—

72-4-565

McCLURE, C la ir W ylie, 1927-EFFECTIVENESS OF MATHEMATICS LABORATORIES FOR EIGHTH GRADERS.

The Ohio S ta te U n iversity , Ph.D ., 1971 Mathematics

ii ?

University Microfilms, A XEROX Company, Ann Arbor, Michigan , ’-'I-

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED

EFFECTIVENESS OF MATHEMATICS LABORATORIES

FOR EIGHTH GRADERS

DISSERTATION

Presen ted in P a r t ia l F u lfillm en t of th e Requirements fo r the Degree Doctor o f Philosophy in the Graduate

School o f The Ohio S ta te U niversity

C la ir Wylie McClure, B*S*, M.A,

* * * * * *

The Ohio S ta te U n iv ersity1971

Approved by

A dviser College o f Education

PLEASE NOTE:| ,

Some Pages have I n d i s t i n c t p r i n t . Filmed as rece iv e d .

!

U N I V E R S I T Y M I C R O F I L M S

ACKNOWLEDGMENTS

My sincere ap p rec ia tio n i s extended to my ad v iser, Dr. Harold C.

Trimble, fo r the valuable a ss is tan ce he gave me throughout the years o f

graduate work as w ell as during th is study . His in te r e s t and encour­

agement l i f t e d my s p i r i t s a t tim es when i t was most needed.

A debt o f g ra titu d e i s a lso owed to the o th e r members o f my

reading committee, Dr. L es lie H. M ille r, Dr. E. Joseph Crosswhite, and

Dr. Jon Higgins. They have been very in strum en ta l in making th e w riting

of th is m anuscript po ssib le by reading th e o r ig in a l d r a f ts and by

making many worthwhile suggestions fo r improvement.

The cooperation o f th e Mercer School D is tr ic t and the Lakeview

School D is tr ic t i n providing the f a c i l i t i e s f o r th is study i s appreci­

a ted . My sincere g ra titu d e goes to the te ach e rs , Mary Wiese and David

G. Cook, fo r the cooperative maimer in which they performed t h e i r con­

tr ib u tin g ta sk s .

F in a lly , t h i s manuscript i s dedicated to my love ly wife and c h il­

dren who have sa c r if ic e d so much during the many years o f my education .

I t i s now my goal to repay them somehow fo r the postponed vacations,

cancelled a c t iv i t i e s , and the many exhaustive hours of typing which a re

so very much a p a r t of t h i s endeavor.

l i

VITA

Bom - G reenville , Pennsylvania

B .S ., T h iel College,G reen v ille , Pennsylvania

Mathematics Teacher, Mercer Area School D is t r ic t , Mercer, Pennsylvania

M.A., The Ohio S ta te U n iversity , Columbus, Ohio

A ss is ta n t P ro fesso r, Mathematics Department, S lippery Rock S tate C ollege, S lippery Rock, Pennsylvania

Teaching A ssociate, Department of M athematics, The Ohio S ta te U n iv e rs ity , Columbus, Ohio

A ssociate P ro fesso r, Mathematics Department, S lippery Rock S tate C ollege, S lippery Rock, Pennsylvania

FIELDS OP STUDY

Major P ie ld t Mathematics Education Dr. Harold C. Trimble

Minor F ie ld s : MathematicsDr. L es lie H. M iller

Higher Education and Teacher Education Dr. H erbert Coon

October 27, 1927. * • •

1950......................................

1951-1963 .........................

1962 .........................................

1963-1967 .........................

1967-1968 .........................

1968-1971 .........................

TABLE OP CONTENTS

Page

AGKNOY/LEDGMENTS.................................................................................................... i i

VITA.............................................................................................. i i i

III ST OF TABIES.................................................................................................... vi

CHAPTER

I . INTRODUCTION....................................................................................... I

Purpose and Design of the Study The Hypotheses to be Tested Need f o r the Study- S e ttin g o f the StudyD if f ic u l t ie s in Studying Mathematics la b o ra to r ie s S ign ificance of the Study Overview o f the Study

I I . RELATED LITERATURE AND RESEARCH............................................... 15

D escrip tions of a Mathematics Laboratory L aboratories in the Elementary Schools la b o ra to r ie s in the Ju n io r High School Research S tudiesThe N ature o f a Continuing Program The Laboratory Teacher Sources f o r M aterials and Devices Summary

I I I . FEATURES AND ORGANIZATION OF THE STUEf................................... 32

The School D is tr ic ts Physical P rovisions The TeachersP reparatory and T ria l Periods Student Samples Class Grouping In s tru c tio n a l Procedures Nature o f the Units CoveredTreatment V ariations w ith A b ility Levels and Units

i v

CHAPTER Rage

SeatingQuestionnaireFeatures Lim iting the Scope of the Study

IV. DATA AND ANALYSIS OF DATA.......................................................... 53

Separation of Schools Control V ariables—Mercer Dependent V ariable—Mercer Independent V ariable—Mercer Analysis o f Covariance—Mercer Another Analysis—Mercer Mercer Questionnaire A nalysis Control V ariables—Lakeview Dependent V ariable—Lakeview Independent V ariable—Lakeview Analysis o f Covariance—lakeview Another Analysis—Lakeview Lakeview Q uestionnaire Analysis

V. SUMMARY AND INTERPRETATIONS...................................................... 78

The Problem R elated L ite ra tu re O rganization of th e Study S ta t i s t i c a l Analysis of Data lim ita tio n s of COVAR fo r th is Study P in a l In te rp re ta tio n s Recommendations f o r F u rther Study

BIBLIOGRAPHY....................................................................................................... 91

APPENDIXA. Experiments w ith Suggested Procedures and Leading Questions 94

B. D iagnostic Tests 114

0. Achievement T e s t s ............................................................................... 124

D. Teacher and Student E valuation F orm s........................................... 133

E. S tudent In te re s t Q u e s tio n n a ire ...................................................... 137

v

LIST OF TABLES

TABLE Page

I , In te llig e n c e Quotient Means o f Classes by Schools . . . 39

2• Assignment o f A b ility Levels to Laboratory Typesper U nit a t Mercer. ......................................... 46

3. Assignment o f A b ility Levels to Laboratory Types perUnit a t Lakeview............................................................. 47

4. Assignment o f Laboratory Types to A b ility Levelsper Unit a t Mercer. ...................................................................... 47

5. Assignment o f Laboratory IJypes to A b ility Levelsper U nit a t lakeview ............................................................. 48

6. R e lia b il i ty C oeffic ien ts f o r D iagnostic andAchievement T e s t s .............................................................................. 52

7. Control V ariable Means by C lass and U nit a t Mercer. . • 54

8 . Mean Achievement Test Scores f o r Each Class byU nits a t M e rc e r ................................................................. 55

9. Adjusted Mean Achievement Test Scores fo r EachClass by U nits a t Mercer.................................................................. 55

10. Adjusted Mean Achievement Test Scores fo r Eachlabo ra to ry Type a t Mercer ............................. 56

11. Analysis o f Covariance f o r Achievement D ifferencesAmong Three Experimental C lasses w ith Assigned Laboratory Types C ontro lling on P rio r Mathematics Achievement and In te llig e n c e on Mercer U nit I .................... 57

12. Analysis o f Covariance f o r Achievement D ifferencesAmong Three Experimental C lasses w ith Assigned labora to ry Types C ontro lling on P rio r Mathematics Achievement and In te llig e n c e on Mercer U nit I I . . . • 57

v i

TABLE Page

13, Analysis o f Covariance fo r Achievement D ifferences Among Three Experimental Classes w ith Assigned Laboratory Types C ontrolling on P rio r Mathematics Achievement and In te llig en ce on Mercer Unit I I I • • • • 58

14, t-S cores Prom Pair-V/ise Comparisons o f Treatmentsfo r U nit I I I a t Mercer* • • • • ....................................... 60

15* V aria tion of the Adjusted Mean Achievement ScoresProm the Adjusted Mean of the Pooled Sample in Terms of Standard Scores a t Mercer* • • • ................... 61

16* V aria tion of the Adjusted Mean Achievement ScoresProm the Adjusted Mean of the Pooled Sample Mapping Mercer Classes Onto Treatments........................... 62

17* Responses to the Student In te re s t Questionnaire byPercentages fo r Each Class a t Mercer........................ 64

18* Control V ariable Means by Class and Unit a t Lakeview. • 66

19* Mean Achievement Test Scores fo r Each Class byU nits a t lakeview • • • • • • • • • • • • * • • • • • • 68

20* Adjusted Mean Achievement Test Scores f o r EachClass by Units a t Lakeview............................ 68

21* Adjusted Mean Achievement Test Scores f o r EachLaboratory Type a t Lakeview .......................* ..................... 69

22* A nalysis of Covariance fo r Achievement D ifferencesAmong Three Experimental Classes w ith Assigned Laboratory Types C ontrolling on P rio r Mathematics Achievement and In te llig en ce a t Lakeview Unit I • • • • 70

23* A nalysis o f Covariance fo r Achievement D ifferencesAmong Three Experimental Classes w ith Assigned Laboratory Types C ontrolling on P rio r Mathematics Achievement and In te llig en c e a t Lakeview Unit I I . • • • 70

24* Analysis of Covariance fo r Achievement D ifferencesAmong Three Experimental Glasses w ith Assigned Laboratory Types Controlling on P rio r Mathematics Achievement and In te llig en c e a t Lakeview Unit I I I • • • 70

v i i

TABLE Page'

25* t-S co res from Pair-W ise Comparisons o f 'Treatmentsfo r U nit I a t Lakeview* • • • • • • • • • • • • • • • • 72

26* t-S co res from. Pair-W ise Comparisons o f Treatmentsfo r U nit I I a t Lakeview* • • • • • • • • • • • • « • • 72

27* V aria tion o f th e Adjusted Mean Achievement Scores Prom the A djusted Mean of th e Pooled Sample in Terns of S tandard Scores a t Lakeview* • • • • • • • • • 74

28# V aria tion o f th e Adjusted Mean Achievement ScoresProm th e Adjusted Mean of th e Pooled Sample Mapping Lakeview G lasses Onto Treatments* .................... 75

29* Responses to th e Student I n te r e s t Q uestionnaire byPercentages f o r Each Class a t Lakeview* • • • • • • • • 76

v i i i

CHAPTER I

INTROIXJ CTIOK

What e ffe c ts do organized lab o ra to ry experiments have upon the

m otivation and achievement of e igh th grade mathematics students? The

number of teachers and adm in istra to rs seeking an answer to th is ques­

tio n has su b s ta n tia lly increased over the p as t few years* The recen t

in te r e s t i n mathematics la b o ra to rie s i s in d ica ted by the number of re ­

la te d a r t i c le s appearing in contemporary mathematics and educational

journals*

The teacher acceptance of the recen t attem pts to re v ita liz e th is

alm ost fo rg o tten teaching technique w i l l g re a tly depend upon the re ­

s u l t s of th e i r attem pts to in te r e s t the studen ts in labora to ry experi­

ments* These attem pts, in tu rn , w ill be influenced to some ex ten t by

recen t rep o rts of educational research w ith labo ra to ry methods. Mathe­

m atics la b o ra to rie s are in a c ru c ia l s tage o f development*

Because of the many in te rp re ta tio n s of the meaning of a mathemat­

ic s lab o ra to ry , numerous questions a r i s e in the minds o f In te res ted

te ach e rs . These questions need to be answered to a considerable degree

o f s a t is fa c t io n before a c r i t i c a l , r e f le c t iv e teacher w ill se rio u sly

attem pt to u t i l i z e mathematics la b o ra to rie s in h is classes*

1

As.answers are sought f o r the obvious questions concerning gen­

e ra l methods o f approach, o th e r questions, more p e r tin e n t to the lo c a l

problems, become the focus o f a t te n tio n .

l i s t e d below are some o f the questions which th is re se a rch e r con­

sidered during the time o f the study.

