book was the spearhead of another new wave of activity-
oriented mathematics instruction to appear later on. Also
in the period the National Society for the Study of Educa-
tion (1951) issued another yearbook on the Teaching of
Arithmetic. This yearbook still reflected the strong
position of the 'manipulative instructional aids' and the
'laboratory method' , and this time it also discussed the

¥

1Stern, Catherine, ”Children Discover Arithmetic,”
New York: Harper, 1949.

 

 

'euiio-vi
and sore j
and aanip
of 98 8111
esnipulat
nterials
greet edu:
Isterials
0f encoum
Leb', The
‘130 made
lateriala

(a) 1

tion a 1
deYiéea fi

l4

'audio-visual instructional aids' which was becoming more
and more popular. It contained a whole section on visual
and manipulative materials with an almost exhausting list
of 98 suitable devices.1 It further recommended that
manipulative objects and certain types of pictorial
materials should be made _i_n_ _t_l_1_e_ classroom, since there was
great educational value in having the pupils make these
materials. This was, in fact, parallel to USMES's principle
of encouraging students to make things in their 'Design
Lab' . The National Council of Teachers of Mathematics
also made available such information on manipulative
materials to teachers in two of their yearbooks:

(a) the 16th Yearbook, “Arithmetic in General Educa-
tion,“ 1941 (See the Section on 'Utilising supplementary
devices and materials),

(b) the 18th Yearbook, "Multisensory Aids in the
Teaching of Mathematics," 1945.

Needless to mention that in this period many articles could
still be found in professional Journals describing various
'activity methods' of teaching mathematics. Typical among
these were articles by Steiss and Baxter (1943), F. M. Burns
(1944), R. Morris (1947) and M. B. Vilson (1955).

To learn mathematics through activities and concrete
manipulations received another impetus in the late 50' s
and early 60's, as a result of the 'New Math Era' , in which
the mathematics content in the curriculum was drastically

1s.s.s.s., "The Teaching of Arithmetic,” 50th Yearbook,
1951, pp. 172-185.

 

     
  
  
   
  
  
   
  

revised. U
of the 30's
activity pr
often with

tive device
Cnisensire

the geo-bo
blocks. 11
to describ.
in Senor-u,
(1962) on c

15
revised. Unlike the learning-through-activity programs
of the 30's which was very flexible, the nthematics-through-
activity programs of the 60's were more specific, quite
often with stated objectives and corresponding manipula-
tive devices for sale. The important ones were: the
Cuisenaire rods, the Madison Project's ”shoe boxes”,
the gee-board, and Z. P. Dienes' multibase arithmetical
blocks. Numerous articles appeared in professional journals
to describe these devices and the 'Mathematics Laboratory'
in general. Typical were the articles by Clara Davidson
(1962) on Cuisenaire Rods, Z. P. Dienes (1962), R. B.
Davis (1960), U. Liedtke (1969), Patricia Davidson and A. 1!.
Fair (1970). ,

Although these activities related to Mathematics
Laboratory are mostly short-ranged, their usefulness in
promoting the students' understanding of mathematics has
been demonstrated beyond any doubt. T. E. Kieren (1969)
reviewed all research works in this connection for the
period 1964-1968 and reported that most of the findings
were in favor of the concrete manipulation or activity
type of learning. .

Mam recent comparative studies were carried out to
evaluate the mathematical learning obtained from abstract/
verbal vs. concrete/games classes. Several games or activi-
ties described were similar to those observed in USMES
classes. The control group in all cases were the abstract/
verbal type. r. L. Coltharp (1969) and 1). Ross (1970)

l6

dealt with the teaching of integers to 6th graders. The
'concrete counting' of their experimental groups was
similar to that observed in the Dice Design Unit. B. M.
Bisio (1971) did a comparative study on fraction at
Berkley, California, and he concluded that even passive
use of manipulative materials was better than no use at
all. This conclusion then supported the USMES position,
since the knowledge on fraction which the children learned
from the Soft Drink Design Unit, Human Proportion Unit,
etc. was all derived from concrete situation. J. J. Bowen
(1970) used games as instructionalmedia (to teach logic),
just like the games involving logical circuits, which some
children played in the Burglar Alarm Design Unit. U. P.
Palow (1970) researched on the ability of children (Grades
3-12) to visualise perspectives of solid figures, and his
finding that such visualisation was impossible for children
under twelve should have direct bearing on the placement
of that USMES unit which involves polyhedra construction.
Several USMES units observed (in Lansing) could provide
excellent opportunity for introducing elementary topological
ideas to children, and E. T. Esty's study (1970), carried
out at Harvard, could throw some light on the subject.
He carefully distinguished between the children's recogni-
tion of 'topologically drawn' and 'geometrically drawn'
figures.

The review of literature on learning mathematics via
activity classes would not be complete if the present

f

   

revolution

 

1?
revolution in English primary schools is not mentioned.
The 'revolution' started slowly and quietly during the
60's when Piaget's research became more well-known, and
the Schools Councill published a paradigm account of organic
growth of mathematical ideas in children, once the environ-
ment became apt. Many teachers took initiative on their
own, and they were officially supported by the Plowden
Report, and practically by the publication of the luffield
Mathematics Project and the Association of Teachers of
Mathematics.2 Instead of fully relying on textbooks,
chalk and talk like in the past, the British teachers are
now using the following guiding principles} for their new
approach:

(1) Children learn mathematical concepts more slowly
than adults realised. They learn by their own activities.

(2) Children pass through certain stages depending
on their chronological and mental ages and their experience.

(3) Their learning can be accelerated by providing
suitable experiences and suitable language development
simultaneously. ‘

(4) Practice is necessary to fix a concept once it
has been understood. Practice should follow, not precede,
discovery.

 

J‘Schools Council, "Mathematics in Primary Schools,"
Curriculum Bulletin No. 1, London: E. M. Stationery
Office, 1965.

2A. T. M., ”Notes on Mathematics in Primary Schools,”
London: Cambridge University Press, 1969.

3Quoted from the Plowden Report, p. 237.

 

Joseph
schools as

'Mathe
school
formal
introd
(connu
schedu
contin
(paral
enviro
and to
activi
Skills

In the
1967: has p'
0! books:

Shape and s.

“Wu. P.
Imdes ' to

textbook; ii
ho. t0 leap,

Edith 1
Shaw of thj

Of DOOkg in
“utto do

 

 

18

1 who visited British primary

Joseph Featherstone,
schools as an impartial non-educationist observer, noted:
"Mathematics is now an important catalyst for
schools making a transition from formal to less
formal teaching....Teachers provide experience and
introduce children to various means of expression
(communication). This takes time and a flexible
schedule, so that children can see a piece of
continuous work through to their satisfaction
(parallel to USMES strategy). It implies an
environment stocked with materials to see, listen
and touch; and it suggests the necessity for varied
activities to suit different tastes, talents and
mllaeeee"
In the U.I., The Ruffield Mathematics Project, since
1967. has published (via John Uiley, New York) a series
of books: I Do and I Understand, Computation and Structure,
Shape and Size, Graphs Leading to Algebra, Environmental
Geometry, Probability and Statistics,..'., to act as
”guides” to the teachers and pupils alike. These are not
textbooks in the traditional sense: the stress is on
how to learn (organically), not on what to teach.2
Edith Biggs, who has been in the forefront since the
start of this 'revolution' , has also published a umber
of books in this direction. Rather than telling the teachers
what to do,‘ she usually fills her book-3 with children's
actual reports, drawn graphs, number patterns discovered
by children, etc.

IPeatherstone, Joseph, “Informal Schools in Britain
Today-«An Introduction,“ MewYork: Citation Press, 1971, pp.

29-300 7
2Quoted from ”I Do and I Understand,“ p. (i).

38“, (i) Biggs, Edith, "Mathematics for Younger/Older
Children,“ 2 volumes), New York: Citation Press, 1971;
(ii) Biggs, E. and McLean, J. R., "Freedom to Learn,“
Reading, Mass.: Addison-Vesley, 1969.

 

{Y

 

 

Teachers
priate d

'In
It
wid
are
of
end
of
mus
stu
Cur
On

19

2. The literature on the Integgation of Mathematics
with Science, Social Science and Other Courses:

Uilliam F. Russell, who was the Dean of Columbia's
Teachers College during the 30's, produced the most appro-
priate definition of "integration" as follows:1

"Integration is not just another educational slogan.
It constitutes a direct answer to the profound and
wide-spread disintegrations that now exist in all
areas of human experience. It is the integration
of experiences that educators should aim at. The
end to be sought is a unified and related pattern
of experience in each child. All subject-matter
must become integral and truly functional in the
student's growth. There must be an integrating
curriculum: the study of all aspects of culture--
on all its fronts, and in all its interrelationships."

Integration of mathematics with other subjects did
take place extensively during this 'progressive' era.
In order to convey the true spirit of teaching according
to an integrating curriculum in that period, details of a
few paradigm cases will be presented in this review
instead of an exhaustive summarised list of articles.
Arretta L. watts (1928) reported the following lively
scene she observed in several arithmetic classes:
"Arithmetic today is an extremely dynamic and
flexible affair. Students do their calculations
on automobile and airplane distances, baseball
records, weather reports, railroad timetables,
averages, and various scientific and technical
reports, all being worked out in decimals. They
also work out a formula for getting around the

city, figuring out a saving account, the number of
calories afforded in a plate of corned beef, or

 

lQuoted from Russell, William 3. , "Education and
Divergent Philosophies,” Teachers College Record, Vol. 39,
December 1937, pp. 186-188.

20

making a graph showing the increase during the last

five years of summer tourists to Europe. By motion

pictures, the students learn that the movement of a

baseball pitcher in pitching his ball included more

than a hundred distinct motions. In the same w”,

the modern psychologist has shown us (teachers) how

to chart the mental motions the mind goes through

in each of the processes of arithmetic. It is not

the aim in modern mathematics to stump the youth

with some unworkable or trick problems, but to give

him the satisfaction that he can do things success-

fully and the message that mathematics is useful.“

Dwight S. Davis (1923) called for 'live problems' in
teaching mathematics. In a long article, he suggested
problems arising from new real-life sources: agriculture,
mechanical arts, domestic science, banking, athletics, radio
and electricity, local bus routes, building and construction,
and the post office. For his own 'Algebra via Agriculture'
class (which included field work), he listed the following
mathematical behaviors observed: formulae in area; distri-
bution over areas; workable formulae for sprays, feeds and
fertilizers, statistical graphs on yearly production,
profits, and weather; the laws of levers; fhrmulae in
Steam and Gasoline Engine”: regular linear equations drawn
from catalogues; percentage; friction; machine efficiency.

R. E. Slaught (1928) compared mathematics with sun-
shine in the sense that, without either, civilisation would
gradually come to a halt. Students could learn a great
deal of science and social studies, as well as mathematics,
if they set out, as a team, to investigate the truth of
the statement "Mathematics underlies present-day civilisa-
tion in much the same far-reaching manner as sunshine

underlies all forms of life."

 

 

    

0! 'period‘
to time he

21

Alfred North Whitehead, in two different books pub-
lished during this period, also hinted at teaching inter-
disciplinary courses from such themes as "rhythm” and "order
of nature." To learn the abstract (mathematical) concept
of 'periodicity', he advised students to examine 'rhythms':
to time heart-beats, breathing, wave-phenomena, music ,
etc. He concluded:

”Apart from recurrence, knowledge would be impossible;
for nothing could be referred to our past experience.

Also, apart from some regglaritz of recurrence,
measurement would be impose e. Tn our experience,
as we gain the idea of exactness, recurrence is
fundamental.“

In another book, Uhitehead wrote:

“The thought organisation of experience is made

possible only by the conviction that there is 'order

of nature' , and by observing the interconnectedness
of all events. This possibility of disentangling
the most complex evanescent circumstances into
various permanent laws (like periodigity) is the
controlling idea of modern thought."

E. V. Sadley (1926) described his 'pleasant approach
to demonstrative geometry' as follows: Given a simple
lesson in central, axial, and plans symmetry and equipped
with a background of knowledge of geometric forms gained
from the students' own experience, the children could, with
the help of picture magazines and field trips, come up
with many interesting themes for 'geometry with social

science ' :

 

thitehead, Alfred North, ”Science and the Modern
world," New York: MacMillsn Co., 1935, p. 47.

2Whitehead, A. M., “An Introduction to Mathematics,"
New York: Henry Holt and Co., 1911, p. 11.

 

“m
«geo

ri
”8T5.
-whe

One girl,
topic "The
(and the t
ancestral
0f geometr
“Vulture,
V. V.
and Cleo”
“thematic

(1938) Poi

(i.e., 8 n

Width and

QIPQQBEQ ’

22

--symmetry of a colonial house and a Greek temple;

--geometric forms in a Gothic Cathedral, beautiful

bridges, furniture, and snow-flakes;

--symmetry in plants, flowers and trees;

«whether the primitive people knew any geometry.

One girl, who visited Europe during the summer chose the
topic 'The Geometry of Sulgrave Manor,‘ and took the class
(and the teacher) on a personally conducted tour of the
ancestral home of the Vashingtons to observe many examples
of geometric forms and symmetry in the architecture,
furniture, and landscapes of the gardens.

U. V. Lovitt (1924) used the easily observed growth
and decay of natural and man-made objects to express the
mathematical principle of continuity. . David Eigene Smith
(1928) pointed out the examples of a dependent variable
(i.e., a mathematical function) in everyday life usage.
"The cost of a piece of cloth depends on its length,
width and quality,"- wrote Smith, ”Similarly, all costs,
expenses, weights, volumes, success, joy, etc. depend on
something else.“ He did not elaborate how to quantify
'success' and 'joy' , which the USMES children (of the Soft
Drink Design Unit) have done.

The National Council of Teachers 'of Mathematics issued
two yearbooks} in 1931 and 1936, to suggest ways in which
mathematics teachers could bring their subject closer to

real life and other disciplines. In the 1931 Yearbook,

 

1a.c.r.n. mathematics in Modern Life," 6th Yearbook,
1931, Chap. 1-2, 5-10, and “Mathematics in Modern Education,‘I
11th Yearbook, 1936, Chap. 1, 4-8.

 

   
  
  
   
   

the chapte
sicence, b
and techni
ntheaatic
is applied
calculatio

 

23

the chapters on the application.of mathematics to social
sicence, biology, and investment gave a survey of literature
and techniques used in those days. Unlike the high-powered
mathematics (Linear Algebra, Monte Carlo Method, etc.) which
is applied to these three fields today, the methods and
calculations in those days‘gggg simple and easy to follow,
Egg _a_.r_g, and will always be, suitable for teaching children
at the more elementary level. The possible hang-up might
be what M. J. Lighthill1 called “the language problem.“
The children hardly knew enough language (vocabulary,
concepts, facts, etc.) in social science, biology, or
investment. And it would be useless to impose verbal
teaching before appropriate real-life experience. In the
same Yearbook, the chapter on the application of mathematics
to physics pointed out clearly the lack of the teaching
of modelling or problempformulation, as opposed to problemp
solving. Again, this was due to the fact that most mathe-
matics teachers _a_t_ £11 13213.; did not know enough about the
language of physics. This problem still persists even in
1975.

In the 1936 Yearbook, a whole'chapter on."Mathematics
in General Education? called for the teaching of mathematics
as a.mcans to facilitate language (communication), to

develop the sense of consecutiveness (sequencing or

 

1Lighthill, Sir James, ”The Art of Teaching the Art
of Applying Mathcmatics,“ Presidential Address to the
Mafihamatical Association, London, Mathematical Gazette,
Vol. 55. June 1971, pp. 249-270.

 

ordering)
combine it
to be imp
or knowle
Chapter 1
in this C
Polyhedro
For insta
iron, gin
1‘hmnbolrueci
anWhite ‘
nichelgng
artist V1
for drawn
from t0p ‘
head to c

24
ordering), responsibility (accuracy) and the ability to
combine ways of thinking (application). All these were
to be implemented by "relating.mathematics to other fields
of knowledge” which was fully explained in the following
Chapter in that yearbook. Several sections of the material
in this Chapter had direct bearing on the USMES units of
polyhedron-construction.and designing for human proportion.
For instance,1 the models for the crystals of magnetic
iron, zinc, lime and salt are octahedra, tetrahedron,
rhombohedron and cube respectively; any sea-shell of the
ammonite type has the spiral as their outer contour;
Michelangelo (1475-1564), following the ancient Roman
artist Yitruvius, established the following 'Cancn' (law)
for drawing a human body: ”If a - the total height of man,
from top of hair line to sole of a foot, then, top of the
head to chest - a/4,chest to upper part of thigh - a/4,
upper part of thigh to knee - a/4, knee to sole of a foot
- a/4, lower neck bone to chin.- a/l6, foot of a.man
- a/6, etc.”

The concern for having too many cut-and-dry 'algebraic'
problems in.the teaching of physics was already voiced in
the 1931 yearbook (of H.C.T.M.). James P. Davis (1937)
spelled it out in.more details and added a new concern

that most physics problems were not 'real' enough. He wrote:

 

1All examples are quoted from the N.C.T.M.'s 11th
Yearbook, ”Mathematics in Modern Education," 1936, pp. 218-
219, 239-242.

 

 

25

”One of the most important goals of physics instruction
is the development of scientific thinking. First,

the students should be given opportunities for dis-
cussing the meaning of the problem, reading its con-
stituent data accurately, using symbols and known
relationships of the parts, and discovering new
relationships. Secondly, the problem should be in

the pupil's world, not the 'brain-teaser' type which
has no place in the student's life. Thirdly, it should
be a real problem, a felt need, and its solution
produces some satisfaction."

After a lull due to the compartmentalized traditional
teaching during the War, the late forties' literature
was once again teeming with activity-oriented integrating
suggestions. Carrie Lou Early (1947) was a typical healthy
example of the period: her students learned a great deal
of mathematics ‘from her Nutrition class. The following
is part of her log, which, in several respects, was similar
to Piaget's account:
“I placed a quart of milk, several large-size glasses
and a pitcher on the table. Three children took turns
to pour a full glass of milk, and a fourth child
poured the remaining milk into a jug.
--'How many full glasses do we have?' (Three).
--'How many meals do you eat every day?" (Three).
--'Uill a quart of milk give you a full glass for
each -meal?' (Yes).
--'Uill that take the whole quart of milk?‘ (No,
we have some left in the jug for breakfast cereal).
--'How, let's pour the milk out for our party.‘
One child counted the number of paper-cups, another
estimated the number of tables required to put the
cups on, others helped to pour the milk....
Her unit also included the actual making of butter and
vanilla-milkshakes in class, and a trip to the dairy. barn.
Apart from milk, she also utilized eggs, cereals, potatoes,

green vegetablés for other 'llutrition' units.

 

The
1944, spP
building
era. The
publicize
rerul ;.
and social
to recognj
ideas and
exPeriehce
Other proc
tion of '0
can”, b.
“WM th.
Phase. of I

The 1;}
educational
mlicatio
fer the St
three "sen

threads-:2

(
pmnzhga‘
30cm 8(tie:

     
  

.39

‘ s 1,
h on:
57th 35-8

26

The National Council of Teachers of Mathematics, in
1944, appointed a commission to set up principles for
building a stronger mathematics program for the post-war
era. The Second Report of this commission was widely
publicized. The important recommendations for Grades 1-6
were:1 to conceive arithmetic as having both mathematical
and social aims, (arithmetic is not a mere tool subject);
to recognize that readiness for learning arithmetical
ideas and skills is primarily the product of relevant
experience; and to supplement paper-and-pencil tests with
other procedures such as interviews, observation, examina-
tion of work products and use of arithmetical ideas in
ordinary happenings. Such recommendations clearly pointed
towards the concept of integrating arithmetic with other
phases of a child's school life.

The theoretical groundwork for the integration of all
educational experiences came forward in 1958 with the
publication of the 57th Yearbook of the national Society
for the Study of Education. Benjamin 8. Bloom listed
three essential means that could be used as ”integrative
threads":2 .

(a) §or ideas or methods, e.g., the scientific
process w c s common o ranches of science and

social science ,

 

1'Quoted "from Bidwell, J. L, and Clason, R. G. (ed.),
"Readings in the History of Mathematics Education,“
Uashington: H.C.T.M., 1970, pp. 624-29. .

28.8.8.3" ”The Integration of Educational Experiences,”
57th Yearbook, 1958, pp. 97-101.

 

(b)
short-ran
several d

(o)
to unify,

seeming 13
been an e
facts obs
science 0
deductive
Postulate

By 1
PI‘OJ'ects 1

threads (:

were long.

 

27

(b) or roblems whose desired solutions, whether
short-range or ong-ranged, demand a careful study of
several disciplines and their interrelationships,

(c) A connected theo , or enc clo edic or anization
to unify, often By a 5155» level 0; aBs‘EracEness, severi'l
seemingly divergent topics. (This has, traditionally,
been an accepted form of applied mathematics, where scattered
facts observed or measured from various topics in physical
science or economics were accounted for by an elaborate
deductive system stemming from only a few well-chosen
postulates.

By late 50's and during the 60's, several integrated
prod ects appeared, relying mainly on the integrative
threads (1) and (2) above. The important projects, which
were long-ranged, were:

«Ham A Course of Study}

«The ninnemast Project,2

--The Man-Hade World , 5

--this 081138 Project,

--The 'Ihvironmental Studies' Prod ect

and, more recently, the People and Technology Project}

4

 

lInitiated by Bruner, J. S., in “Toward a Theory of
mtggciggn,” Cambridge: Harvard University Press, 1967,
PP- ‘- -

2l‘limeesota School Hathematics and Science Teaching
Project, 1965.

amueering Conce e Curriculun Project Polytechnic
Institute of Brooklyn, ew York, 1968. ’

“Initiated by the American Geological Institute at
Boulder, Colorado; typical activities were described by
its director, Samples, Robert B., "Environmental Studies,"
Science Teacher, Vol. 38, October 1971, pp. 36-57.

5Directed by Peter B. Dow, Education Development
Center, 1972.

 

In is
always bee
above pro;
massive oz
one-nan ex

-—A.

net

to

and
6C0}
heac
BOCj

--Ger1

of r
and
“Dorc
aI‘it
8Dec
aqua

the

28
In fact, teaching a more or less integrated course has
alwus been an appealing concept for many teachers, but the
above projects have represented, for the first time, a
massive organised effort instead of the usual piecemeal
one-man crusade. Examples of these individual efforts were:

--A. G. Doherty and L. 1. Shaw, (1945) who integrated
mathematics with music ,

--Bar1 Murray (1944) who assigned students to groups
to analyse the assumptions, intermediate deductions,
and conclusions of statements drawn from advertising,
economics , and governments under the long-ranged
headings: personal living, economic living, and
social-civic living,

«Gertrude D. Hillman (l958)«who taught the technique
of data-gathering, adding, counting, percentage, and
graphs from a school-wide polling on uWhat is your
favorite dessert?” This was very similar to the
'Soft Drink' Unit of USMES,

--James H. Humphrey, (1967 and 1972), who used Physical
Education as a learning medium in the development I
of first mathematical, than science concepts with
and and 5th graders,

«Dorothy Amsden and Edward Ssado (1958) , who combined
arithmetic with the unit 'Fish' in biology using real
specimens in an aquarium. “How many fish will the
aquarium accommodateY'u-“Vell, first let Mary measure

the dimensions of the aquarium while I measure those

29
of the fishes....” Students also drew feed charts,
graphs of the temperatures of the tank, the room
and outside. They also learned a great deal of
biology of fishes.

«J. Richard Suchman (1964) who regarded elementary
science primarily as an 'inquiry training.‘ His
program at Illinois emphasised both on the focusing
points of children's attention and on the freedom
to branch out their own investigations. This writer
was famous for setting up apparent 'discrepency'
in science classes, and asking the children why.

--John C. Archbold (1967) , who integrated graphs with
geography. The class made cylinders whose heights
were proportional to some geographical data (papula-
tion, mineral resources, etc.) and placed the corres-
ponding cylinders at the correct places on a map.

«Henry Lulli (1972) who integrated art with geometry
by guiding the students to construct polyhedra.

This was very similar to some classes of the USHES

'Dice Design' Unit.

Although most of these integrated projects, organized

or individual, specified the appropriate grade level to
try out, closer examinations of the materials and methodology
have strongly suggested that a continuous non-graded -
approach might be more suitable. As a matter of fact,
this non-graded approach is even independent of readability
and mathematical skill, because a genuinely interesting

project w
needed 3‘:
The
siderably
the proce
caning ne.
involves :
ting, Era}
Considera‘:
nore Akin
focus on 1:

Product ,

All a
Elementary

 

30
project will always be a strong motivation to acquire the
needed skills in reading and mathematics.

The 'new' science education in the 60's moved con-
siderably away from scientific facts and information to
the process of scientific enquiry, which, in effect, meant
coming nearer to mathematics, because scientific enquiry
involves inventing a hypothesis, collecting data, experimen-
ting, graphing results, drawing inferences, etc. which demand
considerable mathematical skill. The Hew Science is also
more akin to social science now as the latter begins to
focus on the process of inquiry rather than Just the finished
product.

All the three successful programs in science education:
Elementary Science Study , Science Curriculum Improvement
Study, and Science«a Process Approach, have science
topics thickly intertwined with mathematics components.

The 388 has units like tangrams, attribute games and patterns,
mirror cards, etc. which are essential mathematics. The

SCIS has units like ' position and motion' , 'relativity' ,
'subsystems and variables' which are mathematical in
character. The 3.211 of ms overtly includes the following
units of mathematics: using numbers, measuring, using
space-time relationship, communicating (by charts, graphs,
etc.), interpreting data and infering.

 

l’li‘or a full discussion of the mathematics component
of SaPA, see Mayor, John B., "Science and Mathematics in
Elementary School," Arithmetic Teacher, Vol. 14, December,
19679 pp. 650.632e

f

The
schools i:
no longer
Iixed with
etc.1 The
lost sin 1;
experience
geometry 1
Contact wi
t0 be stud
and sciencq
"1'0 descri

Severe
1"twisted
“cm and
(1971) did
of SCIS Vs.
”haematite

“the“ 111 C S

 

 

31

The 'new' mathematics program of the English primary
schools is interdisciplinary in nature. Mathematics is
no longer taught as an isolated subject, but it is always
mixed with art, dancing, music, recipes, needlework, stories,
etc.1 The Huffield Mathematics Project's first and fore-
most aim has been to integrate the children's mathematical
experience with other aspects of school life. For example,
geometry is to be learned from having visual and tactile
contact with "Shape and Size," graphs and probability are
to be studied as ways of communications in social studies
and science. Many more examples of integrative experiences
were described in the book written by Leonard Marsh.2

Several research studies were carried out to evaluate
integrated or partial integrated programs. Only the more
recent and typical ones are included here. V. J. Coffia
(1971) did a detailed study on the mathematics outcomes
of SCIS vs. regular science teaching program. For five“
consecutive years, the 'SCIS' students scored higher on
mathematics application, but. the 'regular' students did
better on concepts and reasoning. H. L. Gray (1970)
reported a significantly positive effect of an integrated
program on the acquisition and retention of mathematics

 

1See, e.g., Biggs Edith, ”Mathematics for Younger
Children," New York: Citation Press, 1971, pp. 22-28,
34-37. ‘ -

2Marsh, Leonard, “Alongside the Child: Experiences
in the English Primary School,” New York: Praeger, 1970.

 

and scienc
(1971) die
nathenatic
of grades
As fo
me closer:
(1965), di
elaborate
Her {main
a“Humid“.
t° 1“cream

Him.

In 192
as t° be us
toBether a;
[1.3.3

.3, an

 

 

52
and science behaviors for 5th graders. Betty Uillmon
(1971) discovered a total of 475 technical words of
mathematics hidden within 24 non-mathematical textbooks
of grades 1-3.

As for the integrated activities whose main component
was elementary science, and not mathematics, Mary B. Rowe
(1965), did a large-scale experimental study with an
elaborate design on selected task-oriental science programs.
Her finding was that there was some indication that the
accumulation of experiences in various contexts tended
to increase the probability. that a cognitive network would
form.

In 1967, a colloquium on ”How to teach Mathematics so
as to be useful” was held in Utrecht, Fatherland, bringing
together applied mathematics educators from all over Europe,
U.S.S.H. and U.S.A. In the opening address, Hans Freuden-
thall said, in part,

”What humans have to learn is not mathematics as a

closed system, but rather as an activity, the process

of mathematicizing reality....Take, for example,
fraction. In its traditional teaching, the concrete
context is no more than a ceremony which is hurried
through, and then the abstract theory (of rational
number) is applied. The result, intermediate or
final, is not connected back to any previous
experience on a level which fractions have been

introduced.“ .

He, in another part of his speech, commented on the

narrownsss of classical applied mathematics. His remark was:

 

lPreudenthal, Hans, "Why to teach mathematics so as
to be useful,“ Educational Studies in Mathematics, Vol. 1,

Has. 1968. pp. 6:7.

 

 

'It i:
105151
natic:
longs:

This 1
dm wrote:‘

'The 1
as a s

applie
arise
eeelp;
situat
b10105
activi
as we:
One he
Point,
for hi
TEQUir

 

 

 

35

"It is a fact that biologists, economists, socio-
logists are better prepared to apply modern.mathe-
matics than physicists who carry the burden of a
longer tradition (of using calculus alone)."

This remark was in full agreement with Henry 0. Pollak,

who wrote:1

'The traditional picture of the user of mathematics
as a man who looks up a formula in a handbook and
turns the crank is very much out of date. Uhen.one
applies mathematics in practice, the problems which
arise are not exactly like the problems in the book.
...Applications typically begin with an ill-defined
situation outside mathematics-~in.physics, engineering,
biology, economics, or almost any field of human
activity. The job is to understand this situation
as well as possible, and make a mathematical model.
One has had to know his mathematics previous to this
point, and, more important, to know the conditions
for his success. One can.then.examine what is
required to make it fit the new situation (model)."

 

lPollak, H. 0., "Application of Mathematics,"
N.S.S.E.‘ 69th Yearbook: Mathematics Education, 1970,

PP. 328-329.

5.
legal:

curricula

This
points of
View hold
Homo sapi,
longing I:
solve his
“1188?, e1
expGrimm
l1130 streg
Velcomes n

'ClOBede D

as ‘ threa

charlees.

 

34

3. The literature on the philosophical and pszcho-

logical aspects of an activipz—oriented and integrated
curriculum like USMES:

This type of curriculum is based on two philosophical
points of view: Naturalism and Pragmatism. The first
view holds that all knowledge boils down to nature study.
Homo sapiens is part of nature. Man must give up his
longing for assistancefrom supernatural sources, and must
solve his own problems-«drought and flood, disease and
hunger, etc., by 'scientific' method: making hypothesis,
experimentation, verification, and inferences. Naturalism
also stresses the importance of an 'open' person (one who
welcomes new knowledge and challenges) as Opposed to a
'closed' person (one who looks askance at new knowledge
as a threat) because knowledge, like nature , grows and
changes. Pragmatism views knowledge as an awareness of
continually changing relationships not only between man
and his physical environment, but also between man and
society. Changes bring new truths. Knowledge is "true"
when it solves "practical” problems in society and in the
physical world. Seeking new knowledge and integration of
existing knowledge are never-ending tasks.

These two philosophical points of view were put
forward forcefully early this century by people like
Ssnteysna,1 Russel, Poincare and Dewey. But, going back

1Santavana, George, ''Reason and Common Sense," New
York: Charles Scribners and Sons, 1905.

in histor
Hobbes, L
same idea
emphasis 4
the actioz
no innate

organs, p;
the mind.

ones. Inc
their as”
Bu“ Perce
and nemori
”all! of:
distinguis
experience
independen
19d“ have

Mthods O f

 

 

55
in history one would find that earlier philosophers like

Hobbes, Locke, Hume and Kant, expressed essentially the
same ideas. For knowledge of any kind, Hobbes placed
emphasis on the purely physical character of sensation and
the action of the brain. Locke asserted that there were
no innate ideas in men; experience, through the sense
organs, produced simple ideas (figures, numbers, etc.) in
the mind, which then united simple ideas into complex
.ones. Knowledge concerned the connection of ideas, such as,
their agreement or their inconsistency when combined.
Hume perceived impressions (sensations) and ideas (images
and memories) as knowledge, but he added that ideas were
really effects of impression. Kant was the only one who
distinguished between knowledge obtained directly from
experience and knowledge somehow obtained by the mind
independently of experience. These tw0 kinds of know-
ledge have eventually developed into inductive and deductive
methods of learning.1

So one could easily conclude that philosophers, past
and present, have all favored experience and sense impres-
sions as a means to gain knowledge. Verbalized statements
without apprOpriate experiences are, in fact, not knowledge.
The existentialists go even one step further: they demand

 

lFor a more expounded account of this paragraph, see,
e.g., Horne, Herman H., “Philosophy of Education," London:
McMillan, 1924.

aconplet
itthis w

'ln

exis
awar
John
exis
to l
of h
cann
his.
out:

The 1
311011 in 1;

IPro;
slow,
ideas
we at
nectj
whole

"T17
hand 30 o

 

36
a completely open and free experience. R. G. Olson1 put
it this way:

"An education which intensifies awareness is what an
existentialist would strive for. It refers to the
awareness of that subjectivitym-known publicly as
Johnny sitting presently in this classroom, but
existing our soi in another world in which he wants
to live s owF'Tife. It refers to his awareness

of his precarious role as a baseless chooser who
cannot escape choosing, and therefore who must create
his own answers to all questions and problems in or
out of his classroom..."

