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Humphreys, Alan; Post, Thomas V.BINNEMIST Recommendations ror Science and Math in theIntermediate Grades.'Minnesota Univ., Minneapolis. Minnesota SchoolMathematics and Science Center.National Science Foundation, Washington, D.C.7055p.; For related documents, see SE021201-234;Photographs may not reproduce well'MINNEMAST, Minnemath Center, 720 Washington Ave.,S.E., Minneapolis, MN 55414
EDRS,PRICE HF-$0.83 HC-$3.50 Plus Postage.DESCRIPTORS. *Curriculum; Curridulum Planning; Elementary
Education; *Elementary School Mathematics;*Elementary School Science,; *Instructional Materials;*Interdisciplinary Approach; Intermediate Grades;Mathematical Applications; Mathematics Education;Process Education; Science Education; Textbooks.
IDENTIFIERS *MINNEMAST; *Minnesota Mathematics and Scieshde,,,_Teaching Project
ABSTRACTThis volume provides recommendations for teaching
mathematics and science to intermediate students who have been taughtusing the MIVNEMAST materials during the primary, grades. Afterreviewing briefly the goals and content of the primary curriculum,the authors discuss the transitions from the integrated program todistinct curricula in mathematics and science. For each field,several criteria for the intermediate curriculum are defined andalternate models are offered. Text and supplementary materials aresuggested for each of these models. Brief descripticns of therecommended texts are provided. (SD)
0
************************************************************************ _Documents acquired by ERIC include many informal unpublished* materials not available from other sources. ERIC makes every effort ** to obtain the best copy available. Nevertheless, items of marginal ** reproducibility are often encountered.and this affects the quality ** of the microfiche and hardcopy reproductions ERIC makes available *,* via the ERIC Document Reproduction Service,(EDRS). EDRS is not* responsible for the quality of the original document. Reproductions ** supplied by EDRS are the best that can be made from the original. ********************************************4***************************
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ST
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RJA
ME
S H. W
ER
NT
Z, JR
.,
Professor of PhysicsU
niVersity of M
innesota
ASSO
CIA
TE
DIR
EC
TO
R4
RO
GE
R S. JO
NE
SFO
R,SC
IEN
CE
;A
ssociate Professor of PhysicsU
niversity of Minnesota
ASSO
CIA
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DIR
EC
TO
RFO
R R
ESE
AR
CH
AN
D E
VA
LU
AT
ION
WE
LL
S HIV
EL
Y II
Associate Professor of Psychology
University of M
innesota
The M
innesota Mathem
atics and Science Teaching
Projectproduced this booklet under a grant from
theN
ational Science Foundation
0 1970, University of M
innesota.A
ll rights reserved.
RE
CallM
EN
DA
TIO
NS
for Science and Math
in the Intermediate G
rades
by:
AL
AN
HU
MPH
RE
YS
TH
OM
AS R
. POST
EL
AIN
E V
OG
TE
ditorB
EV
ER
LY
SOH
RE
Layout
JUD
Y N
OR
MA
NC
over Design
FOR
EW
OR
DM
athematicians, scientists and educators com
-bined their talents to produce the 29 sequentialunits that m
ake up the MIN
NE
MA
ST C
urriculum.
They have coordinated the teaching of m
athemat-
ics with the teaching of science in kindergarten
and the primary grades.
This coordinated curriculum
provides a firm foun-
dation on which the children can build as they
follow other m
athematics and science non-coor-
dinated curricula in fourth grade and later. We
place special emphasis on the actual handling of
materials, by the students, w
hich leads to a fun-dam
ental understanding of concepts, as opposedto rote learning.
Through actual experience the
children see how m
ath and science serve eachother and how
closely interrelated they are. We
think that mathem
atics and science taught as to-tally unrelated subjects cannot provide as gooda preparation as w
ith this approach.
If your school or your district is not presentlyem
ploying the MIN
NE
MA
ST C
urriculum, w
e heart-ily invite and encourage you to exam
ine it and toconsider its adoption.
A pam
phlet, Questions
and Answ
ers about MIN
NE
MA
ST, and a price list
will be sent free upon request.
Our O
verview,
which provides an illustrated description of the
contents of each unit and our auxiliary books, isobtainable at m
odest cost.Y
ou will find the
Overview
very helpful; 1,ve'think, in any prelimi-
nary investigation of our materials.
Any of our
publications may be purchased singly. M
any areuseful and rew
arding at levels other than thoseindicated in our sequential arrangem
ent.
CO
NT
EN
TS
,
Minnem
ast Publications2
Purpose of this Booklet
4
Chart of C
oncepts and Skills Developed
by Minnem
ast8
Mathem
atics Transitional Statem
ent
Minnem
ast,Mathem
atics Skills and Concepts
28
Science Transitional Statem
ent33
Chart of T
hreads that Run T
hroughthe M
innemast C
urriculum48
Keeping C
urrent50
MIN
NIM
AS
T
Z.
§-ac
yHOG
20
/11011
CO
OR
DIN
AT
ED
MA
TH
EM
AT
ICS
-SC
IEN
CE
SE
RIE
S
I .W
AT
CH
ING
AN
D W
ON
DE
RIN
G
2.-C
UR
VE
S.AN
D SH
APE
S
3.D
ESC
RIB
ING
AN
D C
LA
SSIFYIN
G
4.U
SING
-OU
R- SE
NSE
S
5.IN
TR
OD
UC
ING
ME
ASU
RE
ME
NT
6.N
UM
ER
AT
ION
7.IN
TR
OD
UC
ING
SYM
ME
TR
Y
8.O
BSE
RV
ING
PRO
PER
TIE
S
9..N
UM
BE
RS A
ND
CO
UN
TIN
G
10.D
ESC
RIB
ING
LO
CA
TIO
NS
11.IN
TR
OD
UC
ING
AD
DIT
ION
AN
D SU
BT
RA
CT
ION
12.M
EA
SUR
EM
EN
T W
ITH
RE
FER
EN
CE
. UN
ITS
13.IN
TE
RPR
ET
AT
ION
S OF A
DD
ITIO
N A
ND
SUB
TR
AC
TIO
N
I4.E
XPL
OR
ING
SYM
ME
TR
ICA
L PA
TT
ER
NS
15.IN
VE
STIG
AT
ING
SYST
EM
S
16.N
UM
BE
RS A
ND
.M E
ASU
RIN
G
17.IN
TR
OD
UC
ING
MU
LT
IPLIC
AT
ION
AN
D D
IVISIO
N
18.SC
AL
ING
AN
D R
EPR
ESE
NT
AT
ION
I9.C
OM
PAR
ING
CH
AN
GE
S
20.U
SING
LA
RG
ER
NU
MB
ER
S21.
AN
GL
ES A
ND
spAcr- -----
22.PA
RT
S AN
D PIE
CE
S
23.C
ON
DIT
ION
S AFFE
CT
ING
LIFE
24.1
CH
AN
GE
AN
D C
AL
CU
LA
TIO
NS
25.'M
UL
TIPL
ICA
TIO
N A
ND
MO
TIO
N
26,W
HA
T A
RE
TH
ING
S MA
DE
OF?
27.N
UM
BE
RS A
ND
TH
EIR
PRO
PER
TIE
S
28.M
APPIN
G T
HE
GL
OB
E
29.N
AT
UR
AL
SYST
EM
S
OT
HE
R M
INN
EM
AST
PUB
LIC
AT
ION
S
The 29 coordinated units and several other publications
are available from M
INN
EM
AST
on order.O
ther publications include:
STU
DE
NT
MA
NU
AL
S for Grades
I,
2 and 3, andprinted T
EA
CH
ING
AID
S for Kindergarten and G
rade I.
LIV
ING
TH
ING
S IN FIE
LD
AN
D C
LA
SSRO
OM
(MIN
NE
MA
ST H
andbook for all grades)
AD
VE
NT
UR
ES tlq SC
IEN
CE
AN
D M
AT
H(H
istorical stories for teacher or student)
QU
EST
ION
S AN
D A
NSW
ER
S AB
OU
T.M
INN
EM
AST
Sent free with price list on request
OV
ER
VIE
W(D
escription of content o2 each publication).