1 . Are mathematics la b o ra to r ie s e f fe c t iv e in promoting achieve­

ment?

2. Are mathematics la b o ra to r ie s e f fe c t iv e i n m otivating studen ts

to a g re a te r in te r e s t in mathematics?

5. W ill average and above average s tu d en ts be as in te re s te d as

slower le a rn e rs in the p h y sica l ap p lica tio n s th a t a r ise i n the labora­

to ry?

4. Are mathematics la b o ra to r ie s e f fe c t iv e in changing the a t -

tid u es of s tu d en ts toward mathematics?

5. Can experiments be enjoyable and educational sim ultaneously?

6. I s i t r e la t iv e ly easy to devise experim ents to teach most

mathematical concepts?

7. I s the success o f the lab o ra to ry experiment d ir e c t ly propor­

tio n a l to th e elaborateness o f i t s design? I s ex tensive , commercially

produced equipment e s se n tia l?

8 . Are physical ap p lic a tio n s of m athem atical concepts b e t te r

presented before or a f te r a lo g ic a l , a b s tra c t d iscussion o f the con­

cept?

9 . W ithin the lim ita tio n s o f the av a ilab le school p la n ts and

studen t populations i s i t po ssib le to design a study which w ill provide

some answers to these questions?

10* How must the teachers prepare f o r th is technique o f presen­

ta tio n ? Can every teacher be expected to use the labo ra to ry approach

e f fe c t iv e ly ?

Purpose and Design of th e Study

This study was designed to s p e c if ic a l ly answer two o f th e above

questions r e la te d to mathematics la b o ra to r ie s . Does th e i r use a t the

e igh th grade le v e l r e s u l t in g re a te r m athem atical achievement? Do

e igh th grade stu d en ts p re fe r mathematics la b o ra to r ie s to o th e r methods

o f p re se n ta tio n ?

Two experienced teachers of e ig h th grade mathematics in d is t in c t

school d i s t r i c t s o f Mercer County vo lun teered to p a r tic ip a te in an or­

ganized attem pt to study the m erits o f the lab o ra to ry methods o f teach­

ing* S ix c la sse s o f eighth grade s tu d e n ts , th ree from each school,

were a r b i t r a r i l y se lec ted from those tak in g the u sua l e igh th grade

mathematics program. These c la sse s c o n s titu te d the student sample fo r

the re se a rch . In both schools the ad m in is tra to rs had c lass enrollm ents

p re -e s ta b lish e d by a homogeneous grouping procedure based on o v era ll

achievement. Therefore, the use o f in ta c t groups was n e c e ss ita te d .

The f i r s t s ix weeks o f the 1970-71 school y ear were used as an

acclim ation and refinem ent p erio d . Even though the teachers and the

au thor h e ld weekly o rgan iza tiona l meetings during the spring and summer

months, more time was required by the te ach e rs and th e i r stu d en ts be­

fo re the re se a rch study could be s a t i s f a c to r i ly refined* Following

th is i n i t i a l , in troducto ry perio d , each p a r t ic ip a t in g c lass was ran­

domly assigned a labo ra to ry type approach to be used fo r one u n it o f

mathematics over a s ix weeks time p e rio d . Within each school one c lass

used a p re - la b approach c a ll in g fo r the laboratoxy experim ents before

the c la s s d iscussed the re la te d concepts in reg u la r c la s s r e c i ta t io n .

A second c la s s in each school used a p o s t- la b approach in which concept

re la te d experiments follow ed class r e c i ta t io n s . A th i r d c la ss in each

school was assigned the standard approach commonly used in these

schools in former y ea rs , This was named the no-lab approach.

The Hypotheses to be Tested

The n u ll hypothesis to be s t a t i s t i c a l l y te s te d in s ix d is t in c t

cases i s th a t the re i s no s ig n if ic a n t d iffe ren ce in achievement among

e ig h th grade mathematics c lasse s being taught e i th e r w ithout labo ra to ry

techniques o r w ith one o f the two d ir e c t lab o ra to ry approaches.

The hypothesis to be te s te d by questionnaire i s th a t the m ajority

o f e ig h th grade s tu d en ts , regard less o f th e i r a b i l i ty le v e ls , p re fe r

the use of some lab o ra to ry experiments in th e i r mathematics classroom;

th a t i s , experiments genera te an improved clim ate fo r th e m otivation to

le a rn m athem atics.

The r e la t iv e e f f e c ts o f the th re e approaches in each o f the

schools was compared through an a ly sis o f covariance on the p o s t- te s t

achievement sco res . C ovariates were employed in an attem pt to equate

d iffe ren ces in in te l l ig e n c e and p r io r knowledge o f the mathematical

concepts.

To compensate fo r th e possib le b ias r e la t iv e to the ad a p tab ility

o f experim ents to the p a r t ic u la r u n i t and to the novelty o f the

la b o ra to ry method f o r s tu d en ts and te ach e rs , research d a ta were sim i­

l a r ly co llec ted fo r two ad d itio n a l s ix weeks u n its o f study . Every

c la ss had the opportun ity to experience a l l th ree la b o ra to ry approaches

since the lab o ra to ry assignm ents passed through a complete ro ta tio n

during the th ree s ix weeks periods of th e program. That i s , every s tu ­

dent in the sample experienced one s ix weeks u n it o f mathematics w ith

each type o f la b o ra to ry .

To fu r th e r compensate fo r the v a ria b le o f teach ing techniques,

the d a ta from each o f th e schools were analyzed se p a ra te ly . Thus, fo r

a l l p r a c t ic a l purposes, s ix comparisons o f labo ra to ry method were made

by a n a ly s is o f covariance.

P eriod ic a ttem pts were made throughout the e igh teen weeks to de­

term ine the s tu d e n ts ' re ac tio n s to the methods o f p re se n ta tio n . Exper­

iment r a t in g forms were used but w ith only p a r t ia l success; the major

to o l f o r guaging the s tu d e n ts ' p references and in te r e s t s was a ques­

tio n n a ire completed by each student a f t e r h is c la ss had experienced a l l

th ree types of lab o ra to ry methods,

Heed f o r the Study

The year a ju n io r high school s tu d en t spends in the e ig h th grade

i s a most c ru c ia l perio d in h is search f o r mathem atical m a tu rity . This

y ea r i s no t only a time o f lea rn in g and a ss im ila tin g , but i t i s a time

o f d ec isio n . The e ig h th grade stu d en t i s din the f in a l s te p s o f se­

le c t in g the curriculum he w il l pursue fo r a t l e a s t the nex t fo u r years

o f school. His choice a t th is time w il l d ire c tly in fluence every as­

p ec t o f h is fu tu re .

The natu re o f the app lied lo g ic of a th ir te e n y ea r o ld i s a t

tim es ra th e r shallow . That i s , i t i s o f te n the case th a t the decisions

he makes are more in fluenced by rece n t experiences than by h is t o t a l i t y

o f experience. Thus, th e c u r r ic u la r choices upon which h is fu tu re

p lans w il l stand o r f a l l a re la rg e ly based upon h is experiences in the

e igh th grade. For many s tu d en ts i t i s a make o r break y e a r in the

study of mathematics.

P sycho log ists agree th a t the l i f e o f an adolescent i s very com­

p lex . This confusing conp lex ity o f ten n eg a tiv e ly in flu en ces h is aca­

demic p ro g ress . The v a s t m a jo rity o f ju n io r high school s tu d en ts un­

derachieve r e la t iv e to th e i r mathem atical p o t e n t i a l i t i e s . - I t i s a ta sk

of tremendous magnitude f o r th e e ig h th grade teacher to awaken these

studen ts to the r e a l i ty th a t mathematics, p lays an im portan t, continuing

ro le in th e i r l iv e s .

Primary o b jec tiv es o f the ju n io r h igh school mathematics program

a re to provide the s tu d e n ts w ith th e o p p o rtu n itie s f o r r e la t in g and

coord inating th e i r m athem atical knowledge learned in the elementary

grades, to m otivate the s tu d en ts through w ell conceived, challenging

experiences to d iscover contemporary ap p lic a tio n s o f m athem atics, to

prepare the studen ts w ith a useable knowledge o f the b as ic mathematical

concepts necessary f o r fu tu re endeavors, and to en lig h ten the studen ts

on the magnitude o f the con trib u to ry ro le assigned to m athem atics in

tomorrow1 s w orld.

Many studen ts in th e e ig h th year o f school o ccass io n a lly f in d i t

d i f f i c u l t to concentrate on m athem atics. More o ften than teachers l ik e

to adm it, the e igh th grade sy llabus c lo se ly resembles th a t o f the

seventh grade. In ad d itio n , the s ix th and seventh grade to p ics in

mathematics o ften appear as carbon copies. This d u p lica tio n , or pos­

s ib le t r ip l ic a t io n , o f presented m a te ria ls tends to dim inish the moti­

v a tio n a l aspect required fo r measurable progress in mathematical com­

prehension.

A la rg e percentage o f teachers o f e igh th grade mathematics quickly

become f ru s tra te d by the lack o f d es ire th e i r students d isp lay sh o rtly

a f te r the school year has begun. As a consequence the classroom atmos­

phere degenerates to a pedagogical ro u tin e through which the students

and th e i r teacher must s u f fe r , each hoping f o r long vacations punctu­

ated w ith only b r ie f periods of academia.