The USMES's insistence on long-range projects is very
much in line with Uhitehead's thinking:

"Progress in the right direction is the result of a
slow, gradual process of continual comparison of
ideas with facts. The important criterion is that
we should be able to formulate empirical laws con-
11;ch vegious parts into conceivable interrelated
V O ’0...

”Try it and see if it works" is a familiar expression
heard so often in an USMES class. John Dewey always
advocated such a philosophy of education, as seen from the
following quotation:

“To prove a thing means primarily to try, to test it.
Not until a thing has been tried out do we know its
true worth. Till then it may be a pretense. The
thing coming out victorious in a test carries
credentials with it. So it is with inferences,

too. Every inference shall be tested. We shall
discriminate between beliefs that rsst on tested
evidence and those that do not...."

1Olson, Robert G., "An Introduction to Existentialism,"
New York: Dover Publication, 1962, pp. 17-18.

2Uhitehead,Alfred North, "The Aims of Education,"
New York: McMillan Co., 1929, pp. 156-157.

3Dewey, John, “How to Think,” New York, 1). c. Heath
and Co., 1910, p. 27.

The 1
will now I
with this
It explaix
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interprets
field. Th
itself in

John
philosophi

lCtivities
tOgethEr V

for the ma

 

57

The psychological base of such a program as USMES
will now be reviewed. Gestalt psychology tended to fit in
with this approach more than other schools of psychology.
It explains learning in a large reference frame in terms
of goal, insight, and maturation.1 The 'goal' corresponds
to the interesting realistic challenge or sub-challenge
of USMES. The learning takes place, not through exercises,
but through insight, and the neurological changes are not
modifications in definite synaptic connections (Thorndike's
word) but are differentiations of maturation patterns
conceived as energy patterns within the total psychological
organism. The progress is made through the constant
interpretation of the individual with his environment
field. The learning, dynamically considered, manifests
itself in organized W and not in isolated bond.

John Dewey seemed to supply USMES not only with a
philosophical base (with his emphasis on experience,
activities, child-centered instruction, etc.), but he
together with McLellan, also supplied a psychological base
for the mathematical aspect of an USMES class. They wrote:"’2

”The two methods: Symbols and Things. The first

method is teaching numbers merely as a set o

We, leaves out the objects entirely, or at

least makes no reflective use of them. It lays

the emphasis on symbols, never showing clearly what
they symbolise, but leaving it to chances of future

 

1Dickey, John M., ”Arithmetic and Gestalt Psychology,“
Element School Journal, Vol. 39, September 1938, pp. 47-50.

2N.C.T.M., “Reading in the History of Mathematics
Education," 1970, pp. 158-159. (The article by Dewey and

exper
abet:
tang:
It ne
It al

Both

13876!
the 1

activ
What
now called
the most 1
lee S. Shh
discovery

"The

regul
ities
80met.
but t,
of th
a bet
an en:
date.

 

58

experience to put some meaning into these empty
abstractions. The second method treats numbers as
tangible properties of iject's, and nothing more.
It neglects the mental activity which uses them.
It also makes numbers meaningless.

Both methods are vitated by the same fundamental

psychological error: they do not take account of

the fact that number arises in and through the

activity of the mind in dealing with objects.”

that Dewey and McLellan meant was, in fact, what is
now called “learning-through-discovery,” which is one of
the most important principles for a program like USMES.

Lee S. Shulman explained the psychological aspect of
discovery learning as follows:1

"The child finds in his manipulation of the materials

regularities that correspond with intuitive regular-

ities he has already come to understand. It is rarely
something outside the learner that is discovered,

but the discovery involves an internal re-organisation

of the previously known ideas in order to establish

a better fit between those and the regularities of

an encounter to which the learner has had to accommo-

date.

This is precisely the pedagogical psychology attributed
to Socrates.2 Socrates maintained that he was not teaching
the student anything new, but he was merely helping the
boy to reorganize his thought and bring to fore what he
had alwus known.

More recently, Tallmadge, Kasten and Shearer (1971)

did a comparative study on the inductive vs. deductive

 

1N .S.S.E., ”Mathematics Education,“ 69th Yearbook,
Chicago: University of Chicago Press, 1970, p. 28. (The
article by L. S. Shulman).

aSee Jones, Phillip 8., "Discovery Teaching-Prom
Socrates to Modernity," The Arithmetic Teachjry Vol. 17,
October 1970, pp. 503-510. “

nethods c
contents:
rules (t:
on arbitr
subjects

were that
than the

there was
Content I;
ton-Pd Slit
°1‘ GXtrove

”Perimem

 

59
methods of instruction for two kinds of subject-matter'

contents: Content I based on mathematical and logical

rules (the 'Transportation.Technique' Unit) and Content II
on arbitrary rules (the 'Aircraft Recognition! Unit). The
subjects were Navy enlisted men. Their preliminary findings
were that the inductive method was significantly better
than the deductive method for Content I (p < .01) and that
there was no statistically significant difference for
Content II. Other learners characteristics (attitude

toward such technical units, anxiety level, being introverts
or extroverts, etc.) were controlled in.this preliminary
experiment.

The battle “inductive vs. deductive teaching" has
always been (and will remain) in.the pedagogical world.
Probably it would be best to go back and seek some wisdom
from an old, experienced and highly competent teacher,

J. V. A. Young. His success was really based on a mixture
of both.inductive and deductive technique. He listed his

1 as follows:

pedagogical principles
1. Mathematics should grow.
2. Experimental (concrete) origins or preliminaries
are necessary. .
3. Pupils must abstract their own mathematics.
4. A.moderate amount of drills, wisely administered,
should help.

The word 'drill' used by Young in those days would corres-
pond to the term 'simulation' of today. The following

 

1Young, J. V. A., “The Teaching of Mathematics," first
published in 1924, reprinted in The Mathematics Teacher,
Vol. 61, March 1968, pp. 287-295.

fl

example oi
is, in fac

“CDC
from

via t
and t
to th
and h

The p
like U355

eXperi enc e;

 

40

example of Young's drill-exercise shows clearly that it
is, in fact, learning by simulation:

”On quadratic equations, the pupils should be led
from the solution of

x? + 2x + 1 = 16,
via that of x2 + 8x + 16 = 2,
and that of x2 + 2ax + a2 = b,
to the solution of x2 + 2ax = b,

and hence that of any equation of this type."

The psychological base for an integrated curriculum
like USMES was essentially the principle of 'organizing
experiences.‘ The 57th Yearbook of the National Society
for the Study of Education contained the following delibera-
tions in David R. Krathwohl's article:1i

"Integration can be defined as 'organising experience,‘

that takes place in the learner's mind. It is an

important factor in learning. without such organiza-
tion, learning would only be an unrelated sequence

of perceptions analogous to the snapshots recorded

on camera films. Application or retention would
be impossible.”

Integration of several subject-matter fields have
been psychologically analyzed by George A. Miller‘2 in a
most interesting way. He analyzed the cognitive operations
in terms of Information Theory, which is a mathematical
model for the process of communication. Among other things
studied in this model is the abstraction of 'essence' or

 

lN.S.S.E., ”The Integration of Educational Experiences,“
57th Yearbook, Chicago: The University Press, 1958, p. 45.

2Miller, George A., "Information and Memory," Scientific
American, Vol. 195, 1956, pp. 42-57.

 

'informati
the multi;

"Pilc
(1) safety
\vthistle'l 1
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The E
science, a
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to Z. P. D
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The follow
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9 lo
:31
faces
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I‘Emai
On to

 

 

 

 

41

'information' contained in a certain message. For example,
the multiple-choice question

"Pilot: parachute as boiler: ---—
(1) safety valve, (2) fire, (5) pressure gauge, (4) steam
whistle“ inplies the same relationship between apprOpriate
pairs, rather than aiming at the meanings of individual
words instructed, students could learn from such a test many
topics from many disciplines in a truely integrated manner.

The British are now teaching integrated mathematics,
science, art and reading in 'open' classes, somewhat
similar to the USMES classes observed. They often refer
to Z. P. Dienes when they discuss the psychological base
of their present work. It would be appropriate to examine
Dienes ' thinking about an activity-ori ent ed curriculum.
The following excerpt shows how Dienes connected play with
mathematical learning:1

”Much has been written about play and its function
in a child's life. We are concerned with the func-
tion of play only as it affects the cognitive process
of mathematics learning. A child usually begins with
exploratory-manipulative type of play. He enjoys the
feel of objects, running his fingers down the sur-
faces, putting things together and taking them a art.
He then radually becomes aware of the observed and
abstract properties of the materials, asks questions
about them, and generally moves up along an awareness
time curve. So it is unlikely that the play will
remain purely manipulative for long. He will proceed
on to the two next stages: representational pl”,
and search for regularities.

 

1niches, 2. P., "An hperimental Study of Mathematics
Learning,“ London: Hutchinson, 1963, pp. 21-23.

42

Representational play adds another component into the
activity: imagination. There may be some attempt

to re-shape the objects so that the result resembles
the thing imagined but often it is pure representa-
tion where objects stand for other things by definition
or assignment. In another direction, manipulative
play may quite imperceptibly move over to a search

for regularities or even the abstract rules governing
them. Children are delighted in regularities. The
formulation of a rule-structure is a kind of 'closure'
which ties up all the past experience. Once the
regularities and the rule-structure have been familiar-
ised, they become part of the play. In the next stage,
the play opens with the rules themselves acting as
objects of manipulation, and it is now at a higher
(intellectual) level."

A brief review will now be given about the literature
on the case-study type of research conducted in the field
of mathematics education. Jean Piaget was one of the first
men to do this_type of research and his work has been.so
widely reviewed that it is almost unnecessary to repeat
here. The depth of his case-study results could be seen
from.the following paragraph:1

"The last stage of development of intelligence is the
stage of formal operations. The child becomes capable
of reasoning not only on.the basis of objects, but
also on the basis of hypotheses or of propositions.
Here a class of novel Operations appears which is
superimposed on the operations of logical class and
numbers. The first novelty is a combinative structure:
somewhat like a mathematical structure. It is super-
imposed on the structure of simple classifications.

A second novelty is the appearance of a structure
which constitutes a group of transformations: rever-
sibility by inversion, reversibility by reciprocity,
(e.g. A.) B =9’B < A), and a combination.of both into
a single but larger structure.“

 

1Piaget, Jean, ”The Stages of the Intellectual Develop-
ment of the Child,” in Kramer, Klaas, Problems in the Teachin
of Elementppz Mathematics, Boston, Mass.: IIIyn and Bacon,

309 9 PP-

7

X. Li
KLOOO Pi.
the value
the follo'

'Dr.
hers
withc
littl
such
that
setti
resuj

Last]
carried 01
during the
this kind
K11Patric};

 

 

43
K. Lovell and his associates carried out almost
10,000 Piaget-type experiments. Lovell's book1 testified
the value of this type of research. Barbel Inhelder wrote

the following in the Foreward of this book:

"Dr. Lovell does not accept any of our (Piaget's and
hers) findings or take over anything from our work
without the most scrupulous checking. We were a
little worried when we waited for the results of

such verifications, because they dealt with researches
that were carried out in a rather different school
setting. It is reassuring to see that Dr. Lovell's
results and our own agree perfectly."

Lastly, the reports of the case-study type researches
carried out by the Soviet Academy of Pedagogical Science
during the last 25 years points to the important role of

this kind of work. Commenting on their success, Jeremy

Kilpatrick and Isaak Uirszup2 wrote:

'A.major difference between the Soviet and American
educational research is that the former has used
qpalitative rather than quantitative methods.

erican readers may find the Soviet papers do not
comply exactly to the U.S. standard of design and
statistical analysis. But, by using qualitative
method, the Soviet psychologists have been able to
penetrate into the child's thought and to analyze
his mental processes which the quantitative method
seldom.achieves."

 

1Lovell, K., "The Growth of Basic Mathematical and
Scientific Concepts in Children," London: The University
PPOBB, 1965, pm 5.60

2Kilpatrick, Jeremy and Uirszup, Isaak, (ed.), "Soviet
Studies in the Psychology of Learning and Teaching Mathe-
matics,” Chicago: University of Chicago, 1969, pp. iv-v.

CAI

Fens
naturally
This case-
fill the ma
0! "Soft n
Pmportion
of Dina w
Ilecember‘
that Many
rI'uiti‘ully
checlI‘list
mimics
“Thematic
potential
Chapter.

 

ache other
Profide t

demonstrat

 

CHAPTER 3

CATEGORIZING THE MATHEMATICAL BEHAVIORS
DERIVABLE FROM.FOUR 'USMES' UNITS

Many topics in the usual mathematical curriculum arise
naturally in the course of carrying out the USMES activities.
This case-study is the first attempt to record and classify
all the mathematical behaviors observed in the USMES units
of "Soft Drink Design, Dice Design, Designing for Human
Proportion.and Burglar Alarm Design." During the course
of nine weeks' intensive continual observation (September-
December, 1972), the writer of this Case-Study realised
that many potential mathematical topics could have been
fruitfully discussed if the teacher had had an appropriate
check-list in her class, and thus branched out into
activities or discussions involving these sophisticated
mathematical ideas at the most opportune moment. These
potential mathematical topics will be marked " in this
Chapter. No doubt another observer might have noticed
some other potential mathematical topics and suggested a
different list. The main purpose of this Chapter is to
provide teachers with paradigm cases and a suggestive list
demonstrating how to teach mathematics from a seemingly

44

45
non-mathematical unit like Soft Drink Design. It is not
the purpose of the writer, nor anybody connected with USMES,
to expect uniformity in all USMES classes.

Details of the mathematical behaviors observed in the
above-mentioned USMES classes will be given in Chapter 4.
In this Chapter, the mathematical behaviors of each Unit,
together with its potential mathematical lessons marked ‘*
will be listed under the following headings:

Arithmetic,

Algebra,

Graph and Tabulation,

Geometry, Topology, and Trigonometry,
Application and Practical Mathematics,

Statistics and Probability,
Foundation of Mathematics.

xvmmrwmw

THE SOFT DRINK DESIGN UNIT

(1) Arithmetic: Calculations (addition through
division) involving whole numbers, decimals and fractions
which arose from the measured quantity of sugar, Kool-Aid
powder, soda-water, etc. Detailed study of the iLong
Division' process, e.g. 335322—. ,Multiplicative inverses,
e.g., to put in %-spoonful of sugar three times is equivalent
to l spoonful). Units of measurement (spoonful, standard
cup, f1. 02., etc.) Place value“ (9 cups = l pitcher, etc.)
Percentage (e.g. to calculate the percentage of sugar
present in.a sweetened packet of Ecol-Aid.) Percentage-
increase“ (when the price of a can of soda-pop increases
from 10 to 15¢.) Estimation (of sugar, soft-drink powder
and liquid.)

45
(2) Algebra: A variable amount (of sugar, water, etc.)
vs. a gigpg;amount. Independent and dependent variables
(Popularity of a drink depends on taste, color, etc.)
Composite functions (e.g. popularity depends on taste, and
taste depends on the amount of sugar). An arbitrary system
of assigning scores or marks to one's favorite drinks.
Weighted scores. Matrices (e.g., defined by raters/drinks
rated, or drinks/let, 2nd, 3rd ...choices). Reduction of
the size of a matrix.(by combining several similar flavors
into one flavor). Frequency of being voted lst, 2nd, 3rd...
choices when the drinks were polled. Algebraic problems
in one unknown (e.g., to find the amount of Kool-Aid powder
in.a sweetened packet). The Number Line“ (showing the
scores obtained by the 1st, 2nd, 3rd...drinks). An algebraic
method“ showing that change of grading systems did not
affect the relative ranking of the drinks.

(3) Graphs and Tabulations: Tabulation of experimental
results in mixing drinks, and the results of students' voting
for their favorite drinks. Tabulation of the blindfold

tests' responses.

(4) Geometpz and Topologz: Circular right cylinders (the
soda-pop can). Curved surfaces“ of various bottles. The
circle, diameter, circumference, and the numberhn . (Measur-
ing the circumference and diameter of a cylindrical can).
Simple closed curves" found on various patterns in bottle-

designs. Betweeneness" (exhibited by various scores' points

in me
of f1
Conjc
one 1
Of we
defi;
tiOn

Wide

47
on the Number line). Change of grading systems which
corresponded to translation“ and shrinkage of scales."
Invariance produced by such transformations” (i.e. trans-
lation and shrinkage of scales did not affect 'between-ness'
or relative ranking of the drinks.)

(5) Application and Practical Mathematics: Experience
in measuring sugar, soft-drink powder, water, etc. Rate"

of flow of liquid and measuring time“ with a stop-watch.
Conjectures. Proposing hypotheses and testing them, (e.g.
one boy proposed: % + % a; but later found out that i- cup
of water + % cup of water f g cup of water). Using the
definition to derive something else, (e.g. using the defini-
tion of "soft-drink" to design a questionnaire for a school-
wide polling of soft—drinks).

(6) Statistics: Polling. To quantify the "popularity"
of a drink by counting votes or total scores assigned by
people. Representative samples. Random selection“ of
sample population. Stratified sampling. To control some
variables (e.g. to control the independent variable 'look'
by blindfold tests). Confounding variables, (e.g. the
independent variable ”color of a drink" is confounding
with the independent variable "amount of water added").
Irrelevant variables (e.g. the color of the measuring spoon).
Voting: arrangement of scores in.ascending or descending
order. mean scores.“ Discussion of whether the experi-
mental results (mixing drinks, polling, etc.) were partly

due to chances.

43

(7) Foundation of Mathematics: Ordinal and cardinal
numbers (1st, 2nd, 3rd...choices are ordinal numbers, while
the scores are cardinal numbers). Reversibility of orderings,
(e.g. lst, 2nd, 3rd...choices become the scores of 7, 6,
5,...points). Two mutually exclusive sets (e.g. soft-drinks
vs. non-soft-drinks, to guess right vs. wrong in the
blindfold tests), and the introduction of the binary digits
(0, 1). Discussion of 'equality' in mathematics, (e.g.
drink 'a! = drink 'b' as far as popularity was concerned,
but their other attributes might be different). Logic:
negation.(nonpsoft-drink), implication.(more Kool-Aid
powder=$’deepened color). The continuum in.a real model,
(the color of lime soda changed continuously from emerald
green to light green.while water was slowly added.) Sets
and their elements: defining the set of non-soft-drink by
enumerating its elements, and then abstracting the properties
of this set afterwards. Intersection of two sets (e.g.
the common.properties belonging to both orange soda and

strawberry'soda).

THE DICE DESIGN UNIT
(1) Arithmetic: various methods of counting and

tallying the scores of coin-tossing games, e.g., recording
in groups of 5 strokes, or marking each square of a graph-
paper to mean 5 or 10 points. Place-value (A boy used A
for S, and D for 10 strokes). Practices on multiplication
when counting up the groups of 5 strokes each. The impor-
tant concept that multiplying by an.integer implies repeated

additions.
column of
Closure IA
Partial 81
division I
(e.g. 1%;
differ on:
19110 = (1c
in groups

(2) g
by integez
Sane). Ne
Penalties
Pr081‘9331
concentri
unmet,
“180mm.
The Comb},
When disc

last term.

(3)
3nd T‘300:

mung e~

49
additions, (e.g. 5 x 3 = 5 + 5 + 5). Addition for a long
column of numbers. Commutative Law (adding backwards).
Closure Law (The sum 'of 2 integers is still an integer).
Partial sums or sub-totals. Mental arithmetic involving
division by 2 (One half of 1482 is 741). Approximation,
(e.g. 1&3; s i). Relative Error“ (e.g. 1% and 1%;
differ only by 1 part in 371). Distributive Law, e.g.
19x10 =(lelO) + (9x10) when counting tallies arranged
in groups of ten.

(2) Algebra: Ideas of increasing or decreasing expressed
by integers (e.g. the points gradually won or lost in any
game). Negative numbers“ for counting points lost or
penalties. A set of consecutive integers. Arithmetical
Progression,” e.g. 7, 13, 19, 25... read from the
concentric circular track of a game (See Figure M9). The
arithmetical mean“ and the subsequent derivation of the
algorithm i=%(a1 + a2 + a3 +...+%) for any set {a1} .

The combination 802, 11C2 (finally generalized to nor")
when discussing a tournament for 8 or n competitors. (This

last formula may be more suitable for the High School level.)

(3) Graphs and Tabulations: Tabulation of H-soores
and T-scores obtained by the whole class. Bar-graphs
showing each pair (H,T) as well as the pooled H vs. the
pooled T. Mean values read from a set of bar-graphs put
together. A histogram drawn from the actual data recorded
from exactly 50 throws of coin-tossing. The histogram as

an appro:
oo-ordinz
A skew B:
against E

frequency

(4)
(a) Q
circular
inscribed
m“ inter
conSl'tlenc
thOOrem:
‘3 a unit
(1)) .02
W
Symmetry.
Squares,
find their
tessellat‘

the cube ,

50
an approximation to the Normal Curve. “ Introducing
co-ordinate axes," abscissa, ordinate, and ordered pairs.
A skew Normal Curve obtained by plotting 'frequency of H‘
against H itself in the above data. Locating the maximum

frequency from this (skew) Normal Curve.”

(4) Geometn:
(a) Observed from the activities of children who played

a horserace game: (See Fig. 4.9) Concentric circles (their
circular 'track'), radii, sectors, annuli, chords, the
inscribed hexagon consisting of 6 equilateral triangles.

The internal and external angles of a hexagon." The
congruence” of two triangles (by paper-folding). The
theorem: W = constant, and hence defining “radian“
as a unit for measuring angles.”

(b) Observed from the activities of children who constructed
polzhedra from pre-cut polygon duripg the Apgpst workshop:
Symmetry. The prencut polygons: equilateral triangles,
squares, rhombuses, pentagons, hexagons, and octagons,
and their simple geometrical properties obtained by
tessellation, or direct measurements. Convex polyhedra:
the cube, tetrahedron, duodecahedron, icosahedron, and
octahedron. Intersection of two planes (an 'edge' of a
polyhedron), and the angle" between them. Three or more
planes defining a point" (the 'corner' of a polyhedron).

(5) Application and Practical Mathematics: Use of an
abacus” for tallying as well as adding. Keeping two

records to show “points won'I as well as "points lost" for

checking t
'double-er
‘Jord-probl
motion and
experience

(6) _S_t
abilitiesl
0f the 10

 

straw, and
exIDeI‘iment
in terms c
Pain ‘2
Pendent a:
Normal Di:
PNbabili,

(7) 3
Sets (H vsi
element a
'get‘fiOn

timers m

 

in totalljr

51

checking the correctness of record-keeping similar to the
'double-entry' book-keeping. Principle of book-keeping.”
Word-problems about budgeting." Counterclockwise (circular)
motion and complete revolutions (560°) studied via the
experience on the circular track (Figure 4.9).

(6) Statistics and Probabilitz: Theoretical prob-
abilities of the coin-game (1/2 for E, 1/2 for T), and
of the lot-game consisting of 3 straws (1/3 for the 'short'
straw, and 2/3 for a 'long' straw). Theoretical and
experimental results compared, and discrepency computed
in terms of relative error.” Equally likely events.
P(A1 n A2 A A3 A A4) = P(A1)P(A2)P(A3)P(A4) for the inde-
pendent events of throwing 'heads' for 4 consecutive times.
Normal Distribution inferred from the constructed histogram.
Probabilistic model exemplified by the histogram. Discus-
sion of the Deterministic model.

(7) Foundation of Mathematics: Two mutually exclusively
sets (H vs. T). Complement of a set (e.g. complement of
element H in the set {3,T} is T). Generalization from
"get-«don't get", "won-~lost" etc. me. {am}. Ordinal
numbers utilized in keeping tallies, and cardinal numbers
in totalling. Idea of equality viewed from the fact that
points won by one side equals points lost by the other
side (Reflexive Law). A counter—example of the Transitive
Law: in a certain game, A beats B, and B beats C: does A
necessarily beat C?--(NO!) Symmetry as an example of a

necessary, but not sufficient, condition for a fair die.

One-one c

as 8 near

(1)
egocentri
then the
Accuracy
111., or n
of a Grin
Estimatic
8hoes).
fractions

2577;
1% - 1
(7/16)
L130, c a1

W

on ‘ piec

S2

One-one correspondence: 6, 12, 18, 24, 50, ......

tt i $~1
7, 13, 19, 25, 51, ......

as a means to study A.P."

THE 'HUMAN PROPORTION ' UNIT

(1) Arithmetic: Units of measurement (first, the
egocentric units: hand, elbow-length, the child's own body;
then the scientific units: inch, foot, meter, etc.)
Accuracy of measurements (correct to the nearest 1/16
in., or nearest millimeter). Approximation (e.g., height
of a drinking fountain = 2/5 of a child's height, etc.)
Estimation (e.g. estimating the extra heights due to wearing
shoes). Many opportunities to work on long-division or
fractions, e.g.,

25)76'197}i” when calculating the mean-height of 25 students;

 

1% - «i—é when calculating the error due to measurement:
(7/16)(l%) when enlarging a photo 1% times, etc.
Also, calculations involving the concept of 5113.2 or
propprtionalipz when building a scaled model of the school
on a piece of tri-wall board.

(2) Algebra: A linear function ya; x, when x referred
to any length measured in the smaller photo, and y the
correspondipg length in the enlarged photo. Also, a non-
linear function exemplified by the verified fact that human
height is not a linear function of age. (A 10-year-old
child is not twice as tall as a 5-year-old.) Introducing

arrays of data in the form of a matrix.

(5) E
garteners'
pre-arrana
consisting
ordered pa
and‘b' ind:
(1.2) was
“Wins pi
“19 25 kit
fifth-31‘s:
grid to re
draftg or

tee.te

 

(4) Ga
°mh°80na
No 1301111:
Mint of
‘ Nana,
“'0 horiz
and loves

Scale. 9

human fee 6|

 

53

(5) Graphs and Tabulations: Twenty-five kinder-
garteners' heights neatly tabulated as a 5x5 matrix, by
pre-arranging the children into 5 groups: each group
censisting of 5 children. Identifying each child by an
ordered pair (a,b) where'afreferred to group-membership,
and'b' individual number within that group, i.e., the child
(1,2) was not the same as the child (2,1). Experience in
drawing pictorial graphs and bar-graphs from.such data as:
the 25 kindergarteners' heights, the waist-sizes of 22
fifth-graders, etc. Introducing the Cartesian Co-ordinate
grid to facilitate the drawing of pictures from rough
drafts or given photographs. Playing the game of tic-tac-

toe.“

(4) Geometpz and Trigonometpz: The concept of an.
orthogonally projected height (i.e. the length between

two points obtained by orthogonally projecting the highest
point of the hair and the lowest point of the heel on to

a vertical plane.) Such projections carried out by placing
two horizontal planes (cardéboards) touching the said highest
and lowest points, and being perpendicular to the vertical
scale. 90° rotation of the two horizontal planes and the
vertical scale in order to measure.projected width of a
human face. Measuring a curve-length (e.g. waist size,by

a tape or a straight ruler rotating as a variable tangent
to the curved waist-line). Idea of an envelop or line-
conic." Difficulty involved in.projecting sets of points

54

on a curved surface (e.g. the sets corresponding to the
organs in a human face) on to a plane (a flat sheet of
drawing paper.) Sterographic projection.“ Study of
similar fimges” from two photographs, one being the
enlarged copy of the other, and a ratio of 5:2 in linear
measurements=>9w in area measurements. hperience in
drawing parallel and perpendicular lines for the Cartesian
Co-ordinate grids. Simple trigonometry for calculating
the height of classroom ceiling. Using the slopes of
right-angled triangles' hypotenuses to indicate the relative
'fatness' . Introducing P tan'l(y/x) as another way to

measure 8118198, ’ ‘ when P refers to the principal value.

(5) Statistics: Collecting data about the kindergarteners'
heights, the lengths and widths of 5th graders' faces, and
various measurements of a 5th grader's body. Calculating
the mean, median, mode and range from these data, and com-
paring the mean values of their leg-lengths, arm-lengths,
trunk-lengths, etc. with the measurements of classroom
furniture and fixtures to find out'why some furniture and
fixtures caused inconvenience to persons of children's size.
Writing up a physical profile of a 5th grader."

(6) Application and Practical Mathematics: Measuring
by means of a foot-rule, yard-stick or a tape measure.
Building a scaled model of a school (not completed).
Learning to make allowance when measuring a human body for
the purpose of making a garment (e.g., collar size should
be slightly bigger than neck-size.) Curve-stitching”

(e.g., mal
conics.)

to fit a 5
steel tape

from trigc

(7) g
limits: "
Discussion
5° Ill-p.11.
V0176. etc.
pairs (a,b
“my-town
t0 the rea

chi ldI‘en .

(l) A

mine an
(3,
Q 11111 Of
°’ Manly
finding Drh
thi ‘

 

8 elec
COuld not
Mtgetic
demnfitra i

P08i t1Ve

   
  

55
(e.g., making a hyperbole, a parabola, or other line-
conics.) Making simple furniture from wood and tri-wall
to fit a 5th grader's size." Verifying (by means of a
steel tape measure) the ceiling height previously calculated
from trigonometry tables.

(7) Foundation of Mathematics: An example of upper
limits: "The, library shelf should be at most 5 ft. high.”
Discussion of some lower limits" such as: the speed of
50 m.p.h. on a freeway, 18 years old for being eligible to
vote, etc. As mentioned in (2) above, the use of ordered
pairs (a,b) with ls as 5, 1sbs5 to identify 25 children.
Many—to-ona correspondence when mapping these ordered pairs
to the real numbers representing the heights of these
children.

THE BURGLAR ALARM DESIGN UMIT

(l) Arithmetic: Counting the number of turns when
making an electromagnet. Practice on fractions and decimals:

(Ex, To wind 20 ft. of (insulated) electric wire around
a nail of diameter 1/1: in., it required fifi 405.7,
or roughly 506 turns. Further, if these 506 turns of
winding produced a solenoid of 5", then the diameter of
this electric wire-333 -.01654", whose order of magnitude

could not be measured by a ruler.) The powerful method of

 

arithmetical deduction from indirect” measurements, as
demonstrated above. Scientific notation, e.g. 1.634 x 10-2.

Positive and negative exponents." Discussion of possible

error, (
of .0004
d=.0159
Informat:
'Off' anc
e.g., 0‘01

01' a batt

(2)
(imam
and the g
e.g. Curr
Pemtati
chfilnical
bination 3

thread

(3) LP

 

 

56
error, (e.g. , in the above example, there was a discrepancy
of .000” when checked against a standard Table which gives
d = .0159#",) and ways to improve the accuracz .of such work.
Informational aspect of integers, e.g. #26 wire; 0 means
'off' and 1 means 'on' in the logical circuit. Estimation,
e.g., obtaining a wooden block of comparable size to that
of a battery, when constructing a battery-holder.

(2) Algebra: Algebraic equation of one unknown, e.g.

(5.1416)d= 2:52-13. (This work included both the formation
and the solution of the equation.) Inverse proportion,
e.g. current oC W. The Inverse. Square Law.
Permutation of 5 elements in a circuit diagram: + pole, - pole,
chemical solution (of the battery), bulb and switch. Com-
bination n02 arising from all possible connections, e.g.,

bottom of bulb connected to bottom of battery,

bottom of bulb connected to curved side of battery,

threaded part of bulb; connected to top of battery,
e c.

(5) Graphs and Tabulations: Tabulation showing how the
magnetic force depends on the number of turns in a solenoid.
Study of First Difference to derive the growth-rate and

 

d003, , C e 8 e
No . of N o . of clips First
turns suspended Difference
50 9
100 17 8
150 25 8
200 55 8
250 40 7
300 45 S
350 39 -6

Ordered p:

graph 3' =9

In

Inversed

(4) _C;
the batter
center, d1
0! the co;
atMRS cu
(a bulb) ,
'"inging'
contact.

Relat
and discos
the + and
“mum.
between 2

(S) A

g

10118 dist
Vining e1
Contact (t
Circuits..

 

(6)
Electric c

heating or

en circ

57
Ordered pairs (8,72); (9,81), etc. giving the straight-line
graph y = 9x. Graph of the experimental results about the

Inversed Square Law.“

(4) Geometpz and Topology: Circular cylinder, (e.g.
the battery, the empty coffee can.) The circle: its
center, diameter, circumference =2nr. Cylindrical surface
of the coffee can flattened out into a rectangle. Parallel
strips cut from this rectangular sheet. Spherical surface
(a bulb). Mid—point of a line, (where the mechanical
'swinging' switch'was pivoted.) Discussion of 'geometrical
contact' vs. 'electrical contact'.

Relative positions on a circuit-diagram: connection
and disconnection. A topological line (any wire connecting
the + and - terminals.) Simple closed curve (a complete
circuit). Simple discontinuities. Separation and distance
between 2 planes. (The 'springy' door-mat and the floor).

(5) A lication and Practical Mathematics: Measuring
long distances for [electric wiring. Cutting empty coffee
cans into thin straigpt strips. Soldering intersections.
Winding electromagnets. Gluing springs and electrical
contact (tin-foil) to a door-mat. Manipulating logical

circuits.” Use' of a micrometer” to measure small distances.

(6) msics: Dry cells. Conductor (wire) and insulator.
Electric current, shown by the lighting of the bulb or the
heating of the wire. A complete or closed circuit. Switch.
Open circuit. Short-circuit phenomenon. 'On' and 'Off' .

Positive 2
series.

number of
Ohfs Law,
in series,
a battery.
Thelnver:
electric J

Gram

 

58

Positive and negative poles of a battery. Batteries in
series. Electromagnets. A solenoid. Magnetic force CK:
number of turns in the solenoid. External resistance and
Ohm's Law. Resistance of a wire ocits length. Resistors
in series. Potential Difference. Internal resistance of
a battery. Voltmeters. Ammeters. Magnetic Induction.
The Inversed Square Law. Electric terminals. Domestic
electric wiring.“ Logical circuits.