MIN
NE
MA
ST R
EC
OM
ME
ND
AT
ION
S FOR
SCIE
NC
E A
ND
MA
TH
IN T
HE
INT
ER
ME
DIA
TE
GR
AD
ES
(Suggestions for programs to succeed the M
INN
EM
AST
Curriculum
In Grades
4, 5 and 6)
For additional information, a price list and orders, w
rite to:
MIN
NE
NIA
ST D
irectorN
linneniath Center -
720 Washington A
ve. S. E.
Minneapolis, M
innesota 55414
PUR
POSE
OF T
HIS B
OO
KL
ET
4
M,any thousands of children
are studying or havestudied science and m
athvia the M
INN
EM
AST
Coordinate K
-3 Curriculum
.M
any more w
ill beentering
ur program in the future.
We have had
numerus requests from
educatorsinterested in
cong/nuing MIN
NE
MA
STobjectives and techniques'
thiough theinterm
ediate grades for sugge'stionsconcerning m
aterialsw
e think appropriate.T
heM
INN
EM
AST
staff, too,is m
uch concernedw
iththe kind of science and
math education thatM
IN-
NE
MA
ST children w
illreceive in G
rades 4, 5 and6. W
e should liketo see the positive
attitudestow
ard science and math
that- the childrenacquire
in our program continue
to grow. W
e should alsolike to see the special
concepts and skills thechildren develop by the
end of Grade 3 putto ad-
vantageous use in the nextgrades. W
e believethat the interm
ediategrades offer greatopportuni-
tiesC
TJ
for the application ofour chidren's skills.
For these reasons, staffm
embers have ecam
inedall of the leading
science and math
curricula(both project and
comm
ercial) for theinterm
ediategrades.
Because M
INN
EM
AST
is-trraquein cdordinating
sciencre-alid-Math,
we found no single curriculum
that fulfilledby itself all of.the
MIN
NE
MA
STrequirem
ents; but neither didw
e see any particular problem
s in switching
to non-coordinatedm
a-terials for G
rades 4,5 and 6.
In fact, we feel
that our program gives
children, in thesem
ostIm
portant early years,a better preparation for the
more difficult w
ork of theinterm
ediate grades thancan be provided by
non-coordinated materials.
In this booklet the authorssuggest a num
ber ofalternatives for the continuing
education of MIN
-N
EM
AST
children.O
ur suggestions are intendedto be helpful rather than dogm
atic.C
urricula notm
entioned by us may
prove to be more suitable
for certain local conditions,and educators are
encourageeto survey otherm
aterials for them-
selves to see if thism
ay be so.A
lso, our fail-ure to m
ention any. particular curriculumis by no
means to be'interpreted
as a criticism of it.
We
are merelypresenting here, foryourconsideration,
recomm
el? ations of materials thatseem
to be ad-equate s a`cessors to our curriculum
.
To m
ake clear the criteriaby w
hich MIN
NE
MA
STstaff m
embers m
ade theirselections, w
e presentfirst a short sum
mary of the
project's objectives,techniqU
es and curriculum.
The M
innesota Mathem
aticsand Science T
eachingProject (M
INN
EM
AST
) is the onlym
ajor projectestablished for the express
purpose of coordinatingthe teaching of science and
the teaching of math-
ematics in the elem
entary grades.In recent years
the National Science
Foundation has provided thefunds. T
he result isa curriculum
of 29 unitsex-
tending from K
indergartenthrough G
rade 3.In
addition to the teaching units,Student M
anualshave been provided for G
radesI ,
2 and 3.For
Kindergarten and G
rade 1, variousprinted T
eacherA
ids have also been prepared.(M
any schoolsystem
s have no kindergartenclasses.
In thesecases, the teachers select key lessons
from the
kindergarten materials to
present at the beginning5
6
of the first grade.B
ecause of the greater matu-
rity of first graders, the kindergarten lessonsare
quickly mastered.) W
e have also published two
auxiliary books:(I) L
iving Things in Field and
Classroom
is useful at all elementary le'vels; and
(2) Adiientures in Science and M
athcan be used
by both, teacher and student at different levels toprovide m
otivation and, enrichment.
The principal aim
s of the MIN
NE
MA
STProject are:
to develop competency in both m
athematics
and science,
to provide experiences that involve the stu-dents in challenging and interesting problem
-solving situations,, andto provide settings that enable the children
toexplore, discover and clarify the. relationshipsbetw
z:en Mathem
atics and sciencew
hereverfeasible.
Why coordinate m
athematics and science?
There are m
any natural bridges betweenm
athe-m
atics and science. In fact, the science thatw
e.know
and wish to com
municate to children
todayw
ould not even exist without the clarification
andprecision that m
athematics provides.
And m
uchtheoretical science has evolved,in a sense, frombranches of applied m
athematics
. The dependence
of mathem
atics on science isnot so explicit, for
mathem
atics in its purest form is basically inde-
pendent of science. How
ever, the rhany discov-eries and dem
ands for quantification in the vari-ous sciences provide a great im
petus to the ad-vancem
ent of mathem
atics.B
ecause of thesevalid relations betw
een the two disciplines,
pro-ject planners and w
riters believe thatan early
coordinated approach helps childrensee the in-
terrelationships and develop a stronger and deeperunderstanding of both fields of study. W
ealso
believe that this coordination leads toan earlier
appreciation of the principleson w
hich the world
operates, enabling children to understand, andsom
etimes even predict, various changes
Project writers have been successful in finding
many relationshilas for the children to explore.
The M
INN
EM
AST
treatment of concepts dealing
with m
easurement, sets, sym
metry, slope,
den-sity and system
s', as well as.our developm
ent ofo
techniques for accurate observation and descrip-tion, exem
plify areas that allow the children
touncover and exploit i terrelationships betw
eenthe tw
o disciplines.
The chart on the follow
'cipal concepts and skillevels of the M
INN
EM
be helpful to educatorsthe interm
ediate gradeuse of w
hat MIN
NE
Mlearned.
A m
ore detaiR
un Through the M
INN
Evided near the end of thi ng pages show
s the prin7.s em
phasized at differentT
Curriculum
.It should
in selecting materials for
that build on and rrlakeT
children have alreadyed chart, "T
hreads That
AST
Curriculum
, " is pro--booklet on pp. 48 and 49.
to
Observing
Describing
Classifying
Comparing
,
.
Ordering11....,
Measuring
Communicating
Symbolizing...,.
Representing
Graphing
. .Experimenting-r.. 1 I I
Predicting
Simplifying...
Approximating
. HypothesizingI_ i
.-(- - -:Inferring
Model Buildingj
L___1 r Abstracting
13
CO
NT
EN
TC
ontent topic.; are selected for:
0 we-grade approPilatenes.-
0 concept development
0 procers applicationis
sets and group.shapes
-sym
metry
one to one cotiesporidencedeveloping num
ber conceptequivalent set::
using sensesm
easurement
indoor-outdoor observing
sets - intersections and unionsset and num
ber line interpretationsof addition and subtractiongeom
etry - de:cribing locationsequivalence of setsfl,:..velopinq num
ber conceptordering
measurem
ent of length,area, volum
e, time duration
changing properties of icecube, seeds, plants,m
ealworm
s 'fram
es of referencesym
metry
!Joe iactional units in measurem
entreview
and amending addition and
;ubtractIon conceptsm
ultiply and divide small w
holenum
ber.;scaled diagram
s and models
geometry - review
, angles and theirm
ea.aire, polygon_: and polyhedragraphing
functional relationshipssystem
l, - natural andcontrivedm
easurement of w
eightinterpretation of graphs
review and extend piaci, value
me cairem
entgeom
etry - mapping
emend concept arid. C
omputation
with rational num
bersintroduce the grid to Interpretm
ultiplicationgraphingextending m
ultiplication
condition,. affecting lifem
ortoninvestigation of m
atenals.
..
tti
INO
.41
.J;
Because M
INN
EM
AST
coordinated the teach-ing of m
athematics w
ith the teaching of sc4 nce,m
any of the objectives and methods used w
ereem
ployed in the teaching of both disciplines.T
hese were described in the preceding
pages.B
ut, no doubt, educators who specialize iri m
ath-em
atics will be m
ore interested in this section ofthe docum
ent, while science educators w
ill beconcentrating on the "Science T
ransitional State-m
ent." Because of the sim
ilarity of 6Yrr M
INN
E-
MA
ST, learning/teaching m
ode for both subjects,there w
ill be some overlap in the inforrriation
pro-vided.