F ortuna tely , most o f the studen ts who succeed in enduring th is

p a in fu l p rocess, while su f f ic ie n tly meeting the sometimes meager stand­

ards o f the bored teach er, are required to spend no more than one year

in th is s i tu a tio n before they are graduated to an o ften more in te re s t­

ing form o f mathematics ca lled algebra.

U nfortunately, each year the e igh th grade teacher i s allowed only

th ree months to recuperate before being subjected to another group of

unmotivated eighth g rad ers . A fter a few years of being assigned to th is

comparatively unproductive ro le , the teach er looks fo r greener pastu res

in a lgebra o r geometry and w illin g ly s te p s aside fo r some unsuspecting

neophyte who w ill eagerly attem pt to c rea te an atmosphere conducive to

learn ing and apprec ia ting mathematics. The cycle s t a r t s again . I s

there any way to break th i s cycle?

Get the studen ts involved , i s a cry heard from a l l com ers o f the

pedagogical world today, l e t them see f o r them selves, I t t; them use

t r i a l and e r ro r , l e t them malce some educated guesses, l e t them re f in e

th e i r th ink ing through the experience o f p a r t ic ip a t io n .

As Jo P h il l ip s wrote in 'P u ttin g the Tic in A rlthm e,"

Gone fo rev er are th e days when the mathematics c la ss ­rooms are a book, a p e n c il, and a ream of paper.M ultisensory a id s to le a rn in g , o f ten th ings commonly found in the o rd inary classroom environment, a re es­s e n t ia l to a su ccessfu l contemporary program. The contemporary classroom i s in a l i t e r a l sense a lab ­o ra to ry . . . . U n til a ch ild can make some simple c la s s if ic a t io n s and see some sim ple r e la t io n s , he i s n o t ready to le a rn about numbers (28:216).

The Des Moines P ub lic Schools use mathematics la b o ra to r ie s in the

L.A.M.P. (low Achievement M otivational P ro je c t) . Some of th e i r ideas

are expressed as fo llow s:

P rim arily , a mathematics la b o ra to ry i s a s ta te of mind. I t i s ch a rac te rized by a questioning atmos­phere and a continuous involvement w ith problem so lv ing s i tu a t io n s . Emphasis i s p laced upon d is ­covery re s u lt in g from stu d en t experim entation. A teach er a c ts as a c a ta ly s t i n th e a c t iv i ty between stu d en ts and knowledge ( 2 l ) .

U nfortunately , id e a l s i tu a tio n s e x is t only in the minds o f men,

while r e a l i ty provides the dented and b a tt le -s c a r re d s e t t in g s in which

educational research must tak e p lace .

In recen t years the commercial market has been flooded w ith edu­

ca tio n a l games and : v/s whose m anipulation a lleg ed ly o ffe rs n o t only

p leasure but a lea rn in g experience o f one s o r t o r ano ther. Thousands

of d o lla rs and thousands o f hours have been involved in te s t in g these

toys through customer p a r t ic ip a t io n .

The r e a l educational value of these toys and games may n o t be

h ighly c o rre la ted w ith the f in a n c ia l success o f the marketed product.

The inheren t b e n e f its of an educational natu re are extremely d i f f i c u l t

to measure. This i s p a r t ia l ly due to the f a c t th a t b en e fits from such

an experience may n o t be in d iv id u a lly recognized o r app rec ia ted .

Contemporary schools a re becoming involved on a la rg e sc a le in a t ­

tempting to teach m athematical concepts v ia p h y sica l models o r concrete

examples. This i s n o t to be confused w ith the app lied mathematics

which was emphasized so s tro n g ly fo r the f i r s t few decades o f th i s cen­

tury* A pplications are im portan t, but dem onstrations o f m athem atical

concepts through physical models may t r e a t ap p lic a tio n of these con­

cep ts only as a p o ssib le by-product. The main emphasis i s the concept

i t s e l f , which may have a m u ltitude of ap p lica tio n s as the le a rn e r

matures in h is comprehension o f the concept and of the problems which

must be solved.

S e ttin g o f the Study

Knowing the importance o f the e n t i r e ju n io r high school mathe­

m atics program, many eager teach e rs are co n tin u a lly seeking f o r improved

pedagogical techn iques. During the spring of 1970 a seminar was held

fo r the mathematics teach ers o f th ree Mercer County, Pennsylvania

school d i s t r i c t s . The au tho r was in v ite d to p a r t ic ip a te . The objec­

tiv e s o f the seminar were to c o lle c tiv e ly pool the e f fe c t iv e techniques

used in the mathematics classroom s throughout the County and to d e te r ­

mine which methods could b e s t be used to improve the teaching of mathe­

m atics in the represen ted ju n io r high schoo ls.

10

Mathematics la b o ra to rie s and th e i r e ffec tiv en ess was a p re -a s-

signed to p ic proposed to stim ula te d iscussion . Although every ju n io r

high school teacher a t the seminar had used some models and games to

demonstrate mathematical concepts, none had any experience w ith

teaching in a mathematics labo ra to ry s e t t in g .

The av a ilab le l i t e r a tu r e on mathematics la b o ra to rie s surveyed

p r io r to the seminar d ea lt fo r the most p a r t w ith experiments fo r slow

lea rn e rs in mathematics. Nearly a l l of the la b o ra to rie s described in

these m a te ria ls were w ell equipped rooms, separate from the classroom,

w ith labo ra to ry ta b le s and r e la t iv e ly extensive equipment fo r experi­

menting.

The average ju n io r high school in th is lo c a l i ty cannot affo rd the

expense o f separate lab o ra to ry f a c i l i t i e s nor e labora te experimental

apparatus. However, n e i th e r can they a ffo rd to jeopardize the academic

w elfare of th e i r s tuden ts by neg lecting to take advantage of innovative

ideas av a ilab le through mathematics lab o ra to ry techniques.

Though the need fo r some type o f program to re ju v in a te ju n io r

high mathematics was unanimously agreed upon a t th is sem inar, i t was

no t p ra c t ic a l a t th a t time to make a wholesale adoption o f an un tested

lab o ra to ry program. However, the in te r e s t in the lab o ra to ry experiments

was s u f f ic ie n t to germ inate the id ea o f a research study to determine

the e ffec tiv en ess o f labo ra to ry techniques on the m otivational and

achievement le v e ls o f e ig h th grade s tu d en ts .

11

D if f ic u lt ie s in Studying Mathematics L aboratories

The hodgepodge o f id eas concerning mathem atical la b o ra to r ie s

which a re v erb a lized e i th e r a t seminars o r in the l i t e r a tu r e gen era lly

serve to confuse ra th e r than organize the th in k in g o f the in te re s te d

teach e r. Thus, one fa c e t o f th e problem confron ting p o te n tia l u se rs

o f the la b o ra to ry method 13 to a ss im ila te the numerous op in ions, a f t e r

they have been gathered , f o r th e purpose o f designing an organized

study of la b o ra to ry methods.

To com plicate the problem, the study mu3t be r e a l i s t i c r e la t iv e

to e x is tin g c u rr ic u la r designs a t the involved schools. Research in

education, which i s done w ith the cooperation o f school ad m in is tra to rs ,

most o ften involves a s tu d en t sample in an a c tu a l school s e t t in g . This

au th en tic school environment freq u en tly d ic ta te s c e r ta in r e s t r ic t io n s

to the study due to the im p ra c tic a li ty o f m anipulating s tu d en ts fo r ex­

perim ental purposes.

Another hurd le on th e p a th to a successfu l study i s th e presence

o f so many v a r ia b le s . Of course, every attem pt should be made to m ini­

mize the e f f e c t o f unmeasurable v a ria b le s i f the r e s u l t s o f the study

are to be g en e ra lly m eaningful. Two v a r ia b le s , c lo se ly r e la te d to each

o th e r, a re the teacher and the a d a p ta b ili ty o f the school p la n t to la b ­

o rato ry teach ing .

Every teach er has h is own unique s e t o f c h a ra c te r is t ic s in f lu ­

encing the e ffe c tiv en ess o f h is teach ing . His academic p rep ara tio n in

genera l areas as w ell as i n mathematics and c lo se ly r e la te d su b je c ts ,

h is experiences i n the classroom both as a s tuden t and as a te ach e r,

12

h is perso n a lity t r a i t s which psychogenically co n tro l the degree o f rap­

p o rt with the s tu d en ts , and h is confidence in th e methods he i s using

to teach a lesson are a few o f the many c h a ra c te r is tic s to which no

guage o r measuring device can he app lied with accuracy.

Teacher v a r ia b i l i ty i s accentuated by the d iffe rence in physical

f a c i l i t i e s provided by the school d i s t r i c t s . Continually expanding

school d i s t r i c t s have a very d i f f i c u l t time merely providing s u f f ic ie n t

classrooms and teachers in th e i r overcrowded cond itions. Except in

wealthy d i s t r i c t s where keen fo re s ig h t has produced su f f ic ie n t bu ild ing

programs, rap id population growth p ra c t ic a lly e lim ina tes the p o s s ib il­

i t y o f separa te ly lo ca ted and w ell equipped lab o ra to ry f a c i l i t i e s .

Confounding th is problem i s the s iz e o f the c la ss enrollm ent. Con­

ducting mathematics la b o ra to rie s w ith in the confines o f the classroom

w ith th i r ty to th i r ty - f iv e e igh th graders i s a challenge many teachers

a re re lu c tan t to face .

The e f fe c t o f these ex is tin g d e te rren ts cannot be' overlooked.

I t appears obvious th a t the purpose of learn in g mathematics through the

use o f labora to ry techniques can b es t be served under id e a l cond itions.