Gravitational pull. Spring constant and potential
energy of a compressed spring. Equilibrium or balancing.
Limiting friction.

(7) Foundation.of Mathematics: Abstractien (e.g.

noting relevant elements of an electrical set-up and
abstracting them into a set called circuit-diagram.)
To operate formally (i.e. to permute, or combine symbolic
elements of a circuit diagram, to do calculations by
referring to the diagram alone.) To gain.a.deep impression
of the cardinal numbers '1' and '2', e.g.

--”0NE'battery'd0esn't work; TWO work beautifully."

--”0NE:meta1-strip (in a battery holder) leads to the

short-circuit phenomenon; TWO separate metal-strips are
quite safe."

The J
Presentedl
records o:I
the teachvl
input dunl
um uniJ
the 'Dice:

tiOn' Uni

 

CHAPTER 4

DETAILS OF THE FOUR CASE-STUDIES

The full text of the four Case-Studies will now be
presented. They were written from the chronological
records of the writer's own observation, supplemented by
the teachers' logs in order to clarify some of the teachers'
input during these activity-oriented classes. The four
USMES units observed are: the 'Soft Drink Design' Unit,
the 'Dice Design' Unit, the 'Designing for Human Propor-
tion' Unit, and the 'Burglar Alarm Design' Unit.

59

I.

II.

The 95
Obser

The t
h
and r
activ

etude

' r
Brie-

V1 th
The H
cups
allot
Vedas

 

I.

II.

60

The CASE-STUDY of the “Soft Drink Design" Unit,

Observed at Wexford School, Lansing, Michigan
September 26 to November 28, 1972

The behaviors to be observed: Mathematical content
and mathematical process, which arose from the unit's
activities, class discussions, teacher's advice,

students' responses to questions, and other remarks.

Brief Description of the "Soft Drink" Group: Seven
students (four girls and three boys) were selected,
on a voluntary basis from a combined Grade 5-6 class
of about 90 children under a team-teaching situation
which involved three teachers. The teacher of this
Unit was one of the three, and she had actively
participated in the Soft Drink Unit of the USMES
Summer Workshop here in August. One girl student had
also taken part in that Workshop.

While doing USMES, they had a room of their own
with a kitchen sink, a) drinking fountain, a long table.
The Kool-Aid powder, sugar, spoons, pitchers, paper-
cups and paper-towels were provided. The time
allotted to the unit was twice a week (Monday,
Wednesday 1:50—2:15 p.m.).

h
'd

 

Six 5
and scienc
I

in these ell

problem we

 

61

Six students were at Grade 6 level in mathematics
and science and only one student was at Grade 5 level
in these subjects. No serious reading nor communication

problem was observed.

62

Activity 1: Mixing drinks and tasting these drinks

amogg themselves until each student came up with his
or her own 'brand' of soft—drink.

The children were experiencing a unique classroom
environment which involved a great deal of laughter,
manipulative skills in using household utensils, but
not without mathematical learning. They mixed different
kinds of Kool-Aids with various measured amount" of sugar
and water, tasted the new inventions, looked at each
other's colorful tongues, (the writer hoped that food-
coloring was relatively harmless), and sometimes left
the room hurriedly to avoid the near-nauseating feeling
caused by the grossness of certain inventions. Still,
all was done in good faith, and, after much experimenting
with the variable amount' of Kool-Aids that should be
used for a flag amount“ of water and sugar, every
student did come up with an acceptable formula of his
own. They learned that the success (popularity) of
one's drink depended on two factors: the taste and the
look of the drink. (One girl remarked that a drink

 

‘Throughout this Case-Study, the words marked "
refer to the mathematical behaviors observed, and those
marked " refer to the potential mathematical tapics
which could have been discussed in an USMES class, but
were not included in the observed unit.

which loo

they were

variables

(populari
and, as o
the whole
the drill}:
for disc:
in the re
independe
(1)
nature;
lime, col
(2)
titetiveW
(3)
measured
I‘ealize e.
(a)
by bottl€
The
idea that

 

Vari dbl e

 

63

which looked like vampire blood would not sell.) Hence
they were introduced to the concept of independent
variables‘ (taste, look) and a dependent variable‘
(popularity). The latter was easily quantifiable‘,
and, as one boy suggested, they were going to ppll’

the whole school to measure the relative popularity of
the drinks. The independent variable was brought up
for discussion, and, to the surprise of the two adults
in the room, the children discovered four (not two)
independent variables, as follows:

(1) the kinds of Kool-Aids (This is qualitative in
nature: the variable being strawberry, cherry, orange,
lime, cola flavors, etc.).

(2) the volume of the Kool-Aids used (This is quan-
titative, measured by a 'standard' teaspoon).

(5) the volume of sugar used (quantitative, again
measured by a teaspoon). Here the children did not
realize that there were many kinds of sugar.

(4) the volume of water used (quantitative, measured
by bottles of 16 fl. oz. capacity, or l-pint pitchers).

The majority of the children seemed to possess the
idea that "the color of the drink" was an independent
variable which directly affected the popularity of the
drinks. But, after some experimentation, they were
satisfied that the variables (1), (2), and (4) above
did take care of this extra variable. At this point,
the teacher could have introduced the concept of a

(D

cosgsit

later on.
and (4), 1
on color,
(I). (2) 1
ly, the t«
confoundiz
would be 1
Research 1
the Popule
must not 1
these are
there was
Pendent Va
”50’. they
in “lilting
from emer,
and more 1

Anot;
Vity Was «

Units whi4

 

 

requisite
& studentl
after a J
390011; th
Vill the 4.

 

64
composite function“, which the students would meet
later on. (The color of a drink depends on (1), (2)
and (4), and the popularity of the drink partially depends
on color, and hence the popularity, in turn, depends on
(1), (2) and (4), and perhaps more variables. Alternative-
ly, the teacher could have explained this in terms of
confounding variables“ in Statistics, a concept which
would be useful later on in such USMES units as Consumer
Research and Pedestrian Crossing. (If one says that
the popularity of a drink depends on its color, then one
must not introduce variables (1), (2) or (4), since
these are confounding variables to 'color'). In fact,
there was some advantage of treating 'color' as an inde-
pendent variable, as suggested by the children; in that
way, they could‘ggg,the concept of‘2.continuum“: e.g.
in.mixing drinks of a lime flavor, the whole continuum
from emerald green to light green could be seen as more
and more water was added to the solution.

Another concept the children learned from this acti-
vity was that of irrelevant variables‘. (All USMES
units which involve graphing will demand the identifica-
tion of relevant and irrelevant variables as a pre-
requisite.) Uhen the writer (of this Case-Study) asked
a student what the irrelevant factors were, he replied,
after a momentary reflection, “the color of the measuring
spoon; the kinds of containers used in.testing." "But
will the kind of container be a relevant factor in selling

65
the soft drink?" asked his teacher. "Yes," replied the
student, “because it would change the popularity of the
drink.”

One interesting behavior observed was the way the
children guantified‘ the ingredients used. They did not
use a weighing pan, nor a graduated cup to Judge the
amount of Kool-Aids and sugar; they Just used ordinary
teaspoons. This might sound unscientific, but they had
good reasons. "I have always kept this game spoon every-
time I measured the Ecol-Aids and sugar," said one girl,
"and one teaspoon means this full.” (She demonstrated
the amount.) So she got the basic idea of a 33$}; 9!;
measure‘, (which is, usually, arbitrary).

”But what happens if you want to communicate to
other people who have never seen your spoon?" asked the
teacher.

"Then I will say one-quarter, one-half or a whole
packet of Kool-Aid" replied the girl. 80 she viewed
the subject of fraction‘ from the utilitarian's vantage
point. This was precisely how fractions had been dis-
covered and used in antiquity: 1/2, 1/4, 1/8, ....

During the same period, another student proudly
told his teacher that he discovered a practical way to
'prove' (meaning 'verify') that 1/2 + 1/4 - 3/4, because
he first used one-half, and then one-quarter packet of
Kool-Aid, and he noticed there was a quarter left over,
and so he used up three-quarters of the packet. Earlier

66
that morning, he had been taught that 1/2 a 2/4 and

2/4 + 1/4 = 3/4 in a traditional 'math' period.

An USMES class could provide many opportunities
for practical verifications‘ of this type. The writer
once advised a boy (who ran out of Kool-Aid) to 'play'
with sugar. The boy was asked to divide a certain
amount of sugar on a large plate into 5 heaps, and to
divide the same amount of sugar on a second plate into
7 heaps. When the student added the larger heap of
sugar to the smaller heap, he could tell, Just by looking,
that the sum was approximately a third of the original
amount. This appeared to be more meaningful to the child
than the traditional presentation.l/5 + ll? 2 7/35 + 5/55
a 12/35, which had been shown to him a week before.
Incidentally, such 'playing' (with concrete objects)
would leave a strong impression in the mind of the child
--an impression saying 1/5 + l/7 # 2/12, or generally,
% + e , 8 1°

Also, during this 'mixing drinks' activity, while
the students were measuring water from the drinking
fountain, the teacher could have discussed the gate 93:
$19!” 93; a liguid-n-a very important topic both for
mathematics (rate of change) and for science (property
of fluids). Both a graduated pitcher and a stop-watch
were available in that school, so the (volume) rate of

flow of the water could easily be calculated, and

67
children would enjoy the experience of using a stop-
watch. This is also a very effective way to study

the concept of time."

Activity 2: Evaluating each other's drink:

Seven children produced seven 'brands' of soft-
drinks, and they decided that only the top three drinks
should be used in the general polling. A way had to be
found to choose the top three drinks from seven. After
some discussions, two types of evaluations were proposed
and adopted. (The evaluations would be carried out by
the seven students, plus the teacher and the writer of
this case-study.) In essence, the two methods of
evaluation were:

(1) Counting the,frequencies‘, by which each drink
was named first choice, second choice, third choice,...
and seventh choice;

(ii) Assigning a common maximum score to the drinks
which each rater liked best, and then.descending scores‘
at equal,interval to the next preference, until he
assigned a minimum score to the least-likeddrink. The
ranking“ of the drinks would then be decided by 3133
tgtal‘gggrg‘ each drink received.

They proceeded with the first method (counting
frequencies). At first, the seven drinks were named
Tom's drink, Kathy's drink, etc., but, for the sake of
modesty as well as impartiality, the sample drinks were

re-named
knew the
to be se1
this, a z
the names
column, a
his last
column.

this meth
beam. A
Sample :1:
(or the o
to fill u
rater had
Second 0r
I111ml abou
Peeple mi
Subdectiv
pm‘pOSe 0
drinks, b
thl

 

%‘ i“
b910w) ‘

A

(1

taking a
frOm fire

wwd

 

68
reenamed 'a, b, c, d, e, f, g', and only the teacher
knew the key to these code-names. A tabulation‘ had
to be set up to keep track of everyone's rating. To do
this, a student suggested that each rater should write
the names of his most preferred drink at the top of a
column, and proceed downwards till he wrote the name of
his last choice (seventh choice) at the bottom of the
column. It was the concensus of the class to adopt
this method of recording votes; and so the evaluation
began. All nine people were busy tasting the seven
sample drinks, trying to remember their pleasing flavors
(or the opposite) before walking over to the chalkboard
to fill up his own column of rating. Nearly every
rater had to taste some, if not all, drinks for the
second or even third time before he could make up his
mind about the relative merits of the drinks. Some
people might object that such a rating was highly
subjective, but it should be remembered that the main
purpose of this activity was not to rate home-made soft-
drinks, but to learn some mathematics from the experience.
First, the children here had the opportunity to see.g
matrix" in jag making (the 7 x 9 matrix in Table 4.1
below), an opportunity missed by most college students
taking a course on matrix Theory. Secondly, they learned
from first—hand experience the contrast of ordinal

numbers‘ (used in defining the rows of the matrix below)

 

and card
later or

Uhe

columns,

T5

Rate:

Choice:

lst

 

69

and cardinal numbers“ (used in counting the frequencies

later on).
When the nine raters filled up their respective

columns, a matrix was obtained:

Table 4.1: A Matrix relating the raters to
their preference in drinks

 

 

Raters ‘3
.d
m 3 a.
o «4 'H s: 'H o
e s 52’ a g 2 e a a?
Choices E-4 a M U) 0) 2 F3 5 m
1st c a a c b g a a
and a e c a c a c c b
5rd d c h f a f f g a
4th e g d d e e b b g
5th g f e g f d d f e
6th f b g e d c g f
7th b d f ‘ b g b e d d

 

 

 

 

The next job was to spot winners, especially
winners of lst, 2nd and 3rd places. The children read
from this Table the frequencies’ by which some drinks
were voted lst choice and 2nd choice. Drink 'a' was
t!» times voted to be lst, and 3 times 2nd, while drink
'c' was 3 times 1st and 4+ times 2nd. There seemed to be
little doubt that 'a'l should be declared the winner, and
' c' the runner-up. But the teacher quickly pointed out

70
that 'a' was still a shaky winner, since only 4 people
out of 9 (less than 50%) voted for it as their first choice.
"How many'%’does 4/9 correspond to?" asked a student, and so
they went on to have a mini-session on percentage.‘ Of all
the drinks voted to be lst choice, 'a' accounted for
44% of the votes, 'c' 35%, 'b' and 'g' 11% each.

A big problem arose when they tried to determine
the 3rd place winner. Looking at the 3rd row of Table
4.1, the children found that 'f' appeared 3 times, 'a'
twice, and 'b, c, d, g' once each. It was tempting to
take 'f' as the 3rd place winner. But, fortunately, a
student in the class sensed that it was wrong to ignore
'b' and 'g', because both 'b' and 'g' were voted top at
least once, while 'f' was never chosen top, and, for
that matter, never even for 2nd place. How could 'f'
be ranked before 'b'? Again, from the percentage point
of view, 'f' was put into 3rd place only by 33% of the
voters (3 people out of 9), and yet 3 other persons had
made it quite clear that they preferred 'b' to 'f'. (See
the first 3 rows of Table 4.1 regarding the positions of
'b'). But the f-supporters argued that 'b' was voted to
be bottom of the list 3 times, while 'f' suffered this
fate only once.

After some (heated) discussion, it was resolved that
the arguement 'f' vs. 'b' should be settled by the second
method of evaluation, i.e., by assigning appropriate
scores to each of the drinks according to the raters'

71

likes or dislikes. For example, 1 point could be given
to the drinks in the last row (7th choice), 2 points for
the 6th choice, 3 points for the 5th choice, and so on,
till finally 7 points for the lst choice on the top row.
It was decided by the class that, rather than tasting
the sample drinks again, they would make 932 93 the data“
in Table 4.1. The mere conversion of the data in Table
4.1 into another representation involving scores was a
good exercise in mathematics: it taught the children
one-one correspondence“ and Piaget's "reversibility 93;

ordering " ‘ :

lst choice —-> 7 points

2nd choice ——> 6 points

3rd choice ———> 5 points

7th choice .——.-; 1 point
It also taught the usual arithmetical skills“ _i_n_
multiplication and addition. In the next period, the
children were busy working out the scores obtained by

drinks 'a, b, c, d, e, f, g'. For examples,

 

Drink 'a': 7 points, 4 times 28
6 points, 3 times 18
5 points, twice 12
56 pts.
Drink 'c': 7 points, 3 times 21
6 points, 4 times 24
5 pts. 8: 2 pts,
once each .2

52 pts.

72

 

 

Drink 'f': 5 points, 3 times 15
3 points, 3 times 9
2 points, twice 4
1 point , once _1
29 pts.
Drink 'b': 7 points, 6 points,
once each 13
5 points, 2 points,
once each 7
4 points, twice 8
1 point, 3 times '_3
31 pts.

At this point, the 'b'-supporters clapped; so 'b'
was proved to be better than 'f' and hence 'b' won the
3rd place after 'a' and 'c'. (It was purely coincident
that the three top drinks were labelled like the first
three alphabets.) It could not escape the nOtice of any
observer that the children seemed to show more enthusiasm
to do this sort of arithmeticthan they would in a tradi-
tional 'math' class. When all this arithmetical work
was done, the children recorded all the scores in a

Table like Table 4.2:

Table 4.2: Scores assigned to the seven drinks

 

Drinks a b c d e f g

 

Scores 56 31 52 24 29 29 31

 

 

 

 

Although the dispute 'b' vs 'f' was settled, 'b'
still could not solely claim the 3rd place, because 'g'

75
came up with an equal score of 31. Also, unexpectedly,

'e' and 'f' shared the 4th place, with an equal score of

29. These two instances left a strong impression on

the children about egualitz‘ in mathematics: things

that looked and tasted so differently (such as drinks

'b' and 'g') could be said to be equal when we considered

another attribute (score) possessed by both. This subtle
meaning of 'equality' should always be emphasized in
applied mathematics. The writer has not high school
graduates who did not appreciate the equality expressed
in "Force - Mass x Acceleration.“

Something also seemed to bother the children of
this group.

--"How did you know that the top drink should receive 7
points, and not, say, 10 points?"

--"What made you decide that the last three choices
should deserve any point at all, since they would
probably not sell very well?"

The children's doubts were quite legitimate, and the

teacher should enlighten the children, once for all,

that all ggading systems“ (methods of assigning scores)
were, to a certain extent, arbitrary. A different
grading system, say,

lst choice -——-> 10 points,

2nd choice ——> 9% points,

3rd choice ———> 9 points,

7th choice ——-> 7 points.

74
might very well be applied to Table 4.1, and data quite
different from Table 4.2 would be obtained. This second
grading system would certainly narrow Egg gap‘ between
the best-liked drink, and the least-liked drink, but
would the ranking (ordering)" among a, c, b, g, f, e, d
be an different? It would be an interesting exercise
on multiplflng and adding fractions" to find this out.
(See results in Table 4.3).

Table 4.3: Scores obtained from another
grading system.

 

Drinks a b c d e f g

 

Scores 86% 74 841$ 701$ 73 75 74

 

 

So, the children could have been given the oppor-
tunity to discover that changing from one equal-interval
grading system into another did not affect the ranking
or ordering of the drinks. It would have been even
more interesting if the children were led to discover
why. This would be the children's first opportunity
to meet an invariance u_n_d_.g_g _a_ transformation,“ the key
to all modern mathematics.

[Think of the 7 scores in Table 4.2 as 7 geometric
mints“ on the number-line“. The changing from one
equal-interval grading system into another corresponds
to the translation" of the 7 points, together with the

75
shrinking” 93: the scale. These two transformations do

not affect between-ness” of points on the number-line,
and hence the ordering remains the same.)

The above geometrical” eglanation can easily be
demonstrated in the Design Laboratory, using a fairly
thick elastic band with 7 points marked off when the
elastic is stretched. The band is then bodily shifted
and allowed to shrink. None of the 7 points swap places.

Without changing the theme of this sub-challenge ,
the data in Table 4.1 could also be used as a spring-
board to introduce 'reasoning' by 2.9.33.2 2.; algebra".
That is to say, an algebraic method could be used to
show 3131 the ordering among 'a, c, b, g, f, e, d' did
not change when different grading systems were
assigned, provided equal intervals were maintained
between pairs of scores given to two consecutive choices.

[Let XI, 12, X}, 14, XS, 16’ X7 be the scores assigned
to the lst, 2nd, 3rd, . . ., 7th choices respectively,
with the condition

11> XZ>X3> X4>XS>X6>X7 > 0.

This condition should be 'equal interval', i.e., select
d > 0 such that

12'13’d 12-15+d
no. 01‘ 0.0

16-X7-d 16-X74-d

Score of drink 'a'

Score of drink 'c'

76

4 x1 + 3 x2 + 2 x“

b

+ X + X

3 X1 + 4 12 3 6

=X-X-o-X-X

1 2 3 6

'c' > 0

Therefore, 'a' 'c'

Since '> X

69

Xit> X , and X 'a' -

3

Hence, the score of drink 'a' is always above that of 'c',

of 21,

provided the above condition is satisfied.

regardless of the numerical scores 22, X}, X6,

The ‘b' vs 'f' dispute: Score of 'b' 8 X1 + X2 + X3 +

2Xh + XE + 317

Now XS . x7 + d
Is - x6 + d . (x7 + d) + d . x7 + 2d
and so 1, - x5 + d = (x7 + 2d) + d . x7 + 3d
X3 - x, + d - (17 + 3d) +‘d c 27 + 4d
x2 - :3 + d - (x7 + 4d) + d - x7 + 5d
xi - x2 + d . (x7 + 5d) + d - x7 + 6d
Hence Score of 'b' . (x7 + 6d) + (x7 + so) + (x7 + 4d)
+ 2(x7 + 3d) + x7 + d + 5x7
- 91.7 + 22d
Score of 'f' - 3x: + 515 + 2x6 + x7
. 3(X7+4d) + 3(x7+2d) + 2(xv+d) + x7
- 9x7 + 20d
‘b' - 'r' . 2d > 0. Thus 'b' > 'f']

77
The above algebra.may be a little too difficult for Sixth

graders, but, in view of the fact that USMES will be
extended into Grades 7 and 8 soon, this algebraic work
may be introduced with many subsequent benefits.

Activity 3: Controlling one of the independent variables:
Now that the 'best' drink had been chosen by the
group, mass-production, suggested a‘boy, should be
started immediately so that it could be sold to everybody
in the school. A girl was not too happy about the
suggestion, because drink 'a' had been voted lst choice
only by 44% of a very small group which was not represen-
tative of the whole school (She meant 'target population').
Besides, she felt that the first two choices (drinks 'a'
and 'c') tasted equally good. Drink 'a' had won because
it had an attractive color, and 'c' might have won.if
it were judged on.the basis of taste alone. The teacher
said that this statement (conjecture)‘ could be tested
out scientifically by controlling“ 933 independent
variable (look) and leaving the other independent vari-
able to function alone. One girl who had taken part
in the USMES Summer Workshop in Lansing recalled the
blindfold method to eliminate the variable 'look'. She
suggested to use the method, and the class than pro-
ceeded to rate 'a' and 'c' once again, but this time
each rater was blindfolded when he tasted the two drinks.

78
The grading system was the same as before: 7 points for

lst choice and 6 points for 2nd choice. The results
were tabulated in Table 4.4.

Table 4.4: Comparing drinks 'a' with 'c'.

 

 

 

a
Raters .3
0
use.
efifi’fi'éfiég
Drinks esssezsgss
Drink'a' 66766777658
Drink'c' 77677666759

 

 

 

The class decided that this result was inconclusive,
because

(i) it was a close race; the result 21.5217. b_e_ £133
332 chance‘. (Afterall, the rating was subjective).

(ii) When Table 4.1 was compared with Table 4.4,
it could be seen that there was only one person changing
the ordering of 'a' and 'c'. It was this person's voting
that gave 'c' the lead this time. The majority still
voted as before.

Although the result was inconclusive , the time was
well worth spent. It gave the children some idea how

to eliminate‘ or control a variable. It was also the

”1"

79

first time that the children experienced the formation
of a conjecture or gypothesis‘, conducting an experiment
3:3 1:353“ t_ne_ hypothesis, and deciding whether the results
were conclusive or not.

The teacher also pointed out that it did not really
matter even if this test was inconclusive. In a real
business situation, the store would sell both brands in
any case. Consumers are usually ready to switch to
another brand of the same product, if the brand of their

choice is sold out.

Activity 4: Preparipg a large quantity of the two kinds
of drinks for all their Eera (Sixth graders) to taste

and comment :

 

Apart from the purpose of communicating the progress
of the Soft-drink Unit to their peers who were working
on either Consumer Research or Designing for Human Pro-
portions, the Soft-drink group also wanted to survey
their peers' opinion on the drinks they had produced.
The teacher advised them to make the two top drinks
('a' and 'c') in large pitchers, so that their peers
might taste them and make comments. Thirty-two fifth
and sixth graders (those who were in the same home-room)
were expected to take part. From past experience, it
was estimated‘ that 1/4 of a paper cup of each drink
should be allowed for each person to taste and re-taste.
”How many cups of each drink would be required for 32

80

people?" asked the teachers. This was a problem on
ro ortion‘, and the whole class helped out to find its
answer:

"1/4 of a cup for one person.means 1 cup for 4
persons. 32 . 8 x 4, so we need 8 times as much, i.e.,
8 cups.”

”How much drink will be needed if the teachers of
the other two home-rooms want to take part too?" asked
the writer.

”There would be ample drink for two more persons,”
replied a girl, "because 1/4 cup for each person is
already a lot."

”What Mr. Sompop meant was how to figure it out
mathematically if there were 34 persons instead of 32?"
said the teacher.

”Let me see," replied the girl, "Two more persons
would require 2/4 of a cup extra,“

"What is 2/4?" asked the teacher.

”Two-quarters means one-half" was the final reply.

So the class proceeded to make the drinks by
.measuring out 8 times everything specified in the
previous recipies. First, it meant the pitcher should
be big enough to hold 8 cups of water. A boy suspected
that the pitcher was not large enough, because, as he
put it, {ppg,(linear) measurements‘, 1. e., the width
of the mouth and the height, of the pitcher are only
slightly more than double those of the cup, and they are

 

not 8 ti
it out I
by one,
that the
of watel

At
tionshi;
W
priate e

(1}

block w

 

is asked
every we
6 units
find out
“'0. and

(2)
3 C liné
cylinde;
can are I

take 8 I

 

 

81

not 8 times as much.” The teacher asked him to check
it out by measuring 8 cups of water and pour them, one
by one, into the pitcher. To his surprise, he found
that the pitcher could hold slightly more than 8 cups
of water.

At this point, an informal discussion on the rela-
tionship of linear measurements 955; volume/capacity
measures“ could have been introduced. The most appro-
priate experiences seemed to be:

(1) Starting from a certain cube, say, a Dienes
block whose dimensions are 3 units each, if the child
is asked to double the linear dimensions of the cube in
every way (i.e. to make a new cube whose dimensions are
6 units each from several smaller ones), the child will
find out that he needs eight smaller blocks, and not
two, and we write 8 as 25 or 2 X 2 X 2.

(2) A pop can or beer can should be introduced as
a cylinder”, and an empty (open) coffee can as a larger
cylinder. If the diameter” and heigt“ of the coffee
can are roughly twice those of the pop can, then it will
take 8 pop cans of water, not two, to fill the coffee
can. Again, it is a sort of M relationship"
between a linear measurement and the volume measure.
This second experience is necessary to convince the child
that such a cubic relationship exists not only for 31311”

rectilinear objects, but for 'round' objects as well.

82

The recipe for making one cup of drink 'a' was as
follows: 3 teaspoons of sugar, 1/4 spoon of orange
Kool-Aid, 1/4 spoon of cherry Kool—Aid, and 1/3 + 1/4

spoons of an 'undisclosed' flavor. (The students
insisted that each winning entry should contain an
undisclosed ingredient). As for l/3 + 1/4, they wrote
it this way not because they did not know 1/3 + 1/4 -
7/12, but they thought l/3 + 1/4 was an easier fraction
to work with: they put in 1/3 spoon, and then 1/4
spoon of the same powder. It would be more difficult
to estimate 7/12 of a spoon.

When the writer asked the students how they would
deal with 8 times the quantity (1/3 + 1/4) spoons, one
student explained:

"Four times 1/4 spoons will give one full spoon,
and so eight times (twice as much) will give 2 spoons.
Three times 1/3 spoons give 1 spoon, and six times give
2 spoons; but we want eight times, so we put in 1/3
spoons for two more times."

Apparently, this student grasped the intuitive idea
of a multiplicative inverse", and she also knew how to
use the distributive _l_a_w_‘ g; multiplication 9193 addition,
since she worked with 8): 1/4 and 8 x 1/3, and then added.

During this period of Activity 4, two more mathe-
matical problems arose naturally. Unfortunately, these
two problems were not pursued in details, and no) solutions

to the problems were given in class. The first problem was:

85

"Each packet of Kool-Aid (of the unsweetened type)
weighed 0.15 oz. and dissolved in,2 quarts of water
(sugar to be added afterwards). The teacher just brought
the class another type of Kool-Aid with sugar already
added: this packet (3.30 02.), when dissolved in 1
quart of water, was ready to drink. The teacher asked
what percentage of sugar was contained in the sweetened
type of Ecol-Aid.“

The second problem was:

"A.boy who was mixing drink 'c' in.a pitcher found
that the drink produced according to the recorded recipe
turned out to be too "gross“ (thick). So he poured out
1/10 of the content (of the pitcher into a glass, and
filled the pitcher up with water. He repeated this for
3 times, and the drink then had the clear and attractive
look he had hoped for. He wanted to know what fraction
of the original Keel-Aid remained in the pitcher, and
what amount of sugar must be added to maintain the same
level of sweetness as before."

The writer of this Case-study was of the opinion
that the solutions of these two problems should be
discussed and made available to the students while the
level of enthusiasm was high. Grade 6 is the proper
time to introduce algebraic techniques.‘* The solutions
of these two problems, as given below, would have pro-
vided the best motivation for learning algebra.

*

[Solution to Problem 1’: Let X (02.) be the amount

of sugar in each packet of the sweetened type, and thus

the amount of pure Kool-Aid in the packet - 3.30 - 1 oz.

 

*-

‘It should be pointed out to the children that the
symbol I used in.Prob1em l connotes a different meaning
from that in Problem.2. In Problem 1, the equation is
true only for ONE value of I, while in Problem 2, the
calculations will remain true for all values of I. This
was a point which SMSG decided to emphasize, and it was
one of the first deliberations of SMSG during its first
Writing Session.

Twi

oz. of 1

Thus the

Besides
good pr
also re
Sweeter;

|
I
(97.7%):

C!)

n
\w

I

anOmit
vent th
of {(001,
lst dill

in the +
lifter t

 

84
Twice this amount would function just like 0.15

oz. of the unsweetened type of Kool-Aid, therefore
0.15 - 2 (3.30 - I)
0.15 - 6.60 - 21
2X - 6.45
X - 3.225 oz.
Thus the percentage of sugar in the packet

' '22 x100 -§§§8x loo

' 3%?- - 97.7

Besides algebra, this problem would give the children
good practices on decimals” and long-division". It
also reminded the consumers that, when they bought the
sweetened type, they were paying mostly for sugar
(97.7%), and very little (0.03%) for Kool-Aid.

Solution to Problem 2‘: Let I (oz.) be the original
amount of Kool-Aid in the pitcher. Every time the drink
went through the diluting process, 1/10 of the amount
of Kool-Aid present was poured out, and so, after the
let dilution, X - 1%, X - I31 oz. of Kool-Aid were left
in the pitcher.

,After the 2nd dilution, the amount left

-r%I-rt (1‘3!)
- 'Ig'x - 'Ibgvx - 'I8% 1 oz.

85
After the 3rd dilution, the amount left
- as x - e as an

- «1% X - 18%” X - $883 1 oz.

Hence the new 'clear' drink contained only I853 or
roughly 3/4 of the original Keel-Aid put in.
The same fraction of sugar was poured out, i.e. 1% of
the original amount. Originally, he put in 8 times 3,
or 24 teaspoons of sugar, and so he had to add (18%” ) (24)
or jifglg’ or roughly 2 more spoons.

The same mathematics can be used to discuss some
problems in science, e.g. the problem of pumping out air

from an enclosed space.)

Going back to the activity of preparing drinks 'a'
and 'c' in large pitchers for 32 sixth graders to taste
and comment, it could be observed that both the prepara-
tion and the tasting went on fairly well. But the result
of this polling was again inconclusive. So the class
decided to give up polling on 'a' and 'c', and, instead
they would ask representative samples" from every grade
of the whole school to state their favorite soft-drinks
or pops. This immediately led to the tightening-up
of the definition of 'soft-drink' and using this concise
definition to design a questionnaire for the polling.

86
Activity 5: Defining 'Soft Drink' by examining the
mortise of two mutuallypxclusive sets: soft drinks
and non-soft drinks:

The teacher led the discussion and pointed out to
the children that before they seriously defined "soft-
drink", it would help to list items which were definitely
pp}; soft-drinks. In effect, the children learned of
two mutually exclusive sets‘ (although the teacher need
not say so). A beverage could belong to one of the sets
but it would never belong to both sets simultaneously.
Another concept which arose from the discussion of this
dichotomy was negation‘, and also the principle that the
unary Operation of 'negation' , applied twice, would
produce the original statement again:

“" V‘ P => P

The statement m m p took place when the class
decided against a student's suggestion that cider was
a non-soft drink. The student was somewhat sure that
cider contained alcohol. Although he was not totally
forbidden to drink cider, he was under the impression
that it was not good for children. Many argued that
cider was apple-juice, and although some brands of cider
might be slightly alcoholic, nobody got drunk with cider
yet. So the class decided that cider was go; a gig-soft
drink, and without any difficulty on logic, they con-
cluded that it was a soft drink.

87

The list of non-soft drinks was long, original, and
really interesting. (This might be slightly different
from an adult's list). _According to these children,
soft-drinks or pops did ppp,include the following:
pure water, whisky, brandy, beer, cooking sherry, milk,
tea, coffee, chocolate, ovaltine, fresh orange juice,
fresh fruit juice, soup, vinegar, alker-selzer, ...
(This is one very good way of describing'gflppp: listing
its elements‘).

Then came the first abstraction‘: "Soft-drinks or
pops are thirst-quenching liquids which (i) do not con-
tain alcohol, (ii) are not fully natural."