How
ever, for your convenience, we,are
reiterating here some fades particularly
as theypertain to m
athematics.
In mathem
atics, as in science, the MIN
NE
-M
AST
Project places great emphasis on student
involvement in the learning process.
The adage.
that "children learn by doing" has been takenseriously and it has been 1.ept in m
ind all throughthe preparation of the m
aterials.A
s a result,M
INN
EM
AST
students are, more often than not,
actively engaged in the solution of some problem
'that has been designed to have relevance for them
.T
hey may be attem
pting to predict the growth rate
of plants under various growing conditions,
todeterm
ine the effect that temperature has on the
behavior of goldfish, or to explore the relation-ship betw
een the weight and volum
e of varioussubstances. T
he quantification of such physicalphenom
ena is a powerful tool that m
an has de-veloped to aid him
to understand more fully these
and numerous other relationships that abound in
MA
TH
EM
AT
ICS T
RA
NSIT
ION
PLST
AT
EM
EN
T
12Sr
the world.
The pow
er and utility of this tool isestablished early in the program
.T
hus thereevolves one of the m
ost important, relationships
between m
athematics and science, the
use ofm
athematics to describe and explicate the results
of scientific inquiry.T
he development and im
-plem
entation of this idea has playedno sm
allpartin the developm
ent of the MIN
NE
MA
ST1naterials..
As a result, our students tend to view
mathem
at-ics as both a'"living" subject having
great utilityand as a kind of 'gam
e whose pieces behave
in avery predictable m
anneradhering to rules w
inchthey have been instrum
ental in developing. This
is in contrast to the frequently held student viewof m
athematics as a set of rules to be m
emorized
and applied under appropriate stimuli.
Projectm
embers are hopeful that the m
athematics
programchosen for M
INN
EM
AST
children in the interme-
diate grades will have inherent in it m
uch of theform
er emphasis and little,
if any, of the latter.
Webster defines arithm
etic as "a branch ofm
athematics that deals w
ith real numbers and
computations w
ith them; i.e. com
putation, cal-culation, " w
hile mathem
atics is definedas "the
science of numbers and their operations, inter-
relations, combinations, generalizations and ab-
stractions and of space configurations, configu-rations and their structure, m
easurement, trans-
formations and generalizations."
Applying this
definition, the MIN
NE
MA
ST Project has been pri-
marily concerned w
ith the provision ofa m
athe-m
atics program although arithm
etic implications
have not been ignored.
Our project staff believes that the prim
arygrades are the tim
e for the implantation and
pre-lim
inary development of appropriate m
ajor mathe-
matical icleas,\ a tim
e for the child to experiencem
athematical concepts,
a time to observe, tb
wonder, to explore avenues of interest and
ap-peal, a tim
e to be involved in the rudiments of
inductive thinking derived-ofrom experience w
ithconcrete m
edia.A
ccordingly, we have placed
emphasis on these areas rather than
on computa-
tion and drill.T
he MIN
NE
MPST
Project has ,de-voted its
,efforts to student understandings offundam
ental mathem
atical (not merely arithm
eti-cal) ideas.
The im
portance of computational fa-
cility is recognized, but it is felt that the devel-opm
ent of these skills prior to the provision form
ultiple student opportunities to experience theconcepts in concrete form
is unwise from
botha
mathem
atical and .pedagogical viewpoirit.
As a result of this philosophy, M
INN
EM
AST
students have been exposed to elements of
num-
ber theory, set theory, Eudlidean and projective
geometry, transform
atfonal geometry and the in-
tuitive idea of a mathem
atical function. Besides,
MIN
NE
MA
ST students through the' third
gradelevel have w
orked with m
ost of the arithmetical
concepts found in traditional primary m
athematics
programs.
We believe that by pfbviding considerable
breadth in matheiaatics content and by continually
placing emphasis on Jtudent discovery and
un-derstanding of logical relationships, M
INN
EM
AST
represents a closer approximation to a true. m
ath-
CO
orl
13
MA
TH
EM
AT
ICS
TR
AN
SITIO
N C
RIT
ER
IA
14
ematics curricula than the
vast majority of pro-
grams available.-
In searching fora reasonable
successor to its primary m
athematics
program w
ehaie placed a high
priority on:the suitability of included
topics relative tocontinuation and
expansion of fundamental
concepts that MIN
NE
MA
ST has
beguh to de-velop
the degree to which
student involvement is
encouraged
the extent to which
students are encouragedto, discover and develop
mathem
atical ideasfor them
selves
the degree of variationof m
athematicaltopics
the precision with w
hichthese topics are in-
troduced and developedthe degree of priority
given to the development
of computational facility.
Realizing its obligation
to provide specificsuggestions for the interm
ediategrades that con-
tinue to deVelop the m
athematical
ideas it hasbegun, the M
INN
EM
AST
Project has conducteda-
careful examination of the
more popular m
athema-
tics .comm
ercial text series.T
he criteria statedabove have form
ed thebasis upon w
hich thesem
athematical program
s have beenevaluated.
As m
entioned previously,the M
INN
EM
AST
Project recognizes the importance
of student' in-volvem
ent in ,learningactivities.
Com
mensurate
with this recognition
is our belief that children
should have the opportunity to develop ideas tra-ditionally transm
itted by rote..Probably no pro-
gram has been successful in developing a totally
discovery-oriented program, but M
INN
EM
AST
-stu-dents do have m
any opportunities toformulate-im
-portant ideas for them
selves.
"Developm
ent of insight through discovery, "."conceptual
approach emphasizing discovery"
and "problem-solving as discovery" ate typical
program descriptions found, in the introductory
pages of comm
ercial text series. Our investiga-
tions have indicated that pupils are not involvedin true discovery activities to the extent thatsuch statem
ents suggest. Discovery is a lauda-
tory aim that is often m
inimized.
Editorial pres-
sures, standardized achievement tests, as w
ellasccincern for adm
inistrative and comm
unityanx-
iety, are factors that tend to limit severely the
,,production of a truly process oriented comm
ercialprogram
. In attempting to select m
aterials thatoffer continuing problem
- solving challenges with
emphasis on individual investigation and discov-
ery, MIN
NE
MA
ST suggests three m
odels for yourconsideration.
These m
odels are recernmended
for use in Grades 4 through 6 by children w
hohave used M
INN
EM
AST
mater -rals through G
rade3. Y
ou will notice that each m
odel includessup-
plementary m
aterial from the, M
adison Project.A
lthough supplementary, this m
aterial is consid-ered an im
portant part of any of the three models
we have proposed.'T
herefore a description of theM
adison Project is provided, along viitha list of
some of its m
aterials, before the com
mercial text
16
series are described. The three
models w
e aresuggesting are listed here in alphabetical
order:M
odel I
a) E1'em
entary School Mathem
aticsA
ddison Wesley C
ompany (1968)
b) MadiSon-Project.(S.upplem
ental)M
odel 2
a) Sets andN
uMbers
Singer Com
pany (1968, 1969)
b) Madison Project (Supplem
ental)M
odel 3'
a) 'SRA
Elem
entaryM
athematics Program
Science Research A
ssociatgs, Inc.(1968, 1969)
b) Madison Project (Supplem
ental)
We do not intend that:com
mercial
series notm
entioned abcive should beconspicuous by their
absende.It is entirely possible that
comm
ercialm
aterials other than those named
will be m
orein keeping w
ith local objectives.`W
e would en-
courage locals-currictilum com
mittees
to examine
additional materials to), determ
inew
hether or notthis is .the case.
The-M
INN
EM
AST
staffis, obli-
, gated to state, however, that the three textseries
Mentioned above w
ere not selectedat random
.O
f all series examined,
Singer, AddisontW
esleyand SR
A w
ere found to be them
ost consistent with
the philosophy, aims, and the
teaching-learningm
ode of the MIN
NE
MA
ST program
--
MIN
NE
MA
ST m
athemati&
al skills' and con-cepts are listed, unit by unit, on pages 28 through
.32. T
his fist can be used not only for assistance.
-'in--determining w
hich math program
to select forthe interm
ediate gralso for selecting
individual MIN
NE
MA
ST units to tea-ch-at-other
levels where it is felt there are certain gaps
---,-the students' m
athematical education.
The list
can be helpful in,choosing units to include forstudy in m
ath education courses, too.