S ignificance of the StudY

The ju n io r h igh schools in Mercer County, Pennsylvania, have

problems w ith achievement and m otivation in th e i r e igh th grade mathe­

m atics c lasses which a re no t unique. I t i s po ssib le th a t the r e s u l t s

of a research study conducted in th is lo c a l i ty w il l have con tribu to ry

ap p lica tio n fo r ju n io r high schools in many o th e r regions of the coun­

t r y . Even w ith a l l the inheren t v a r ia b le s , w ith in and between the

13

school d i s t r i c t s across th e nat±onf involv ing teach er p rep ara tio n and

pedagogical methods, urban and suburban d es ire s and needs, and oppoiw

tu n i t ie s fo r advanced education in co lleges o r tra in in g schools, th e re

e x is ts a common problem, the underachievement of unmotivated ju n io r

high school studentB. Any and a l l e f f o r ts to solve t h i s shared problem

should be of unanimous in te re s t*

I f organized a ttem pts a re made to seek so lu tio n s to the problems#

in one school s e t t in g , whether they a re s a tis fa c to ry o r n o t, th e re i s

m erit i n p resen ting th e planned in p u t, the re su lt in g ou tpu t, and the

conclusions in a document av a ila b le f o r a l l in te re s te d p a r t ie s to

examine* I t does no t lo g ic a l ly fo llow th a t every workable p lan in one

school d i s t r i c t w il l be successfu l f o r ano ther. Por th a t m a tte r, i t

may n o t be advantageous f o r another te ach e r w ith in th e same school sys­

tem to attem pt i t* However, even a cursoxy reading o f the fin d in g s i n

one study may r e s u l t i n a gleaned id e a con trib u tin g to an easing o f th e

r e a d e rs ' r e la te d problems*

The s ig n ifican ce of th i s study la rg e ly r e s ts upon the v a l id i ty o f

the above asse rted prem ises.

Overview o f the Study

Chapter I I reviews the h is to r ic a l and contemporary l i t e r a tu r e re ­

la te d to mathematics la b o ra to r ie s and examines fo r relevancy the lim ­

i te d amount o f re la te d resea rch dea ling w ith lab o ra to ry methods o f

teach ing mathematics.

Chapter I I I p re se n ts d e ta ile d d esc rip tio n s of the environmental

asp ec ts o f the school d i s t r i c t s , th e c h a ra c te r is t ic s o f the studen t

sample, the experiences and p reparatory backgrounds o f the teach ers ,

th e lim ita tio n s o f the physica l f a c i l i t i e s , the events during the p re ­

p a ra to ry and t r i a l p e rio d s , ru le s o f procedure fo r th e labo ra to ry

types, k inds o f records m aintained, con ten t o f the u n i ts covered, and

th e l im ita tio n s o f the study*

Chapter IV inc ludes th e num erical d a ta co llec ted and an an a ly sis

o f these d a ta by covariance* .Also, analyzed are the s tu d e n ts ' opinions

expressed through the s tu d en t in te r e s t questionnaire*

Chapter V summarizes th e preceding chapters,' draws conclusions

based upon the a n a ly s is o f d a ta , s ta te s th e scope and lim ita tio n s o f

the study , and p resen ts r e la te d problems f o r fu r th e r re sea rch .

CHAPTER I I

RELATED LITERATURE AND RESEARCH

The primary purpose f o r searching the l i t e r a tu r e was to find ma­

t e r i a l re lev an t to the p ra c t ic a l problems confronting the Mercer County

ju n io r high school mathematics teach e rs . Hopefully, some reported ex­

periences and some completed research s tu d ie s involving mathematics

la b o ra to rie s would have conclusions which a re applicab le to the lo c a l

s i tu a tio n and to o thers o f a s im ila r n a tu re .

This primary purpose, then, d e lim its the scope o f the l i t e r a tu r e

review. The th e o re tic a l considerations th a t underlie the mathematics

labora to ry movement are minimized. Such la rg e scale movements as the

N uffie ld P ro jec t and the School Mathematics P ro jec t in England, which

have been widely discussed in recen t y ea rs , are not in th is chapter.

Rather, the l i t e r a tu r e search was d irec ted toward p ra c t ic a l reportB of

p ro jec ts in lo c a l i t i e s s im ila r to the Mercer County school d i s t r i c t s .

D escriptions of a Mathematics Laboratory

One o ften fo rg e ts what he has heard , ignores o r d isregards what

he has seen, but somehow tends to remember th a t which' he has touched.

To handle, f e e l , bend, tw is t , s tre tc h , cu t, sp in , throw, m anipulate,

guide, s te e r , co n tro l, or ju s t simply touch seems to have a power which

increases the re te n tio n spans of those involved.

15

16

No one re a liz e s th i s more than a mother who watches h er pre­

school ch ild de lig h t in the touch o f a sp ec ia l toy o r snuggle in the

comforting warmth of h is p a re n ts ' aims. School teachers of K - 6 have

always been cognizant o f the g rea te r in te r e s t th e i r p u p ils have in th e i r

work when they can p h y sica lly p a r tic ip a te in a learn ing experience.

Gimmicks and games, b a l ls and s tr in g , cubes and spheres, sc is so rs and

paper, and models o f paper mache can be found in the elementary c la ss ­

rooms o f nearly every school in the country a t one time o r another.

A small boy o f f iv e came in to the k indergarten one morning w ith ra d ia n t face and spark ling eyes, cry - out in jo y fu l tones: "I have something fo r you!I t ’s hard and long and has fo u r edges and two ends!"!Dhe precious ob jec t was held behind him, while he danced around in fond a n tic ip a tio n o f the pleasure he was about to give h is teacher, o f whom he was very fond. "What can i t be?" she answered, en te r­ing sym pathetically in to h is p leasu re . "Do show i t to me." In proud triumph the hand which held the treasu re was extended, and in the palm lay a burnt match. And the k indergarten teacher accepted i t as a g i f t o f va lue , fo r had i t no t helped to unlock the g rea t world o f form and i t s elements—faces , co m ers, and edges?

(Prom a n ineteen th -cen tury k indergarten te a c h e r 's rep o rt) (19:372).

A d e f in itio n , o r a t le a s t a general d esc rip tio n , o f mathematics

la b o ra to rie s may help w ith the decision whether o r no t to use them in

our c la sse s . P iaget has worked w ith ch ild ren o f a l l ages attem pting to

open th e doors of th e i r in t r ic a te minds and discover how children le a rn .

The in d iv id u a lis t ic n a tu re of a c h i ld 's learn in g experiences, as empha­

sized by P iag e t's re p o r ts , makes the labo ra to ry methods even mom im­

p o rtan t as i t may provide the ch ild ren w ith easy opportunity to choose

experiments appropriate to th e i r p a r t ic u la r stage o f development.

Clarkson expresses a s im ila r approach (5:495).

17

\7hat i s th is classroom environment ca lled the mathematics laboratory? There are many k inds of math la b s , but one of the most common i s simply the p rov ision , by the classroom teacher, of ma­te r ia l s and some time when ch ild ren may choose to work on a mathematical problem th a t in te r e s ts them, e i th e r alone o r w ith a p a rtn e r o r sm all group o f ch ild ren sharing a s im ila r in te r e s t . The d is tin g u ish in g fea tu re s o f such la b periods are the independence of ch ild ren from la rg e group or teach er-d irec ted lessons, and the p o s s ib i l i t ie s fo r in d iv id u a lis t ic and ac tiv e so lu tions to a wide v a r ie ty of problems. Needless to say, independence and ac tiv e p a r tic ip a tio n produce an extremely high degree of m otivation in the ch ild ren . Anyone who has observed labora to ry sessions can a t t e s t to th is fa c t!

A complementing d escrip tio n of la b o ra to rie s i s given by Spencer

and Brydegaard (32:5)*

The classroom fo r mathematics should be a learn in g lab o ra to ry . This does no t imply th a t equipment or fancy gadgets in a room o r th ings th a t ch ild ren bu ild make a b ea m in g la b o ra to ry ." R ather, a classroom becomes a learn in g -lab o ra to ry when i t produces mental and physical a c t iv i ty th a t r e s u lts in experim entation; th is in tu rn should lead to form ulation o f procedures and to g en era liza tio n s based upon re l ia b le and s u f f ic ie n t inform ation.The m a te ria ls f o r the lab o ra to ry are w ith in the reach o f every teacher. The m ateria ls c o n s is t, fo r the most p a r t , of th ings th a t ch ild ren and teachers bring in to the classroom fo r the lessons under considera tion . Cups, g la sse s , b o t t le s , cans, ja r s , boxes, la b e ls , ca rtons, s tr in g , measuring s t ic k s , and innumerable o th e r th ings to use in ex­periments w ith measuring should be a p a r t o f every mathematics classroom.

The fo rce behind the scenes fo r a lab o ra to ry fo r learn ing i s the classroom teacher. . . . He i s the type o f teacher who can teach without textbooks.

I f through experiments in the classroom the teacher can succeed

in g e ttin g the studen ts in te re s te d , y e t even ex c ited , then the atmos­

phere fo r learn in g i s more f e r t i l e . Perhaps e s s e n tia l in te re s t in

18

mathematics can be b es t generated and m aintained through experiments

which sim ultaneously provide f o r en terta inm en t, increased computa­

tio n a l s k i l l s , and deeper in s ig h t in to the underly ing , u n ify ing con­

cep ts , This i s no t a re c e n tly made conjecture* Marx re p o r ts on a book

w ritte n two cen tu rie s ago (23*123)*

Elementary education , mathematics and geometry— because they a re th e most a b s tra c t—are u su a lly the d u lle s t su b je c ts from many ch ild re n 1 s view­p o in t; y e t these su b jec ts could be made as ex­c i t in g and in te re s t in g as a r t s and c ra f ts i s now to the b r ig h te r p u p ils w ithout expensive equipment i f we take some in s p ira tio n from a book published ju s t about 200 years ago in Europe* Generously i l ­lu s t r a te d w ith wood-cuts the volume contains among more than 500 item s o f in form ation (and m is -in fo r- mation) numerous p ra c t ic a l experim ents which were provided f o r th e so lu tio n o f what were then im­p o rtan t everyday questions* These experiments are s t i l l as in te r e s t in g to perform today as they were i n 1764, and because they a re based on th e basic m athem atical and geom etrical p r in c ip le s they should make fo r some in te re s t in g c la s s sessions in a school o r be fun f o r a s c ie n t i f i c a l ly minded p a ren t to t r y w ith th e ch ild re n . Because mathematics and geometry a re here brought down from th e i r a b s tra c tp lane to a le v e l where we deal w ith everyday ob­je c t s , r e a d ily measurable q u a n t i t ie s , and concepts e a s ily understood, ch ild ren who a re normally c u r i­ous w il l be s tim u la ted to take a g re a te r in te r e s t in these so im portant subjects* I f s im ila r experi­ments would have been a p a r t o f the curriculum inth e w r i te r 's own school days, re te n tio n , in te r e s t and probably grades would have shown a s u b s ta n tia l improvement 1

To understand the a b s tra c t na tu re o f mathematics i t appears no t

only d e s ira b le bu t, f o r most o f u s , necessary to consider th e ideas as

they r e la te to m a te ria l phenomenon* Use o f a model to e s ta b lis h a p a t­

te rn , to provide d a ta , and to demonstrate a concept should n o t be under­

estim ated*

19

In a recen t Issue o f the Mathematics Teacher, l ic k a lso described

the dual na tu re o f mathematics (20:86).