The condition (ii) above precluded milk, soups,
tomato juices, fruit-juices, etc., for being soft-
drinks or pops. The children also discovered one subtle
implication‘ derived from condition (ii): "Soft-drinks
or pops must contain artificial flavors and sugar." It
was this condition that differentiated between fresh
orange juice and orangeade soda. Also by this implica-
tion, tea and coffee and the like were precluded from
being called 'pops'. When the properties of the Non-
soft drinks were fully examined, it was comparatively
easy for the children to formulate the definition of
soft-drink , or m:

88
"A soft-drink or pop is a beverage which contains

soda-water or water (i.e. which is fizzy or not fizzy),
sugar, and artificial flavor(s)."

The (organic) formulation of a definition by the
children themselves after a great deal of pertinent
experience (i.e. mixing and tasting soft-drinks) was
another instance of the healthy method of teaching
mathematics and science in this program. Usually, in a
traditional 'math' or science class, definitions were
thrown at the children, who were then ordered to memorize

as many as they could.

Activity 6: Using the definition of "soft-drink" to
desig a questionnaire for general polling:

In learning mathematics and science, it is very
important to realize that the mere formulation and
memorization of some definitions are utterly useless,
unless ‘_t_hq_e_ definitions‘ 2;: subsequently mg _t_q derive
sometgpg _e_l_s_e_. Here, the definition of ”soft-drink”
could be put to use immediately in designing a question-
naire for polling sample population of the school. Since
'fizziness' and 'artificial flavor' were the two main
features of the "soft-drink" defined by the children,
the first two items in the questionnaire were:

(1) Do you like carbonated (fizzy, bubbly) drinks?

Check: ( ) Yes ( ) NO

89
(2) Read the following list, and write '1' for your

best-liked flavor, '2' for your next favorite, '3' for
the next one still, and all the way down till you write
'15' for your least-liked flavor.

Cherry Lemon, Raspberry

‘___ Grape ___.Root-beer Creampsoda
Pepsi Cola ___.Orange Ginger-ale
Coca Cola Blackcherry 7-up

Royal Crown Cola Strawberry Lime

: Other flavors

At first the children phrased item.#2 this way:

(2) Please fill in the blank space below the flavor
of the pop or soft-drink you like best: .
Soon it was apparent that, if they wanted to construct
a.preference/flavor matrix (like that of Table 4.1),
the sheer,piqu of this matrix might be prohibitive to
carry this work any further. The sample-population‘
was about 30 in number, and this matrix would be of the
size 30 x 30 if everyone happened to have a different
'favorite' flavor. By presenting the sample-population
with the above list, they could limit the size of this
matrix down to 16 x 16. (In the next Activity, they
discovered that even this size was still far too big,
and, in the last Activity of this unit, they used the
blindfold test to convince themselves that many of the
flavors above were, in fact, the same, and the size of

this matrix could further be reduced to 10 x 10).

90

The questionnaire also contained two more items,
both of which could lead to some meaningful mathematical
exercises. These items were:

(3) What is the convenient size for a bottle or a
can of pop?

( ) 8 fl. oz., ( ) 12 fl. oz.

(4) How much are you willing to pay for a 12 fl. oz.
bottle or can of soft-drink?

( ) 10¢. ( ) 15¢. ( ) 20¢

Item (3) should have furnished the most opportune
moment to answer the question "gps_t pp: 9112 i3 1 .f_l. 9_z_.?"
which must have bothered many children’s (and adult's)
minds for some length of time. First, many children
still think that this is a unit of weight, which is not
true. Secondly, all too many children, by rote learning,
verbalize '16 fl. oz. . 1 pint' without any appropriate
experience in dealing with the liquid content of 1 pint,
or 2 pints, or 8 fl. oz., etc. The writer introduced a
one-pint Coca Cola bottle into an USMES laboratory one
day. He asked a boy to fill half the bottle with water,
and said '8 oz.‘ He then asked the boy to half the
content twice, and the boy said, on his own, '4 oz.,

2 oz.‘ respectively. 'Now, you have to use your imagina-
tion a bit,‘ said the writer, 'Half this amount is what

we call 1 fl. oz.’ 'Oh, I see now,‘ remarked the boy.

Alternatively, any' teacher could have introduced

91
perfume bottles of 1 fl. oz., so that children might see
the actual content of this‘pp;p_measure.‘

The terms '8 fl. oz.', '12 fl. oz.‘ as introduced
by the questionnaire should also be called oneehalf pint
and two-thirds pints in order to have some practice on
fraction. For the boy mentioned above, '2/5 pints' had
a deeper significance than '12 fl. oz.', because '2/3
pints' produced in his mind an image of some liquid in
a one-pint Coca Cola bottle which was two-thirds full.

During the discussion on.pop bottles, mathematica-
artistic creativity“ could have taken place with regard
to the design of new kinds of attractive bottles, and
the approximate measurements of such designs. All kinds
of bottles could be brought into the classroom for the
children to examine qualitatively the curved surfaces“
of some bottles, and to study quantitatively their
diameters, heigpts, and capacities.

Another kind of container of soft-drink, the can,
could also be used in many ways as an apparatus for
teaching elementary mathematics. The prototype cans are
circular,pigpp cylinders.“ It would be easier to wrap
a piece of metal wire round a cylinder than to measure
the circumference“ of a two-dimensional circle by using
a piece of thread.

Hence, by using metal wires and a soda-pop can, the
attempt to discover,pgp.number‘3;“ would be less painful

92
than the usual method of using a thread and a 'flat'

circle on paper.

Finally, the last item in the questionnaire about
the prices 10, 15, or 20¢ for a bottle of soft-drink
could be used to generate a discussion on inflation which
is usually quoted in terms of percentagg increase.‘

The middle-class children may not be too thoughtful
about an increase of 5¢, but it represents an increase
of 50% (from.10¢ to 15¢) and 33fl% (from 15¢ to 20¢)
respectively. In Thailand, the inflation.has more or
less forced the coins of l, 5, and 10 cents out of
circulation (l U.S. 8 . 20.80 Bahts - 2,080 Thai cents).
The businessmen use the dirty trick of increasing the
retail prices in steps of 25 cents. It is not healthy
to accustom children to thinking of increases in steps
of 5s while only an increase of 1¢ is justifiable.

Activipy 2: An attempt to poll the entire school with

the questionnaires:
The original plan was to poll the entire school,

 

but as soon as the discussion about this ambitious
project started, they realized their limitations: the
quantity‘ of ditto paper required, the insurmountable
task of countipg‘ and analysipg‘ the returns, and the
pipq‘ it would take to carry out a large-scale‘ survey.
In any case, what was the use of acquiring information
pertinent to one school only? Two children hit on the

93
correct approach of getting the opinion of a cross-

section‘ of this school population from.K to 6, and hoped
that the cross-sections of other elementary school
populations were similar. Although the teacher did not
specificly give a formal lesson on stratified samplipg‘,
the children did learn the essence of this topic from
this experience. As events turned out later, the
responses to the questionnaires did come from a fairly
even cross-section of the school population.with respect
to grade levels and age-groups. It would have been
even better if these stratified samples were randomly
selected“. The writer sincerely believed that the
excellent co-operation between.the office and the class-
room teachers at Wexford would have permitted random
selections (pulling names out of pro-arranged bags) to
take place.

The survey group came across one difficulty when
polling the samples from kindergarten, grades 1 and 2:
the primary-grade children could not read most of the
writing in the questionnaires. This difficulty was
overcome by the service of three volunteers from the
Soft-drink group who patiently talked to the younger
children (K-2) about fizziness, the flavors of the drinks,
and the price, and these volunteers tried to record their
verbal responses as accurately as possible. One girl
remarked afterwards that it was delightful to talk in
'kindergarten! language once again, (e.g. 'bubbling' for

94

carbonation). But she was seriously doubtful whether
the younger children really knew some particular flavor
she was talking about, unless they had tasted the £221
t_h_ipg before, and in such cases it would be meaningless
to ask the question whether they liked that flavor or
not. So, even a sixth-grade student can deduce that
experience reigns supreme in education: It was unfor-
tunate for the overall result of this polling that, when
the younger students did not know any flavor, they
marked them as 15th choice (least liked) instead of

'no comment,‘ and hence the 15th choice here did not
necessarily reflect the quality of the drink. (See the
15th column of Table 4.5).

The sample population‘ was 32, but only 26 of the
returns (if counting both verbal and written returns)
were analysable. The class decided to use the Frequency
Method (See Activity 2) to analyse the returns with
regard to the p0pu1arity of different flavors. The
result is tabulated as follows: (See Table 4.5 on
following page).

It took the children.five class-periods to establish
Table 4.5, and they were somewhat tired at the end.

This feeling was not altogether bad, because it forced
them to take another good look at the 15 x 16 matrix

they had constructed, and to find an.offective way pq_
reduce‘ £13 pip: q_i_‘_ 331; matrix. (See the next Activity).

 

1!- .

'in. Fri:

Ii l.\: l| I .i

F. f‘ll

II: ’ "I.I

, 2nd, . . .l5th choices.

which each flavor was voted lst
was for checking whether the frequencies

votes)

(total‘

Pro quency by

The last column

Table 4.5:

were counted properly.

Total

 

 

 

 

\o u>xo m>xo u>xo u>xo u>xo u>xo u>\o
Votes NNNNNNNNNNNNNNN
N0 cu W\r443 c>.4 c><3 wxca enc: nice on
Comment
15th oaooomooaoommom
14th OOHHHMNOOOmHHNm
13th ooammmoasmaosmm
12th o1c>.4 c>cu N\r4 mseirn mtratu enc:
11th Haaaemoemosooom
10th NHHNNHHHOMNMHHM
9th OHMOHNHHNNHr-cmerm
8th HNOONNemmmaoaw-qm
71in mmoaaaommaammao—I
6th wxuxrn c>r4¢3 «\rn F'CUaH 01¢: K\r4
5th HNMMMONHCMHMNHFJ
4th r4!“ olin wxrq wxc>.4 04.4 n1.4 N\<D
3rd e~<3 alcu s-c: m~c>.a xxcu c><3 01¢)
2nd 01.4 d-Ln c>cu s-Cu N\r4(3 ri<3 r1<3
let .4 # «\rn otc><u M\Fir4:4 #‘Fiffi o
m
3
'H m
m a:
g 8 a? n gt, 3
P as o H
.3 8° BeBB‘am'.
fin Eon-llvgfi boa-33.050
‘4de 0950mm. gm
masseuse” ‘35
ommomnmommmowuq

 

nail

1105'

chc

96
With regard to the other two items in the question-

naires, it was reported that

(i) 20 had voted for carbonated drinks, and 6 for
non-carbonated drinks.

(ii) 14 had voted for the price of 10¢ per bottle
and 12 for 15¢ per bottle.

The way the children interpreted Table 4.5 was
interesting. On the criterion of using maximum‘ lst
choice frequency, they decided that Grape and CreamPSoda
(lst choice frequency - 4) might be the two most popular
flavors. But they were not too sure because Pepsi and
Coca-Cola ranked lst choice with frequency - 3 ppg.2nd
choice with very high frequencies compared with the and
choice frequencies of Grape and Cream-Soda. Orange and
7-up were definitely runners-up with lst choice frequency
. 3. Ginger-ale seemed to be least liked by this sample-
population because it ranked 15th choice with frequencies
- 6. Perhaps the majority of this population.had never
tasted this flavor before. The case of Cream-Soda
seemed to confirm this pypothesis.‘ While CreamPSoda
ranked lst choice with frequency 4, it simultaneously
ranked bottomrchoice with frequency 6. It meant that
some children who had not tasted Cream~Soda before wrote
down 15th choice instead of 'no comment'. The same
reasoning‘ could be applied to the cases of Coca Cola
and.Pepsi. Every child in the sample group seemed to
have had experience with these flavors before. The

97
result was that Coca-Cola and Pepsi enjoyed high fre-

quencies for lst and 2nd choices and zero frequency for
bottom choice. The teacher pointed out to the class
the tremendous effect of advertising, which at least
helped to introduce certain products to the public.

Just like Activity 2, the next step was to assign
some scores corresponding to the lst, 2nd, ... choices.
This time the children knew that the scores could be
assigned arbitrarihy‘, so long as they were consistent‘
_i_n_ 12113;}; calculation. They seemed to be proud of the
fact that they could assign scores freely, and they
decided to give 5 points for the lst choice, 3 points
for the 2nd choice, 1 point for the 3rd choice and none
for the rest. The teacher called this system‘q'weigpted
gqqpq‘, because it was tilted in favor of the top choices.
The children then proceeded to calculate the total
score for each drink. The list of top drinks, with
their respective scores, is shown in Table 4.6.

As expected, Coca Cola and Pepsi turned out to be
the top two, having scores much higher than those of
Grape and Cream Soda. The scores for Coca Cola and Pepsi
would have been even.higher if the 4th choice were given
some points as well. In fact these scores could be
regarded as a measure‘ of how well-known.a certain drink

was among the population sampled here.

98
Table 4.6: The 8 most popular flavors,
arranged according to their scores.

 

 

Flavors Scores
Coca Cola 32
Pepsi 29
Root-beer 25
Grape 23
Cream Soda 23
Orange 21
7-uP 20
Cherry 18

 

 

 

 

The children then discussed the common features of
these two top flavors. They all agreed that Coca Cola
and Pepsi were very similar to each other: both were
carbonated, sweet and both had a tainty 'Cola' taste.
The girl who participated in the Summer Workshop
immediately suggested the blindfold test to prove that
nobody could tell the difference between Coca Cola,
Pepsi Cola, Royal Crown Cola, and indeed spy Cola (such
as Faygo Cola, Shasta Cola, etc.) The rest of the class
did not believe her, and so the blindfold test was going
to be the next and last activity of the Soft-Drink Group
before the Christmas vacation.

99
Activipy 8: The blindfold test to determine the

qquivalence of all Colas:

This activity was one way to introduce the concept
that two things are equivalent‘ if one cannot be dis-
tinguished from the other. It is an important concept
in geometpy“, e.g., since a circle and a square cannot
be differentiated from each other if we consider only
the property of simple closed curves,“ then a circle
and a square are said to be topqlogically“ e uivalent,
and the properties associated with simple closed curves
are called topological properties. By the same line of
ggasoning, since it could be established by the blind—
fold test that all Colas could not be distinguished from
one another if only the variable 'taste' was considered,
all Colas should then be equivalent as far as taste was
concerned.

All students and an invited teacher from another
class took part in this experiment. Each took turns to
be blindfolded and then given 3 kinds of Colas (Pepsi,
Royal Crown, and Coca Cola) in 3 different cups, and
asked to supply the correct names. Re-tasting was
allowed and everyone was urged to think seriously of
all the previous experiences one had with Colas, so
that guesswork or playful answers were cut down to the
minimum. The experiment, whose result was tabulated in
Table 4.7, revealed that the majority of this group could

DUO
not give even a single correct response, and everyone

gave at least two incorrect responses.
Table 4.7: Results of the blindfold test: V referred

to correct responses, otherwise the names of the
incorrect responses were shown.

 

 

 

Correct ‘
Responses

Emperi- Coca Cola Pepsi « R.C.Cola
mentors
Allan R.C. Coke Pepsi
Tom R.C. Coke Pepsi
Eddie v/ R.C. Pepsi
Kathy Pepsi R.C. Coke
Shari R.C. v’ Coke
Sheri Pepsi R.C. Coke
Debbie R.C. Coke Pepsi
The Teacher R.C. ./ Coke

 

 

 

 

 

In fact, Table 4.7 gave more details than required.
(It showed which drink was indistinguishable from which
one.) A simplified matrix, using 0 for incorrect res-
ponses and l for correct responses, would serve the same
purpose, (Table 4.8), and it would immediately lead to
the discussion of binqpy“ digits (0, l) which the
children might have heard before in connection with
computers. The binppy system could easily be followed
by playing with Dienes' Multibase blocks,“ and study-

ing place value.“

101

Table 4.8: Scores given for correct identifi-
cations of the drinks

 

 
  

Experi- Coca Cola Pepsi R.C. Cola

 

menters

Allan 0 O 0
Tom 0 0 0
Eddie l 0 0
Kathy 0 O O
Shari O 1 0
Sheri O O 0
Debbie 0 O O
The Teacher 0 l O

 

 

 

 

The teacher pointed out to the class that, although
no guesswork was implied, the 22211 percentage‘ of
correct responses might well be due to chance. The
writer of this Case-Study informed the class that, in
Montery, California, the Soft Drink group had done the
same blindfold test on 4 Colas (Coca Cola, Pepsi, Royal
Crown, and Shasta Cola), and they, too, had obtained
only a small percentage of correct responses. Only one
student (from a class of 23) gave all 4 identifications
correctly; the majority (10 of them) got the names all
wrong, 8 of them.made three mistakes, and 4 made two
mistakes.

The children were consequently satisfied that all

Colas were equivalent as far as taste was concerned.

102
One boy further remarked: "Since you can't tell one Cola

from another, and since the TV commercial says one of
them is the real thing, all of them would be real things
then." The writer would like to add: "Another way to
look at it is that none of them could be the real thing,
since they all are artificial, when referred back to

the definition."

Now, the children could start working on the inten-
ded objective: to reduce 2.9.2 si_§_e_ pf 329. matrix in
Table 4.5 by cutting down repetitious items in the
questionnaire. First, Pepsi, Royal Crown, and Coca-Cola
could be grouped into one: the Cola flavor. The child-
ren further grouped 7-up, Lemon, and Lime into one
flavor, and also cherry and blackcherry into just one
flavor. There were now only ten flavors left, ranking
from 1 to 10, and hence there were eleven choices
including the 'no comment' column. The size of the
corresmnding matrix‘ was thus reduced to 10 x 11. A
student suggested further that all the other sweet fruit
flavors: grape, strawberry or even raspberry could be
grouped with the cherry set, since the taste-difference
among these fruit flavorings was minimal. Only the
colorings differed somewhat. In fact many firms sold
these products as one item: the mixed fruit flavor, or
Hawaiian punch. Although many in the class agreed with
this idea which would result in reducing the matrix even

further, it was almost the concensus of the class that

103
orange should not be combined with the cherry/strawberry

group, because a good orange drink should have a dis-

tinctive 'orange' taste and a remarkable 'orange' smell.
The discussion on the different characteristics of

the orange and strawberry drinks unexpectedly led to

the mathematical topic of 'intersection 2f Egg _s_e_t_g'.‘

The children listed the characteristics of the two drinks

as follows :

 

Orange drink Strawberry drink

May be carbonated or May be carbonated or
non-c arbonated non-c arbonated
Containing sugar Containing sugar

Having reddish-yellow color Having bright red color
Having strong orange smell Having faint strawberry smell

"You can, in fact, write the common properties only
once instead of writing twice," said the teacher. So
the children wrote the two common pr0perties above in
the middle between the two columns:

Orange drink Strawbergy drink
Carbonated or non-carbonated

Containing sugar
Having reddish-yellow color Having bright red color

 

Having strong orange smell Having faint strawberry smell

The first line of this revised presentation tended
to promote the misunderstanding that the orange drink

was carbonated and the strawberry drink was not. To make

104
things clearer, they circled each side with a different
color-chalk, and obtained a perfect Venn-diagram:

' a -
' Orgge drink ; 1 Strawberry drink

Carbonated

.or Non-c arbonat ed
I

l

{Containing sugar

l

 

 

Having reddi sh- Having bright
yellow color ' red color
and | and
» Strong orange: Faint straw-
smell ;
l

J berry smell

 

T L _____________________________ J

The teacher told the class that the common part
in the middle is called ph_e_ intersection 9; 35333 Egg _sqpq.
This concluded the major types of activities of the
Soft-Drink Unit during the observed period (September 26 -
November 28, 1972).

Conclusion: It seemed hardly possible that some
forty items of mathematical behaviors plus some twenty
potential mathematics lessons should have arisen from
such a non-mathematical theme like 'Soft Drink Design. '
But a reader who actually counted up the mathematical
behaviors marked ‘ and the potential mathematics lessons
marked “ could verify that this vast amount of mathe-
matical learning did take place. Horeover, a reader

105
who thoroughly read Activities 1-8 would easily see

ghy;this mathematical learning took place. The whole

1 but care was

I

approach was organic and evolutionary,
taken so that the approach was not revolutionary
against the traditional content of mathematics. many
of the topics marked ‘ and “ were 'traditional' mathe-
matics in every sense of the word, and, hopefully they
were conventional enough to make it possible for a
conscientious student to pass the usual Stanford
Achievement Test or any State-wide Assessment Program.
Needless to mention that a conscientious student would
acquire more practice on his own with such important
topics as fraction and percentage. The Soft Drink Unit
activities were arranged to motivate the truth-searching
spirit, and to relate mathematics to one of the child's
most enjoyable habits (drinking soda-pops), but pqp,to
replace the hard work demanded by any course in mathe-
matics, even at the elementary level. But hard work plus
enjoyment is a surer way toward success than hard work
alone.

Hard work does not mean regimentation. As a
matter of fact, the USMES philosOphy allows divergence
within the same unit. If this Case-Study is compared

 

1The words 'Evolution, not Revolution' were used
by E. H. Moore in.his 1902 Presidential Address to the
American Mathematical Society, reprinted in The Mathe-
matics Teacher, April, 1967, pp. 360-374.

106
with the reports of other Soft Drink groups, say, the

one in Mbnterey, California, one striking difference

is that the Monterey group did a great deal of work on
graphs, while the Lansing group did not have time to do
any graphing in the Fall term, because they did con-
siderably more on tabulations and calculations. The
Lansing children would hopefully graph most of their
results in the following term.

The "Soft Drink Design“ is a good choice to motivate
the children. The British also include the polling of
favorite soft-drinks in their 'presentation of mathe-
matical challenge to Primary School children,‘ usually
done on a series of work cards. The following, quoted
in full, is one such card.

"Mathematics Vbrkshop Lgard2No. 178

Make up a list of ten flavours, such as
cocoa, strawberry, lemon. write them on a large
sheet of paper and give every child in the
school one vote for his or her favourite flavour.

when you.have counted the votes, arrange

them.in.order with the favourite first. Now
illustratg the choice of children by means of

a graph."
The content of this card is very similar to the descrip-
tion of part of an USMES Soft—drink unit, but the British

work card seemed to be more directed.

 

2Pethen, R. A. J. "The Workshop Approach to Mathe-
matics," Toronto: McMillan Co., 1968, p. 8.

107

The CASE-STUDY of the "Dice Design" Unit,
Observ33_at—W3§Tord School, Lansing, Michigan,
September 25 to November 27, 1972

I. The behaviors to be observed would include children’s
remarks while tossing coins, drawing long-short straws,
choosing left-right hands, and playing various horserace
games; the ingenious ways of recording (tallying) results
in Experimental Probability; the childrenis methods of
graphing, particularly their special method of building
up a histogram; and, time permitting, the children's
manipulation of various polyhedra to be used as dice in
this Unit.

Any mathematical topics arising from class discussions
and students' responses to questions put by the teacher
or the writer would also be recorded.

II. Brief Description of the 'Dice Desigp' Group:
Twenty-six 3rd graders from an entire wing of 'Team 2'

(i.e. Grade 3-4) took part. The time allotted was Monday,
Wednesday, Friday, 1:50-2:30 p.m., but cancellation of
classes due to Parent-Teacher Conferences, concerts, etc.
effectively reduced USMES activities down to twice a week
on.the average. The teacher had taken part in the USMES
Summer WorkshOp in Lansing, but, as it turned out, she
conducted the class by using her own originality (about

108

coinpgames) rather than repeating the techniques and
materials (about polyhedra) she had picked up from the
Workshop.

This was not an all-volunteer class, and so a few
children's overt and covert boredom with the coin-games
was to be expected, but even these reluctant students
did learn at least addition, multiplication and graphs
from this Unit. There was another drawback: two children
were observed to have reading and communication problems,

but they did try to learn something from the coin-games.

109

Activity 1: The games of (i) drawing_straw§,
(ii) choosing hands, and (iii) flipping coins; and the
various tallying methods devised by the children:

The initial discussion on how to choose just one
person from two equally 'good' (meaning 'qualified')
people led to the suggestion of a fair method of selection
by means of a fair game. After eliminating a few unfair
practices such as

- fighting,

- reaching for the highest shelf in the library,

(although this was a favorite game for third graders),

the children gradually converged on fairer types of games
such as drawing the shorter straw out of Egg partially
concealed straws; choosing correctly the hand that held
a.t1ny object (while the other hand was empty); and flipping
a coin. It was delightful to hear that they did not say
outright these were perfectly fair games. They only
conjectured‘ that these might be fair, and they set out
to verify‘ their conjectures. The class then split into
three groups, fetched the necessary instruments (long
and short straws, tiny counters taken.from.'Monopoly' sets,
pennies and tallying sheets), and proceeded with these
three games. The writer asked one pair of children (who
‘were about to toss a penny) whether a thumbtack might be

 

t

Through the Case-Study, the underlined words which
are also marked ’ refer to the mathematical behaviors observed,
while those marked “ refer to otential mathematical topics
that could have been discussed EH class.

110

used instead of a coin. "No," replied the children firmly,
“because a thumbtack is not flat pp 32229 flag; it has
something sticking out on one side." "It is not balanced,"
said another child. Hence the children learned that
‘gzppgppzf or a balanced gpgpg‘ is a necessagz’condition‘
for fairness.

In another group who were drawing straws, one pair
of children wanted to be different from the rest: they
used 5 straws (two of which were 'long') instead of the
usual set of two straws (1 long, 1 short). It did.not
take them long to realize that the game was not fair.
The holder of the straws had a 'better chance' of winning
than the drawer. Theoreticall ,“ the probability‘ of
winning:= 2/3 for the holder and 1/3 for the drawer, and,
as a matter of fact, the result obtained from this game
came remarkably close to the theoretical values, i.e.,
in 30 trials, the drawer succeeded to get the shorter
straw 9 times.

Although most couples in these games kept only one
tallying sheet, a few pairs hit upon.the ingenious idea
of checking‘ppg accuracy‘ of the scores by keeping two
tallying sheets. Each.student simultaneously recorded
his own winning score ppd_the other side's losing score,
or vice versa. At the end of the game, the points lost
by one should pgpplf the points won by the other unless
one of them cheated. This was an instance in which egualitz‘

occurred in a non-trivial manner. This method of checking

111

debit/credit items (double gpppz'book-keeping‘) had, in
fact, been invented by Simon Stevin around 1600 A.D.,
and has gradually developed into the modern accounting.
During this activity, the elementary principle of 2225‘
keeping“ could have been introduced, by considering on
one side the teacher's (real or hypothetical) incomes
from the usual salary and summer jobs, and, on the other
side, the grocery bills, the mortgage payment, the insurance
premiums, etc. Alternatively, the teacher could have
introduced budgeting“ for such activities like camping
which, again, involved incomes and expenses. (The children
sold apples to realize some income for the 'Camping' group.)
Now, back to the games. First, it was interesting
to observe different tallying methods. Most of them
seemed to know intuitively that the results to be tallied
were divided into $3.12 mutually exclusive _sp_1_:_s_‘ (heads-
tails, or winning-losing). Only a few students recorded
in terms of mixed data like HHHTTHTTTHHTH... These
children would find it hard to count accurately afterwards.
Some children drew ”smiling faces” for 'heads' and ”fish-
tails" for 'tails'. But some tried 33 abstract“ the
acts of winning and losing by writing one I for a point
won, and one 0 for a point lost, and then counted the Pa
and 0's afterwards. Negative integers” could have been
introduced to these students by writing -1 for the first
point lost, -2 for the next point lost, then -3, -4, --5
and so on. One girl, Lori, did use the gap 9; integers‘

112

for recording her results of flipping a coin, but she
used it to record the total number of throws rather than
separating the throws into 2 sets: positive integers
for heads and negative integers for tails. Had she used
positive and negative integers, she would not have to
count the numbers of heads and tails afterwards. But
Lori's method had the advantage that, if she wanted 50
throws exactl , she would know when to stop.

Since no student in this group used positive and
negative integers to record the scores of the two sides,
they all had to undertake the next task of counting' the
two sets of tallies to find the (total) scores for 'heads'
and 'tails', or points 'won' and 'lost' -- as the case
might be. Most of them wrote the tallies in each set

into gpoups pg 2", as the following examples:

Example 1. Throwing a coin:

 

Heads Tails
Group 1 ///// /////
Group 2 ///// /////
Group 3 ///// /////
Group 4 ///// /////
Group 5 ///// /////
Extra // ////

 

 

 

 

113

The student counted: 5, 10, 15, 20, 25 and then 26, 27
points for 'heads' and 28, 29 points for 'tails'. This
presentation showed that the two children here knew the

Multiplication Table’ for 5.

Example g. Choosing hand:

 

Points won Points lost

IIIII 5 00000 5
IIIII 10 00000 10
IIIII 15 00000 15
IIII 19 00 17

This pair of children did not just verbalize 5, 10,
15, ..., they actually wrote down what they said at the
end of each line. This was one of the best ways 32
prevent g miscount.‘ It also shows very clearly that
multiplication _a_; integers‘ is successive addition.
(HereSX3 = 5+5+5).

The activity of counting up the tallies turned out
to be the best motivation for learning multiplication.
Many surprising results followed, for example, the use
of Eaph-papers“ in conjunction with counting in groups
of 5. This method of putting 5 tallies in each square
of the graph-paper and considering 10 squares at a time
was most efficient, and it enabled some students in the
'coin-tossing' group to record scores as high as l96.
The method is demonstrated 33 follows, with part of the
graph-paper shown:

O
O\
H

O
U\
H

i
i

8 8
...

E

 

 

 

 

 

i
i

 

 

i
i

ii%§
§§%%
iiii
%§§§
i§%§

i§§§

 

%§§%\

§%%%%

 

 

 

 

 

LE

 

 

 

 

 

 

115

The brute-force method of counting would certainly have
led to some mistakes at some stage, while this gystematic
method‘ of counting could be done accurately up to several
hundreds.’ The subject of place-valueu could have been
discussed here. In conjunction with this idea, a boy at
North School, Lansing, did even better: he drew a small
rectangle to represent 5, and a small triangle to represent
50. He kept on tallying until he got ten rectangles

before he changed them into a triangle. This is how he
finally wrote 65 heads and 64 tails:

A [_j Lfi [_fi heads, A :1 (:3 //// tails.
The exchange of ten rectangles into a triangle should

 

have prompted the discussion of different denominations"
of the U.S. currency. The discussion, focused on the
one-dollar, ten-dollar, and a hundred-dollar bills,

would in turn.have led to very vivid explanations of the
old favorite subjects of 'carrying' and 'borrowing'.
('Exchange"‘ is probably a better word for both these

terms.)

Activipz 2: Abstract some conce ts in Probabili
from the results of plgying the three tzpes of games:

All the three types of games mentioned in.Activity 1
provided a common experience to the children: they all
learned the vocabulary of a dichotomous situation cone

stituted by two mutuallz exclusive sets,‘ and this

116

situation was the realistic counterpart of the theoretical
probabilig‘ of value 1/2. In the coin-tossing group,
at first it was a certain James vs. a certain Stacey,
but eventually they realized that it was simply a matter
of heads vs. tails. Moreover, James and Stacey could
easily switch sides (because the scores“ for 'heads' and
'tails' were nearly the same), and thus there was no
advantage in monopolizing 'heads' by one person or the
other. For the other two groups, the teacher carefully
explained that the game was not really about one child
betting against another, but was more concerned with the
chance‘ (probability) of choosing correctly when somebody
had to choose one thing from a set of two, (either two
hands or two straws). The children apparently understood
the implication of the games, and they by themselves
changed the egocentric names in their tally-sheets from
Kim vs. Lana, Peter vs. Dee, etc. into more abstract
names such as

Get -- Don't get,

Right -- Wrong,

Yes -- No,

In -- Out.
The final abstraction‘ occurred when a child wrote H vs. T,
and the whole class agreed that from then on the two
scores on their tally sheets would be re-named H and T
respectively. This was an example of generalization’.
Here the scores generalized H = 19 and generalized T = 17,

etc. in Table 4.9 below need not refer to heads and tails,

117

but they might refer to the scores obtained by the two
sides playing the games of choosing hands or drawing
straws. It could be seen from this Table that the scores
obtained by these children, especially the pooled scores,‘
were not too far off from theoretical values." Another
USMES class at North School, Lansing, did coin-tossing
only, and their results were also close to the theoretical
values.

Table 4.9 The actual data recorded from the
H/T types of games.

 

 

Uexford School North School

H-points T-points Heads Tails
27 29 65 64

19 l7 7 9

141 119 24 25
176 196 8 8
196 196 55 39

19 25 29 26

21 19 12 16

10 8 27 29

27 21 15 20

64 66 58 40

45 43 30 50

18 21

745 73? 308 527

 

 

 

 

 

 

To find the gppg' of the two columns of H and T
became a formidable task for the third-graders at Wexford
School but it was no problem at all for the sixth-graders
at North School. At Vexford, several methods were

118

attempted by the children: adding two numbers‘ at a time,
PM 5 graph-paper‘ to regulate the place-values“ while

column addition was carried out, and there might be more
ideas. But no student got the correct answers, and
finally the correct sums had to be supplied by the teacher.
But they did learn something during various attempts,
for example, ”adding two numbers at a time" was a real-
life instance of experiencing the closure 13:} in algebra.
Column-addition was a happier event at North School: the
teacher reported that the children merrily 'chorused the
addition' as they produced partial pppgf along the column
(for each place-value). But she did not report whether
they checked the answers by addingbackwards.“ If they
did, that would be an.example of using the commutative
lppf‘ for addition.

The mathematical learning derived from Table 4.9
was considerable. The teacher (at Wexford) asked the
children to find the total number of 'throws' (generalized
throws) of the pooled results. Most students could do
this without difficulty:' they just added up the two
sub-totals‘ (745 and 757) to get the gpppdnpgpgl’ of
1482. In an attempt to introduce the statements:

Probability of getting H =: 1/2,
Probability of getting T = 1/2,

the teacher asked the class how much was one-half of 1482.
Surprisingly, quite a few students gave the correct

answer of 741. It showed that they knew the Multiplication
Epic] for 2, and the process of division ‘91 g‘. Then the

 

teachc
halve:
half 1
this .

err."