The M
INN
EM
AST
Project was unable to locate
a comm
ercial text series that fully reflects thephilosophy of m
athematical learning to w
hich itsubscribes.
Therefbre, the desirability of sup-
plementary m
aterials was realized. T
he Madison
Project materials have been designed to supple:-
ment "(approxim
ately 40 minute's per w
eek) existing m
athematics program
s at the intermediate
grade level.T
he objectives of the Madison Pi-o-
ject were found to be largely consistent w
ith theM
INN
EM
AST
philosophy from both a content and
pedagogical viewpoint. T
he Madison Project
ma-
terials are designed to allow schools to m
akecer-
tain modifications in their m
athematics
programs.
The m
aterials can be used for the following
pur-poses:- to provide "a foundation lot- developing'aprogram
that unifies 'arithmetic, algebra, geom
e-try, and som
e science- to shift the tone and em
phasis of the school'sprogram
iway from
'rote learning and toward learn-
ing by processes
r.
N.
SUPPL
EM
EN
TA
RY
MA
TE
RIA
LS17
to move tow
ard a greateruse of physical ma-
terials and multi-sensory
experien.Ces in m
atheLm
atics classesto create greater
opportunities for sinall-groupw
ork and individualized instructionto use ,a specific teaching
strategy based ondiscussion of activities
the children, havedone
during an explorationperiod
to create a more
receptive environment for
student initiative,,especially w
here unexpected(but correct)
responses are made by students
to open the doorto a reconsideration
of thegrade-level placem
ent ofm
any topics- to open the door to a non-graded
program- 'to m
ake available the simplest
possible pro-gram
for students who
are not experiencing-StIC
,7cess w
ith mathem
atics- to m
ake available a more
sophisticated anda
more advanced m
athematics
program for those
students who can benefit
from it.
Besides the sim
ilarity in teachingm
ethodsin the M
INN
EM
AST
and Madison
Projects,the
.specific topics developed in theM
adison Projectm
aterials are a natural extensionof m
any threadsthat the M
INN
EM
AST
Project hasinitiated. A
lso,the M
adison Project hasrealized the necessity
in many instances of providing
specific instruc-tional suggestions for
teachers not familiar w
iththe m
aterials.A
n extensive teacherin-service
program com
plete with
films and teacher w
ork-books accom
panies thestudent m
aterials. These
in-service materials provide
detailed suggestionsand illustrations of the
way in w
hich the student
material can be m
ost effectively, taught.T
he.M
adison Project greatly simplifies the problem
ofin-service teacher training by providing the basiccom
ponents of a complete instructional package
for both teacher and students.In particular, the M
adison Pr Oject's C
urric-ulum
(3(B
eta) is recomm
ended.T
his curriculumis m
ade up of a combination of lessons developed
by the Madison Project, the E
lementary
ScienceStudy (E
SS) and the Nuffield Project of G
reat Bri-
tain and by individuals suchas M
arion Walter,
Lauren W
oodby, Leonard Sealey, Z
. P. Dienes,
Edith B
iggs and- Geoffrey M
atthews.
Curriculum
Beta com
bines elements of
car-pentry,
social studies,art and com
munication
skills with m
athematics and science.
It is in-tended. to reach a great diversity of children'andappears to be able to do so.
Certain parts of C
urriculum B
eta can servequite different purposes because of. the varietyof its content.
The curriculum
employs units
'developed by the '.'Madison Project (especially
those emphasizing arithm
etic &lid geom
etry)as
well as units. devised by other individuals and
groups.Som
e of the borrowed m
aterials havebeen m
odified; others are used just as theyare.
For those schools wishing an interm
ediatelevel m
athematics program
that isa logical sue-,
cessor to the MIN
NE
MA
ST m
aterials, the Madi-
son Project is felt to be an essential element in
any transitional model.
It is hoped that schoolpersonnel responsible for this type of curricular
20
decision will seriously atternpt.to integrate
thesem
aterials with w
hatever Com
mercial
program is
ultimately adopted.
A m
uch more com
plete descriptionof the
Madison Project is provided in V
olumes
I and IIof its Final R
eport, "Modern
Mathem
atics Programas it Pertains to the Interrelationship of ,M
athe-m
atical Conterft; T
eaching Methods
and Class-
room A
tmosphere." H
ere we shall
content our-selves w
ith adding only thatC
urriculum B
eta isa non-text program
using materials from
a wide
variety of'sourdes. A partial H
itt ofthe M
adisoProject m
aterials follows.
(Selections -of supplementary
materials should
be made and ordered w
ellin advance of the dates
they will be needed.
For additional information,
write to the addresses provided
in this list.)T
he Madison Project
.9 18 Irving A
venueSyracuse, N
ew Y
ork1,3210
(Telephone - A
rea Code 315, 476
- 3768 or 476 -`554)
X2336)
lActual classroom
lessons ofC
urriculuth Beta
can be seen in the following film
s:eom
etry Via C
oncrete Objects
luing and Stamping
Using G
eoboards with Second G
radersA
n Introduction to Geom
etry viaN
ailboardsA
Sixth-Grade L
essonon Place-V
alue aumerals
The C
oncepts of Volum
e andA
rea.
The C
las.sroom D
ivided intoSm
all Groups
Counting, V
olume and R
ation.41 Approxim
ationsSm
all-Group Instruction
Signed Num
bers, Rational A
pproximations, and
Motion G
eometry
Outdoor M
athematics
A L
esson with Second G
radersSecond L
essonG
uessing FunctionsIn-Service C
oursefor T
eachers (combines
films and printed m
aterial)
In addition to the films, C
urriculum B
eta of-fers units and devices of its ow
n, as well
asthose borrow
erL, or adapted from
other sources.N
ote the following typical' credits:
Units have been borrow
ed from the N
uffieldM
athematics Project;. M
irror Cards and attri-,
bute blocks from E
SS; and Informal G
eometry
from M
arion Walter. C
uisenaire rodsare also
used.
Units developed by L
eonard Sealey, such asthose on
pronouncing' andw
ritingnum
bernam
es ;are-in connection w
ith Dienes'
MA
B blocks; and units
ydith B
iggs, suchas that in w
hich geometric-shapes are classi-
fied by using an assortment of
comm
on card-board boxes, are also used.A
ewide variety of calculators is used.
These
include desk types suchas the- L
agomarsine,
the Monroe, ten-key, full-lkeyboard, double-
keyboard (often used by statisticians), ,thosethat print on paper strips, inexpensive plasticones., m
anuals and electrics;
C..
2.'1
T1
.1r(t
22
Lauren W
oodby's Outdoor M
athematics,
which
emphasizes m
easur'nmen't, ratio
and proportionis a part of C
urricull4m B
eta.
Uses for D
ienes' MA
B blocks, other
than -thosealready m
entioned, are included.B
eryl Cochran's developm
ent ofplace-value
numerals by the use of beans,
tongue depres-sors, and so on, is a part of the curriculum
.Included is the sine-generating
machine de-
veloped by the Cam
bridge Conference
on SchoolM
athematics during the sum
mer of 1967.
The sim
ple rough study of periodicfunctions,
such as temperature at various hours
of the day,w
as suggested by Professor Andrew
Gleason
of Harvard U
niversity.L
'Publications
Inquiry in Mathem
atics via the Geo-B
oard,by
Donald C
ohen.(A
vailable' from W
alker Co.,
720 Fifth Avenue, N
ew Y
ork,N
. Y.
100 r9.)E
xplorations in Mathem
atics: AT
extforT
each-ers
Explorations in M
athematics: Student
Discus-
sion Guide
Discovery in M
athematics: A
Textfor T
each-ers
Discovery in M
athematics:
Student Discus-
sion Guide
(All by R
obert B. D
avis.A
vailable from A
ddi-son-W
esley Publishing Com
pany,South Street,
Reading
, Massachusetts
01b67. )
I Do and I U
nderstandM
athematics B
eginsPictorial R
epresentationB
eginningsC
omputation and Structure
,2, 3, 4
Shape and Size1,,
2, 3E
nvironmental G
eometry
Probability and StatisticsG
raphs Leading to A
lgebraProblem
s: Green Set and R
ed Set(A
ll of the aboveare products of the N
uffieldProject of G
reat Britain.