Mathematics i s a b s tra c t; mathematics i s no t n a tu re .However, the key to the study of n a tu re and n a t­u ra l phenomena i s the concept of the mathematical model. That i s , a mathematical system may be so chosen th a t i t s terms and assumptions have some meaningful r e la t io n to the physical s i tu a tio n .This i s another b e a u tifu l aspect of mathematics; in one instance i t may be used as a 'too l w ith ap p li­ca tio n to models rep resen ting physica l phenomena, and in another i t may be an a b s tra c t d isc ip lin e in and o f i t s e l f .

Laboratories in the Elementary Schools

Based on h is study o f the comparative m erits of a m anipulative

approach w ith second g raders, Nasca concludes:

A procedure which provides ch ild ren w ith the op­p o rtu n ity to perform operations w ith concrete m a te r ia ls and encourages them to abandon such models in favor o f mental o r "a b s tra c t" manipu­la t io n can obviously provide su p e rio r gains in achievement (25:225 ) .

Sensory m a te r ia ls have o ften been developed by good teachers o f mathe­

m atics because they found them necessary. S tudents en terin g school fo r

the f i r s t time' may know how to count in a ro te manner, but they have

considerable d i f f ic u l ty understanding what th is process accomplishes.

The author re c e n tly observed a twenty two month o ld ch ild co rrec tly

id e n tify the l e t t e r s o f the alphabet and the d ig i t s when h is a t te n tio n

was drawn to them on a calendar and in new sprin t. T elevision programs

of today provide numerous opportun ities fo r ch ild ren to observe and

le a rn . However, co rrec t id e n tif ic a t io n and r e c i ta t io n of the symbols

may no t imply th a t the r e a l meaning of these symbols i s understood.

20

Understanding r e s u l t s in the combined uses o f the concrete and the ab­

s trac t* Simple use of the n a tu ra l numbers to count a s e t o f o b jec ts

req u ires these combinations* The basic computational exerc ises req u ire

th a t students can f re e ly th ink in an a lte rn a tin g manner between the

concrete and the abstract*

One o f the most popular to p ic s fo r d iscussion and research in

bridging the gap from the concrete to the a b s tra c t i s the use of

Cuisenaire rods.

Uuring the process of a b s tra c t computation, we ignore the concrete th in g s , and then when the computation i s completed, we apply i t to the con­c re te th in g s which we have tem porarily ignored.We begin w ith the concretion , then tu rn from i t while we compute a b s tra c tly , and f in a l ly bring these ab s trac tio n s back to the concretion (8:317).

Cuisenaire rods are a physical model f o r ra tio n a l numbers and fo r a ra tio n a l f i e ld . This makes them valuable fo r studying the commutative, a sso c ia tiv e and d is tr ib u tiv e laws. And by using a sp ec ia l convention, the p ro p e rtie s of s e t a lgebra are made p a r tic u la r ly ev ident. Other mathematical re la tio n ­sh ips explored w ith Cuisenaire rods include fra c ­tio n s , p lace value, d if fe re n t number bases and p o s itiv e and negative in te g e rs . Because the col­ored rods help make mathematics re a l to s tu d en ts , they understand and enjoy d iscoveries made from m anipulating these b righ tly -co lo red models (7*4).

H o llis (17*683) compared the e f fe c ts o f using the C uisenaire-

Gattegno method w ith a tra d it io n a l method over a two year period en­

compassing f i r s t and second grade. At the end o f the f i r s t year, pu­

p i l s taught w ith the Cuisenaire-Gattegno method achieved as w ell as the

con tro l group on tra d it io n a l te s ts and had acquired ad d itio n a l concepts

and s k i l l s . Ey th e end o f the second year both achievement and concept

knowledge were su p e rio r fo r the Cuisenaire-Gattegno c la sse s .

21

Dealing w ith retarded ch ild ren with In te llig en c e quo tien t le s s

than 80, Callahan and Jacobson re p o rt,

A d is t in c t advantage o f the rods i s the a b i l i ty to see the problem! I t d id not take the ch ild ren long to v isu a liz e the number concepts which heretofore had been i l lu s iv e a b s tra c tio n s . •• .Gains were made in th e area of t r a n s fe r . This was evidenced in o ra l (mental) a rith m etic . Mathematical s to r ie s and games, and a lso in th e i r workbook examples. . . . Throughout the experiment the teacher reported th a t the in te r e s t remained a t a high le v e l (5*12).

Nasca s tu d ied the m erits o f a m anipulative approach to second

grade a rith m etic , choosing C uisenaire rods as the stru c tu red concrete

model. With due caution d ic ta ted by the design o f h is study, he

s ta te s ,

I t i s safe to conclude th a t primary studen ts re ­ceiving in s tru c tio n based on pup il exp lora tion of concrete models can achieve tr a d it io n a l ly es­tab lish ed goals in mathematics (25*221).

Fassy (27:440) repo rts a negative e f fe c t from use of Cuisenaire

rods. Thus, i t i s ev ident th a t m a te ria ls alone do no t produce miracu­

lous r e s u l ts . Many o ther devices are now av a ilab le fo r teachers in te r ­

es ted in using labo ra to ry methods.

• Laboratories in the Jun io r High School

Experiences which a re good fo r elementary students may a lso be

good fo r ju n io r o r sen io r high school s tu d en ts . I f lab o ra to ry tech­

niques increase the achievement le v e ls o f elementary c la sse s , there i s

no r e a l reason to doubt th a t the same th in g w ill happen a t the eighth

grade le v e l . At the p resen t i t appears th a t the elementary schools

are lead ing the secondary schools in th e i r u b o of labo ra to ry

22

techniques. However, the id e as o f the discoveiy approach can d e f in i te ­

ly be enhanced a t a l l le v e ls , k indergarten through graduate school,

through e f fe c t iv e use of the mathematics la b o ra to r ie s .

Glowing rep o rts of successes w ith mathematics la b o ra to r ie s fo r

below average p u p ils have been published in magazines and jo u rn a ls

during the p a s t few y ea rs . Ruth Irene Hoffman, U n iversity o f Denver,

wrote i n The B u lle tin o f The N ational A ssociation o f Secondary-School

P rin c ip a ls , A pril 1968, concerning "The Slow le a rn e r - Changing His

View o f Math" (16:86-89), Hoffman c ite d p ro je c ts and case s tu d ies to

show the co n s tru c tiv e e f fe c ts of the lab o ra to ry approach upon low

achievers in Grades K through 12.

E ighth graders in Bingham High School (Kansas C ity) spend from th i r t y minutes to tw o-and-a-half hours a week p a r t ic ip a t in g in an in s tru c tio n a l dialogue on supplementary and enrichment to p ic s in grade 8 science and m athem atics. They use the IBM 1500 In s tru c tio n a l System.

To accommodate th e varied backgrounds of t h e i r s tu ­d en ts , th ree Kansas C ity secondary schools have adopted modular scheduling. Under th is p lan s tu ­dents meet in la rg e groupB fo r p resen ta tio n o f new m a te ria l by the teach e r, f o r summaries o f s tu d en t re p o r ts , f o r speakers from in d u s try , or fo r t e s t ­in g . lab o ra to ry periods provide fo r study w ith teachers o r o th e r studen ts and f o r lea rn in g by d is ­covery through many re la t iv e devices o r through games o r puzz les.

In B e llp o rt Middle School, B e llp o rt, N. Y., s tu d en ts were en­

couraged to e n te r in to a p ro je c t o f bu ild ing such th in g s as toboggans,

k idd ie ro ck ers , to o l boxes, and wagons from a sp ec ia l type o f cardboard

(35:209). These construc tions were teach ing a id s f o r o th e r sub ject/

areas such as mathem atics. Freedom to work w ith the m a te r ia l , to make

e rro rs , to co rrec t m istakes and d iscover new ideas were an im portant

23

p a rt of the p ro je c t. A summary of some o f the advantages c ite d by

T in ti inc ludes l ) S a tis fa c tio n by handling and experimenting, 2) Cre­

a t iv i ty o f expression, 3) S e lf - re a liz a tio n in constructing , and 4 )

Bridging the a b s tra c t to the concrete.

The N ational Education A ssociation i s a lso aware of recen t a t ­

tempts to incorporate labo ra to ry methods in mathematics c la sse s

(26:50).

Basic ideas are approached in tu i t iv e ly through d is ­covery and experim ent. . . . This includes operations on the abacus; use o f cu isenaire rods fo r re in fo rc in g the concepts of number re la tio n s ; r a t io , and f ra c ­tio n s ; p rac tice in use of the hundred board, the num­ber l in e , the geo-board; the use of meaningful lo g ic puzzles; graphing games; and topo log ica l puzzles.

In Washington, D.C. ju n io r high studen ts meet fo r two labora to ry

periods a week in small groups. They work w ith e le c t r ic c a lcu la to rs ,

tapes and earphones, f i lm s tr ip s , overhead p ro jec to rs , and mathematics

fames.

C learly , fo r some ch ild ren , these programs have added a dimension and provided in s ig h t not read ily av a ilab le from the reg irlar chalkboard and p rin ted page (26:51).

In Winnetka Public Schools, Y/innetka, I l l i n o i s , mathematics lab­

o ra to rie s are held to help ch ild ren become independent le a rn e rs

(24:501).

As th e pupils work w ith concrete m a te ria ls , they record th e ir observations and then make g enera liza­tio n s from the d a ta . . . . The main idea i s to keep studen ts so m otivated by th e i r own work th a t they want to le a rn more. There i s no f a i lu r e , because a l l students are f re e to ask questions whenever they need h e lp . The atmosphere of the la b o ra to rie s i s a carn ival with a purpose.

24

The responses of the studen ts quoted in th is a r t ic le in d ic a te

th a t Winnetka students are motivated not only to do more mathematics,

but to l ik e doing i t . They look forward to the labora to ry periods w ith

an tic ip a tio n .

Research Studies

As th is chapter in d ic a te s , most of the l i t e r a tu r e on mathematics

la b o ra to rie s co n s is ts of general program d esc rip tio n s , su b jec tiv e eval­

u a tio n s, and considered opin ions. In a recen t research review by

Kleren the in d ic a tio n i s th a t serious researchers in education are

giv ing more a tte n tio n to m anipulative a c t iv i ty in mathematical in s tru c ­

tio n (lQ:22B). Among the questions which Kieren proposes fo r con­

s id e ra tio n a re the follow ing:

Does m anipulative experience f i t a t a c e r ta in le v e l in to a p a r tic u la r h ierarchy in the promotion of learn in g ?What are the best q u a l i t ie s of m anipulative exper­ience in promoting learn ing?