 

 

119

teacher explained: "If you split the grand total into 2
halves, about one-half will go to H, and also about one-
half will go to T." The children did not quarrel with
this small discrepency, but, at this point, the subject

of relative error“ could have been discussed.

CW: Score for H = 745

One-half of the grand total == 241
Error == 4

Thus the relative error is 4 parts in 741 parts,
or, approximately, 1 part in 185 parts,

which is a small error.)

This activity also gave the students a first—hand
experience about the well-known fact that a small relative
error is to be expected whenever theoretical ppg_e§peri-
mental results‘ppg com ared,“ like the present case:
Iggg' against the theoretical value of'%. In any case,
such approximations" as fig; 2: $- and 7%:- 2 2'60 are
valuable facts to note. The writer often recalls such
answers to center-of-gravity problems as “1&3; ft. above
the base," which should, in reality, be approximated by
évft. Strange enough, the elementary students in these
USMES classes did not find it difficult to accept
1%; c: i» or Egg 2: é», although they knew very little
about fractions. While playing coin-tossing or similar
games, they were keenly aware of the fact that this
activity was a means to figure out the chance‘ of getting

 

one OJ
'heads
these

"one <

been 1
three
coins
H and
might
as:

VS,

*.‘0

event

&I‘ec

 

Bic:
W!
or be
only
then
While

as 9/

 

 

120

one out of two possibilities. Since they obtained 745
'heads' out of 1482 throws, or 508 heads out of 655 throws,
these two probabilities had to be approximately equal to
”one out of two".

During this activity, another concept could have
been discussed: that of equally likely events.“ The
three games of choosing hands, drawing straws and flipping
coins will each generate two events (called generalized
H and T), both of which are equally likely. Children
might be asked to suggest many more opposite-pairs, such
as: to be healthy vs. to be ill; rich vs. poor; genius
vs. idiot, etc., which are not pairs of equally likely
events, and the probabilities" associated Lip]; pkg-gig £133
are certainly not 0%, %). Take for example the dichotomy
'Hich vs. Poor'. In many countries, the poverty line is
clearly defined and evidently visible, the probability
of being born in.a poor family can be calculated. If
only 10% of the population live above the subsistant level,
then the probability of being born 'rich' is only 1/10,
while the probability of being born 'poor' is as high
as 9/10.

The dichotomous situation also lends itself to the
discussion of complement" pg _a_ universal gap. In the
coin-tossing game, the complement of 'head' is 'tail'.
In.the set of human beings, the complement of 'men' is
the subset of 'womenf; the complement of‘the rich' is

the subset of poor people. This last example illustrates

 

graph

0f co

 

 

121

that the complement of a set may have a larger or smaller
cardinal number“I than the original set.

One last point about Activities 1-2: an abacus“
could have been used both for tallying and for adding

integers, especially those arranged in a column.
Activity 5. Graphinggthe data obtained from the games:

So far the children.in this class had been using
graphppaper for keeping tallies, facilitating the process
of counting, and other purposes, but not for graphing.

It was apparent that such a rich collection of H and T
data could be graphed’ in a meaningful way. The teacher
at the point motivated the class to graph their results.
First, she asked what was the purpose of spending so many
USMES periods playing the H/T types of games, and the
children replied: "To prove (verify) that tossing a
coin, or playing other games like it, is a fair'means to
choose one person from two equally good people.“ The
teacher then.asked the class how to communicate this
important message to other people by means of graphs.
The following graphs were suggested by some children:

(1) A pair of bar-gpaphs* for each couple of (H, T)
data: the two columns were to be placed side by side
but colored differently, say red for H and green for T.
Nearly every red column had approximately the same heigpt‘
as the adjoining green column. PeOple could then tell

 

 

 

122

at a glance that, in most cases, the chance“ of getting H
is approximately the same as getting T. (See Figure 4.2).

(2) Putting all the (red) H-columns on one graph-
paper and all the (green) T-columns on another sheet:
the first graph-paper then contained a patch of red area,
and the second a patch of green area. Using scissors to
cut out the two 335$” and lay one on top of the other
could demonstrate that the overall chances“ of getting
H and T were approximately the same (provided all the
columns had the same pic-11:33)

The writer would like to add that, in (2) above, the
5332“ of H-columns (already cut) could have been arranged
in ascending p_r_de_1;" according to heights, and thus the
median" of H could have been spotted directly from this
arrangement. Similarly, the median of T could have been
found. (See Figure 4.3).

 

NC
A ‘

0C

“175'

his“, he. an, Earth a .
2

W2?
‘ 101'
T75

1
F

125

[\No. of tosses

1 200

 

 

 

 

 

 

 

1.175

-150

\

 

 

 

 

 

 

 

in most cases, nearly equal

getting T.
Data were taken from Table 4.9.

A visual presentation to show that the chance
Scale 1cm - 10 tosses).

of getting H is,
to that of

(

Figure 4.2:

[2:] Tails

Heads

\\\\\\\¥

4.!

 

of tosses 124
200

No.

7F

 

 

 

 

W

 

 

Median

g§§§§§§§§§§§§§§

§§§§§§§
§§§§
§§§§x§

175
150
125

100

75

50

 

 

prolonged game, the overall chance for

A visual presentation to show that, for a
Hz that of T.

Figure 4.3:

=56.

Median for '1‘

:56:

Median for H

12S

Activity 4. Discussing the difficulty in keepipg
the tallies for tossing a coin EIACTLY_5O times:

The teacher then introduced a special kind of bar-
graph which would show that relatively few players
obtained an excessive number of heads (or tails), while
the majority of players obtabned almost equal numbers
of heads and tails. The students were told that, to get
such a graph, everyone had to use exactly‘ the same
number of tosses (no more, no less). The class was also
informed that, at Eaton Rapids, Michigan, an experimental
USMES group had tried this idea with 20 tosses by each
student, and got a beautiful graph that looked like "a
tower with several layers of broader bases." (Normal
Distribution’). But the teacher added that the more
tosses, the better. The children volunteered 100 tosses
each. Having read the report of the ”Dice" group at
Lexington, Mass., the writer informed the class that 100
tosses might prove too boring, or at least it was so for
some people. So the children settled for 50 tosses each.
It turned out to be a wise choice: nobody got bored,
and moreover, they unexpectedly faced a new problem in
tallying exactly 50 tosses. The solutions which these
children found for this problem would enhance difficulty
in counting H and T separately, if the number of tosses

was too big.

 

methc
, .
tall;
tall

tn
:1
(1'

them
and,
was

2 :1
Quit
atte;

were

 

\

net

 

126

The new difficulty arose from the fact that the older
method of tallying (as used in Activity 1) did not work
any more. Many children started by the old method: they
tallied in two separate columns, H and T, grouped the
tallies in sets of 5, and meant to count them up afterwards.
But, by the time they did the actual counting, most of
them discovered that they had done page 3139.3 52‘ throws,
and, since no history of the tally-marks were kept, it
was impossible to figure out which marks were the gippp
59.throws.‘ Hence many had to start all over again.
Quite a few made the same mistake again in the second
attempt, and even in third attempt. Obviously there
were better and more systematic ways of recording the
tallies that would take care of the ppppl,number’ of
tosses as well as the numbers of H and T. About one
quarter of the class were lucky (or bright) enough to
come up with a system that served the above triple
purposes. Examples of such systems were:

(1) Lori's Method: This girl came up with so
ingenious a method that her teacher and the observers
decided to xerox:her work and send a copy to the USMES
headquarters in Boston, Mass. Her secret was to record
not only the number‘ of tosses, but the histopy‘ of all
the 50 throws. In.this way, there was no question that
she might go over 50 throws, because her method could
stop at any n, when n.= some natural number.‘ The record

of her work is fully reproduced here:

 

Ha
rath

the

 

Velo=

 

 

NAME: Lori H T
8 1
9 2

10 5
l5 4
14 g
16 7
20 ll
31 12
4 1%
26 19
27 22
29 25
go 28
53 54
57 55
2e 32
45 42
46 44
47
48
19"
5O

 

Figure 4.4. Lori's Method of Tallying
Exactly 50 Tosses.

This is one of the most delightful ways of‘pgipg,natural
numbers.‘ She utilized the ordinal‘ aspect of natural numbers
rather than the usual cardinal‘ aspect, which is implied in
the 'word-problems' found in.most arithmetic texts. Her
thinking was very much in line with the interpretation of
pppl’numbers“ in Classical Physics: in a classical problem,
it is usually required to determine the position and/or
velocity of a particle at time t==5, say, what they really
mean is to follow the spatial behavior of the particle at
the end of the 53d,second (or some other time-unit) which
is an ordinal number.

 

tang:
ordi:

|
the c

8X8“?

unti.

ofI

enc o1
thi 8

futu

Of H
line
25 ‘

 

128

The marvelous thing about Lori was that nobody
taught her Classical Physics before, nor any usage of
ordinal numbers. This is another instance to substantiate
the claim that the interplay between the ordinal and
cardinal aspects of natural numbers is innate within the
human race. Hence in the older days (before Peano,

Cantor and other pedantic logicians), such interplay
between the ordinal and cardinal interpretations used

to be the foundation“ on which addition“ was defined.
In the opinion of this writer, this i5 still the best

way to define the addition of two natural numbers, for
example: 17 + 25 should imply the result of countipg“
until you reach the twentyefifth natural number after 17,
and you will unmistakenly land on the natural number 42.
The writer once had the experience of teaching 'addition'
to adults (by using the above definition) for the Program
of Illiteracy Eradication, implemented in certain.remote
parts of rural Thailand, and the result seemed to be
encouraging. (No comparative studies were done to support
this claim yet, but one would be carried out in the near

future.)

Going back to Lori's work: in counting up the number
of H, she used the 'grouping in 5' method, i.e. she under-
lined every fifth integer, and counted 5, 10, 15, 20,

25, 26 and 27 for H. Similarly it was 5, 10, 15, 20, 21,
22 and 25 for T.

11811

the

 

 

129

(2) A girl wrote down 1, 2, 5, ..., 50 first, and
then placed either H or T (as the case might be) under-
neath each natural number, and she carefully counted up
the H's and T's afterwards. This method was inefficient
in the sense that a miscount could have easily occurred,
and it was almost impossible for a Egg number p; tosses,‘
say 500.

(5) A student kept the tallies for H and T in the
usual manner, i.e., tallying in two columns, and wrote
the tally-marks in groups of 5. But he 1:311; 3 second 513.333;
to record the papal number‘ of throws all along. This
method had the snag that it was quite easy to forget
tallying the second sheet. This boy had to do the
experiment twice, because, in his first trial, he found
that the two sub-totals for H and T gave the sum of 52
instead of the required 50.

(4) Just like (5) above, minus the second sheet.
Using the fact that 8 times 5 = 40, a student could
have given himself the first 'warning shot' when he got
8 groups of tallies, say 5 groups of H and 5 groups of T,
or vice versa. (Such deviation“ £135 Normalipy as 6 or
more groups of H is usually rare). Care was to be exercised
after the 41st throw until the student completed the 9th
group or the set of 45 throws. After that he could have
used Method (2) for the last five throws of the game. ‘

150

It was a pity that nobody in class used Method (4),
or at least the writer did not observe any. This is a
relatively efficient method, for, with the slight modifi-
cation, 1t can be used to record H and T up to the total
of 200 throws or more. For an assigned task of 200 throws,
one starts with 19 times 10 ==19O, and hence one could
proceed leisurely for the first 19 groups. (Again, by
normal distribution, the scores should appear as 10
groups of H and 9 groups of T, or vice versa.) Then one
uses Method (2) for the last ten throws.

Incidentally, this method would clearly exhibit the
distributive lggf‘. In the first numerical example
mentioned above, 40 could be viewed as either "5 times 8"
or "5 times 5 plpg 5 times 5”. In the second numerical
example, 190 could be viewed either as "10 times 19” or
"10 times 10,2;22 10 times 9."

Activity 5: Building up a Estogam by using the
data collected from the_50-throw games; and making spans

taneous comments about the Probabilistic Model:

Of the 26 children in this class, only two did not
succeed to record the tallies of H and T for the particular
game involving exactly 50 throws, and so there were 24
sets of data to build up a histogram? on the chalkboard.
The teacher began by drawing a horizontal‘ggigf and put
50 dots (spaced at 33221.1nterva1’) on the axis. She

then.wrote ordered pairs‘ of natural numbers underneath
these dots:

b0}(
the
hav

Sue I

 

131

L A 4‘ _L
v1 fi 1

O 1 2 5 . . . 47 48 49 50 H
50 49 48 47 . . . 5 2 1 O T

Her ordered pairs were written in the form (E), where H
referred to the number of 'heads' and T the number of
'tails'. Although this was an unconventional notation‘
for an ordered pair, which is usually written as (H,T),
there should be no doubt about the children's internaliza-
tion of this important concept in.mathematics. During
this activity, the children could differentiate between
(32) and (ii).

Each.child took a turn.to march forward to the
chalkboard, proudly announced his scores for H and T
(whose pppglf was checked by the class to make sure
that the total = 50), and then the child put a box above
the corresponding‘ ordered‘pgip on the axis. If there
was a box there already, he just put another box on top
so that his was still above that particular ordered pair.
The class nearly went wild when a.boy mistakenly put his
box above (32) instead of (32) because he had previously
announced H = 26, T ==24. (Of course he quickly corrected
the mistake.) The teacher should be congratulated for
having transmitted the concept of an ordered pair so
successfully.

Gradually, the central part of the axis began to
look like multi-story towers. Although the highest

152
'tower' (maximum freguency‘) did not fall on the point

 

corresponding to (3;) as expected, the data obtained here
clearly communicated the concept of normal distribution‘
to the children. (See the histogram in Figure 4.5). Most
'boxes' clustered around the center, and a few off-center

boxes were found on both sides.

 

10 15 19 21 25 24 25 26 27 28 44 H
40 35 31 29 27 26 25 24 23 22 6 T

Figure 4.5. The Histogram obtained from the
5 throw games.

Children made several interesting comments about
this histogram and one boy even commented about the
Probabilistic Mpggl‘ in general. For example,

, - " (3;) means a p;p;‘ five people got a tie because
there are five boxes above (3%)."

-- "Most boxes are around the middle, and so this
game (coin-tossing) is fair."

-- "This picture (the histogram) should be called
the 'USMES Board' because we got it during the USMES

period."

135

-- "It is an unusually nice bar-graph." (Perhaps
the child referred to the pattern of normal distribution’).

Teacher: "We have two boxes (:8) and (42) a long
way off the center. Are they wrong?"

A student: "No, it just happened that way."

This last sentence, in a naive way, summarized the
underlying philosophy of the Probabilistic‘gpdgl’, or the
Mathematics p§_Uncertainty. It is so different from the
Deterministic 3.1.9.82? usually found in the 'word problems'
of arithmetic and algebra. The data-collection carried
out in.this Activity was a convincing example of the
Probabilistic Model of 1/2. Instead of getting one and
only one exact result of (3;), as given by the Deterministic
Model, the children now learned to appreciate the Prdbabilistic
way of thinking: one should not expect an‘pgppp answer; one
could only expect the answers to cluster around £9352 314133
and one even makes allowance for a few off-target values.

Another mathematical behavior could have followed
immediately: the continuous‘pppyg". The teacher could
have joined the tOps of the bar-graphs together by means
of a continuous curve (See Figure 4.6) because this
histogram is an approximation to the Normal Qppyg“.
Although the curve obtained here was slight gkp!,"
it could have been.a good introduction.to the subject of

Normal Curve, which children would meet later on.

88..

118

he'

’5'

Dec

1
'a‘

154

 

Figure 4.6. The histogram as an approximation
to the Normal Curve.

A comment should be made about the unusual way this
teacher labelled the horizontal axis of Figure 4.5.
Instead of writing (58), (4%), (43), etc. for values (g),
she could have used the easier abscissas“ 0, 1, 2, ...
by showing the values of H alone. This should be possible
because T = 50 - H, and hence, once H assumed a certain
value, T could assume only one corresponding value.

The way she used 'boxes' laid above the abscissas
to denote the freguencies' (her'y-co—ordinatea) for
various (3) was a good method in introducing ordinates‘
or y-co-ordinates. If she had introduced a vertical
gpgg" at the left of the diagram, and labelled this axis
"frequencies", then the whole idea of a co-ordinate plppg“
could have arisen by plottipg 'freguencies' against g."
This plane would consist of the pgp‘gg points“ defined
by the ordered pair (H,f), where H referred to the number

of 'heads', and f referred to frequencies or the heights

of t;
this

the 1

 

Of +
Chi:

and

con:

01‘s

 

the

 

135

of the 'towers' in Figure 4.5. Significant points“ in
this set are shown in Figure 4.7, and they also approximate

the Normal Curve.

frequency

1,1 0 ' o

 

 

1‘5 1‘9 23 2% Es 28 114 H
Figure 4.7. The set of ordered pairs (H,f) visually
presented.
If one were not happy about the approximate nature
of the graph in Figure 4.7, one could have asked the
children to graph
T = 50 - H
and one would have obtained a perfect straigpt‘lgpp,‘
(Figure 4.8). In.either case, the following generalization“
could be made:
A‘gpgppf is a visual presentation of the set of
ordered pairs (x, y), once the (horizontal) x-axis, and
the (vertical) y-axis have been clearly defined.

136

50-
40 ~

30 "
2O “

10 “

 

 

 

0 16263646 ’11
Figure 4.8. The graph of T'= 50 - H, drawn from

typical ordered-pairs: (O 50), (10,40),
(25925), (28922) and 4496)

Activity 6: Devising horse-race types of games to
be plgyed in conjunction.with coin-tossing:

The children were now satisfied that tossing a coin
is a fair means to decide between two persons, and they
could conceive of many interesting games whose moves
depended on the results of tossing a coin. Obviously
the children enjoyed this period which was a welcome
change from a very structured Reading Program just before
it. Several horserace types of games were devised by
the children themselves (with very little help from the
three adults in that classroom). By far the most interesting
game Observed was the game whose 56 moves were constrained
to the sectors‘ and annuli‘ of six concentric circles;‘

each circle was divided into six parts by six radii‘,

157

making an angle’ 2; g9? to each other at the common

center.’ (See Figure 4.9). The 56 spaces were numbered

1 through 56 in a counterclockwise’ direction. At the

start of the game, all the players placedtheir counters

at position’ '1', and the main idea was to reach position
'56' before the opponents did.

52

 

Figure 4.9. The circular tracks in which the
counters moved.

158
Normally each player would 'go' one step at a time

by getting a 'head' or a 'tail'. If he got four
consecutive’ heads, he would be awarded bonus points,
and allowed to move his counter radially’ outwards, i.e.,
to gain an increase’ of 6 points. If he got four con-
secutive tails, he would then be permitted to move his
counter to 'the other side' of the same circle (to the
space diametrically gpposite’), i.e., to gain an increase
of 5 points. The 5th consecutive head or tail would,
however, be awarded one point only, and this should be
taken as a warning of the penalty to follow soon. The
6th consecutive head or tail would bring a penalty of
-12 points or mpg 233 .13 112223.: (If any player was
_lp_s_s_ 151132. 2.3. £17pr from 'start' , he had to move his
counter back to 'start'). The children explained that
the 6th consecutive head or tail would be more likely
effected by cheating than by luck. The probabilipy”
pf; getting 6 consecutive page}; could have been discussed,
at this point. On the whole, this Activity provided a
rich experience for mathematical learning. It could act
as a catalyst to spark off the following mathematical
lessons:

(a) The discussion on the probability of getting
2, 54 4 ... consecutive heads: Although it would not
be difficult to verify by coin-tossing that the
probability” 2!; getting _2_ consecutive 31929.9. = «5- . $- = élr,
it would be rather tedious to do so, and it was certainly

159

not advisable to experiment with.the next cases of 5,

4, 5, 6, ... consecutive heads. It is a big question

mark whether the teacher should explain the principle
P(AnB) = P(A) .P(B)

to elementary school children or not. It involves this

difficult concept: the joint occurrence” of two events

1

which are independent” of each other. Piaget contended

that children before age 11 would not appreciate probability
in terms of ' proportion of favorable outcomes,‘ but Yostl
disputed this contention on several grounds especially
the following:

(a) Piaget's experiment seemed to confound 'preference'
with 'expectation'.

(b) Piaget's experiment relied too heavily on the
verbal skill of the children -- a doubtful quantity
before age 11. Yost re-designed the experiment so that
it involved mainly non—verbal responses from the children,
and his finding was that children between 7 and 10 931d,
in fact, learn elementary concepts of probability if the
learning environment was appropriate. Yost' s conclusion

was later supported by two studies, carried out by Davies2

‘A

1See Yost, A.,. "Non-verbal Probability Judgements

by Young Children. Child Development, Vol. 50, December
1962, pp. 769-780.

2Davies‘,’ 0., "Development of the Probability Concept
in Children, Child Develo ent, Vol. 56, September 1965,
pp. 779-788.

140

and Goldbergl

than Iost by concluding that children above the age of 9
could learn not only the non-verbal tasks in probability

respectively. Davies went even further

but also the verbal ones, if these tasks were properly
managed; while Goldberg re-affirmed that the rudimentary
concept of probability could be learned even earlier, i.e.
by 5-year-old children. It could be inferred from
Davies' and Goldberg's studies that the probability of
the joint occurrence of independent events should be
within the grasp of upper—grade children in elementary
schools.

(b) The concept of angle measurement derived directly
from Figure 4Q: Both units of angle-measurement, the
degpee” and the radian”, arose naturally from the
activity of designing Figure 4.9. Take the radian first:
the children would have a good chance of discovering by
practical means that _t_h_e_ arc-lepgph increases Lg _t_hp gape
proportion 9.3 3113 radius, _i_f flip vertical ppglp 13 k_ep1_:

 

constant.” For example, using a copper wire or a fuse,
the children could verify that the arc-length” separating
'7' and '15' is twice the arc-length separating 'l' and

'7 ' when the radius of the 2nd circle is twice the radius
of the innermost circle. Similarly, the arc-length
separating '15' and '19' is 5 times that separating '1'

 

1'Goldberg, S., ”Probability Judgements of Preschool
Children," Child Development, Vol- 57. March 1966,

PP- 157-167 .

141

and '7' when the radius of the 5rd circle is 5 times
that of the innermost circle.

ll

 

- 19
15
.2 7
3 1
4 6
5

Figure 4.9, partially reproduced.

Hence the children would have the Opportunity to
associate” a constant $13” with the constant ratio:
W. The teacher may or may not verbalize this
association or correspondence.” If she does, the word
'radian' can be introduced. The idea of measuring angles
in degrees is somewhat easier and perhaps more suitable
for children at this level. A complete revolution’ (like
moving the counter along positions '1' through '6') is
assigned 560° (an arbitrary number). This was divided

142

into 6 equal parts, as shown above, and hence each
0
angle = 229 = 60°.

(c) Introduction of the e uilateral tri 1e and

 

the hexggon: Six triangles can be formed by joining
together the points A,B,C,D,E,F on the innermost circle,
as shown in Figure 4.10.

 

Figure 4.10. The six equilateral triangles and
the hexagon formed inside a circle.

By direct measurement,” or by consideration of M,”
the six triangles can be said to be page}; _13 _a_1_l_ respects,'”
and moreover, each is equilateral.” Paper-cutting and
paper-folding would help tremendously in uncovering the
above facts. After cutting out the six equilateral
triangles from paper, the children should be encouraged
to try to do the same on cardboard, and this time they
would be asked to produce a greater number of equilateral
triangles. "Why?" may be the children's reaction to
having so many equilateral triangles in front of them.
At this point, the major challenge of the Dice Design
Unit could be put forward: "How can you design some

145

other kinds of dice, besides the cube, which are fair,

and easily tested to be so?" The children, playing with
the equilateral triangles in front of them, might come

up with a tetrahedron.” If several pairs of hands were
helping to stick the triangles together into some sizeable
5D objects, they might even discover the icosahedrgp”
and the octahedron,” which are fair dice. Obviously
they would also produce a host of other convex polyhedra,”
which might not act as fair dice. Holdenl listed eight
Possible convex polyhedra which could be constructed
from equilateral triangles alone. They are: the tetra—
hedron (4 faces), the octahedron (8 faces), the icosahedron
(20 faces) and the remaining five contain 6, 10, 12, 14
and 16 faces respectively. The experience in constructing
Polyhedra would help to clarify many concepts in 5D
Geometry later on, e.g., £93 angles between £132 _o_f_
planes,” (children called them the 'corners' of a die),
and 2.9.9. intersection 9; H9. planes” (an 'edge').

As for 2D geometry, the learning resulting from this

Activity would be equally spectacular. When a child

lays an equilateral triangle casually on top of another
and rotated until the tw0 coincide, he would learn that
the acts of lifting, moving and rotating a triangle do
not change its sides nor its angles. (Lengths and

angles are invariant” under the Euclidean $119.”)

When a base angle (of the top equilateral triangle) is

 

rHolden, Alan, "Space Visualisation," usrms Workin
Pa er, Serial No. D5 in the 'Dice Design' 05.115, 1969, p. 5.

144

made to coincide with the vertical angle (of the lower
triangle) after some rotation, it carries a message

that pppp;ang1e of either triangle is 60° by the argue-

ment in (b). Hence the sum” of the 2;internal angles = 180°
for the case of the equilateral triangle, and the child
should then be encouraged to speculate the result (after
several direct measurements and addition) for the general
case. He would soon find out that the sum of the internal
angles of ppy'triangle is 180°.

Figure 4.10 also demonstrated the plane figure of
the hexagon.” Various activities could follow the
recognition of this figure, e.g., the investigation of
the size of its internal angle (each= 60° + 60° = 120°),
and the sum of all its external angles ( = 180°, and why?),
not forgetting the probable construction of polyhedra
using both equilateral triangles and hexagons.

(d) Introducgpg Arithmetical Progression” and
Arithmetic Meanz” Each of the six sectors of the largest
circle in Figure 4.9 could have provided the best intro-
ductory example of an.Arithmetical Pro ression,” (in
this case, the common difference”==6). Children are
naturally curious and attracted to patterns, including
number patterns”. (The writer still remembers vividly
the sight of a 4-year old Thai baby-girl counting the
numbers of flowers and octagons appearing in the floral-
patterned sarong (skirt) her>mother was wearing that day.)
Referring back to Figure 4.9 again, one could justly

14S

say that the sequence 6,12,18,24,50,56 might not generate
too much excitement (because that was merely the Multiplica-

 

3.9.2325. for 6), but all other sequences,” such as
7,15,19,25,5l would be a challenging pattern to the
children if. a teacher asked such timely questions as:

-- "Will an arbitrary number, say 85, be included”
in the above sequence? How do you check it?"

-- "Now if you call the next number (57) the £132;
1:_e_r_ip,” and one next to that (45) the seventh term, etc.
What number is going to be the 100th term?"

Surely there could be found some children who would
keep on adding 6 and tally every stage of the addition
until they finally reached the 100th term, and meanwhile,
11’ all the partial sums were correct, they would hit on
the number '85' somewhere along the line. Although
this was a good practice on addition, a more efficient
method did exist, and all it needed was the concept of
One-one correspondence ’ ’:

7. 13. 19, 25. 31, ...

11¢ 11¢

6, 12, 18, 24, 30, ...
The writer asked the 'Soft-Drink' Group (6th graders)
to solve the above two problems for the 'Dice Design'
Group (5rd graders) and some positive reinforcements
(Rare postal stamps from Thailand) were promised to be
Given to any 6th graders who could communicate the ideas

146

or at least the correct answers to the 5rd graders. All
the seven 6th graders got the rewards. After informing
the writer of the (correct) answers she obtained, one
6th grade girl said she was going to explain to the 5rd
graders in the following way:

"Look at these two lines:

7, 15, 19, 25, 51, ...

6,12,18, 24,50, ...
Do you see that the top line is always ONE more
than’ the bottom line? — (Yes). Do you also see
that'the bottom line is just the Multiplication
Table’ for 6? - (Yes). Now take the last number
In the bottom line (50); call it by another name,’
(the 6th term) is 56 or 6 x 6. The seventh term
is going to be 6 x 7, and you always do that for
the bottom line. Its one-hundredth term==6 x 100
= 600. The top line is always ONE more, so the
answer is 601."

"Why don't you try out your teaching method now? The
boy concerned is in the Design Lab perfecting his
'circular' track," said the writer.

The meeting was cordial and the boy did pay much
attention to her explanation, because it flattered him
to think that somebody else should take interest in his
circular track (Figure 4.9). But the 6th grader spoke
so fast that the writer was not sure if her listener
could absorb any part of her explanation except the
final answer '601' which he took for granted. In any
case, it $§_difficult for a third-grader to deal with 5
almost simultaneous mathematical operations: treating

100 as an ordinal number,’ multiplying’ 100 by 6, and
adding’ 1 to 600.

147

As for the first question: "Is 85 a member’ of the
ppp’ 7,15,19,25,51,57,...?" the sixth grader again dealt
with that on the basis of the one-one correspgndence’ with
the set 6,12,18,24,50,56,... They multiplied successively
all the natural numbers by 6, until they got 14 x:6 = 84.
The next product 15 x 6 ==9O was too big, so they went
back to '84' and added ONE to it. So they concluded
that '85' is in the set 7,15,19,25,50,... Apparently
it did not occur to this group of children that the two
inversed Qperations”: subtracting 1 from 85, and dividing
by 6, could have provided them with an easier test.

The A.P. 7,15,19,25,5l would give an arithmetic
93.22" of g- (7 + 13 + 19 + 25 +31) or 19 which would
carry a deeper significance than the mere addition and
division by 5. For an A.P., its arithmetic mean is
always a median’: the 'middle' number in the proper
sense of the word. For example, the '19' above is mid-way
between"7' and '51', between.'15' and '25' etc. Unfor-
tunately, the 'mean' has been generalized to include
results obtained from applying the algorithm 1
311- (a:L + a2 + a3 + ... + an) on _a_ny set of numbers

{a1} , not just A.P., and thus the 'mean value' gets
considerably distorted from the 'median value'. In this
writer's opinion, the best way to introduce mean-values
or averages to children is to use examples from Arithmetical
Progression, so that the children may see the ideas behind
its algorithm.

148

Watching the children playing this 'circular track'
game was enjoyable. Two children, A and B, used two
pennies to play the game and they were very excited
whenever bonus points or penalties were about to come up.
B won when he was 'home' in 55 moves. He was obviously
lucky: he got four consecutive heads twice and four
consecutive tails once out of 55 throws, while the
theoretical probability for getting four consecutive
heads or tails is only 1/16. The chronological movements
of A's and B's counters are shown in Table 4.10 (The
integers in this Table refer to the positions of the
counters on the circular track.)

S0 A lost the game by 56-52 or 4 points. Out of
the 55 tosses, B got 18 heads and 17 tails, both of which
were remarkably close to the theoretical result of a
perfect coin. It should be noticed that both B and A
suffered the penalty of (~12) points once in their 55
tosses. The theoretical probability” of getting this
penalty is (1/2)6 or 1/64, but neither the number of
throws nor the size of the circular chart was big enough
to test this theoretical value. So this experience
should be taken as a means to introduce and utilize
negative numbers’ more than anything else. The writer
also questioned the children why they gave 6 bonus points
for '4 heads' and only 5 bonus points for '4 tails'. They
replied that they just wanted two different methods of

149

Table 4.10. Chronological results of a two-player
game using the circular track in Figure 4.9; the
integers shown in boxes refer to either bonuses
(if increasing) or penalties (if decreasing).

 

 

 

 

H T H T

4 1 1 2

5 2 ; F€%I
{its 3 e : s:
:12: 8 I 9'
“{Z' 1? it:
I I I
:20: F72”: L_1_J
:21: m :2:
Li?" 1 £19.: F3:
12 5 I
{141 r--. I13:
I 15 ' | 17 | :16_:
:16: :18! ~-
L22] 25 :19:

25 24 |_2.§__| 26
26 27 27

28 29 28 29
32 30 31 30

31 32
33 34
35 36
No. of
tosses 2O 15 18 17

 

 

 

 

 

 

jumping on the circular track. Although they learned
earlier that the chance for getting 4 heads is the same

as that of getting 4 tails, the two different bonus systems
should bring home the truth that equal chances do not
always bring equal rewards in real life.

150

Another boy in this group was playing by himself on
'championship': he was tossing a coin to find out the
semi-finalists, the pair of finalists, and the champion
from 8 competitors or 8 teams. His arrangement was
simple: the 8 competitors or teams were grouped arbitrarily
into gpppupggpg,’ and he flipped the coin 4 times to get
the 4 winners or winning teams of the first round. These
were again grouped arbitrarily into two pairs, and he
tossed his coin twice to select the finalists. He then
flipped the coin.for the last time to choose the champion.
The process of successive elimination’ was something like

the following diagram:

A

B A

C A

D D

E A
F E

G G

H G

It should be pointed out to the child that the above
method of elimination is unfair, although such method
is actually used in many tournaments, and also in selecting
candidates for political offices. The transitive.lgg”
does not hold when one discusses human ability. There
have been numerous counter-examples” taken from tennis
or chess tournaments, or any fields of human activities
whose records showed that A beat B, B beat C and yet C
beat A (if C was allowed to challenge A.) A fairer

151

arrangement is to have each competitor playing against
all the rest, one by one. Two points will be awarded
for winning, one point for a draw, and none for losing.
For 8 competitors, there will be 802: gi—E = 28 games
altogether in the first round and the maximum points
earned = 2(8-1) = 14. There may not be a second round
if only one competitor gets the higpest 172121”. (Here
the highest total need not be the maximum score of 14,
because there might be some draws.) This is another
instance of using mathematics to create a fairer situation
to all concerned. It also provides the best motivation

to learn the Combination formula”

 

 

The n02 method has another advantage: it can be
used for _a_ny number 93; competitors,” while the child's
method of elimination described above can be used con-
veniently only for 8,16,52, or generally 2]: competitors.