Available in the
United States from
John Wiley and
Sons Pub-lishing C
ompany, 605 T
hird Avenue,
New
York,
N. Y
.10016.)
OD
Other Film
s
Three film
s not produced by the Madison Pro-
ject are also highly recomm
ended for expressingthe m
athematical concepts ver>
clearly and.for
relating mathem
atical ideas to otherareas of
study. They are:
I Do and I U
nderstand (Available from
Radim
.Film
s, 211 East 43rd Street, N
ew Y
ork, N. Y
.10017.)
Maths A
live (Available from
National A
udio-V
isual Aids-L
ibrary,2 Paxton Place, G
ypsyR
oad, London, S. E
. 27, England.)
'Classroom
s in Transition (A
vailable from M
aryL
ela Sherburne, Education D
evelopment C
enter,Inc. ,
55 Chapel Street, N
ewton, M
assachu-setts
02158.)23
RE
CO
MM
EN
DE
DC
OM
ME
RC
IAL
TE
XT
SER
IES
29
Text Series: E
lementary School M
athematics
Addison W
esley Com
pany (1968)A
uthors:E
ich'oly, O'D
affer
Unusually w
ell illustrated, the A
ddison Wes-
ley materials represent perhaps the
most com
pleteinterm
ediate mathem
aticsprogram
that the MIN
-N
EM
AST
Project has examined and found
to be anacceptable successor to its prim
arym
aterials:A
side from the Student T
exts, this series contains
a mathem
atical development of each topic cle-
signed for the teacher's use,a suggested list of
supplementary m
aterials for each chapter,objec-
tives for each lesson, suggestions forthe evalu-
ation of each chapter, a short list ofappropriate
supplementary books as w
ellas detailed sugges-tions
for lesson presentation.T
opics in thefourth grade m
aterials thatare felt to be particu-
larlyappropriate include the geometry of the
circleand three-dim
ensional geometric form
s,estim
a=tion,' m
easurement, num
bertheory, coordinate
geometry.
Their approach to rational num
bersclosely parallels that of the M
INN
EM
AST
Project.M
ultiple interpretations of mathem
aticalconceptscoupled w
ith suggestions for theuse of m
anipu-lative m
aterials enhance the viability ofthis pro-
6grani. The A
ddisonWesley
Program provides both
a rich variety of activities and a sound mathem
a-tical approach.
The M
adison Project materials
can provide tie additional amount of the 'kind
ofexplorative activities w
e feelare so essential to
effective mathem
atical learnings.
C2
Tekt Series: Sets and
Num
bersSinger C
ompany (1968, 1969)
The Singer C
ompany Series, Sets and
Num
-bers, provides a M
athematical prO
grainthat can
be successfully adopted to succeedthe M
INN
E-
MA
ST K
-1 mathem
atics curricula. The
authors ofSets and N
umbers recognize the need
for activestudent participation in the learning
process andhave designed m
anyexperiences accordingly.
An attem
pt has been made
to shift the responsi-) bility for learning from
the teacherto the sttudent.
Problem situations are som
etimes
designed sothat the student is encouraged
to play a centralrole in the developm
ent ofthe m
athematical
ideas. The authors of Sets and
Num
bers realizethat learning' takes M
ace fromthe concrete to the
abstract.T
wo-dim
ensional models
are exten-sively used to provide the
concrete framew
orkfrom
which m
athematical ideas
can ultimately be
abstracted. Multiple em
bodiments
are suggestedfor m
any of the concepts considered.It is hoped
that schools adopting thisseries w
ill attempt to
procure many of the m
anipulative aids depictedin the textual m
aterials.
The M
INN
EM
AST
K-3 m
athematics
program.
has been concerned with
many of the topics
treated in the 1-3 materials of the
Singer Series.M
INN
EM
AST
studentsw
illtherefore possess
many of the skills and concepts required for
suc-25
cessful involvement in the fourth
grade book ofthe Sets and N
umbers series. T
helist of m
athe-m
atical concepts and skills thathave been devel-
oped by the MIN
NE
MA
STProject (pages 28 through
32) should be examined carefully
so that discrep-ancies -betw
een these and the SingerC
ompany
Series can be discovered.T
ests would be help-
ful in diagnosingareas w
here additional activi-ties could be used to advantage.
Tests w
ouldalso reveal' the extent to w
hichsuppleM
entaryactivities m
ight be required.v-4
Text Series: SR
A E
lementary
Mathem
atics ProgramScience R
esearch Associates,
Inc.(19680. 1969)
Com
mitted
to variation of approach, the SRA
Series generally presentsa single m
athematical
idea in a number of different
contexts. Realizing
the importance'of the role that
visual and mani-
pulative materials play in the
effective learningof m
athematics, the
authorshave developed
Multim
edia Manuals to correlate
with each text.
26
,1.
Althoughthe m
aterials suggested do not representa com
plete picture of the breadth of materials to
which the M
INN
EM
AST
Project would subscribe,
these itgre certainly a step in the right
direction and they offer useful suggestions forsensory activities in the m
athematics program
atthree levels of ability (rem
edial,average, en-
richment). T
he teachers' editionsare quite thor-
ough from a pedagogical standpoint. T
hey offerdetailed suggeStions for appropriate m
ethods .ofpresenting '19ssons.
The teachers' m
anuals donot develop m
athematical ideas for the teacher
'beyond those suggested fortne students.
Ingeneral, the m
athematics is presented
as a care-fully developed finis hed 'product and students
. areseldom
encouraged to develop theirow
n, rules-andprocedures.
The topics considdred are
conven-tional in nature but are presented in
a mathem
a-tically acceptable m
anner.A
s indicated above,students, are often given m
ultiple embodim
ents ofim
portant mathem
atical concepts. Coupled w
iththe M
adison Project materials,
we feel that the
SRA
Series can provide a sound mathem
aticalprogram
for children in the intermediate grades.
MIN
NE
MA
ST M
AT
HE
MA
TIC
S SKIL
LS
AN
D C
ON
CE
PTS
Kindergarten
Unit
I, Watching and W
onderingIntroductory unit
no specific math skills
Unit
2, Curves and Shapes
.l
Redognition of: curves
N.
/simple
non-simple
openN:closed
closed/
penN
.1
regions and boundaries
includes circle, triangle,square, rectangle
Unit
3, Describing and C
lassifyingPlacing objects in sets and subsets
according topropertiell
Defining sets by listing
Equivalent sets
One to one correspondence
Set comparison (m
ore, fewer)
Unit 4, U
sing Our
SensesN
o specific math skills
Unit
5, Introducing Measurem
entL
ength; comparihg lengths, ordering,
reference objectA
rea: comparing areas, superposition,
orderingV
olume: com
paring volumes, ordering
28T
ime: com
paring durations ofevents
.1
Unit
6, Num
erationO
rdering setsE
quivalent setsC
ountingI
to 10Introducing 0N
Um
erals 0 - 10
Unit
7, Introducing Symm
etryR
ecognition of:rotational sym
metry; repeating patterns
, bilateral symm
etry
Unit
8, Observing Properties
Intersection of sets.G
rade I
Unit
9, Num
bers and Counting
One to one correspondence (m
ore, fewer)
Num
erals from 0 to 20 (counting)
Ordering w
ith symbols: <
, >,
<11
Unit 10, D
escribing Locations
Introducing point, line, segment, locations
Nam
ing of each of the aboveIntersections of lines
..U
sing a grid, betweenness, sim
ple maps
Unit 11, Introducing A
ddition and SubtractionA
ddition (1 digit by 1digit), union-of sets ,' arrays
Num
ber line, introducing simple frabtions
Num
eration, 0 - 100
Unit 12, M
easurement w
ith Reference U
nitsC
omparing lengths, ordering
Measuring length w
ith. standard units suchas inches and centim
etersIm
precision of measurem
ents.