In ad d itio n to K ieren1 s questions the question of a t ti tu d e changes has

also been asked in two research s tu d ies .

Bruner suggests th a t le a rn in g mathematical ideas over a period of

years involves enactive , ic o n ic , and symbolic rep resen ta tio n s (2:10-

l l ) . R espectively these th ree rep resen ta tio n s are m anipulation of an

object using the sense o f touch, organizing what one sees by observing

p a tte rn s , and f in a l ly sym bolically coding the a c t iv i ty fo r compacting

and condensing the re s u lts fo r easy reference . The v a lid ity o f the

action-image-symbol sequence was considered in numerous s tu d ie s . Arm­

strong , working w ith m entally re ta rded ch ild ren , found a c t iv i ty to have

25

a pronounced e f fe c t on le a rn in g ( l ) . Fennema ( l l ) , Trueblood (34-)}

and Vance (35) found th a t normal school ch ild ren in grades 2, 4, 7»

and 8 can achieve os w ell a t the symbolic le v e l as a t the m anipulative

level*

Whether o r no t a s im ila r sequence o f p re sen ta tio n should be f o l ­

lowed in teach ing a s in g le concept over a sh o rt period o f one o r two

lesso n s has n o t been in v e s tig a te d . The design o f the p resen t study in ­

volving d if fe re n t orderings o f experim ents and r e c i ta t io n s may provide

re le v an t inform ation.

C uisenaire rods, p rev iously discussed in th is chap ter, provide an

example of a m anipulative device g en era lly possessing the proper qual­

i t i e s to promote le a rn in g . Sources fo r o th e r m a te ria ls and devices

which may have these d es ired q u a l i t ie s a re mentioned l a t e r in th is

ch ap ter.

There i s l i t t l e doubt th a t people o f a l l ages perform ta sk s , sim­

ple o r com plicated, w ith much more success i f they e n te r ta in the proper

p o s it iv e a t t i tu d e s toward th e i r assignm ent. Conversely, success very

o ften genera tes contentment and p le asu re . For some, hoeing the garden

can provide as much re c re a tio n a l p leasu re a s p lay ing a round of g o lf

w ill f o r o th e rs . The d iffe ren ce su re ly i s a t t i tu d e . P leasu re i s a

r e la t iv e s ta tu s o f s a t is f a c t io n and joy, measured in d iv id u a lly by p e r ­

so n a lly e s tab lish ed stan d ard s.

Higgins made a f iv e week study invo lv ing 853 s tu d en ts of 29

e ig h th grade teachers i n Santa C lara County, C a lifo rn ia a t the be­

g inning of which the s tu d en ts were assigned to a t t i tu d e groups based on

the r e s u l t s o f a b a tte ry o f t e s t s (14). The m a te ria ls on teaching o f

26

mathematics through science prepared by SMSG were stud ied . I t was

found th a t d iffe re n ce s in a t t i tu d e p a tte rn s among groups i s n o t re ­

f le c te d in s ig n if ic a n t d iffe ren ces in e i th e r a b i l i t y o r achievem ent.

In ad d itio n , only 8 p e r cen t o f the studen ts made a p o s itiv e change in

th e ir a t t i tu d e s toward mathematics while 6 p er cen t developed an un­

favorable a t t i tu d e toward the content o f th i s u n i t .

A s im ila r study was made by Y/ilkinson invo lv ing achievement and

a t t i tu d e s o f s ix th graders studying geometry under e i th e r v e rb a lly d i­

rec ted o r discovery o rien ted techniques (36). The ch ild ren w ith aver­

age and below average in te l l ig e n c e in d ica ted a p o s itiv e change in a t ­

titu d e toward mathematics w hile above average stu d en ts seemed to appear

p assiv e . No s ig n if ic a n t d iffe re n c e s in achievement were found.

The Nature of a Continuing Program

At the time o f the rev o lu tio n in mathematics during the 1960 's ,

educators were h ig h ly c r i t i c a l o f 'ju n io r high school c u rr icu la em­

phasizing q u a n tita tiv e concepts and procedures which did no t appear to

serve th e i r proposed o b je c tiv es . Such to p ics a s l i f e insurance, d is ­

counts, commissions, banking, and ta x a tio n in th e e igh th grade were

o ften condemned as having too l i t t l e to o f fe r in bu ild ing so lid founda­

tio n s f o r e i th e r the term inal o r th e co llege bound s tu d en ts .

Many modem programs emphasize a b s tra c t s tru c tu re and form to

provide fo r p ro fic ien cy in the use o f mathematical concepts, p r in c ip le s ,

and s k i l l s and have been c r i t i c iz e d f o r leaning too f a r in the opposite

d ire c tio n .

The author be liev es th a t an in te l l ig e n t and lo g ic a l approach f o r

average studen ts o f e igh th grade i s one o f compromise* True, i t i s the

re s p o n s ib ili ty of the elementary school to la y the fundamental ground­

work o f basic concepts upon which a r ith m e tic a l s tru c tu re i s bu ilt*

However, the ju n io r high school should n o t assume 100 p e r cent e f f i ­

ciency on the p a r t o f the elem entary school* Apparently a continuing

process o f foundation bu ild ing must be evidenced a t a l l le v e ls i f math­

em atical growth toward m atu rity i s to take place* Mathematical problem

s itu a t io n s presented to provide f o r s treng then ing concepts and p rin ­

c ip le s and a t the same time lending themselves to a p ra c t ic a l ap p lica ­

t io n , may be most e f f ic ie n t ly made in a lab o ra to ry s e t t in g . The r e s u l t ­

in g so lu tio n s may be experienced through sev era l o f th e senses as w ell

as tab u la ted and recorded* C alcu lations may be meaningful beyond

s a tis fy in g the te ach e r o r agreeing w ith the answers in the back o f the

book* I t i s presumed th a t a program so designed w il l b e t te r serve con­

temporary so c ie ty provided enough tim e, e f f o r t , and f l e x ib i l i t y a re em­

ployed in i t s construction*

The Laboratory Teacher

Every stu d en t should be f u l ly challenged i f p o ss ib le , but p r io r

to the challenge the teacher must n e c e ssa rily have the a t te n t io n and

the in te r e s t o f the student* Thus, the teach ers o f ju n io r h igh school

mathematics who search f o r improved techniques, w ill f in d in c reas in g

need to be w ell informed i n mathem atical su b je c t m a tte r, h is to ry of

mathematics, psychology of in d iv id u a l d if fe re n c e s , e ffe c tiv e techniques

o f p re se n ta tio n o f m a te r ia ls , and eva lu a tio n of re su lts*

28

When teachers are provided w ith in s u f f ic ie n t time f o r in se rv ic e

p rep ara tio n and adjustment to the labora to ry methods o f in s t ru c t io n ,

extraneous fa c to rs in the experience may g rea tly in flu en ce the de­

c is io n to continue the endeavor. Following a one y ea r program a t David

J r . High School in Cleveland to e s ta b lish a sound lab o ra to ry o rien ted

program f o r seventh g raders, Woodby recorded these sta tem ents sum­

m arizing the find ings (37) •

(1) O rganization fo r sm all group in s tru c tio n o r in d iv id u a lized in s tru c tio n i s d i f f i c u l t .D isc ip line was a major concern.

(2) The goals of mathematics in s tru c tio n o f te n g e t lo s t in the mechanics o f the lab o ra to ry a c t iv i ty .

(3 ) B e lie f o f the te ach e r in a discovery approach i s n o t s u f f ic ie n t to accomplish the ap p ro p ria te behavior. Teachers w il l re v e r t to t e l l in g , ex­p la in in g , and showing stu d en ts . There i s a wide gap between teach er b e l ie f and teach er behavior.

(4 ) The teachers expected too much o f the m a te r ia ls ; i t was assumed th a t th e m ateria l would provide the m otivation.

(5 ) Rewards fo r the teachers were d if fe re n t from th e i r ex p ec ta tio n s. They had a n tic ip a te d g re a t increases in the achievement by the group as a whole. The rewards turned out to be unusual accomplishments by in d iv id u a ls and those were in frequen t and unp red ic tab le .

(6) The teachers became more concerned w ith how studen ts learn ed than w ith the achievement; questions asked by studen ts became more im­p o rtan t to the teach e r l a t e r in the sem ester than they were a t the beginning.

(7) The teachers worked longer and more in te n s iv e ly than they d id before the p ro je c t. Even i f they had much more time f o r p reparation they stayed l a t e and u su a lly took work home to (a) organize fo r in s tru c tio n , (b) devise a c t iv i t i e s and w rite in s tru c tio n s , and (c) evaluate r e s u l t s .

( s ) Teachers in th i s le a rn in g s i tu a tio n need some­one to ta lk to . Supervisors and consu ltan ts a re im portant to the teacher in th is s i tu a t io n .

/

29

(9 ) Prom the teacher po in t o f view, the learn ing was more n ea rly guided, discovery than true discovery.

(lO ) The teachers became b e t te r teachers because o f what they learned about studen ts learn ing .For example, they ta lked le s s and lis ten e d more a t th e end than they had a t the beginning.

The two Cleveland teachers Involved gleaned some valuable in fo r­

mation from t h i s experience with low achievers in the seventh grade.

Through a summer tra in in g session conducted by the two p a r tic ip a t in g

teachers, fo u r te e n o ther teachers worked w ith the m aterials i n the lab ­

oratory and gained some in s ig h t in to the proper uses of lab o ra to ry

techniques.

Y/oodby l i s t s ten d esirab le behaviors expected of a lab o ra to ry

teacher (3 7 )-

(1 ) The teacher asks questions th a t cause explora­t io n and in q u iry by the s tu d en t.

( 2 ) The teacher devises and uses tasks th a t r e la te to fundamental mathematical concepts and tech­n iques. A good example i s Rosenbloom’s simu­la te d computer in which the student discovers th e d is tr ib u tiv e p rin c ip le .

(3 ) The teacher uses m ateria ls o ther than the te x t­book.

(4 ) The teacher provides ind iv idual and small group a c t iv i t i e s o f an exploratory natu re th a t re­s u l t s in th e student try in g something, gathering d a ta , analyzing data, and te s tin g conclusions.

(5 ) The teacher uses cues th a t come from the stu ­dent in making teacher decisions about ques­tio n s asked o r tasks assigned.

(6 ) The teacher plans fo r and uses the basic s tra teg y o f student discovery.

(7 ) The teacher employs the s tra teg y o f asking th e studen ts to make decisions on the basis o f observation of events.