This discussion brings to a close the major types
of mathematical behaviors that arose, or could have
arisen, from the Dice Design Unit during the observed
period (September 25-November 27, 1972).

Conclusion: Despite the name 'Dice Design Unit,‘
the activities undertaken by the observed children were
far from designing a die, or even playing with the usual
6-sided commercial dice. They spent almost the whole

152

term investigating problems concerning the amplest 'die':
the coins They also extended the game of coin-tossing
to include drawing straws and choosing hands, and then
generalized the head/tail concept into a pair of equally
likely events. Some forty mathematical behaviors arose
naturally from the children's determined effort to answer
such demanding questions as:

-- how to devise an effective method of tallying
H and T so that anyone could quickly and conveniently
count them up afterwards,

-- how to graph the results,

-- how to devise an enjoyable horserace type of game
to be played in conjunction with coinptossing.
The above implied that a great deal of mathematical
learning occurred around the themes of Probability and
Graph, both of which had been termed 'basic' or 'unifying'
themes by the National Council of Teachers of Mathematics.
The Council's 24th Yearbook (1959) presented a list of
seven “most basic mathematical themes which should be
central to the entirety of a modern mathematical curriculum."1
The word 'entirety' was later elaborated to mean ”recurring
and varied contact with these fundamental concepts and
processes so that understanding might grow within children
throughout their school careers." Two years later, the

 

ls.c.r.n., "Growth of Mathematical Ideas, Grades K-l2,"
24th Yearbook, 1959, pp. vi and 2.

153

N.C.T.M. issued an important pamphlet,l

stressing ten
"unifying themes" for teaching modern mathematics. On
both occasions, Graph and Probability were included in
the list. By introducing Graph and Probability in an
activity-oriented atmosphere, this USMES unit is a
positive step towards the idealism deliberated by the
N.C.T.M.

Some people may doubt the real value of such intui-
tive ideas on Probability as introduced by this USMES
unit, but William Feller,2 an experienced teacher in
this field, gave the assurance that intuition would
serve as background knowledge and guide towards deeper
theory and more sophisticated applications.

These USMES children had another advantage over the
children who learned Grade 5 mathematics in.a traditional
class: the former had the first-hand experience in
recording an experiment (by tallying) and counting before
they performed any piece of the resulting arithmetical
work. Such arithmetical problems immediately became
more relevant and meaningful: they were‘ppp just
another sheet of paper, full of numbers, to be handed to
the teacher aide who would grade it the next day, and to

be cast away soon after that.

 

1N.C.T.M., "The Revolution in School Mathematics,"
A report of regional orientation conferences in mathematics,
1961, p. 22.

2Feller, William, "An Introduction to Probability
Theory and Its Applications," New York, John Wiley and
Son, 1968, p. 20

154

Professor Lomon and Mrs. Beck1

ably described
four kinds of coin-games most suitable for this Unit.
It had been a real concern of this writer that the
teacher might have taken the USMES Manual like a cookébook,
and just prescribed the 4 games to her class, but fortunately
this did not happen. In fact the children of the observed
class had ample opportunity to show originality, from
creating ingenious tallying methods to constructing
circular horserace track on tri-walls, so long as such
originality did not involve gambling with money.

The Wexford group concentrated on learning Graphs,
Probability and other derivable mathematics from coin-
games. The writer was informed that, in the following
term, the children would have the opportunity to construct
and play initially with the cube, the tetrahedron, and
the icosahedron, in response to the challenge: "How
would you design dice for 5, 4 and 5 players?" Some
'Dice Design' groups2 elsewhere did begin the unit by
building regular polyhedra from precut polygons. This

is also a healthy approach. The children in such classes

 

1Lomon, Earl and Beck, Betty, "Coin Games," usrms
Working Peppy Serial No. D10, 1972, p. 1-18.

2Take, for example, the 'Dice' group at Champaign,
Illinois. The teacher introduced two cubes and a dodeca-
hedron.as early as the second class meeting. Scarcely
had the third meeting started when the children were on
tgziway to construct various polyhedra and test their
rness .

155
will.have some valuable first-hand experience about the

space of 5 dimensions (particularly space perception).

They can also get familar with various terminologies

and concepts in Solid Geometry such as, the angle

between two plgpgg (faces), the intersection between

two planes (i.e. an edge of the polyhedron), the altitude

of a tetrahedron, etc. The precut polygons themselves

offer many valuable theorems in plane geometry.
Throughout the history of mathematics education,

the 'Dice Design' unit is probably the first practicable

blueprint bringing together two divergent disciplines:

5D Geometry and Probability. But, on reflection, one

would see that both disciplines really represent different

parts of man's efforts to study Nature as a whole. It

would be apt to close the Case-Study of this Unit with

the following lines:

"All Nature is but Art, unknown to thee,
All Chance, Direction, which thou canst not see..."

Alexander Pope (1688-1744)

156

The CASE-STUDY of the "Designing for Human Proportion" Unit,

Observed at Pleasant View School, Lansing, Michigan
October 5 - November 50, 1972

I. The behaviors to be observed would focus on the use of

numbers to describe a human body in contrast to the use

of such vague terms as fat/skinny, tall/short, etc.

Children's involvement with these numbers would, hOpefully,

not be confined to arithmetical work on “word-problems“

about body measurements, but would also include the actual

experience of measuring (not excluding the discussion on

the accuracy of measurements); the recording and tabulating

of data; graphing and the calculation of the mean, median,

mode and range; extensive applications to the designing

of clothing, furniture and house fixtures; application to

Graphic Art, especially the drawing of a human face. A

small group of children here posed another branch of

challenge: "What is inside a human body?" which would

lead to an elementary study on anatomy.

II. Brief Description of the 'Human Proportion' Group:
Fourteen 5th graders left their usual classroom every
Tuesday and Thursday (1:50 - 2:45 p.m.)_to work on USMES
in the School Library. Measuring tapes, strings, yard-
sticks, foot-rules were provided for the class. The
library had a beautiful carpet and children found it most
convenient to work on the floor, because the measuring
activities often involve kneeling down, sitting on the
floor with their backs against the wall, or even lying flat

157
on the floor. The round tables (seating 5 or 6 children)

in the library were most useful, because children, while
graphing and drawing pictures, could compare each other's
work and get immediate feedback. The library also cone
tained many things worth measuring and comparing with the
children's own heights, for examples, book-shelves, coat-
racks, drinking fountains, slide-projectors, film-loop
projectors, etc. Geographical globes were there to
provide a case for measuring on a curved surface. Ency-
clopedias were available for the small group interested
in simple aspects of human anatomy.

All the children were in the traditional Grade 5 level

of mathematics and science. No reading nor communication

problem was observed.

158

Activity 1: Usinggnumbers to describe people;

Measuring children's bodies with specific purposes such
as making garments, and desigpipg furniture and other

things for children's use:

The teacher posed the following set of questions to
the class: "How can your mother or a tailor make garments
to fit you? How can a carpenter make chairs, desks and
other furniture to suit children of your size? How can the
shoe-factories manage to produce shoes thatwill somehow
fit most children?" The answers to all these questions led
the children to the conclusion that measuring’ various parts
of a human body‘gp a very useful and essential activity of
life. They seemed eager to have the experience and practice
of measuring people, because they felt that qualitative
terms like 'tall/short', 'fat/skinny', etc. were not
adequate for many purposes, especially to answer the above
questions. Measuring would provide them with gpantitative
9233? to work on. They could then use numbers’ to describe
people, e.g. "He is a 180 lb. man," instead of "He is fat."

Three or four measuring tapes were then passed along,
and the children merrily measured each other in small

groups. The teacher warned the class about two things:

 

Throughout this Case-Study, the words underlined and
followed py,’ refer to the mathematical behaviors otserved
a group, while those marked ” refer to the potential
mathematical tapics that could have been discussed.

     

159

first, each group should measure with a specific purpose,
and secondly, the accuracy’ of each measurement should be
checked by at least one of the peers in the group. The
children then organized themselves into 5 groups: the
first group pretended to be tailors, and shoe-makers,

(but one girl did produce a pair of sandals from tri-wall
and strings); the second group pretended to be carpenters,
and the third group to be builders who planned facilities
for a new elementary school, e.g. drinking fountains,
bathproom basins, coat-racks, etc.

1. The Garment Grppp: After some discussion, the
group decided to concentrate first on.measuring and checking
the accuracy of each measurement. It was hoped that they
would be provided with needles, thread, and inexpensive
material later on to try out their measurements like a
real tailor. A girl said she would definitely learn how
to use the sewing machine afterwards. Each person in the
group then fetched a piece of paper and was responsible
for recording’ his or her own‘ggpg’obtained from tape-
measurements. These would-be tailors seriously measured
the parts of their peers' bodies and checked the data
before recording. Most children in the group included
the following measurements’: collar size (which is slightly
bigger than the measurement round the neck); width of
shoulders (distance from left shoulder to right shoulder);
arm-length; breast size or chest size; width and length
of the back; waist size; hip size; pants length; skirt

160

length or dress length; length and width of feet; depth
of a shoe.

At first they wanted to measure to the nearest 1(8 of
ppflpppp,’ but this high degree of accuracy only resulted
in too many arguements and the standard had to be relaxed
down to the nearest pplgflépgp,’ Little did the children
realize that they were measuring curved length’ when‘ppp
pppyg’ itself was only defined vaguely. This was particularly
true when.measuring hip sizes and waist sizes. Watching
these children measuring each other made the writer realize
that a tailor's measuring never involved straight-line
measurements, and the children here were quite realistic
about it. For example, the length of the pants leg has
to be longer than the distance between heel and waist,
measured by two marks on a vertical wall. The distance
between the marks on the wall would give.g straight-line
measurement, M £3, py Euclid's pips; ppstulate,
shortest,’ and hence no allowance would be made for the
movement and bending of the knee.

This activity gave the children an opportunity missed
by most children in a traditional 'math' class: the measuring
of curved distance. Too many 'word-problems' in arithmetic
provide measurements which.involve only straight lines.

It was not surprising to find a girl in this group trying
to measure her girl-friend's waist size by means of a

straigpt ruler. Her method was ingenious, though.

161

2nd Starting
Pt' Contour of
lst the waist
Starting
Pt.
Pt. 0
contact The a?
ruler

Figure 4.11: {Measuring a curve by means of a
straight-edge ruler.

She started with one end of the ruler on her friend's
left side and gradually swung the ruler round the waist,
making sure that the ruler was always a tangent"ppuppg
m. The ppligp 2; contact’ was, in this case, a
variable ppgpp, varying along the length of the curve as
well as the length of the ruler. The last point of contact
(the point at the other end of the ruler) became the
second starting‘ppgpp’ in order to continue the measuring
along the curve. She completed her measuring when the
ruler finally swung back to the first starting point on
the left side. Her result (12 + 4% inches) came remarkably
close to the tape-measured result (17 inches), although
she complained that the ruler slipped too easily and too
often. This activity should give the students a concrete
experience on.tangency to a curve, and the "envelope"’
outlined by a set of (variable) lines. The activity of
curve-stitching (using needlework to envelope line-conics

like the arabola, ellipse and pyperbola”) as suggested by

14

12

10

162
both the Schools Council of U. K.1 and the National Council

of Teachers of Mathematics2 could have been introduced at
this point. Figure 4.12 below shows the simple stitches
outlining the line—conics” parabola and hyperbola.

23 22 21 2o 19 18 17 16 15 14 13
o l 2 3 4 5 6 7 8 9 10 ll 12

O l 2 5 4

16 1718 19 20 21 22 23
1514 13 12 ll 10 9 8 7 6 5

    
 

 

 
     
     
      
   

AIIIV

  

‘lil'

4"
4‘IF'

  

4'
4"

i.

  

-.

I'I1g’
:’
'f'

   

‘
{‘m-
i"
\

      
 

‘
‘\\‘- _
\

 

 

 

 

Figure 4.12: The line-comics parabola and
hyperbola.

 

1Schools Council: "Mathematics in Primary Schools,"

London: H. M. Stationary Office, 1965, p. 85.

T. M., "Multisensory Aids in the Teaching of

O G.
Mathematics," 18th Yearbook, 1945, p. 79.

165

The tape-measuring of any curve-length should be
coupled with an important approximation,” which is used
so often in practice. It is to approximate the length
of any curve by those of a series of short chords.” For
examples, in navigation, a set of 'rhumb lines' often
replace part of the great-circle‘gpg” connecting two
sea-ports on a globe, and, in laying a railroad track
around the bend, engineers use a series of gradually
deviating tracks, (each of which is straight but short)

instead of a geometrically curved rail (See Figure 4.15).
A D

A

Figure 4.15: A series of chords approximating
a curve.

The wealth of data generated from the measurements
of this group could have been graphed” to find the
median” for each kind of physical measurement (waist
size, hip size, etc.) or could have been pooled to find
the‘ppgp” for each item. The median or mean would have
provided a profile of an average 5th grader. The pgpgg”
for each item could also be calculated, and within this
range, several sizes of garments would be recommended for
production, with maximum” production on sizes around the
mean. The children might even write to the garment manu-
facturers about their findings because the accuracy of their

tape-measurings had been checked several times.

164

2. The Furniture Group: Each member of this group
undertook to design his own desk and chair, temporarily
ignoring the measurements of the desks and chairs provided
by school. The greatest divergence of opinions seemed to
center on the height’ of the desk; some wanted the desk-top
to be half-wgy’ between eye-level and seat-level, some
wanted it lower, and some higher. There was, however, the
agreement that it could not be too low, otherwise the drawer
might interfere with the knees and thighs. So they learned
the existence’ of a'lpgppwppppd’ for the vertical heigpt’
of the desk-top. The criteria for building one's own chair
were easier to agree on: everyone wanted the height to
be determined by one's‘lgg lengph’ (from.knee to heel),
everyone wanted a deeper seat (than that of a school chair),
and the depth of the seat was to be determined by one's
thi -len h.’ Nearly all children in the group complained
about the seat-back: it was too short. They wanted the
back to be three-fourths’ of each one's trunk-lengp ’
(from.neck to hips). Each student also measured his or
her own arm-len th,’ bent at the elbow at his or her
preferred an le,’ just in case one might be allowed the
luxury of having arm-rests in a classroom chair:

It would be unwise to suggest the calculation of
'pggpg and medians for the data of this group, because
that would result in.a set of desks and chairs comfortable
to nobody, (either too big or too small). But the children
were pointed out that, from the furnituredmanufacturers'

165

point of view, it was the mean or median that determined
the size of their products.

5. The School Design Group: This group started with
an ambitious plan of designing a model for a new elementary
school. They wanted to make scaled models’ for classrooms,
offices, corridors, store-rooms, restrooms, the playground,
trees, etc., and their "lot of land” was a 5 ft. xpgfi ft.’
tri-wall ppppd. They used yardsticks and a steel-tape
measure to determine the length’ and‘gggpp’ of typical
rooms, corridors, the paths and the parking lot outside
the school building. At first they thought of using a
39219: pf _l_:2_, but after measuring the length and width
of a typical classroom (18 ft. x 16 ft.), they decided to
change the scale of the model to 1:100. The height of
the model was approximated by that of the ceiling. They
looked up the USMES "How to” cards and learned to use
simple trigonometpy’ to find the heigpt’ of the ceiling
(8% ft.). This figure was checked by means of a yardstick
plus the custodian's ladder. Most of them seemed to enjoy
the experience of measuring distances’ and working out the
corresponding lengths’ on their model. This involved a
lot of calculations on proportion.’ These were the benefits
derived from this experience, although the model was never
completed. Moreover, this group of children were the only
ones that were exposed to the concept pguppppp’ while the
other groups (the Garment group and the Furniture group)

dealt only with linear measurement.’

166

When the childrenfs interest changed from building a
school model to interior design, they spent the rest of
the time measuring’ various things inside the school:
library shelves, drinking fountains, coat racks, heights
of chalkboard and bulletin boards, dimensions of the work-
bench in.the Design Laboratory, etc. They recorded and '
made suggestions for improvement. For example, their
measurements revealed that the top shelf of the library
was too high even for the tallest child in class (4 ft.

11 inches), and the children recommended that the spaces
between the 5 shelves could be reduced’ by 5 inches at
each level, thereby the top shelf could be lowered by 9
inches. They also said the things used by children should
be at most 5 feet high. This represents the concept of’an
pppppfllgpgp’ which the children saw before, e.g., on road
signs. In fact, the children could have been asked if
they had seen _lpggp M’ written somewhere, and the
children should be able to cite several examples: 45 miles
per hour on.a free-way, 18 years old for voting, 850 for
opening a bank account, etc.

They were surprised to find that the height of the
drinking fountain (2 ft. 2 in.) was much 12:93 phpp’ they
thought. One student said this extra low height might
be designed for the sake of kindergarteners. This led
the whole class into guessing or estimating’ the height of
a typical kindergartener. One girl said jokingly: "I am

167
10 years old, and a kindergartener is 5 years old, or
one-half’ my age. My height is 4 ft. 10 in., and a
kindergartener should be one-half my height, so typically
he is 2 ft. 5 in.--just the right height for the drinking
fountain." Everyone in the class knew that she was only
joking, because the attributes’ 'age' and 'height' were
different. It could have been pointed out that 'height'
is not a linear function” _a_; M (age). However, the
class seriously desired to know whether her estimated
height of a kindergartener was correct or not. So, in
their next Activity, they were going to measure the heights
of a sample population’ taken from the kindergarten classes.

Activity 2: Measuring the heights of a sample
population of kindergarten classes, and graphing the
results:

The measuring instrument was a simple device con-
structed by these 5th graders in the Design Laboratory.
It consists of a graduated’ vertical’ tri-wall board
about 4% ft. high, and a piece of glgp’ cardboard used as
a "height indicator" to be put horizontally’ on top of the
kindergartener's head pp.pigpp.angles’ to the vertical
scale. One boy stood by the vertical board and used
ppgflgg: corner’ of a.hard-bound book to check whether the
cardboard indicator and the vertical triwall were really
perpendicular’ to each other. Further it was agreed that
the kindergarteners could keep their shoes on while being

168

measured, so that the data gathered were really the natural
heights pgpp’ the heights of shoe-heels. (The kindergarteners
were measured with their back leaning against the vertical
scale.) Three volunteers measured the heights of the shoe-
heels as the kindergarteners marched in, five at a time.
They found that the heights of the shoe-heels were roughly
1/2 in. in most cases, and so they were going to subtract’
1/2 in. from every item in the collected data. This should
be good enough because the overall heights were measured

pp _t_h_e_ nearest _1_L2_ ip.’ only. The additional heights due
to hair-styles presented a bigger problem. Some hair-
styles could easily add an inch to the natural height,

and many wanted this, too, to be subtracted from‘ppg
relevant 1.393? in the data, because, these people argued, ‘
the class was primarily interested in designing facilities
like drinking fountains, coat-racks, etc. Other children
disagreed and they wanted to measure "from hair to heel"
whatever type of hair a person might possess, because,

they argued, the data should be accurately describing
people how they really looked at that moment. The class
finally decided to go along with the latter because the
additional height due to one's hair was difficult to
estimate’ in most cases. The children were very thorough
in their measuring and recording: they recorded the names
as well as the heights of the kindergarteners, so that their
data could indeed be used "to describe people," (i.e., to
store data for identification purpose) if so desired.

169

When they presented the data in tabulated form,’ they used
code-names like Al, A2, ... A5, B1, B2 etc. instead of

real names, and the students interested in the Describing
People Unit held the key to these code-names. The children's
recorded data are shown here in Table 4.11.

Table 4.11: The heights measured from a sample
population of kindergarteners.

Group A Group B Group C Group D Group E

#4 5'7%' #1 5'8" #1 5'7%“ #1 5'8" #1 5'4"
2 5'8%" 2 5'7" 2 5'8” 2 5'8" 2 5'10"
5 5'8" 5 5'11%" 5 5'8" 5 5'8" 5 5'7"
4 5'5" 4 5'llfifl 4 5'11" 4 5'6%" 4 5'10“
5 4'0” 1 5 5'9%" 5 5'9" 5 5'8" 5 5'8"

 

 

 

 

Many mathematical topics could be learned from this
Table. Kindergarteners A5, B1, 02, D1, D2, D5, D5, E5 all
corresponded to height 5'8", and this was a real-life
example of mppyfto-one correspondence.’ The labels A1, A2,
etc. were, in fact, ordered airs,’ the first label being
used for group-identification, and the second label for
individual-identification within a group. This notion
was brought out very clearly when a boy tabulated the above
data as Group 1, Group 2, etc., and named the kindergarteners
(1, l), (1, 2), ... (l, 5), (2, l), (2, 2) etc. He quickly
realized that kindergartener (1, 2) was'ppp the same as

kindergartener (2, l). The class also had a chance to see

170
2 matrix _ip the making’: one student shortened Table 4.11

above into the following matrix from:

 

Group

Individual\\\\\
123%" 3'8" 3%" 3'8" 3'4"
3'8h" 3'7" 5'8" 5'8" 3'10"
5'8” 5'1115" 5'8" 5'8" 3'7"
3'5" 3'11”" 3'11" 3'86" 3'10"
4-0" 394" 3'9" 5'8" 3'8"

A B C D E

 

\I'I'PUJNH

 

 

 

 

 

 

 

 

The sample population were not randomly selected,”
which was a pity. If the procedure of random selection”
were explained to the class beforehand, the children would
certainly have carried this out and learned from the
experience.

It should also be pointed out to the class that the
data in Table 4.11 were projected heigpts”, and not the
curved distance measured by means of a tape from skull
to heel. Instead, a projected height is the distance
between the higpest ppg'lowest points” when.a human figure
is projected’ on to a vertical p;3p_.’ The British model
text on.modern mathematics teaching1 suggested a similar

activity: students lying down on,a large sheet of paper or

lSchools Council (of U.K.), "Mathematics in Primary
Schools,” London, H. M. Stationary Office, 1965, p. 119.

171

cardboard, while the peers nearby drew the contours of
their bodies from heads to heels and then measured the
distance between the two extremities” of each contour on
paper. This is also another way of projecting a human
figure on to a plane, provided the pencil that draws the
contour is always perpendicular” pp'ppphplppg of the paper.
Many measurements, such as widths of the neck, arm, wrist,
waist, hips, etc., could be taken.from this 2D diagpam,”
and checked with the earlier tape-measurements. The immediate
application” seems to be the designing of a comfortable
sleeping-bag for camping, or a comfortable bed.

Next, the teacher asked the children to put the data
of Table 4.11 in,a‘gpgpp.’ The emphasis was that the
children had to think of some methods of graphing by
themselves (without any help from the adults in the class-
room), but the teacher gave the following hint: "The main
idea is to communicate the important aspects about the
numbers in Table 4.11 to other people." After working
for some time, two students came up with the following
graphing methods, which the teacher showed to the whole

class:

172

l. A bar- ra h u side-down showin, the distribution of
heights among the sample population:

3'4" 3'6" 3.8:; - 3.019". 4'
> Lkh

 

 

 

 

Figure 4.14: Heights shown in a bar-graph.

Apparently this student was interested in tallying
the number 91 people’ against 939;; possible heigpt’ in
Table 4.11. He labeled the various heights in‘gp increasing
sequence’ (5'4", 5'4 ", 5'5", 5'5 ", 5'6", etc.) along a
horizontal line (pgpg’). He wrote an x for each person
tallied, and put the x's beneath the correspondipg’ heights
of those persons. The teacher suggested the graph would
look clearer and better if he put a box around each x. "The
resulting picture is called a‘barsgraph,’" said the teacher,
"and the height (5'8") corresponding to the longest column
is called the‘ppgp.’" In fact, the median” could have
been read directly, and the concept of Normal Distribution”
could have been discussed. One girl, interested in coat-
rack design for kindergarteners, proceeded to calculate the

175

“average" (pepp’) of the data in Table 4.11. She did not
merely add up the 25 items and divide; instead she used
the distributive £9.33 and wrote:

9 persons of height 5'8" : 27'72"

2 persons from each of the groups
5'7". 5'716". 5'10". 5'11”"; ,
Twice 12'56" : 24'72"
1 person from each of the rest;
304:, 3‘5", 3'6”", 5'8””, 509",
5'915". 5'11". 4' : 25'5536"

 

7619715"

 

She seemed to have some trouble with lppg division’
(the divisor being 25). The writer asked her to think of
the exchange of Quarters ippp pennies.’ She said 5 quarters
=75 pennies, and sofigspggflgfiplrppiiyp’ and the
remainder’ = l.

25 ) 76' 19226"
5. one

The writer watched carefully to see whether she made the
usual mistake in the next stage by forgetting the remainder l .
But she avoided the mistake by writing ‘

197%
+ 12
25 1 209%
At this stage, the writer helped her along by saying
1 dollar = 4 quarters = 100 pennies,
and she echoed: 2 dollars = 8 quarters = 200 pennies.

1.6. 8125=25x8=200

174

So 25 could go into 209% eight times, and the remainder=9 ".
The writer gave her further encouragement (B. F. Skinner
would say reinforcement) by saying that it was correct so
far because 9 quarters= 9 x 25 =225 pennies which is beyond
the dividend’ of 209%. She nodded in agreement, but then
she did not know how to combine such a diversity of results
to produce the final answer. In the writer's opinion,
adult assistance was essential at this point, otherwise
she would be so muddled up that she might never reach her
objective and might soon lose all her interest in the
problem. The writer reminded her that her original objective
was to find the pepp’ by dividing’ 76'19716" by 25. This
she accomplished in two stages: the first yielded a
partial answer’ of 5'; the second a partial answer of 8"
with a little "left-over". This left-over was 92 par__t_§
£0,111 pl; 22’, or proportionately’ 19 parts out of 50
(2 x 9% =19), and the writer introduced the notation of a
rational number’, 19/50, to the student. Combining these
partial answers as indicated, the student could now write:
5' and 8" and 19/50 inches,
which, finally, became 3' 8%”.

175

2. A Pictorial Graph or a visual presentation of range and
differences in heights recordgd in Tgple 4.11:

4
.4' Q

5 I 10"

» /\/

A1 A2 A3 A4 A5 Bl 32 B5 B“ B5 ... ... El E2 3 4 5

 

 

 

 

 

 

 

 

 

 

 

Figure 4.15: A Pictorial Graph

Apparently the girl who invented this graph was interested
in the chronological’ presentation of these measurements as
well as visual clarity to see at a glance the differences’

(in the individuals' heights) and the pppgg’. The vertical
pgplg’ on the left helped to achieve this visual clarity.
Nobody would fail to see that there was a difference in

176

heights between, say, A2 and 31’ The range, determined
by the difference in heights between A5 and E1, could be
read directly:
Range=4'0" - 5'4" = 8"

There was one strong criticism about this graph:
the inconsistency’ over the ppgpmp£_heigpt’ in the vertical
‘pypg’. She enlarged the scale between 5' and 4' without
enlarging that between 0 and 5', and so the 5'4" kinder-
gartener looked only about two-thirds’ of 4'-kindergartener,
while the correct proportion’ should be 40":48" or 5:6.
Her emphasis on visual clarity of individual differences
was done at the expense of correct proportional graphing.
This was the sort of ”Chop-off-theébottom" bar-graph
referred to Huffl in his amusing book, and it was used often
in the popular press to give the readers the false impression
of a pig difference’ between pairs of individual bar-graphs.

At this point the teacher formally introduced.ppp¢
graphs’ by distributing reprints of a journal for elementary
school children. The journal was Scholastic Young Citizen,2
Vol. 57, No. 7, October 50, 1972, which contained an article
called 'A healthier me'. This article used a colorful

bar-graph to show the increments’ in life expectation of

 

1Hurt, Darrell, "How to lie with Statistics," N. Y.:
W. W. Norton.and 00., pp. 62-65.

2Scholastic Youn Citizen is published weekly, from
Septemter tfito ugh ME? IEcIusIve, by Scholastic Magazine,
Inc., McCall St., Dayton, Ohio 45401.

177

mankind from prehistoric time till 1970. (See Figure 4.16)
The children examined this Figure carefully and then each
produced a set of bar-graphs, arranged in ascending‘ppggp’,

from such data as 'the waist sizes of our group', 'the

trunk lengths of our group', etc.

 

 

 

7‘7“
01
£1
“2
O
-——- s»
.5
__.L‘.
eéoam
mgm 0
ages:
QNMLA
H MM
a b c d e f

 

 

 

 

 

 

 

 

Figure 4.16: The average life span of mankind has consider-
ably increased since ancient times: a) pro-historic time,

b3 2,000 years ago, Home, c) Middle Ages, England,

d 18th.century, New England, e) 1900, U.S., f) 1970, U.S.

178

Activity 5: Measuringgthe dimensions of a human face

and other facial features in order to draw better pictures:

Nearly all the children in this class liked drawing, and
they wanted to tie this USMES activity with Art, particularly
with the crayon drawing of a human face, which seemed to be
a difficult task for them. They thought that the numerical
description’ of the facial features would perhaps make the
job easier. So they started to collect quantitative gppg’
about the faces of the students in.c1ass. This was an
admirable and enterprising task, especially when it was
initiated by such young students as fifth-graders. Great
maSters in the past (Leonardo di Vinci, Albrecht Durer,
and others) had long advocated the quantitative study of
‘pppgp,proportion” and projections” as prerequisites to
good painting. Durer himself had written four books on
Human Proportion, one of which contained the following

quotation:1

"First, give the figures a rigpt ro ortion” according
to the canon (laws of physio ogy and anatomy), arrange
them orderly,” lay out the outlines, give the effect

of dep y ers ective,”...see that every limb be
made right in.aIE tEe smallest as well as the greatest
things...."

.By means of a tailor's tape, the children measured

each other's facial features in considerable details:

 

1From "Egg Writingg of Albrecht Durer,” as quoted by
Henrietta Midonick, "The Tieasury of Mathematics," N. Y.,
Philosophical Library, 1965, pp. 545 & 548.

 

179

‘ppdpp’ and‘gpppp’ of forehead, distances’ between eyes,
distance between an ear and the nearer’ eye, shape and size
of the nose, width of the lips, shape and size of cheeks
and chin, etc.

After gathering these data, the children used the
ppglgflpg‘lpg’ to put all the facial characteristics on
paper and began to draw. Much to their disappointment,
they found that even such a carefully planned method of
drawing produced only distorted pictures, because the tape-
measured distances of various organs on the face were
distances pp p curved surface’, and it seemed geometrically
impossible to project’ a curved distance of a fixed "length"
on to a plppg distance’ of the pppg "length". At this point
the teacher could have pointed out to the class that mathe-
maticians had already discovered that distances are not
invariant” under projective transformations.” A discussion
on Geometgy pp _a_ curved surface”, or even the elementary
aspects of Riemannian Geometpy” could have also been initiated,
because the children were highly enthusiastic about knowing
ppyithey got distorted pictures. One simple but profound
activity would be to measure the distance between any two
cities, the gpgg” of a certain country, etc. on a spherical”
globe, and then compare with the distance, area etc. printed
on a 2D map. The more enthusiastic students might be intro-
duced to stereographic projection” which is the usual
method of transforming a contour” from a sphere” to a plppg”.

180

An easy-to-handle activity-oriented version of stereographic

projection1 is as follows:

 

 

 

Figure 4.17. Stereographic Projection

A piece of vertical” cardboard is fixed gp'contact
‘pgpp” a spherical globe at some point A on the eguatorial
pppyg.” The center” 0 is the theoretical‘pppppup£,projec-
pégp”. B' and C' are imgges” of B and C respectively.

In practice, the theoretical point of projection (inside
the globe) is not essential to locate B' and C'. To draw
the line OBB', a student can use a toothpick, or a small
stick to create a line through B, perpendicular” pg ppg
spherical surface. (All lines perpendicular to the sphere

will go through center 0.) ‘This line, OBB', meets the
vertical board at B'.

1See, for example, Morris Kline: "Mathematics in
Western Culture," N. Y.: Oxford University Press, 1955, p. 152.

 

181

Two or three examples of pppjecting” points like B
and C on to their images B' and 0' should be sufficient to
convince the children that, when two end-points” 2; _a_ m
(say, the two eye-balls) are transferred from a pppyg surface”
(the human face) to a.plppg” (the paper), distance is not
preserved,” and hence the tape—measured distance between
the two eyes could ppp be used to draw the picture of the
two eyes on a flat sheet of paper.

It was then apparent that the tape-measured data collected
thus far offered very little help in drawing a better picture
of the human face. The teacher then distributed magazine
cuttings that bore photographs showing the front view (not
side view) of human faces. The children examined them,
and, after a long discussion (not without adult input),
they realized that the distances between any two organs on
a face, ,a_s appeared pp php hoto ra h, were really vertically
projected distances.’ (See the definition of "Projected
distance" in Activity 2). So the children decided to measure
projected distances instead. To start with, they could
measure the projected lengths and widths of the faces of
students in class. The vertical scale on triwall board,
invented for Activity 2, was being brought out once again,
but this time the scale was subdivided’ further to allow
readings pp 3133 nearest _lfi _ip.’ Unlike Activity 2, which
measured‘gpgp ground‘lgygl,’ this new activity would involve

the vertical measurement from chin to forehead and horizontal

182

measurement from left ear to right ear, and so two pieces
of cardboard indicators would be required this time. One
cardboard was placed horizontally £11 £1.19. _l_¢_ey_e_l_’ of the chin,
and the other at the level of the top of forehead, and both
cardboards were placed perpendicular"pg,the vertical scale.
The difference’ in heights from ground level gave the
projected length’ of the face. Both the triwall board
(graduated scale) and the two cardboards (indicators) were
then turned around 99_degpees’ in order to measure horizon-
tal distances. The two indicators were vertically’ placed
.111 contact m’ the two ears, and the projected M’ of
the face was read off from the triwall board. (See

Figure 4.18).