Perimeters
29
Grade 2
Unit 12, C
ontinuedC
Om
paring areas, superposition, referenceobjects
Distinguishing area from
lengthC
omparing volum
es, 'displacement, standard
unitsC
omparing, ordering and m
easuring durations
Unit 13, Interpretations of A
ddition andSubtraction
Addition and subtraction on num
ber line, sliderule
Properties of addition and. subtraction, placevalue, num
eration
Unit N
kiE
xploring Symm
etrical PatternsR
oCational, translational and bilateral
symm
etries
Unit
Investigating Systems
No specific m
athskills
Unit I6, N
umbers and M
easuringO
rdering numbers, objects
Approxim
ate nature of measurem
entsFractional unitsC
ircumferences, diam
etersT
-notation, place valueA
dding 2-digitnumeralS
on slide ruleT
he abacus, numerals through 999
Base 4 num
ization, Rom
an numeral's
Measuring w
eight
Unit 17, Introducing M
ultiplication and Division
Multiplication as tepeated addition
Multiplication on the num
ber line, on parallel number lines
Arrays
Multiplying 1
digit by ['digitIntroducing division by m
eans of 'simple fractions
30U
nit 18, Scaling and Representation
Math-related m
easurement, m
aps
Unit 19, C
omparing C
hangesT
ime - duration and clock reading
,r
Plotting ordered pairs and weight/volum
e data
Unit 20, U
sing Larger N
umbers
Place value to 999A
ll addition and. subtraction factsIntroducing addition and subtraction algorithm
sE
stimation
Unit 21, A
ngles and SpacePoints lines
,line segm
entsR
ays; anglers, angle m
easurement w
ith clock protractorPolygons: classifidation-by properties such.as num
ber of sides,convexity'or concavity, regularit_y_or irregularity
Similar triangles, congruence
TesselatiO
ns (distinguishing patterns)Polyhedra (transition to 3-dim
ensional shapes)
Unit 22, Parts and Pieces
Distinguishing betw
een counting and amount
measures '
Fractional parts of objects and of setsA
dding and subtracting fractions on the number line
Fractions greater than I,
mixed fractions
Unit 23, C
ondition's Affecting L
ifeB
ar graphs, doordinate graphsInterpreting graphs
Unit 24, C
hange and Calculation
,C
omplete addition and subtraction algorithm
-sPlace value w
ithout limit' (T
- notation)G
raphing ordered pairs
Unit 25, M
ultiplication and Motion
Graphing ordered pairs (tim
e/distance relations)
Grade 331
Unit 25, C
ontinued.Interpreting graphsR
eview of m
ultiplicationas repeated addition, as arrays,
as Cartesian products
Multiplication using a graph w
ith lines ofcertain slope
Understanding m
otion as the relationbetw
een time and distance (evident
inthe slope of a graph)
Unit 26, W
hat Are T
hings Made O
f?G
raphing volume /w
eight relationsInterpreting graphsM
odel building of regular polygonsfor angle m
easurement
Transition from
clock protractor to standardprotractor
Unit 27,. N
umbers and their
PropertiesD
eveloping standard multiplication
algorithm for m
ultiplying I- and2-digit
numbers in colum
n formPractice w
ith basic multiplication fapts
Unit 28, M
apping the Globe
'.Measurem
ent of area, angles, lengthM
odel buildingProperties of transform
ations
Unit 29, N
atural Systems
Recognizing patterns and sym
metries
32
It has been stated that the aim of the M
IN-
NE
MA
ST Project is to exploit the interrelated na-
ture of mathem
atics and science in a way that
helps children understand the quantitative natureof observed phenom
ena. We believe that through
this approach youngsters will learn about their
environment and w
ill also learn to recognize pat-terns, regularities and uniform
ities in the envi-ronm
ent.T
his is important because understand-
ings based on his own observations and data can
help the child make valid predictions. T
he MIN
-N
EM
AST
PrograM presents the study of science
as both knowledge of the environm
ent .and skillsin eliciting answ
ers from -it. T
he children inves-tigate :
series of carefully planned science prob-lem
s,gain, understan'ding of each problem
and'learn procedures for problem
solving.
We believe
that this method of teaching science is consistent
with m
uch of the contemporary research related
to learning.It goes beyond the acquisition of
facts toward the developm
ent of useful, lifelongattitudes about the endeavor of m
an that is called"active scientific investigation."
A's an outcom
e of this approach, we believe
students gain a clearer understanding of what
learning is and make this a part of their behavior
pattern.It is then incum
bent on-the teacher,principal and curriculum
director to recognize theexistence of this carefully nurtured problem
-solv-ing potential in children w
ho have used the MIN
-N
EM
AST
material. M
aterials for Grades 4, 5 and
6 should offer continuing problem-solving chal-
lenges and pedagogic 'skills should be used toexpand this ability.
33
SCIE
NC
E T
RA
NSIT
ION
AL
STA
TE
ME
NT
GE
NE
RA
L SU
GG
EST
ION
S
34
We w
ould hope that a curriculum selected
to follow the M
INN
EM
AST
Program w
ould beone
complem
entary to theteaching-learning mode al-
ready established. We w
ould hope that thepro-
gram w
ould provide continuing opportunities foryoungsters .,to
investigate and discover. We
would hope that the selected program
would pre-'
sent topics as open-ended, investigative, andquantitatively designed.
The M
INN
EM
AST
K-3 m
aterials providean
adequate background of information and skills
sothat the follow
ing transition illustrations shouldpresent few
, if -any, science transition problems
.T
he sugge'sted programs are representative of
aw
ide range frorri which schools
can make selec-
tions. There is no program
existing today, otherthan M
INN
EM
AST
,' that providesa coordinated
approach to mathem
atics and science. 'We be-
,,lieve that w
ithin a fewyears such program
s will
be developed and published. Until that tim
e, theteacher m
ust look to programs that offer
strongpossibilities for investigation and quantification.
From a survey of m
any comm
ercialscience
programs,
itbecorries evident that provisions
have been made in alm
ostall cases. for the de-
velopment of a teaching-learning m
odethat, at
the discretiOn of the teacher, C
allrange from
textbook reading and occasionalend-of-chapter
questions to active involvement of students
inthe m
anifold operatidns of science.T
hese in-clude apparatus- m
anipulation, problem solving,
question-asking and individual andgroup science
activities and investigations.
7,,f- 11;t1
Consider -the w
idely used text series, Con-
cepts in Science, as an example of com
mercial
,materials that m
ight be used.T
he books-can beread and end-of-chapter questions can be
an-sw
ered by the children; or the books can serveas the nucleus for an active, investigative sci-ence program
.T
he several skills expected ofyoungsters'(instrum
ent reading, graph interpreta-tion, equipm
ent manipulation and som
e under-standing of sam
pling_ procedures) have beer ade-quately developed in the coordinated activitiesof the M
INN
EM
AST
Program.
Concepts in Sci-
ence, for example, offers a related set of equip-
ment packages for individU
al and group experi-m
entation, has a related collection of films and
film strips, and includes a package that provides
suggestions for independent investigations. The
teacher can organize -and develop the, suggestedactivities in such a w
ay that quantitative rela-tionships are perceived, and a quantitative ap-proach to the teaching of science_ is continuedeven though the book-dem
ands for quantification_m
ay. be n.inimal or absent. T
his series will be
,mentioned again as specific transition m
odelsare discussed.
A second kind of elem
entary science programhas been produced in this:idO
untry in the past tenyears. T
his kind eniphagizeda quantitative ap-proach to the understanding of natural phenom
enaactivity program
s that involve a wide and excit-
ing variety of investigatfons,, and-thaCconsider
science as much a,w
ay dr Prockss of finding outas it is a collection of facts representing a sur-vey of parts of the vast body of know
ledge called
SCIE
NC
E T
RA
NSIT
ION
CR
ITE
RIA
36
science. ImpliC
it in each of theseprogram
s is adevelopm
ent' scheme that reduces the
extent ofthe survey, and uses a sm
aller number ofselected
topics to develop a depth of knowledge
throughactive participation in investigative
activities bystudents.' Program
s that follow this,pattern in-
clude SCIS (Science
Curriculum
Improvem
entStudy), E
SS (Elem
entary Sciehge Study),SA
PA(Science A
Process Approach, developed
by theA
merican A
ssociation for the Advancem
ent of Sci-ence), and a collection of
mathem
atics-orientedm
aterials calledM
easure and Find Out
(Scott,Foresm
an and Co.).
In assessing science programs for continuity
of the principles used in MIN
NE
MA
STand evi-
dence of a teaching-learning mode
in which stu-
dents have developed competencies, these
fac-tors w
ere the prime considerations:
Open-endedness
.D
esirable lessons are thosethat set the stage for the children
to find theirow
n answers., T
his is in contrast to lessonsthat consistently present' facts and
do littlem
ore than support these facts with illustrations
and the printed version of otherpeople's
reasoning.A
ctive involvement of the students.