30

(8) The te a c h e r provides s i tu a tio n s f o r the s tu ­dent to p lay 0X1 a c tiv e ro le in learning} r a th e r th an a passive one.

(9 ) The te a c h e r c rea te s a new problem o r ta sk th a t i s e a s ie r o r more fa m ilia r to the s tu ­dent when d if f ic u l ty occurs, then allow s the stu d en t to re tu rn to th e i r o r ig in a l problem.

(10) The te ach e r provides th e student w ith a means f o r determ ining w hether an answer i s r ig h t o r wrong, independently o f the teach er o r the tex tbook.

A statem ent in th e preface of Laboratory Manual to Elementary

Mathematics by F itz g e ra ld , Bellamy, Boonstra, Oosse, and Jones, pro­

v id e s a summary o f the ro le of the lab o ra to ry te ach e r (12).

The essence o f the lab o ra to ry concept in lea rn in g mathematics i s the fo s te r in g o f inqu iry and in ­te rn a l m o tivation to seek answers to questions.I t i s a f a i r g e n e ra liz a tio n th a t te ach e rs , a t a l l le v e ls , t a lk too much. A lab o ra to ry in s t ru c to r must be wary o f the tem ptations to provide exces­s iv e , d ire c tio n f o r the s tu d e n t, and th u s rob the s tu d en t o f th e experiences of find ing answers f o r them selves. And some s tu d e n ts w ill demand ex­cessive d ire c tio n , which may be evidence o f the lack o f the u se of lab o ra to ry techniques in schools in the p a s t .

Sources fo r M a te ria ls and Devices

An annotated bib liography o f m anipulative devices used in United

S ta te s and England i s p resen ted by Davidson . (7*509) under the general

headings of b locks, c a lc u la to rs and computors, cards, construc tion ,

drawing to o ls , geo-boards, measuring dev ices, models, num erical games,

puzzles, shapes and t i l e , s tra teg y games, and studen t in s tru c tio n a l

m a te r ia ls . The su p p lie r s ' names and addresses a re also l i s t e d .

31

A current l i s t in g o f Mathematics la b o ra to ry m a te ria l has been

' compiled by Thomas P. Hillm an through the d is t r ib u t io n o f a question­

n a ire to mathematics educato rs throughout th e country . This l i s t may

be found in the June 1968 is su e of School Science and Mathematics (15:

488).

Summary

In summary, the reviewed l i t e r a tu r e which seemed p e r t in e n t to the

p a r t ic u la r problem of organ iz ing a study reg ard in g e f fe c t iv e labo ra to ry

experiences, led to the follow ing conclusions:

1 . Every mathematics classroom i s a p o te n t ia l la b o ra to ry .

2. The teacher i s probably the most im portan t element i n the

laboratory s e t t in g .

3. Essential p h y s ica l to o ls may vary from expensive, commer­

c ia lly produced item s to homemade devices of paper and s tr in g .

4. One objective i s to in te r la c e the a b s tra c t id eas o f mathe­

matics and th e concrete m a te ria ls a t hand to weave a useable

fab ric of concepts understandable by th e s tu d e n ts .

5. laboratory experiences w ith physica l models, f o r both e le ­

mentary and ju n io r high school s tu d e n ts , have very o ften

motivated s tu d e n ts to be more in q u is i t iv e toward mathematics.

6. The implementation of labo ra to ry methods in th e mathematics

classrooms should be preceded by a c a re fu lly p lanned, pre­

paratory p e rio d including o rg an iza tio n a l fe a tu re s and in ­

structional methods.

CHAPTER I I I

FEATURES AND ORGANIZATION OF THE STUDY

The School D is t r i c t s

The two neighboring school d i s t r i c t s , Mercer and Bakeview, from

which the po p u la tio n samples were s e le c te d are located in northw estern

Pennsylvania. They a re consolidated d i s t r i c t s with a large m a jo rity of

th e ir school popu la tion bussed in from surrounding townships. The

school communities a re ru ra l in nature w ith few in d u strie s lo c a te d

w ith in th e i r boundaries. A large percen tage of the adu lt popu la tion

liv in g in these d i s t r i c t s commute f i f t e e n o r twenty m iles to work in

the Shenango V alley s te e l m ills or the Y/estinghouse Corporation t r a n s ­

former p la n t . Another considerable percen tage of the population own

and operate farms w ith in the d is t r ic t s . Mercer i s a county s e a t which

means th a t many p ro fessio n a l people l i v e nearby and send th e ir ch ild re n

to one o f th ese schoo ls. Both d i s t r i c t s have junioxvsenior h igh schools

w ith the t o t a l popu la tion of children i n grades 7 - 1 2 between 1 ,000

and 1,500 s tu d e n ts . They both have th e problems of being overcrowded

due to abnormal popu la tion growth. T his i s p a r tia lly due to th e rece n t

completion o f two In te r s ta te highways in te rs e c t in g w ithin the County

boundaries. There has been an in flux o f trucking in d u stries w ith in the

County l in e s and many employees have moved in to these school d i s t r i c t s .

32

33

A dditional S ta te law enforcem ent o f f ic e r s were req u ired because o f the

highways^ and the S ta te P o lic e barracks i s located i n M ercer. The

m ajor problem of p rov id ing classroom space fo r the o ffsp r in g of these

new res id e n ts has a r is e n in th e elem entary schools. However, the

ju n io r high school was a f fe c te d a lso . A building program a t Mercer

fu r th e r complicated the space problem s in c e some form er ju n io r high

school classrooms were absorbed in an expanded l ib r a ry before the new

classrooms were a v a ila b le f o r occupancy.

This inform ation i s re le v an t to th e study only i n th a t separate

lab o ra to ry rooms were n o t a v a ila b le and th e classes were generally

overcrowded in both schoo l d i s t r i c t s .

The ad m in istra to rs o f these d i s t r i c t s were extrem ely cooperative

and are open-minded tow ard innovative id e a s in teach in g . Yet th is re ­

ceptiveness to new id e a s i s balanced w ith su ff ic ie n t cau tions to avoid

th e ex p lo ita tio n o f th e s tu d e n ts fo r th e sake of experim entation . The

teaching r e s p o n s ib i l i t ie s were s t r i c t l y assigned to th e reg u la r teachers

during th is study, a lthough a student te a c h e r was ass ig n ed to each co­

operating teach er during p a r ts o f the second sem ester. L i t t le fan fa re

was involved and the a u th o r was s t r i c t l y a behind th e scenes d ire c to r

and observer.

Physical P ro v is io n s

The school p la n ts a t Mercer and Lakeview were n o t constructed

w ith mathematics la b o ra to r ie s in mind. Classrooms a r e o f standard

s iz e s according to S ta te bu ild ing codes; there are no work tab les no r

i s the re room fo r any to be moved in ; s to ra g e cupboards are

34

conspicuously m issing and window s i l l s a re often used f o r book shelves}

expensive labora to ry equipment i s perhaps av a ilab le i n sm all q u an tity ,

bu t most devices are o f paper or wood which can be conven ien tly tucked

in the corner of a desk drawer o r p laced on the sm all s h e l f o f the

teacher* s locker. This i s not a d e sc rip tio n of an a ty p ic a l e ig h th

grade classroom. N either i s i t a d esc rip tio n of a classroom in which

mathematics la b o ra to rie s can be conveniently conducted.

During the b r ie f period of p repara to ry time f o r t h i s study l i t t l e

could be done to a l le v ia te the problems w ith the school p la n ts . Per­

haps th i s i s fo rtu n a te f o r i f id e a l conditions e x is te d f o r an experi­

ment of th is n a tu re , i t would be so ou t of the o rd inary th a t i t s re ­

s u l t s would ixot be g en era lizab le .

The Teachers

In each of the school d i s t r i c t s , Mercer and Bakeview, one teach er

has the re s p o n s ib ili ty o f teaching a l l o r a major p a r t o f th e e igh th

grade mathematics. Both teachers have f iv e o r more y ea rs o f teaching

experience and thoroughly understand the problems fa c in g e ig h th grade

mathematics teach e rs . Because of t h e i r experience and knowledge they

w illin g ly volunteered to p a r tic ip a te i n organizing and adm in istering a

lab o ra to ry type program. Games, m odels, and gadgets had been used in

th e i r classrooms many tim es before. P a r tic u la r a t te n t io n to in d iv id ­

u a liz e d programs w ith in the c lasses re su lte d in the Lakeview teacher* s

having prepared b a t te r ie s o f achievement te s ts which must be succes­

s iv e ly passed before completion o f th e e igh th grade. The Mercer teach e r

attem pted to keep a l l studen ts o f a g iven c lass to g e th e r , bu t in d iv id u a l

35

classes were p erm itted and encouraged to le a rn a t th e ir own r a te s o f

comprehension. Both teachers e s ta b lish e d similes? minimum stan d ard s f o r

s a tis fa c to ry achievement based on th e i r p ro fessio n a l experiences.

P reparatory and T ria l Periods

A sh o rt time a f te r the County seminar i n th e spring of 1970, -the

two cooperating teachers met w ith the au thor to estab lish some g ro u n d

ru les fo r th e p rep ara to ry period . A g rea t deal o f planning was n e c e s ­

sary before a study of any consequence could be in it ia te d and c o m p le te d

during the n e x t school y ea r. Weekly m eetings during the summer m o n th s

were scheduled when vacations and m il i ta ry ob liga tions did not i n t e r ­

fe re . During th ese meetings the content o f th e f i r s t semester o f

eighth grade mathematics was reviewed w ith i t s p o ten tia lity f o r a d a p t ­

able experim ents in mind.

Catalogues o f devices and p e r tin e n t l i t e r a tu r e from methods b o o k s

and mathem atics jo u rn a ls were searched fo r id e a s rela ted to the c o u r s e

content. By th e beginning o f school in September some ideas fo r . ex ­

perim ents r e le v a n t to s e ts , the decimal system , and other number b a s e

systems had been form ulated. However, the methods for presenting t h e s e

experiments and the type of records to keep f o r analyzing the r e s u l t s

were no t a t a s u f f ic ie n t ly re fin ed stage to begin the research. I t w as

rea lized th a t th e methods of mathematics la b o ra to rie s were new to b o t h

teachers and to a l l students* A time o f adjustm ent and refinement w as

requ ired . Xhe f i r s t s ix weeks o f the school y ear were set aside f o r a

continuation o f th e planning perio d .