 

 

l"‘1"'l'*'l"'l"'l"'l" I 1"'l"'l"'l
Scale
I ‘ Triwall
ear—t Head ear Board
indicator-—» indicator

 

 

 

Figure 4.18: How to measure the Projected Width
of a human face.

This activity taught the children many valuable concepts
in‘§plgg.Geometpy’: the two cardboard indicators were
concrete examples of parallel planes’; each plane is per-
pendicular to the triwall scale, and there the children

could see the concept of the perpendicularipy pg 2 planes’;

185

when a cardboard indicator was put in contact with the
outermost ppppp’ of the curved ear, it demonstrated the
concept of tangency’ between 3 29'; ppg p LL‘BJ and,
finally, when the triwall scale and the cardboard indicators
were turned around 90°, the children experienced a concrete
example of a finite rotation.’

The data collected in this Activity are shown in
Table 4.118.

Table 4.11s. Projected lengths and widths of 12
fifth-graders' faces.

 

Length6677 76666;;

efw AR»
Nth wt;

a. .31.

Width 16.55%4} 4%5555

 

 

 

 

Once again the median, iii—623;, £993: _a_ng _rppgg” could
have been calculated from each‘pgp” of the above data,
giving part of a 5th grader's physical profile. Other
interesting profiles might also be suggested: how much
soft-drink he consumes a week, how concerned he is about
pollution, etc.

The teacher wanted to compare the size of a child's
face to that of an adult's. So she herself and the writer
of this Case-study took turns to kneel down and let the
children measure the projected lengths and widths of their

faces. The results were:

184

Teacher's face: Length 8 in., width 6% in.,

Writer's face: Length 7 in., width 6%-in.
The measurements taken from the two adults in class gave
the students some idea about the plpgflpppp_p£_gpowth’ of the
human face once the stage of childhood was over.

The projected length and width of a face would help
to fix the boundgpy'ggppg” of a co-ordinate 5339’: ruled
on rough sketches with a view to improving the likeness of
the picture and the subject. The use of a co-ordinate grid
to improve a drawing was going to be the main theme of the
next Activity.

 

Activity 4: Using the ideas of proportionate enlarge-
ment and co-ordinate grid to draw better pictures of a

human being:

While the teacher distributed the magazine cuttings
bearing the pictures of human beings, she asked everyone
in class to choose a picture one liked best and draw that
picture using anything one is capable of: color pencils,
crayon, water colors, or even oil paint. The emphasis was,
just for once, ppp_on creativity or subjective perception,
but on the degree p£.1ikeness’ between the given picture
and the drawing.

The children seemed to enjoy this period of "USMES Art"
because they drew only the pictures they really liked.

But, despite the intensive effort of everyone, no student

in class could produce a drawing anywhere close to the

185

original photographic picture he had chosen. The big

problem was that things (objects) in the drawing were mostly

pppnpg proportion.’ The children themselves noticed this:

when all the pairs of original pictures and drawings were

placed side by side, the children made the following comments:
--"The girl's face (in one drawing) is too large."

--"The furniture - tables and television sets - look
crooked. The lines should be straight."

--"The man's body is not big enough, compared with
the pack of Pepsi he was carrying."

--"The pants legs look unreal - too long." (This
cutting was an advertisement on fashion.)

--"This (drawn) picture is OK except that the man's
shoes look awfully big."

The teacher then explained that photographic pictures
looked much more realistic than those drawn by free hand
because the camera could capture the correct proportion’
of things as they really were except minor shrinkage’ of
objects further away due to perspectivigy.’ To clarify the
statement about the correct proportion enjoyed by the image
of every object seen in a photograph, she showed the children
two copies of the same photo: one copy was enlarged’ and

was 1%»times as big’ as the other. It was the photograph

of a lO-year-old girl holding a cat, and standing by a table.

 

On the table there were soda-pop bottles and glasses, a
flower vase, and a solid state radio set. The teacher
pointed out that all the objects appearing in the enlarged
photo were li-times as big, and she asked the students to

186

verify this. The two photos were then passed from one
child to the next, and most children could be observed to
measure at least one item (say, the girl's face) from one
photo and, after multiplyipg’ ph_e_ length py 1%, checked
the length of the corresponding’ item in the—other photo.
One boy measured the length of the girl's face in the smaller
photo and recorded 7/16 in. (His ruler could measure 33 _t_h_e_
nearest lg 3333). He then multiplied’ by means of the
distributive l_a_g’:
(7/16)(1§) = (7/16) + (7/16)(1/2)
= 7/16 + 535/16
= 1016/16 in.
He then measured the corresponding length’ (i.e., the
girl's face) in the larger photo and the result was lO/l6 in.
The usual manipulation pp fraction”:
(7/16)(1§) = (7/16>(3/2) = 21/32
was not observed, and so two topics: (1) the method of
converting a 323% number’ like 1%- into a single rational
number” 5/2, and (ii) the principle of multiplflng _t_wp
rational numbers”
(a/b)(<:/d) = ac/bd
could both have been introduced, because the motivation
was there already. But one interesting mathematical behavior
was observed here: this boy used a geometrical method’ to

determine (7/16)(l/2)= 5%ll6. (See Figure 4.19).

187

 

A n C B
o 3 T4 7 16
5%

Figure 4.19: How to find (7/16)(l/2) geometrically.

AB was a length, measured 16 units, and AC measured 7 units,
so that 7/16=AC/AB. By the commutative lag},
(7/16)(1/2) = (1/2)(7/16) = 1/2 of AC/AB
=AM/AB, if M is mid-point’ of AC
==53’2/l6, since M is half-way between 5 and 4.

Another girl measured some other things in this pair of
photos. She used a ruler, correct pphppp.nearest 118 in.’,
and measured the side-way length of the table and the height
of the flower vase. Measuring from the smaller photo, she
reported 2" for the table-length and 1+1/8" for the vase-height.
The teacher encouraged her to express the results as 2:8» ,
or 16:9 when both numbers were increased by a factor‘pg'g’.
This is a very good way of introducing the subject of‘pppgg’.
Next, the girl measured the table-length in the enlarged
photo, and she got 5", which was correct, because

2 . (1%)=2 + g of 2)
==2 + l ==5

When she measured the vase-height in the larger photo,
she obtained 1%”, and hastily she expressed the pair of
numbers as a ratio. She wrote

5 : 1% = 12 : 4 + 5, increasing by a factor of 4

= 12 : 7

188

She was now perplexed because she was expecting the answer
of 16:9, since the teacher had said earlier that, in the
larger photo, 211 things (including the table and the vase)
should increase their sizes by the w Lroportion’, and
hence the ratio

table length : vase height
should remain the same. In fact, this student worked out
the ratio 5:1; too hastily before she checked the validity’
of her pg! 29.322." of 1%". The vase-height in the smaller
photo was 1%" (assumed correct), and the writer showed her that

(1%)(19) = (9/8)(3/2) = '27/16 = 1%

and so the theoretical 9993;: predicted’ a height of 1% in.,
not 1%". The student went to fetch a better ruler (which
could measure to the nearest 1/16"), and this time, the
vase-height, did turn out to be 1%" when she ignored the
blurred part in the bottom. So the 3:33;: in her previous
measurement was

1% - fit; = (3/4) - (ll/l6) = (12/16) - (ll/16) = l/l6 in.
The writer comforted her that it was only a small error,
because the relative _e_§__x-_o_p’=l/l6 : 1% a. 1/16 : 27/16 = l : 27,
or 1 part in 27 parts.
No measurement by rulers will ever be perfectly correct,
and most rulers have been graduated to measure up to a certain
degpee p_f_ accurac ,’ say, within l/16 or 1/52 in.

While the children were comparing the enlarged picture

against the original copy, it was the most opportune moment

189

to introduce the subject of Similar Figures” in geometry.
Both rectilinear and curvilinear figures” in the two
photographs were similar to each other. Their linear
measurements” were proportional”, the smaller length

being always 2/5 of "the larger one (since the enlargement
factor=l%). The children would have enjoyed the exciting
discovery that the gr_e_s, measurements” of any two similar
figures above were in the £92151 9;; 4: ”, the smaller surface

being always 4/9 of the larger one.

Next, the teacher introduced the idea of using a
co-ordinatejgrid’ to help towards further improvement of
the drawing of a human face or any still-life. The teacher
asked a boy pp pulp g co-ordinate 5131’ over the smaller
photo. _Eppp square’ in the grid was chosen to be é-"x «in,
and the origin’ (starting point) was, in this case, the
, top-left corner’ of the photo. Drawing such a grid was a
good exercise on drawing parallel’ and perpendicular’ M,
and gave the children a feeling that graph-paper’ had been
invented with a definite purpose. The teacher also asked
another Student to rule a proportionally’ bigger g1_'j._d_ over
the larger photo. This time the length of each side of a
square in the grid was measured (%)(1%), or 5/4 in. Care
had to be taken to put the origins in the top-left corners
in both cases, otherwise a contour’ inside one set of

squares’ in the smaller grid would not be similar 53’

the contour inside the corresponding set’ of squares in the

190

larger grid. When both grids were done, the two ruled
photos were passed on to the children who obviously seemed
to enjoy tracing similar contours on corresponding sets of
squares in the two grids. The children themselves later
deduced’ that this was a‘pyptematic‘pgy’ of improving the
drawing of a human face when a ruled photograph or even a
(ruled) rough-sketch of the subject was available.

In the next period, students brought old photographs,
magazine cuttings, and rough sketches into the classroom
and ruled a co-ordinate grid over the original picture,
using the intercept-distance’ of either 1/4 in. or 1/2 in.
between parallel lines. They then drew enlarged pictures
of the original copies on graph-paper’ whose squares were

0!
é- x '5'" each. Those who drew from ruled photos did not

seem to have good results, because the photo-paper was often
bent (curved) while the graph-paper was perfectly flat.
This was, once again, the problem arising from the ppgpg—
formation’ of a.ppp.p£_points’ from a curved surface’ onto
a.plpp_.’i

This last activity was one of the best ways to lead
to the formal introduction of the Cartesian Co-ordinate
System.” While drawing a contour or a curve which covered
many squares in the co-ordinate grid, the children would
soon sense the difficulty of keeping‘ppppk’ and counting’
the correct numberlpg'sguares’ involved. Sooner or later

they would discover that an easier way was to select certain

191
kpyrpositions’ on the contour or curve in the original copy
and to draw on the graph paper by jpining’ together~ppp_
points corresponding to those key positions. Once a pair
of mutually perpendicular’ Co-ordinate Agpg’ was set up,
these key positions would be easily identified by an ordered
‘pgipf (x,y), where x and y referred to the perpendicular
distances’ from the vertical‘pgip’ and horizontal pgip’
respectively. Since any two adjacent pgggg’ of a photograph
are perpendicular, they can serve as the Cartesian Co-ordinate
Axes. The lp£p_vertica1'pggg’ and the bottom horizontal‘pggg’
are strongly recommended so that all points in the picture
are situated in Quadrant I’ of the Co-ordinate Plane.

 

é

Picture

 

 

 

Origin ‘ I-axis

Figure 4.20: Cartesian Axes defined by two edges
. of a picture.

At this point the gppgflgg Tic-tac-toe” could have
been introduced. After preliminary playing with 2D
Tic-tac-toe, the children should also be encouraged to play
with the 5D version of the game. This is one way of intro-
ducing the concepts of ordered triplets”, and ppppg_mutually
peppendicular‘gggg,” as well as consolidating many general

concepts about co-ordinate systems.

192

This concluded the major types of activities of this
Unit during the observed period from the beginning of
October to the end of November, 1972.

Conclusion: This USMES Unit was originally an off-shoot
of the 'Describing People' Unit, and.a.new branch of challenge
has been established: how to design clothing, comfortable
furniture, household fixtures and other utensils appropriate
to the body size of each age-group. Although this Lansing
school started with the same challenge about designing
clothes, furniture, etc., to fit the physical proportion
of 5th graders, their activities soon branched out into
Graphic Art: how to apply this knowledge about human
proportion to draw better pictures of human faces and full-
size human beings. A small group of children here also
embarked on another branching out into the study of elementary
aspects of physiology and anatomy. This arose from one
boy's inquisitive question: "Since my waist size is different
from Anne's, although we are both 10 years old, do we have
different things (organs) inside our waists?" After some
lengthy search in the library, the children did have a
chance to examine photographs and diagrams of the main
organs in the vicinity of the human waist. With the writer's
help, they managed to draw the following simplified

diagram.

195

 

‘ I U I 2
Liver *9 j ’ 1 Q'F'mvspleen

A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Stomach ”‘1: ‘ ‘ j,
x
L. Intestine-*- “‘»«~- I
‘1‘ Large
Small Intestine- ‘\,:&::> _445‘ Intestine

Caecum
Appendix I
To Orifice "'5 ' 1

 

Figure 4.21. The main organs around the front
waist-line.

It should be noted that there should be more 22;;5‘ of
intestines than displayed in Figure 4.21. The small and
large intestines together measure up to 40 ft., spirally
curled up within an internal volume’ of l x l x‘% cubic feet.
This experience would be reinforced by such activities as
looking at a spiral staircase in some museum, the spring
inside a clock, or the magnified picture of the wiring inside
a transistor. All of them would represent the idea of
saving space whenever a lengthy object is put in spiral
2.92.9.3

The names used in Figure 4.21 were all common names
because it was agreed that the Latin names appearing in
some encyclopedia were too difficult. To the delight of
the writer, this small group of children were observed to
be interested not only in the positions, but also the

194

different functions of these organs. Given enough time for
guided studies in the library, they would have learned and
drawn the entire anatomical set-up of man.in a somewhat
simplified manner: the skeleton, the breathing system,
the circulation system, the muscles, etc. It is still an
open question whether a 5th grader should be allowed to
dissect specimens of animal organs. But a group of 6th
graders at Wexford School, Lansing, did dissect some frog
legs when they became inquisitive about the variable piggp’
of the thigh and calf muscles while moving their legs.

For those children who have learned even the simplified
version of man's anatomical structure, they would find it
an inspiring example of a well-organized society. The
society's work forces on food-production, transportation,
commerce, communications, waste-treating, defense, etc.,
all have their counterparts inside a human body. In many
ways, the organs of a.man function more efficiently than
the corresponding units of a society.

John Dewey once urged people to consider education'pp
growth‘gpggrowth._a_geducation.1 This USMES unit has grown
from "using numbers to describe people," to the designing
of clothing and furniture, to Graphic Drawing, to the study
of simplified anatomy and its social implication. Every stage
of this growth is some tangible motivation for the study of

mathematical, biological and social sciences.

 

1Dewey, John, "Experience and Education," Ontario:
Collier-McMillan, 1969, p. 56.

195

One observation should be noted about this group of
5th graders. They measured things in inches and feet from
the start, and hence did not experience the gradual evolu-
tion.of "ppippflp£,measurement"’ as suggested by Piaget.1
Piaget advocated the use of a child's body (from shoulder
to feet) as his own "measuring unit" first, before the
formal introduction of commonly acknowledged(but arbitrary)
units like 'foot' and 'meter'. An USMES teacher, Charlotte

2 at Eaton Rapids, Michigan, did take up this Piagetian

Hayes,
idea and encouraged each child to use his body or parts of
the body to be units of measurement. Many delightful
equations’ were consequently noted, for example:

2 lengths of arms = length from toes to chest,

4 times around the foot = once around the thigh.

etc.

This enterprising teacher also suggested a very good way
to introduce the right-angled triangle’, and the‘plppg’
9; _a_ straight 1.193 by using the data collected from human
bodies.3 Each student took measurements of his own.height

(in inches) and weight (in 1bs.). He then drew to scale

 

1Piaget, Jean, "How Children Form Mathematical Concept,"
Scientific American, Vol. 189, November, 1955, pp. 81-82.

2Hayes, Charlotte, "Teacher' 3 Log, " USMES Workin. Pa er
on."gesigning for Human Proportion,“ Serial No. C6, 1572,
pp. -5-

3Hayes, Charlotte, "Teacher' 8 Log," USMES Working_Paper
on."Describing People," Serial No. C5, 1972, p. 6.

196
a right-angled triangle OAlBl, with

length 0A1: the number of lbs. from weight measurement,
length A131: the number of inches from height measurement.
On the same diagram, other right-angled triangles, such as
OA2B2, CABBS’ etc. were similarly drawn from his peers'
measurements. (See Figure 4.22). The slopes of the pypotenuses‘
obtained would indicate whether a person was relatively fat
or skinny.

A.Height

 

 

 

 

 

3'
A2 Weight

 

3 1
Figure 4.22. The 310pes of the hypotenuses of a series
of right-angled triangles telling the relative

body sizes of people.
Besides learning the interpretation of a graph, the children
here were also introduced to a new way of measuring anglesz'
by means of the ratios BlAl/OAI’ B2A2/OA2 etc. It should
be emphasized to the children that such angular measures
are constantly used in more advanced mathematics, where
they are represented by the symbols P tan-1(B1Al/OA1), etc.,
with P referring to the principal range‘ of values between

 

197

The CASE-Study of the 'Burglar Alarm Design' Unit
Observed at Pleasant View School, Lansing, Michigan
October 5 - December 5, 1972
And also the Summer Workshop's Afternoon Sessions
(with children participating)
from August 25 to September 1, 1972

I. The Behaviors to be observed would be focused mainly on
the children's participation in settingguplsimple direct-
current circuits (and thereby learning from scratch the basic
concepts of current electricity), in abstracting the relevant
physical objects into symbols or elements of a circuit diagram,
in.making and experimenting with home-made electromagnets
(and hence learning about magnetic forces and magnetic induc-
tion), and, finally, in hookinggup electric alarm-bells to
doors; windows, desks, door-mats, etc. This last activity
would give the children the experience of measuring large

distances, because they wanted to hide the alarm-bell circuits

at obscure places well away from the 'bugged' doors and windows.

II. Brief Description of the Classes Observed: The two
groups at Pleasant View School were 3rd graders and 5th
graders respectively. Both classrooms were very near the
Laboratory which contained all the electrical apparatuses

as well as measuring devices, e.g. yardsticks, and workshop
devices for soldering, metal cutting, wire-bending, etc.

On the average, the USMES period was twice a week: Tuesday
and Thursday, 12:30 to 1:40 p.m., but sometimes an additional
period was arranged for Fridays. The children participating
in the Summer werkshop's afternoon sessions were mostly 5th

or 6th graders from.Wexford School.

If

 

 

 

198
Both teachers at Pleasant View School took part in the

Summer Workshop. One of them is a male teacher who is quite
familiar with all types of electricians' work, and the lady
teacher of the other class was quite good at arranging a
suitable environment for her jrd grade students to learn from
scratch the concepts of electricity and electromagnetism.

No reading nor communication problem.was observed in

these groups of children.

199

Activity 1: Playing with batteries, wires and bulbs;
and learning from scratch the fundamental ideas of current

electricity.

This activity was observed in the 3rd grade class. The
teacher posed the main challenge of this Unit by talking
about thieves' breakings into stores and homes, and asking
the class what could be done to discourage them. Apparently
most children in this class had heard of burglar alarms
(perhaps from television movies), because they suggested
to install such instruments, but, on closer examination,
the teacher realized that, although these children talked
about burglar alarms, few had seen a pggl.one, and fewer
knew (even vaguely) how it worked. The trouble was that
even the few children who had seen a burglar alarm did not
have a chance to see the entire set-up: they had been
exposed to only part of an alarm system (usually, a bell
or a buzzer). So the real challenge of this Unit became:
how to find out the 'mysterious' (hidden) parts of any
burglar alarm system, or, better still, how to construct
an alarm system that would be equally mysterious to outsiders.
The teacher gave the children a hint that an easy-to-make,

and yet effective, alarm was an electric* one. This hint

L

1"Throughout this Case Study, the underlined words marked
‘ refer to the significant topics in Mathematics or Physics
observed in these classes, while those marked " refer to the
pgtential lessons in these two fields.

200

turned out to be a kind of pretest, because the children's
responses revealed a high degree of ignorance about electric
circuits.‘ They had heard of electricipzf; (one boy even
said "lightning‘ is electricity"); they knew they could get
electricity from batteries (dpzflggllg‘) as well as from
wall sockets, but none of them was aware that it required
a complete circuit‘ for electric current‘ to exist. The
children were thus recommended to go and fetch batteries,
copper wires (properly insulated), bulbs, crocodile clips,
etc. from the Laboratories, and they were going to play
'Light up your heart' because one girl said that the bulb
represented the 'heart' of the system, and a few burned
fingers would later testify that the bulb was indeed the
‘heart (regulator) of the system in this case.

It was gratifying to watch children picking up the
ideas about an electric circuit from scratch. Most of
them only knew that somehow the battery could light up a
bulb (e.g. in a torch light). So they began by putting
the bottom of the bulb on the top of the battery, but
nothing happened. One boy said, "Daddy always talks about
'electric wires' and wiring; so let us try a wire." The
addition of a third element (the wire) did not present any
permutation‘ problem, because the children had the intuitive
idea that the wire, being a connection‘, was to be placed
between‘ the battery and the bulb. The first trial:
connecting the bottom of the bulb to the top of the battery

(by means of a wire) produced no result again. But the

201

children could see and did try other combinations‘,
such as:
bottom of bulb connected to bottom of battery,
bottom of bulb connected to curved side of battery,
threaded part of bulb congggted to top of battery
It would be interesting to ask the children _h_9_w_ £9.22" distinct
combinations they could try. Besides, they learned many
ggometricalpidggg‘ from this experience: (1) the battery
was a real-life circular‘pigpp_cylinder which they could 5

see and manipulate, (2) the top part of a bulb was approxi- E

 

mately a hollow s here,‘ and (3) the wire was topogicallz
equivalent‘ to a straight lipg‘ in their first set of
experiments. In this case the wire was topologicalkz
different‘ from.a.lppp,‘ As a matter of fact, the children
discovered electric current‘ only when one of them abandoned
the straight line and thought of the next element in the
topological hierarchy: the simple closed 23313,‘ or loop.
Once the idea of bending the wire into a loop set in, all
children would in due course experience the sensation of

some unwelcome has}? (on the wire) and a pretty l_i_gh_t‘ (on
the bulb). But it usually took a long time (nearly an hour
for this class) to discover the correct spatial relationship‘
among the battery, the bulb and the wire. Once the bulb

was properly lit up, it meant success as far as learning

to establish a simple electric circuit‘ was concerned.

In fact, most students who learned by discovery had established
a mischievous, though legitimate, circuit before that: the

202

short-circuit‘ obtained by connecting the bottom of the
battery to the threaded part of the bulb while this threaded
part remained‘gg contact‘ with the top of the battery.

It should be noted that this experience was contrary
to all their previous experiences: they had seen Mother
using a wire to connect an electric iron or vacuum-cleaner
to a wall socket, or Father using a wire to connect a lamp
to a switch. But this was the first time something (i.e.
the battery) had to be connected to itself in order to bring
about the phenomena of heat and light mentioned above.
This generated a lively discussion among three students
because they by now suspected that there were really 322
things inside the battery. "The battery is not really
connected to itself, but we Just connect together the two
things inside the battery: one is at the top knob and the
other at the bottom," explained one child. So the concept
of positive gpg,negative,pg;g§‘ began to take shape. Further,
since no heat or light was observed in the events of such
ordinary connections as the connection of two things by a
string, the connection of a dress to a coat-rack by means
of a coat-hanger, etc., this connection between the two
poles of a battery was indeed a special event. Something
happened to the wire, the bulb and the make-up of the
battery itself, and this special "something” was the children's

idea of the abstract concept of an electric current.‘

203
Children also tried to connect the two poles by means
of an ordinary string, a sticky tape, two pencils, etc.,
nothing happened. So the class of insulators‘ began to form.
Besides copper wire, they would later discover that other
metal wires, tin foil and aluminum foil could be used,
giving them the other complementary} set: the set of

conductors‘.

Activity 2: Making a switch from an empty coffee can,
and defining an open or closed circuit hypinsertingpthe switch:

One child suggested to solder together the battery,
the wire-loop, and the bulb to have a permanent light,
but other children objected.

"The battery will soon be run down," said one boy.

"If the light is going to be an alarm," said another
student, "we want it 'ppf‘ only when there is a burglary,
otherwise the light should be flpgg'.‘ We should work out
an automatic switch.‘"

This second remark came exceedingly close to the 29933-
lyipg,principle‘ of the Burglar Alarm circuit: "The whole
thing is one big switch, whether it is mechanical, pptical,
or electromagnetic."“ The idea of a switch.naturally led
to the concept of an.gpgp‘ or closed‘ circuit, and other
relevant terminology like breaking‘ or closing‘.g circuit.

The teacher suggested that the students should 23kg_

a switch themselves by using metal strips cut from an empty

coffee can, and she explained her rationale: "To make a

switch yourself will enable you to see its mechanism‘ in a

204

constructive manner (in contrast to seeing the internal
mechanism of any toy by tearing it into pieces). This is
better than just hooking a commercial switch to your circuit,
because a commercial switch.has everything concealed inside
a case."

The writer could spot many mathematical behaviors from
the children's activities of cutting coffee cans and drawing
up plans for circuit switches. First, the bottom of a can
is a circle.‘ The children used an ordinary can-opener to
remove the bottoms of these cans, and they competed to get
the most perfect circle. Secondly, the side of a can (where
one gets most of the metal) is a gylindrical surface.‘ The
children had to flatten‘ this curved surface into a.plgpgr
figppe‘ in the form of a rectangle‘ after cutting along the
side of the cylinder. A more traditional teacher might
have objected to the deafening sound of children's hammering
the cut cylindrical surfaces into rectangles. The students
then measured‘ and _I_'u_l_e_d_‘ phi rectangular ir_e_a_ into thin
strips‘ and cut accordingly. While they were cutting metal
sheets into strips, it should drive home vividly the concept
of a straight ling} and parallel l_ip_e_g‘. "Hey, you don't
cut it straight. Follow the line," said one boy to his peer.
The teacher advised the children to fold over the sharp
edges (and corners) of each strip, so that it would be safe
to work with these strips later on.

The next job was to make a switch from the metal strips.
The teacher told the class that they had to think and

205

reason 933‘ how a switch might fit into the complex of the
battery, copper wires and the bulb, before deciding on the
mechanism of the switch. Working individually at first,
the children did not meet with much success, and so the
teacher suggested that drawing §_diagram‘ might help to
figure out the (spatial) relationship‘ of the switch to
other things in the complex. The diagramatic representation‘
of a physical situation is indeed mathematical learning:
the students had to abstract all important junctions (here,
the positive and negative poles, the bottom of the bulb)
into geometrical points‘ (shown as 'boxes' in Figure 4.25)
and the connecting wires into topological lgpgg.‘ They
also had to discard irrelevant variables‘ such as length
of the wires, colors of the battery case, shape of the
bulb, etc. To sum up, a diagram is a kind of mathematical
abstraction,‘ showing only relevant factors such as
relative positions,‘ connections‘ and disconnections.‘

On their first attempt to draw the circuit diagram,
the children learned one more thing: a diagram is £123
necessarily the photographic copy of a physical phenomenon.
A diagram may reveal some internal or subtle connections
or relationships which the eye cannot see. For example,
it took these children a long time to realize that, in a
simple circuit, the 'wire-loop' connecting the negative
pole to the bulb and the positive pole was, in fact, 2gp
a loop at all, but a topological line. Such a conceptual
scheme for relative ppsitions‘ of the three things implied

206

that, in the diagram to be drawn shortly, the bulb had to

lie between‘ the negative and the positive poles. (See

Figure 4.25) Of course, there had to be a loop somehow,

otherwise it could not be a complete circuit. The 'missing'

link in the diagram was, in fact, the internal connection

via the chemical solution, (represented here by the dotted
line.)

 

 

 

Copper Copper

_ Positive ”ire Bulb I ”ire [Negative
Pole l [ Pole

 

 

 

 

'- ------------------- Chemical Solution ------- <— --------- _,

Figure 4.25: An abstract diagram showing a simple circuit
without a switch.

It was, at this point, relatively easy to insert a

switch into the above circuit because the students could

operate formally‘ on the diagram by means of pencils and

erasers. The teacher reminded the class that the idea of

a switch was to break or make a complete circuit whenever

one wanted. A girl quickly suggested: "We can break the

circuit anywhere on the 'copper wire' part of the diagram

and insert the switch there." The children by now began

to see the significance of the metal strips out earlier,

as one boy remarked: "When two strips touch‘ each other,

the light is on; but when they are moved apart,‘ the light

is off." The teacher reinforced that the remark was correct,

207
and added that the light is on or off, when the circuit

is said to be M‘ or 922‘.

Discussions later led to the conclusion that one
moveable‘ metal strip touching two 9523‘ Fahnestock clips
would be preferable to two moveable strips. The children
drilled a hole near the mid-point‘ of the metal strip,
put a screw through it to serve as a pivot, and mounted
the metal strip and screw on a tri-wall board. Two
Fahnestock clips were then fixed to touch the ends of the
‘ strip when it looked like a horizontal ;i_1_i_g‘ (i.e.
parallel 12‘ an edge of the board), and this was 'on'.
When the strip was rotated to make an _a_ng_l_g_‘ with the
horizontal line, it was 'off'. (See Figure 4.24).

Metal Fahnestock

+ Strip / Clip
1301. Q” g: r-—-—fi ..
Fahnestock Pivo Bulb Pole

Clip

 

 

 

 

 

l
l
i
'- ----------------- Chemical Solution -- — --------

L--

Figure 4.24: A mechanical switch added to the
simple circuit.

Activipy é: Mg a batten-holder.

Another activity related to cutting empty coffee cans
into metal strips was the construction of battery holders.
This was to be accomplished in three stages: (1) obtaining
a wooden block or a tri-wall board of comparable 3553‘ to

208

that of the battery, (ii) nailing a 2" x l" rectangular‘
piece of metal strip to the wooden block or triwall board,
and bending both ends of the strip upwards into half-a~
cylinder‘ to fit the curved 3393‘ of a battery laid
horizontalgy,‘ (This was enough to prevent it to 2232.§£§2f
‘ggy§‘), (iii) fixing something close to the ends of the
battery to prevent it from moving lengphwise.‘

Stage (i) involved the measuring‘ of the length‘ and
diameter‘ of the battery and then the drawing of a rectangle‘

 

whose dimensions were numerically‘gqul to the measurements
just obtained. This activity also involved the children
in sawing wood or using a saber-saw to cut tri-wall into a
rectangle slightly bigger ppgp‘ the one just drawn.

Stage (ii) involved an estimation‘ of the middle-
portion's length of the battery and the semi-circumference‘
of the cross-section. The bending of a rectangular metal
strip into half-a-cylinder gave the children the unique
experience of making‘g.curved surface‘ in contrast to the
traditional experience of lookipg‘gp_one. Many children
made this curved surface almost three-fourth‘ of a cylinder,
because it seemed to hold the battery better. These
children also experienced the 'springy' reaction‘ of the
curved surface as they pushed the battery down. (See
Figure 4.25s).

In stage (iii), most children made the mistake fixing
only 2513 piece of metal strip (about 5" x )6") pp 535113 angles‘
to the first strip and then bending the ends upwards to stop

209

lengthwise slipping.‘ Sooner or later the ends of this
lengthwise strip would pppgp‘ the positive and negative

poles simultaneously,‘ and a violent short-circuit phenomenon
resulted! (See Figure 4.25b). The children learned their
lesson quickly, though. One boy remarked: "Two separate‘

pieces of metal strips are needed, because one continuous‘

 

piece does not work." So the children started to nail two

Lpshape metal pieces to the wooden block, the distance‘ between

 

the two vertical planes‘ being approximately the length of the
battery. Some children achieved the same objective by
removing the middle portion‘ of the lengthwise metal strip

and fixing an extra nail to the now loose metal strip on

the left (See Figure 4.250). They explained: "This is to

disconnect‘ the mischievous 'wire' (lengthwise strip)."

 

 

f

 

 

 

 

 

n

r I

Nail Wooden Nail Nail Nail
block

(a) (b) (c)
Figure 4.25: A home-made battery-holder;

(a) Viewed from one end,
(b) Side-view, short-circuit phenomenon,

(c) Side-view of the improved model. (Cylindrical
strap in the middle is not shown.)

210

The short circuit mentioned earlier gave the children
a very deep impression of the distinction‘ g£_ppg cardinal
numbers '1' and '2'. ("ONE metal strip was disastrous,

TWO were quite safe," said one child.) All too often

students in a traditional methematics class (including College
students) just verbalize the numbers 1, 2, 5, 4, ... etc.,
without having too much impression about 322 events involving
ppggg.numbers.‘ Recently, an Economics professor at Michigan
State University spent a greater part of his lecturing time
analysing meat prices, and one student remarked afterwards:
"It is the first time these weird things called numbers mean
anything to me."