Research
evidence substantiates 'the need forchildren
to be involved in doing operations. The
abil-ity to ask useful questions, to think
of ways
to answer these questiO
ns, to design and don -
duct, experiments and to
collect data andassess its m
eaning, is, we believe,
as much
a purpose of science teaching as is the learn-ing of facts.Q
uantitative idea development.
The m
athe-m
atics that children learn should be usedas a
tool in their investigative activities.
Inquisitive' attitudes.It is only w
hen thechild discovers that he
can do something,
succeed at it, and find out for himself
thathe can build an "I can do" attitude.
Failureis often the alternative w
hena child does not
know how
to learn and has notexperienced theexhilaration of finding things
out for himself.
Depth vs. breadth.
Content that represents a
selected collection of conceptual schemes de-
veloped in depth, as contrasted with shallow
surveys of a multitude of topics
,is the final
criterion on which w
e feel decisionscan best
be made.
Schools that are just beginningto use the
CH
OIC
ES O
FM
INN
EM
AST
Curriculum
will have
certain free-M
OD
EL
Sdom
s of choice in their K- 6 program
develop-m
ent not enjoyed by those alreadyusing it
Foreeach, general and specific
suggestions for tran-,sition procedures are offered
for guidance pur-pose.For schools planning to use M
INN
EM
AST
ma-
terials in Grades K
- 3, there are flexibilities andalternatives. In particular, the third grade M
IN-
NE
MA
ST m
aterialscan lie stretched into G
rade 4,
and portions of the selectedsuccessorcan be
introduced in Grade 3.
37
38
The independent science packages
of the ESS
Program represent the first chbice
of MIN
NE
MA
STin science transition m
aterials.It is suggested
that several of the ESS units
be inserted duringG
rade 3, and the MIN
NE
MA
STProgram
be extendedinto G
rade 4. A second procedure
involves theuse of the M
easure and Find Out
materials.
Through judiciously com
biningthe M
easure andFind O
ut program and selected
ESS units, a bal-
aribed program that
satisfies transition criteriaas stated can be developed.
A second m
odel involves theuse of one or
more com
mercial text series.
Certain problem
sarise having to do w
ith teachingm
ode and mate-
rial appropriateness for studentsin the interm
e-diate grades. T
hree comm
ercialprogram
s illus-trate transition m
odels thatcan satisfy transition
criteria. Those included
as illustralive are Con-
cepts in Science, Experiences
in Science andScience T
hrough Discovery.
We feel that w
hencom
mercial series pare selected
it would be ap-
propriate to-omit certain selections
that do notlend them
selves to concreteactivities, and to
substitute activities from E
SS`units,M
easure andFind O
ut or others specifiedin the illustrations.
We suggest such alterations only
after consider-ation of the m
aterials and of thelearning patterns
of the children. These suggestions
are made w
ithreference to research in learning,
and this pro-ject's extensive experience w
ithm
aterials inclassroom
s'. Children in the interm
ediategrades
are in the learning period designated byPiaget as
"concrete. " The im
plications ofthe research of
Piaget and co-workers have been interpreted. by
the MIN
NE
MA
ST Project as supportive of
a sci-ence program
that provides ,continuing manipula-
tive and investigative activities by students.U
nits That require active involvem
ertshould be
given preference overthose treated ina vicarious
reading and discuss_ion format.
Further, certainconcepts require form
al intellectualization.A
c-cording to Piaget, m
any children in the intermedi-
ate grades are not mature enough for form
aloper-ations. T
hese concepts-efid,conceptual schemes
cannot be handled in a mode even approaching
experiment or investigation.
FOnthis reason the
illustrations suggest only minim
al treatment of
molecular theory and the fundam
entalparticulatenature of m
atter.In their place w
e suggest theuse of such E
SS units as Kitchen Physics, G
asesand A
irs and Mystery Pow
ders. There
is a secondreason for reducing em
phasis on the development
of concepts related to theories ofthe particulate
nature of matter:
when som
e of these studentsare in junior high, they w
ill be ableto operate
on information in a form
al way.
At this tim
e, a-m
ong programs available, schools
may select one
called"Interm
ediate Physical Science"(IPg)
.C
oncepts of the particulate nature ofm
atter aredeveloped through a carefully planned
series ofexperiences and activities. T
he studentis given
a chance to develop and consider general infer-ences about the nature of m
atter and is led througha historical-logical developm
ent of atomic theory.
The aim
s of the IPS program,
which include
treating the learner as an activeinvestigator,
would be difficult to achieve if
through prema-
39
r
40
ture telling without the benefit
of investigation-,-
the answer w
ere given beforethe question w
asasked.T
his same om
ission policyis suggested for
such topics as tree farming
and molecular gene-
tics.T
he importance of the topic
is, again, notbeing questioned. W
e would
recomm
end substi-tution of such E
SS unitsas B
rine Shrimp and
Small T
hings or one of the genetics:units devel-
oped at Webster C
ollege (WIM
SAProject) entitled
"On the Fly" and "E
ven FliesR
eme 1-..ber. "
Many of the units of the
comm
ercial textseries deal w
ith topics thatcan .be handled in a
quantitative, investigative mode. It
will be nec-
essary for the teacher to go beyond thetext les-
sons in some cases.
These units, for exam
ple,are treated in a straightforw
ard, descriptivem
an-ner: Path of the M
oon, Stars andM
achines. Ineach, m
easurements
can be made by youngsters
and investigations can be conducted.T
he Illi-nois A
stronomy Project also iias
units that would
provide the teacher with
many suggestions for
making m
easurements and treating
astronomical
topics in a quantitative, -investigative modd.
Schools planning to use SAPA
or SCIS should
use these programs in their entirety (K
- 6).In
general, both of theseprogram
s have been de-signed to provide a unified and
comprehensive
sequence of science experiences forelem
entarychildren.
It is possible that certain somew
hatindependent M
INN
EM
AST
units ,-suchas those on
symm
etry, would offer schools
an opportunity totailor program
s to their specificneeds. L
iving
Things in Field and C
lassroom, the M
INN
EM
AST
Handbook, is highly recom
mended as a valuable
addition to any elementary science program
thatis selected.
The follow
ing five models represent, in the
opinion of this author, acceptable transitionalprocedures. T
hey are intended to illustrate theM
any ways schools, using m
aterials appropriateto their philosophy, can capitalize on the inves-tigative strengths that ,have been developed inchildren through M
INN
EM
AST
Program activities.
The m
odels are intentionally terse, for we believe
that the best programs are developed w
hen con-cerned, know
ledgeable teachers, subject matter
specialists, 'psychologists and children all work
together to share ideas and build programs.
Transition m
odelsI.
The E
SS Program2.
Measure and Find O
ut and ESS
3.C
oncepts in Science and selected insertions4.
Science.:through Discovery and selected in-
sertions5'.
Experiences in Science and selected inser-
tionsM
odel ItT
he ESS Program
; Webster D
ivision,M
cGraw
-Hill
Grade 3
MIN
NE
MA
ST U
nit 23C
onditions Affecting L
ifeE
SSL
ight and Shadows
MIN
NE
MA
ST U
nit 26W
hat are Things M
adeO
f?E
SSM
ystery Powders
42
ESS
Clay B
oatsM
INN
EM
AST
Handbook L
iving Things
in Field andC
lassroomG
rade 4E
SSIce C
ubesM
INN
EM
AST
Unit 28
Mapping the G
lobeE
SSB
rine Shrimp
ESS
Small T
hingsM
INN
EM
AST
Unit 29
Natural System
sE
SSC
rayfishE
SSPeas and Particles
MIN
NE
MA
ST H
andbook Living T
hingsin Field and
Class room
Grade 5
"zJ4
ESS
Batteries and B
ulbsE
SSW
here is the Moon?
ESS
Rocks and C
hartsE
SSA
nimal A
ctiE
SSPond W
aterE
SSPendulum
sM
TN
NE
MA
ST H
andbook Living T
hings in Fieldand
Classroom
Grade 6
ESS
Behavibr of E
arthworm
sE
SSG
ases- and Airs
ESS
Kitchen Physics
ESS
Senior Balancing
ESS
Colored Solutions
MIN
NE
MA
ST H
andbook Living T
hingsin Field and
Classroom
.