This short-circuit phenomenon also made quite an
impression on children about the difference between a
continuous gppyg‘ (metal strip) and two separate‘ curves
(strips) involving simple discontinuipy.‘

Lastly, the children in this class found out that it
was misleading to look at a commercial battery-holder (without
careful examination of the parts). In,a commercial battery-
holder, the two metal strips touching the top and bottom
of the battery respectively were separated‘ by a very 325p}
rubber strip (an insulator‘) which was barely visible. "This
gives people the wrong impression that it is one piece of
metal there, but in fact they are two pieces," said one
child.

 

211

Activity 4: Discovering a way_to have a stronger
purrent;_discussing the anatomy of the electric bell; and

windingelectromagnets:

A girl in the 5th grade class wrapped a sticky tape
round the bottom portion of a battery and continued, with
the same tape, to wrap the top part of another battery,
so that the two batteries were joined together with the
positive pole of the second battery touching‘ the negative
pole of the first. It did not take her long to discover
that such a "series" arrangement‘ of two dry cells was an
effective way to provide.§ stronger current‘ to any circuit:
she connected this 'lengthwise‘ combination' (as she called
it) of two batteries to her previous circuit involving a
bulb and a switch, and this arrangement gave her a brighter‘
light than the single battery alone. She was puzzled about
the fact that a commercial battery—holder (which could hold
two batteries) had the batteries laid side by side instead
of joining them lengthwise. It was indeed a good exercise
in topology‘ as well as in ppysics‘ to figure out the
equivalence‘ of the two simple closed curves‘ (current lpppg‘)
in (a) and (b) of Figure 4.26 below:

o 3’

      

(a) The girl's arrangement' (b) The commercial arrangement

Figure 4.26: Two batteries arranged in series.

212

She went on to join together 5 and 4 batteries 53 series‘
and obtained a brighter and brighter light until the rest
of the class gathered round her desk to see what she had
done. This discovery turned out to be crucial for the whole
class. Before that, several students who had connected
commercial alarm-bells to their circuits tried in vain to
get them to ring, or at best they rang only faintly. "I
have got it:" exclaimed one boy, "we need more batteries:
Since two batteries made the bulb light up pgigg‘ as bright,
they would make the bell ring twice as loud as before."
His statement was not altogether correct; two batteries
would make the electric bell sound as it should, and certainly
not louder than usual. (The commercial electric bells
provided were designed for a potential difference‘ of
5 yplpg,"whil ‘gpgp_battery generated about 1.4 volts only.)
Besides a bulb, a buzzer might be a better current-indicator‘
than a bell.

The class now knew how to set off the electric bell
and in the next ten minutes the whole room was filled with
the ringing noise of electric bells. Soon the children
got interested in the anatomy of the electric bell. This
interest sprang from the request of a boy who wanted to
take apart the different pieces of an alarm bell system,
but the teacher instead advised him to study ppgf pp
electric pgllwgppkg,and then make a buzzer from coffee-can

strips by using the same principle‘ as an electric bell.

213

"What are the parts of an electric bell?" the teacher
asked the class. I see a spring, hooked to a tiny hammer
that strikes the bell," one student responded. "What is
the function of the spring?" asked the teacher. A boy was
observed to take a careful look at the spring and hammer;
he diSplaced‘ the hammer slightly with his finger and the
hammer pgppgflppgk,‘ striking the bell once. (This demon-
strated the gppk‘ done via the potential energy‘ of a Spring.)
The student reported this finding to the class. The teacher
remarked further: "All right, the spring causes‘ the hammer
to swing back whenever the latter is displaced, but what
caused the displacement‘ of the hammer in the first place?
Surely we are not going to use our fingers every time."
This was a timely remark because it made the children look
at their electric bells carefully once again. After several
wildly speculative and incorrect statements such as

--"the spring works both ways: it pushes as well as pulls."

--“the hammer strikes the bell too hard, it bounces back."

etc.

finally one boy did spot a valuable lead, though the whole
set-up was still a mystery to him. He asked the teacher:
"Could it be that the thing which looks like a "cotton reel"
(the electromagnet“) attracted‘ the hammer away from the
bell?" The teacher responded in a wise manner: he did
not say 'yes' right away, but, instead, he advised the whole
class to 2552 electromagnets‘ by first winding an.insulated
electric gipg‘ round a nail, a bolt or any piece of iron,

214
and then passing pp electric current‘ through that winding

wire. "If your home-made electromagnets attract metals,"
said the teacher, "then the electromagnet looking like

'a cotton reel' in an alarm bell will certainly attract
the hammer nearby."

This activity of winding electromagnets offered many
relevant mathematical learnings about circular cylinders‘
and spirals‘. First, if the iron bolt was treated as a
right circular cylinder, one complete pppp‘ of the copper
wire = 2n13units, or nearly 5 times the diameter‘ of the
2233‘. The next complete turn represented the beginning of
a circular spiral‘ path, if there was no overlapping‘ with
the first turn. If a fine copper wire (coated with an
insulating chemical) was used, the children could wind an
incredible length‘ of this wire round cylindrical bolt or
nail of about 5 inches long or even shorter. At Roxbury,
Mass., an USMES class managed to wind 29 £933" 32 11933 of
copper wire around a nail, and to add to the excitement,
they measured“pp_the length of the wire before or after
winding. This gave the children two valuable experiences:
(1) a clear-cut view of saving space by means of setting
up a spiral, (ii) the practice of measuring sizeable distances‘.
(The writer has to concede that throughout his school and
College training he has never had a chance to measure any
length greater than 10 feet, nor to play with a wire or
string of any sizeable length.) At this point, the use of

a trundle wheel“ to measure great distances could have been

introduced.

215

This activity could also have been an interesting and
highly relevant exercise on fractions“ and decimals“ as
exhibited in the following sample calculation:

By direct measurement, e.g., by a screw-gauge or even

a ruler, the student should be able to note that

The diameter of the cylinder = 1/4 in.
The length of 1 turn of the wire = (5.14)(l/4)
= .785 in.

To wind 20 ft. of wire, it requires $3§§g or 505.7,

i.e. roughly 506 times. (The quotient 20x12

worked out fully: €8§§a = 20x%§§1000 = giggg9.=:505,7)

' 3

should be

 

 

 

 

 

Figure 4.27: Winding an insulated wire round a
cylindrical iron bolt.

If the #26 AWGl

wire is used, and if the turns are
really _t_i-ggfi and closed 332 3933 93323, 506 turns will cover
approximately 5 in. of the length of the cylinder.

Hence the diameter of this fine electric wire==386'in.,

or roughly 1/61 in. This is equivalent to 0.01654 in., or,

 

1A.W.G. stands for 'American Wire Gauge'. AWG Standard
copper wires (bare and insulated) ran e from #0000 (diameter
0.46 in.) to #40 (diameter 0.0051 in.). See, for example,

Lister, Eugene C. "Electric Circuits and Machines, N. Y.,
MCGraw-H11-1 00., 1966’ p. Sme

216

as an engineer would say, 16.54 pilg“ (l in.= 1,000 mils).
This simple calculation would reveal to the students the
powerful method of mathematical deduction“ from indirect
measurements. Nobody expects a 5th grader to use a
micrometer“ to measure anything which is i3 Eh_e_ 9_I_:_d_e_r_“
‘2; l/61 in., but, by indirect measurements and calculations,
he can, in fact, determine small magnitudes of that order,
or something even smaller. (He can usually measure things
correct to 1/16 of an in. by a ruler and never anything
which is more refined). After the calculation, he could
check the result with standard AWG Tables, which give the
diameter of #26 wire as 15.94 mile.

The small discrepancy of 0.01654-0.01594, or 0.0004 in.,
should not become a source of discouragement for this kind
of calculation, but it should encourage the students to
investigate further into the reasons for this discrepancy:
first, the approximate“ nature of the solenoid“ length
(approximately 5"), the number of turns (approximately 506),
and n = 5.14, and secondly, the imperfect winding (i.e.,
the turns might be less close to each other than appeared
to the eye). Such an investigation would have been a good
exercise in learning Scientific Process“ and systematic
improvement for accuracy“ in measurement. In any case,
the children who went through the above process of calculating
the thickness of #26 wire would have learned at least one
topic: the ppdgpflp£_magnitude“, for 0.01654 in. and 0.01594

in. really belonged to the same order of magnitude, i.e.

217

1.6 x 10"2

in. (in Scientific Notation“). The concepts
of positive“ §pd_negative exponents“ could have been
discussed here in conjunction with topics in which children
were normally interested, e.g., the large distances involved
in space travels. ’
An arithmetical problem, similar to the one described
in this Activity, arose during the USMES Summer Workshop
at Lansing in September 1972. A 5th grade girl wanted to
make an electromagnet out of a 2" nail by winding #26
wire round it. At the request of the writer, she counted
up the number of turns, and she reported 92 turns when she
used up exactly 5 ft. of the #26 wire. The writer then
formulated a 'nice' arithmetic problem for her to think about:
"92 turns of #26 AWG wire wound around a 2 inch nail
measures exactly 5 ft. What is the mean diameter of
the nail?"
This was a 'nice' problem in several respects:
(1) it contained two irrelevant numbers: #26 and 2 inches;
(the girl could spot them immediately, though),
(ii) the word "mean diameter" was used in a practical
sense, not just an 'average' calculated from some fixed algorithm,
(iii) it introduced the child to the important fact that
l turn=21rr ,
(iv) it introduced a simple first degree algebraic equa-
tion, and its solution,

(v) it was a good exercise on decimal fractions,

(vi) the final answer could be verified by means of a ruler.

218
[Solution: 1 turn = 1rd = (5.1416)d=

Hence d=gzigém = 0.124"

Direct measurement gave d==l/8 in.]

5x12

Activipy 2: Experimenting with electromagnets:

After most children had made their own electromagnets
from nails and #26 wire, they connected the ends of the
wire with a battery, and began to play the enjoyable games
of testing which articles could or could not be attracted‘
by these magnets. For many children, the phenomena of
magnetism‘ was not altogether new to them: they had played
with bar magnets or horseshoe magnets before. Many toys
today also utilize the properties of attraction and repulsion
of magnets, although the magnets are often concealed within
the toys. But the following could be observed to be 93!
experience on Magnetism for these 5th graders:

(1) Increasing the strength of an electromagnetjgy
winding more turns of wire around it: The children quantified‘
t_h_e_ strength of their magnet by counting‘ _t_h_e_ number 2;; @-
‘plgpg it could pick up and hold them steady in.mid-air
for some length of time (i.e. by balancing the magnetic
‘gppgg‘ against the gravitational‘ppll‘ on the paper-clips).
Most children's recorded results were almost the same,
plus or minus 1 or 2 paper-clips at most. Table 4,12 below
is a typical result for the circuit consisting of only one
battery:

219

Table 4.12 Showing how the magnetic force depends on
the number of turns of the winding wire.

 

 

No. of turns No. of clips suspended

50 9

100 17

150 25
200 33
250 40

500 45

350 59

This result, whether it was graphed‘ or not, showed clearly
that as the number of turns increased during the first
stage of the winding, the magnetic strength also increased '
'gp.g constant 5333‘ equivalent to the weigpt‘ of 8 paper-
clips for every 50 turns of wire. But, after the 200th
turn, there was a decline‘ in the growth-rate‘ of magnetic
strength, and, finally, the magnetic strength itself began
to drop after the 500th turn. The children were very puzzled
by this unexpected decline both in the growth-rate and
in the magnetic strength itself.

At this pOint, a simple (qualitative) explanation based
on.9pp;puggp“ could have been offered to these children
who had already been motivated by the above puzzle. The .
reason was that, as the electrical resistance“ of the wire
gradually increased (1.13.1.1. 39.2 increasing length“), £1.12.
current gradually decreased.“ Since the magnetic strength
depended 2215.9. 93“ the number of turns and the current, a
decline in the growth-rate of the magnetic strength could

be observed if the current began to decline. (This was an

220
example of a function g; two independent variables“).

Finally when the (external) resistance“ due to the wire was
nearly the same as the internal resistance“ of the battery,
the current was drastically Mg“, and the increasing
number of turns could not make up for the loss of magnetic
strength due to the Lap decline“ in the current, and hence
a solenoid pg 529 M‘ or more would pick up less and less
clips than before. (See Table 4.12). The more enthusiastic
students could now experiment with ammeters, voltmeters“ ,
etc., and study the quantitative aspect of Ohm's Law:

E
I:

 

H+r

Problems based on Ohm's Law are always good exercises on
fractions“ and solving simple algebraic equations“
involli_ng one 93; two unknowns .

(2) A 'paper-clip' egeriment demonstrating the essence

of the Inversed Square Law:
Several children were playing with their electromagnets

in the following way: each moved his electromagnet nearer
and nearer to a paper-clip, until suddenly, under the £939;
25 attraction,‘ the clip jumped towards the tip of the
electromagnet (with about 20 turns of electric wire in

most cases). The writer asked each student to take note

of the distance‘ between the clip and the magnet when the
'jump' took place. "low, M‘ that distance,” said the
writer, ”and test to see how many more turns of wire you

221
would have to wind around the nail-«in order to make the

paperclip jump as before."

”253.33 pp 2551‘ , naturally, " remarked one boy.

“Let us all try it," said the writer.

To their disappointment, the children found that after
doubling the number of turns, the magnets in all cases were
still not strong enough to attract the clips placed twice
as far as previously. Some even tripled‘ the number of
turns, but the paper-clip in each case still failed to jump
as expected. Most children were on the verge of giving up
when the writer suggested the last brute-force attempt:
to increase the number of turns py _a_ $3339; _a_; g'. and if
it still did not work, then nobody would be able to blame
them for changing activities. But, fortunately, it did
work: all their electromagnets had now enough strepgph‘
(wens £93939 to overcome the limitipg frictional
_f_9_l;_c_e_‘ between the clips and the table, and the clips
jumped toward the magnets as expected.

The writer then asked each child to prepare _tw_o_
electromagnets, having either 8 and 72 turns, or 9 and 81
turns, or 10 and 90 turns, and tried out the same experiment
with the 9.91 condition‘ that the distance between the clip
and the 'stronger' magnet (having more turns) was 5 m:
as much as the distance between the clip and the 'weaker'
magnet on the other side. "What is the significance of
these number-pairs like (8,72), (9,81), etc.?" asked the
writer. “The stronger magnet is always having 9 times the

222
strength of the weaker magnet,“ one boy replied, after a
long and deep thought. At this point, the mathematical

function‘
8 = 9W

(3 stands for 'Strong' , and W for 'Weak'), or generally
y = 9x could have been introduced.

The children carried out the suggested experiment.
They found that , when only one magnet was used, there was
always a critical m: for the stronger magnet was always
5 times that of the weaker magnet. When both magnets were

 

simultaneously placed on the two sides of only one clip,
and if critical distances were used in both cases, then the
clip would not move: it was _ip eguilibrium‘ under tw0
mg; 3333 opppsite £33333) Only two boys out of the
whole group achieved this spectacular equilibrium, or
balanc ‘, which was a very difficult task to manipulate.
The writer introduced the following two columns to
the children:

 

 

If you increase the then you have to
distance between clip increase the strength
and magnet by of magnet by

2-fold 4-fold

5-fold 9-fold

 

 

 

 

Unfortunately most children in this group were not
accustomed to thinking of 4 as 22 and 9 as 52, and so no

225
mathematical relationship evolved from this experiment, nor

was there any sign of awareness about its implication of

a very important law of Physics. The writer did not draw
any conclusion from the above two columns, nor did he
attempt to verbalize the Inversed Square £92“ Hopefully,
one day in the future, when some High School teacher teaches
these children Physics verbally, they might recall this
earlier experience and learn the Inversed Square Law in a
more meaningful way than those who have not been exposed

to this experience.

(5) The phenomenon of mqgetic induction:
In Experience (1) described earlier, the children had

the opportunity to see a great number of paper-clips (See
Table 4.12) being suspended by an electromagnet. 0n

closer look, the children realised that most clips did not
cling to the magnet directly, but they clung, instead, to
other clips which were nearer to the magnet. It could have
been pointed out to the children that the m: clips really
became temmrqu W‘ ‘ and this phenomenon was known as
Lagetic induction“. Several children here‘b discovered that,
if these clips remained in contact with an electromagnet for
a length of time, they could retain a certain £93293. q;
Lagetizstion‘ even) after leaving the electromagnet.
Incidentally, an electromagnet with several layers of
Paper-clips clung to it was an effective instrument to

study spatial relationshi ,“ especially such concepts as
innermost or outermost elements“, something lying My

224

within“ something else, volume“ and surface area“, etc.

Activity 6: Connecting alarm-bell circuits to doors,
windowsI desk-drawers, door-mats, etc.

Apparently most children in these two classes knew by
now how an alarm circuit could be set up, and so the next
job was to connect these alarm-bell circuits to doors, win-
dows, etc. to achieve the original objective of preventing
burglary. The children spent a great deal of time on this
Activity, but some were quite successful even in their
first or second attempt. The more successful ones are
described as follows:

(i) The 'positive' and 'negative' terminals‘ (i.e. the
two sides of the circuit switch) were connected to two tin-
foils (conductors) separated only by a thin piece of card-
board (insulator). A tight string tied the cardboard to a
(closed) door or window. As the door or window was forced .
open, the string pulled the cardboard away from the tinfoils
whose contact M‘ 3:33 circuit‘ and set forth the alarm.

(ii) Several children made use of the _sp_ri_pg‘ in a
clothespin to ensure the contact of the tin-foils when the
insulator (e.g. a cardboard) was pulled away by the door
or window. Two pg; copper wires could be used in place
of tinfoils in this case: the wires were wound round each
side of the jaws of the clothespin, and the insulator was
inserted in-between. The electrical principle employed

here was exactly like (i) above.

225

(iii) A group of students out out a 2 ft. x 1 ft.
triwall board and labeled it 'Welcome Mat. ' Four springs
were fixed at the four corners of the mat. A large sheet
of tinfoil (about 1-1/2 ft. x 5/4 ft.) was nailed to the
bottom side of the mat, and it was secretly connected to
the 'positive' terminal of an alarm-bell circuit. When the
mat was placed on a floor, the 4 springs held it up (about
1" above ground level). A second tinfoil, connected to
the 'negative' terminal of the same alarm-bell circuit,
was placed on the floor imediately underneath the mat,
and thus was separated from the first tinfoil by 1325 312
ll. When somebody stepped on the mat, the springs were
depressed, the mat was lowered, and the two tinfoils came
into contact, setting off the alarm.

Activity 6 was, indeed, a sort of climax of the
Burglar Alarm Design Unit. It partially answered the
Unit's main challenge about designing a burglar alarm
which would give effective warning. It also brought
together all the physics learned in previous activities,
especially Activities 1, 2 and 4.

The children in one class agreed that it was a good
idea to have the bell or buzzer hidden well away from the
doors, windows and the 'Welcome Mat' which were "bugged.“
So a considerable M‘ of wire was to be used. The
children tried to do a bit of 'advance planning' : they,
by a means of a steel tape measure or a yard stick,

measured‘ the distance between the door. or window to the

226
obscure corner of the room where the alarm-bell circuit
was hidden. Many students forgot 3:3 W‘ _t_h_e_ measured
distances (for the purpose of wir ‘ 3 complete circuit)
and so their advance planning was incorrect, e.g., when a
boy came forward and asked the teacher for 14 ft. of
electric wire to hook up his hidden alarm circuit to a
window, he actually needed 28 ft. of wire. This kind of
realistic distant M‘ also gave the children a realistic
sense of the problem of electrical resistance‘ which gradually
M pp‘ to a sizeable magnitude. The current was con-
siderably reduced‘ as judged by the faint ringing of the
bell. (This was a real-life example of inversed proportion‘,
i.e. current 0C 1 ). The children had already

resistance
learned how to remedy this: adding more voltqge‘ by

 

connecting one or two more batteries (_i_n_ _s_q_z_~_i_._e_s_‘) into
the existing circuit. (See Activity 4).

This concluded the major types of activities of the
Burglar Alarm Design Unit during the observed period from
early October to early December 1972.

Conclusion: It was already implied in the short
description of Activity 6 that, although the main challenge
of the Burglar Alarm Design Unit provided a kind of focusing
point within an otherwise divergent set of activities on
basic electricity and magnetism, opportunities for learning
some forty different topics in Mathematics and Physics arose
mainly from these divergent activities, and fewer appor-
tunities spring from the main challenge itself. The

227
closest analogy was that, although an orchestra conductor
provided directions for focusing on one or more groups of
instruments at a time, the pretty sound of music came from
a divergent set of instruments, and not from the conductor
himself.

All USMES Units have encouraged the _use of real
objects, realistic experiments, real measurements and
actual counting, etc., in teaching mathematics and science.
The Burglar Alarm Design Unit seems to be more so than
other USMES units observed thus far. Teaching children
by means of real objects, and arranging children to have
some real experience before verbalization of any type, are
in fact not now. One and three-quarter centuries ago,
Johann Heinrich Pestalozzi published: How Gertrude '
Teaches Her Children, a classic in the theory of "Natural
Education.” But it should be emphasized that to learn
from real objects does not mean plwing aimlessly with
these objects; sense impression, observation and intuition
should lead naturally to concept-formation, the art of
abstraction and classification, etc. which Pestalozzi
called "the child's developed powers." The following
wordsl were written by Postalozzi himself:

”All instruction...is only the Art of helping Nature

to develop in her own way; and this Art rests
essentially on the relation and harmony between

 

J’Pestalozzi, Johann B., How Gertrude Teaches Her
Children, (translated by L. E. HoIIand E F. C. Turner),
$acuse: Bardeen, 1900, p. 26.

228

the impressions received by the child and the exact

degree of his developed powers. . It is also necessary

that the beginning and progress (of these impressions)
should keep pace with the beginning and progress of
powers to be developed in the child."

Pestalozzi's practice was followed and greatly
enriched by Johann Friedrich Herbart who augmented
Pestaloszi's Anschauung (intuition due to sense impression
and observation) with association, generalization and the
"method."1 Herbert's "method‘ meant the development of
general principles for dealing with generalized systems,
as well as the applications of existing principles to new
situations. Both Postaloszi and Herbart voiced regrets
at the separation of theory and practice in traditional
teaching.

Those teachers who would like to add some "Modern
Math" flavor to this Unit could easily do so by branching
out into the study of logical circuits involving only a
battery, switches and bulbs. This is probably one of the
best ways of introducing mathematical logic to elementary
school children, because they could check the validity of
each logical statement by practical means. Figure 4.28
below shows how to set up simple circuits for starting

off this new set of activities.

 

1Dunkel, Harold B., Herbart and Education, New York:
Random House, 1969, pp. ‘75 m 116.

229

 

 

___,$ ‘ _3

 

 

 

 

 

 

L1- __[__
(a) (b)
Figure 4.28. The circuits displaying the consequences

of the logical operations p“; pB and p‘v p3.
Statements like "switch A is on" and "switch A is off"
are abbreviated to be PA and.»pA respectively. Various
joint operations upon switches A and B would result in
one of the two possibilities: the light is either pm;
or 93;, written briefly as L, ornL. Sooner or later the
children will be able to abstract the following basic

statements from their practical experience:
(a) p“ p3=91oa (app/x («p3)=>-1a;
(iv-Pfin PB =54“ P‘ A (spa) =$"'L-
Similarly four other basic statements could be attracted

from (b). Usually the children could easily spOt the first
two possibilities and verbalize "Both 'on' means 'on',

and both 'off' means 'off'.‘ The teacher had to step in

at this stage and asked them to think of more possibilities
until all four were exhausted. Many logical games can then
be played, and points are scored by correctly supplying
logical symbols for the question-marks in the following

250
complicated circuits, e.g.

PA
/

fl \

_SL

‘ 9 ‘
rif j «L 7 L ?

fi—J ? PE
[ I I

Figure 4.29. Logical circuits for a kind of guessing
game.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

These games can.be made openeended by allowing children

to change freely the logical symbols in the given circuits
into opposites, or better still, to devise new logical
circuits of their own.and play accordingly. The only
important rule is that they must not chest by operating
the switches to get the answers; they should figure out
the answers first before operating the switches to verify
their answers which have been arrived at logically; it thifl
point, the kind of elementary logic advocated by the
Cambridge Conference1 could have been introduced: open
sentences and their true/false solutions, equivalence as a
'double implication,‘ contradictions, indirect proofs, etc.
These tepics were recommended for classes between Grades

4 and 6, depending on their interests and maturity.

 

1Cambridge Conference Report: Goals for School
Mathematics, Boston: Houghton MifflIE 00., 1963, pp. 51-41.

231

To sum up, although Symbolic Logic can be introduced
here, this USMES Unit is pqp,primarily designed for that
purpose. Instead, it should be viewed as an effective means
to provide children with first-hand experience about many
principles and concepts in mathematics and physics which will,
unfortunately, be presented gymbolicalgy,in the next rung of
the educational ladder. They will soon meet with a sea of
symbols in Junior High Mathematics:1r, sq. rt., m(AB), :5, etc.
plus a wider sea of symbols in Physics:

n for number of turns of a solenoid,

H 1'5? magnetic field,

I Tor'current

E To? voltage

f To? a dry-cell

mm a resistance
SEE.

Hopefully the experience they gain from this Unit will help
to put some vitality into an otherwise boring manipulation
of symbols followed by meaningless calculations. The writer
showed the script of this Case-Study to a personal friend,
an instructor in Psychology. After reading it, he exclaimed:
"What a _n_e_! and revolutionapy way of teaching math and physics:
It's good for the kids, though. If I were taught this way
when I was little, I might have ended up with a Ph.D. in
Physics, and not Psychology like the one I've got." The

writer reflected afterwards: 'revolutionary' was the wrong

word. To put vitality into mathematical symbols has been

252

practiced by good teachers for centuries. The following words

are quoted from one of them, Harold Fawcett:l

"Behind every symbol is an idea. It is the idea
which is important, and it is familiarity with the
idea which puts life in the symbol. It is,
therefore, of the greatest importance that the
idea be developed before it is symbolized, for, to
give a child a symbol for an idea he does not have
is to encourage him to take a long tragic step
toward intellectual disaster."

The same sentiment was also implied by work of another
retired teacher, F. L. Griffin,2 and it can be concluded

that good teaching knows no temporal boundaries.

 

1Fawcett, Harold P., "Reflection of a Retiring Teacher,"
The Mathematics Teacher, Vol. 57, November 1964, pp. 450-456.

2Griffin, F. L., "Some Teaching Reminiscences," The
American Mathematical Monthly, May, 1969, pp. 460-467.

CHAPTERS

DISCUSSION AN D CONCLUSION

The foregoing description of mathematical behaviors
arising from the four USMES units shared considerable

m (“‘1’-i3 (In?

common ground with the mathematical content for I - 6
proposed by the 1965 Cambridge Conference for School
Mathematics. The following points of similarity between
the asses Units and the Cambridge Report1 are important:

1. The role of‘physical equipment is emphasized.
Whether one thinks in terms of pro-mathematical experience,
or in terms of aids to effective communication, or simply
as attractive objects that increase motivation, the con-
clusion is inescapable that the children can study mathe-
matics more satisfactorily when each child has abundant
opportunity to manipulate suitable objects.

 

I? . Thai-

2. The bed-rock foundation for elementary mathematics
consists of counting and physical interpretation of fractions;
symmetry and invariant properties of geometrical figures;
real experience in collecting data; use of graphs and other
{land displays of data; and the vocabulary of elementary

o c.

5. Students are encouraged to seek methods for checking
the correctness of their solutions or answers without
recourse to the teacher.

4. Long-range projects undertaken by individual
students or small groups of students are preferred so
that students may have experience with the extended aspects
of mathematical learning involving many types of modeling
and mathematical techniques.

 

 

1Cambridge Conference for School Mathematics, "Goals
for git-1:31 Mathematics,” Boston: Houghton Mifflin Co., 1965,
PP- .

235

254
Nevertheless, some mathematical topics recommended by
the Cambridge Conference were not observed in the four USMES
units, for examples,
(a) Arithmetic of signed numbers,
(b) Inequalities and absolute values,
(c) Prime numbers and factoring,

(d) Use of straight-edge and compasses to do geometrical
construction,

(e) Interpolation.
0n the other hand, these USMES children learned many

[FFW-Qe-fmmv “MK-.0196 «an

sophisticated concepts in Probability, Topology and Founda-
tion of Mathematics which were conspicuously absent from
the Cambridge Report. This Case-Study would, hopefully,
help future USMES teachers to locate and utilize the
potential mathematical behaviors inherent in these Units'
activities. No doubt the list of mathematical behaviors
(in Chapter 5) will grow as future groups of USMES teachers
and students discover more and more mathematical topics

in the course of their work.

The writer of this Case-Study fully realized that his
graduate training in Mathematics was a positive asset in
doing this research, because it helped him "to see mathe-
matics in.action” whenever such an opportunity arose.

But he also realized that graduate mathematics alone would
not be sufficient for this work. It required another
component: he had to be interested in the children's

235
development in the cognitive, affective, and psychomotor
domains.

In doing this Case-Study, the focus was on:

1. The integration of several kinds of educational
experiences pertinent to mathematics learning, and the
integration of activity learning with the usual paper-and-
pencil work,

2. The relation of mathematics to some real-life
long-range problems, not just the short-range problems
involving only symbol manipulation,

“ I?

5. Students' opportunity to acquire problem-solving
ability and to learn the scientific method,

 

4. The mathematical behaviors arising naturally from
these USMES Units or their sub-tasks.

V'

The first three points mentioned above are, in fact,
concurrent with Harold Fawcett's thinking.l He wrote:

"To meet the responsibilities of citizenship, the
mathematics program should be designed to promote
problem-solving process, to define and deal with
problems of concern to the society, to gather
and organize relevant data, to detect underlying
assumptions of the system, and to draw valid
conclusions by both inductive and deductive
reasoning. Mathematical principles should be the
outgrowth of experience rather than the basis of it.
Operational skills should serve a recognized and
useful purpose."

The USMES activities motivate and introduce a student
to certain mathematical ideas, say fraction and percentage,
but it is up to the student to practice more on those topics

if he wants to achieve such.mathematical skills. There are

 

1Fawcett, Harold P., "Mathematics for Responsible
Citizenship," Mathematics Teacher, Vol. 40, November 1947,
pp. 199-205.

256

students who genuinely like expository teaching to guide
them along in these sub-tasks. Others continue to explore
and discover their own solutions to these sub-challenges.
At this point diversity is desirable. (Eventually the
results of these sub-tasks will be combined to answer the
major challenge.) The most appropriate underlying principle
here seems to be: "Teaching should increase the diversity
of students' achievement rather than the uniformity."1

Conclusion: The following mathematical behaviors were
observed to have permeated throughout all the four USMES
Units here: counting and measuring which lead to meaningful
usage of whole numbers, decimals and fractions; place value;
the four fundamental operations of arithmetic; percentage;
ratio and proportion; estimation; collection and tabulation
of raw data; graphs; proposing hypotheses and testing them;
elementary logic; abstracting concrete situations into
mathematical relations; sets and their elements; ordinal
and cardinal numbers; ordered pairs; simple geometrical
shapes like triangles, rectangles, and circular cylinders.

In addition, the following mathematical behaviors were
observed in the individual units described below:

The 'Soft Drink Design' Unit: Independent and dependent
variables; composite function; multiplicative inverses;

matrices and reduction of the size of a matrix; frequency;

 

lFitzgerald, William M., "On the Learning of Mathematics
by Children," Mathematics Teacher, Vol. 56, November 1965,
PP. 517-521-

 

a"

237
,points on the Number Line and transformations of these
points; invariance under a transformation; confounding
variables.

The 'Dice Design' unit: Theoretical probability;
approximations; closure, commutative and distributive
laws; arithmetical mean and progression; the combination
n02; the histogram; the Normal Curve; plane geometry of
the triangle and the hexagon; construction.of polyhedra
from pre-cut polygons; the 'double-entry' principle of
book-keeping; one-one correspondence.

The 'Human Proportion' Unit: Examples of linear and
non-linear functions; curved lengths and curved surfaces;
orthogonally projected heights; rotation; tangency; simple
trigonometry; calculation.of mean, median, mode and range;
many—to-one correspondence, parallel and perpendicular
lines; Cartesian Co-ordinates.

The 'Burglar Alarm Desigp' Unit: Arithmetical deduction
from indirect measurements; scientific notation; inverse
proportion; inverse square law; permutation of five elements;
first difference; spherical surface, a topological line;
simple closed curves; simple discontinuity; separation;
distance between two planes; direct proportion; informational
aspect of cardinal numbers.

The following potential mathematical topics could
have been included: algebraic problems involving one or
two unknowns (Soft Drink and Burglar Alarm Units); sets

 

P?“

257a

which are topologically equivalent (Soft Drink, Human
Proportion and Burglar Alarm Units); random selection;
stratified sampling (Soft Drink Unit); equality (Soft
Drink, Dice Design Units); relative error (Dice Design,
Human Proportion, Burglar Alarm Units); intersection of
two planes and the angle between them (Dice Design Unit);
envelop and curve-stitching (Human Proportion Unit);
various measures of angles: degree, radian, and tan"1 (y/x)
when referring to principal values (Dice Design, Human
Proportion Units); positive and negative exponents (Burglar
Alarm Unit) and logical circuits (Burglar Alarm Unit.)

This Case-Study could serve as a guideline or recommended
procedure for teaching mathematics in.an.USMES. The list
of potential mathematical topics would help teachers to
initiate mathematical sub-tasks whenever an opportunity
arises, so that these USMES Units might be utilized to
their full potential in generating activities and discussions
which are pertinent to mathematical learning.

 

BIBLIOGRAPHY

 

WT-..5~"

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a . e
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m e I
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e
I v. ~ -
. e e O
a r r a .
r a e .
. , .
1‘ “... o
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. v a
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' r
e
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. o _
1 ‘ O 0 e
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m a o e
a a

   

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