Model 2:
The M
easure and Find Out Program
,Scott, Foresm
an and Com
pany; andE
SS Units
.
Grade 3-
MIN
NE
MA
ST U
nit 23M
FO B
ook IM
INN
EM
AST
Unit 26
MFO
Book I
MFO
Book I
MFO
Book I
MIN
NE
MA
ST U
nit 28M
INN
EM
AST
Handbook
Grade 4
MFO
Book II
ESS
MFO
Book II
MIN
NE
MA
ST U
nit 29M
INN
EM
AST
Handbook
Grade 5
MFO
Book II
ESS
ESS
ESS
MIN
NE
MA
ST H
andbook
Conditions A
ffecting Lire
Activities
I6
What are T
hings Made
Of?
Activities 7 - 17,
Activities 18
23A
ctivities 2832
Mapping the G
lobeL
iving Things in Field aad
Class room
Unit
I
Cardboard C
arpentryU
nits 2 and 5N
atural Systems.
Living T
hings in Field andC
las sroom
ti
Units 3 and 4
Peas and Particles.
Small T
hingsPond W
aterL
iving Things in Field and
Classroom
43
44
Grade 6
MFO
Book III
Unit I
ESS
Anim
al Activities
MFO
Book III
Units 2
- 5M
INN
EM
AST
Handbook L
ivingT
hings in Field andC
lassroomA
n Ecology Study developed
from Science T
hroughD
iscovery, Book 5', C
hapter.7; W
ater and its Pol-lution, B
ook 6, Chapters
6 and 7 '(Air Pollution
andE
cology) and theM
INN
EM
AS.T
Handbook, L
ivingT
hings in Field andC
lassroom.
Model 3:
Concepts in Science, H
arcourt,B
race,W
orld
Grade 3
MIN
NE
MA
ST Program
omitting U
nit 26M
INN
EM
AST
HandbO
ok -Living T
hingsin Field and
Classroom
Grade 4
CIS
Units
I through 4E
SS"Sm
all Things
CIS
Unit 6
MIN
NE
MA
ST U
nit 28M
apping the Globe
CIS
Unit 8
Augm
ent with
Illinois Astronom
y mate-
rials, Book 1
MIN
NE
MA
ST H
andbook Living
Things in Field and
Classroom
Grade 5
CIS
Unit
I
ESS
Gases and A
irsC
ISU
nits 3 through 8
MIN
NE
MA
ST H
andbOok L
iving Things in Field and
Class room
Grade 6
CIS
Units
I through 3.E
ither CIS
Unit 4
MFO
,IIIU
nit 5or E
BB
.K
itchen PhysicsE
SSB
ulbs and Batteries
MFO
IIIU
nit 2C
ISU
nit 8 -Augm
ent with O
nthe Fly and E
ven Flies Re-
mem
ber (WIM
SA Project)
CIS
Unit 9
MIN
NE
MA
ST H
andbook Living T
hings in Field andC
lassroom
Model 4:
Science Through D
iscovery,Singer/
Random
House
Grade 3
MIN
NE
MA
ST Program
Units 23 -29------
MIN
NE
MA
ST H
andbook Living T
hings in Field andC
lassroomG
rade 4ST
DU
nitsI
- 4ST
DU
nit 5- A
ugment this
with m
aterial from M
FOIII, U
nit 4.ST
DU
nit 6M
INN
EM
AST
Handbook L
iving Things in Field and
C las sroom
45
Grade 5
STD
.E
SSE
SSST
DM
INN
EM
AST
Handbook
Grade 6
STD
ESS
MFO
STD
.
MIN
NE
MA
ST*H
andbook
Units
I- 4
Batteries and B
ulbsG
ases an;' Airs
Units 6 and 7
Living T
hings in Field andC
lassroom
Unit
I
Kitchen Physics
MFO
, Book III,
Units 3
and 4 (See above)U
nits 3 - 7_L
iving Things in Field and
Class room
Model 5:
Experiences in Science, W
ebster Divi-
sion, McG
raw-H
ilG
rade 3M
INN
EM
AST
Unit 23
EIS
EIS
MIN
NE
MA
ST U
nit 26
Conditions A
ffecting Life
Earth, Sun and Seasons
Heat
What are T
hings Made
__...
Of?
EIS
SoundE
ISL
ife His4ories
'MIN
NE
MA
ST H
andbook Living T
hings in Field andC
lassroom
Grade 4
EIS
Adaptations
,
MIN
NE
MA
ST U
nit 28M
apping the Globe
EIS
Atm
ospheie and Weather
EIS
Geological Process
ESS
Gases and A
irsM
,INN
EM
AST
Unit 29
Natural System
sE
ISE
cologyM
INN
EM
AST
Handbook L
iving Things in Field and
Classroom
Grade S
Com
plete EIS Program
MIN
NE
MA
ST H
andbook
Grade 6
Com
plete EIS Program
"MIN
NE
MA
ST H
andbook
Living T
hings in Field andC
lassroom
Living T
hings in Field andC
lass roomI
47
G
K23
nits
PRO
CE
SSES
OB
SE
RV
AT
ION
OB
S. -G
EN
. LINK
GE
NE
RA
LIZA
TIO
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60 4. 4,.
1W
atching and Wondering
_**
Curviesiand S
hape' sI
**
4*
**
*I
bit
Id3D
escribingan
Classifying
*.*
4*
**
t '6 IS I
s
4- Using bur S
enses'
*4
**
*1
14
4
5 Introducing 'Measurem
ent'
**
**
**
I1
t1-
f6 N
umeration
.r*
**
11
11
1
7 Introdubing SY
mm
etry"flit
**
**
*.
8 Observing P
roperties*
**
**
**
.-*
*I
1I
1
9 Num
bers and Counting
**
**
**
.
10 Describing Locations
**
,
**
**
*t
ttitt.11
Introducing Add. and S
ubt.*
*12 M
easurement w
ith Ref. U
nits1
**
**
**
*I " "
cf:1i
t'd S
'13 Interps. of A
dd. andubt.
.*
*14
Exploring S
ymm
etrical Patterns
**
**
**
15 Investigating System
s.16
**
**
.*
Num
bersr:IIM
JasuIring*
**
**
*17 Introducing M
ult. and Div.
*"
4i
II
1
18 Scaling and R
epresentation*
-,.**
*19 C
omparing C
hanges*
**
**
*.*
**
*.*
*IIIIIII
20 Using Larger N
umbers
**
**
**-
21 Angles and S
pace*-
**
**
*-..
*It'll!!
22P
arts- and Pieces
**
**
**
iv*
fit tlt.23 C
onditions Affecting
Life*
,-**
**
**
**
*.*
**
**
**
*.
VD
in.
24 Change and C
alculations*
**
**
**
**
**
**
**
*25 M
ultiplication and Motion
**
**
**
**
**
**
**
It
fItt26 W
hat are Things M
ade Of?
*.*
**
**
**
**
**,
**
**
*27 N
umbers and T
heir Properties
**
**
**
**
*28 M
apping the Globe'
**
*-1**
**
**
**
**
**
t,I
ti
- .
.29 N
atural System
s*-
**
**
**
**
**-
*-
**
*
0100 MINN MIN MEN MEV OM MEISE.ME MEMOS-MOM.
0 00001011111 MIUM0001111
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KE
EPIN
G C
UR
RE
NT
50
1Perhaps one of the m
ostsignificant contributions
that the MIN
NE
MA
STProject has m
ade to educa-tion is that its curriculum
.demonstrates
the fea-sibility and advantages of
coordinating the teach-ing' of m
athematics and
science.A
t any rate,present interpstin such coordination
is high. The
authors believe that when sufficient
federal fundsare again available for this
purpose, significantam
ounts will be allocated to the
development of
coordinated curricula.
At present there are tw
oprojects concerned w
iththe integration of m
athematics
and science. You
may w
ish to keep abreast of them
aterials theyare producing:
USM
ES (U
nified Scienceand M
athematics in the
Elem
entary School) ProjectE
ducation Developm
ent Center
55 Chapel Street
New
ton, Massachusetts
NU
FFIEL
D Project of G
reatB
ritainM
aterials are handled in theU
nited States by:John W
iley and Sons PublishingC
ompany
605 Third A
venueN
ew Y
ork, New
York
10016