36

One class In each school was a r b i t r a r i ly assigned to a type of

la b o ra to ry approach i n which the experim ents precede th e discussion of

th e re la te d concept i n c la s s r e c i ta t io n s . Another c la s s was a rb i t r a r i ly

chosen to have a la b o ra to ry type in w hich the experiments would follow

th e c la s s re c ita tio n s . Hie th ird c la s s in each school was to have the

t r a d i t io n a l approach w i th no organized laboratory experim ents. These

th re e approaches were re sp e c tiv e ly named pre-lab, p o s t- la b , and no-lab .

During 1he f i r s t s i x weeks t r i a l period the d ec is io n was made

w ith mutual agreement t h a t regularly scheduled lab o ra to ry experiments

were no t convenient. E a th e r , experiments would be conducted when the

re sp e c tiv e classes w ere ready for them. The teachers a lso experimented

w ith and refined th e d i f f e r in g teaching and questioning techniques used

to d istingu ish the p r e - l a b and p o st-lab approaches. At the conclusion

o f th e t r i a l period th e p lans and procedures for an organized study had

been formulated.

Student Samples

Of the eighth g ra d e classes tak in g the regular mathematics pro­

gram , three were a r b i t r a r i l y selected by each teacher to make up the

s tu d e n t sample. The homogeneously'grouped classes were ranked by the

te a c h e rs as high, medium, and low in expected achievement levels . How­

e v e r , no special e f f o r t was made to s e le c t the h ighest o r the lowest

o f th e school-ranked c la s s e s .

The combined t o t a l o f eighth grade students in both schools i s

approximately 350. Of th e se students, about 180 were involved in a t

l e a s t a part of the la b o ra to ry methods program while 80 Mercer students

37

and 66 lakeview s tu d en ts ac tually completed a l l phases o f the work and

a re therefore Inc luded in the d a ta , For reasons such as fam ilie s

moving from o r in to the d is t r ic ts o r studen ts having prolonged absences

due to illn e ss du rin g the periods when m a te ria l was presented and te s ts

were adm inistered, I t was impossible to Include a l l 180 s tu d en ts in the

analysis. However, a l l students who a c tu a lly completed th e e n t ire

study program a re included since no o th e r process o f e lim in a tio n was

employed a f te r th e i n i t i a l se le c tio n o f the s ix c la sse s ,

The s tu d en ts were not informed th a t any kind o f study was in

progress. They, o f course, re a liz e d th a t some c la sse s were using ap­

proaches to the su b je c t areas d if fe re n t from th e i r s , but they were not

made aware of any p a r tic u la r assignment design . This inform ation was

in ten tiona lly w ith h e ld and the a u th o r 's v i s i ta t io n s to the classrooms

were minimized to avoid any possib le Hawthorne E ffe c t.

Class Grouping

The 146 s tu d e n ts included in th i s study were members of s ix

e igh th grade c la s s e s , three each from Mercer High School and lake view

High School, M ercer County, Pennsylvania, On the one hand, the tru e

c la s s enrollments a r e s ligh tly h ig h e r than the sample in d ic a te s , but

only those s tu d en ts who were p h y sica lly p resen t to complete the e n t ire

study are included i n the data. On the o ther hand, a l l o f those s tu ­

dents completing th e program were Included in the s t a t i s t i c a l d a ta .

The range of in v o lv ed students p er c la s s was 18 to 27,

One experienced teacher from each school tau g h t the th ree c lasse s

involved and conducted the labora to ry se ss io n s . In both schools the

38

studen ts were homogeneously grouped according to o v e ra l l achievement

ra t in g s . The th re e a rb i t r a r i ly se le c te d c lasses d id no t include th e

c la s s o f h ig h e s t achievers in the e ig h th grades s in c e th e i r course work

i s enriched to inc lude more a lg eb ra than i s g en e ra lly studied in th e

e ig h th grade a t these schools.

As i s o f te n tru e when a study involves s tu d e n ts from a r e a l

school s i tu a t io n , the format must be designed to u t i l i z e the e x is t in g

in ta c t groups • This i s necessary to avoid unwanted in te rfe ren ce w ith

the schoo ls ' programs which cure a lread y e s tab lish ed . ' The im p ra c tic a l-

i t y o f having th e school ad m in is tra tio n and te a c h e rs c a te r to the r e ­

sea rch e rs1 w ishes makes i t necessary f o r the re se a rc h e r to design h i s

study according to the ex is tin g c la s s e s and th e i r c u r r ic u la .

In each school the th ree homogeneous c lasse s included one o f

r e la t iv e ly h ig h ach ievers, one o f average ach iev e rs , and one of below

average ach iev e rs as ra ted by the te ach e rs . The re a d e r i s reminded

th a t the f i r s t s i x weeks o f the 1970-71 school y ea r served as a t r i a l

period fo r some o f the ideas which were developed by th e author and th e

two teachers du rin g the summer months o f 1970.

Beginning w ith the second s ix weeks and con tinu ing through th e

fo u rth s ix w eeks, the study wa3 organized to provide c r i t e r i a which

when analyzed would determine the e f f e c ts of la b o ra to ry techniques on

achievement i n mathematics a t th e e ig h th grade l e v e l . The n u ll hypoth­

e s is to be t e s te d i s th a t the re I s no s ig n if ic a n t d iffe ren ce in

achievement between eighth grade mathematics c la s se s being taught

e i th e r w ithout lab o ra to ry techniques o r with one o f the two d ire c t l a b ­

o ra to ry approaches.

39

Because i t was h ig h ly im p rac tica l to se lec t th e students of p a r­

t i c ip a t in g classes randomly from the e n t ire eighth grade population,

th e re a re d e fin ite c l a s s d iffe ren ces i n In te llig en c e quotien t means and

in average pre-knowledge o f the concepts to be ta u g h t.

{The schools' re c o rd s show th a t th e Lakeview s tu d en ts were admin-

i s t e r e d in te llig e n c e t e s t s (O tis MAT AS) in 1970 and the Mercer s tu ­

d en ts were also te s te d th i s year f o r in te llig e n c e q u o tien ts (SRA High

School Placement T e s t , 1970 e d i t io n ) . Thus, reasonably accurate mean

in te l l ig e n c e q u o tie n ts were re a d ily av a ila b le (see Table l ) .

TABLE 1

INTELLIGENCE QUOTIENT MEANS OF CLASSES BY SCHOOLS

C lasses by T eacher Bankings*

In te llig en c e Quotient Means Mercer Lakeview

C lass 1 — High 111.27 111.89

G lass 2 — Medium 107.56 102.44

C lass 3 —• Low 96.63 102.95

*Note th a t th e teach e rs ranked the learn in g p o te n tia ls of the c la s s e s in the o rd e r o f the num erical c la s s i f ic a t io n s in Table 1 based on t h e i r experiences w ith them during the t r i a l p e r io d . This ranking corresponds w ith th e In te ll ig e n c e score rankings i n Mercer, but w ith th e closeness of th e mean sco res fo r c lasse s 2 and 3 a t lakeview, i t happens th a t the te a c h e r rankings a re num erically reversed . In th i s p ap e r the rankings o f h igh j medium and low w ill rep re sen t the teach er rank ings corresponding re sp e c tiv e ly to c lasses 1 , 2 , and 3*

In s tru c tio n a l P rocedures

One c la s s of the th re e in each school was assigned a p re - la b ap­

proach f o r th e f i r s t u n i t o f work on f r a c t io n s , r a t io s , and p ro p o rtio n s .

This type o f labora to ry was designed to see what th e s tu d en ts could do

w ith predesigned experiments involving a p a r t i c u la r concept and to a l ­

low them freedom to make conjectures concerning g en e ra liz a tio n s from

the s p e c if ic experiments performed. A ll o f t h i s took p lace before th e

concept was discussed in th e usual manner d u rin g c lass r e c i ta t io n ,

n a tu ra lly th e re was some carryover from r e l a t e d ideas explored in p re ­

vious y e a rs . I t was hoped th a t th is p re - la b approach would serve tos.

bridge any e x is tin g , re te n tio n gaps between seventh and e ig h th grade

when th is con tinu ity o f concepts was in v o lv e d .

fhe studen ts in a p re - la b were g iven th e physical m a te r ia ls f o r

the experiment and th e e s s e n tia l in s tru c tio n s fo r performing the f i r s t

s tep s of th e experiment. Xhe nature of th e p re - la b experiments was

such th a t th e teacher explained re la tiv e ly sim ple procedures to i n i t i a t e

the experiment and then th e students were encouraged to apply th e i r own

ideas in expanding and extending these p ro ced u res .

For example, consider Experiment D ( s e e Appendix A) invo lv ing th e

ro l l in g o f a b a ll down an inc lined p lane. I n th i s case, th e s tu d en ts

may work i n committees o f f iv e or six s tu d e n ts , each committee having

i t s own s e t o f m a te r ia ls . I f a trough i s n o t av a ila b le down which to

r o l l the b a l l , any smooth board w ill s u f f i c e , though i t may take more

t r i a l s to keep the b a l l on th e board f o r th e e n t i r e le n g th .

41

Tho s tu d e n ts were in s tru c ted to s e t up the in c lin ed plane and

time the descen t o f th e ro ll in g b a ll f o r a predetermined d is tan ce .

The te a c h e r may promote student p a r t ic ip a t io n in the form ulation

of the succeeding s te p s o f th is experim ent by asking q u es tio n s l ik e

those s ta te d below.

W ill th e tim e o f the b a l l 's descen t be any d if fe re n t i f the b a l l

i s re le a se d above th e f i r s t mark so t h a t i t has a running s ta r t ?

Now th a t we have one recorded tim e f o r the descent o f the b a l l ,

i s the re any method we can use to get a b e t te r estim ate o f th e ac tu a l

time i t ta k e s to r o l l from one mark to another? How do we f in d the

average o f 5 read ings o f the stopwatch?

I s th e speed o f th e b a ll constant f o r the en tire d is tan ce down

the in c lin e d p lane? How f a s t was i t go ing the in s ta n t you re leased i t ?

I f th e d is tan ce i s known and th e tim e of descent i s known, can

we f in d th e average speed o f the’ b a l l? What i s a formula involv ing

time, speed o r r a te , and distance?

What u n i t s o f measure fo r the tim e and the distance should we

use? What u n i t o f measure applies to th e average ra te o f descent?

I f we know the speed of the b a l l i n f e e t per second can we con­

vert thiB measure to m iles per hour? I f a moving ob jec t goes 88 f e e t

per second, i t i s t r a v e l in g a t a speed o f 60 m iles p er hour. Can we

use th i s re la t io n s h ip between un its o f r a t e measure to convert fe e t per

second to m ile s p e r hour in our experim ent? How?

. What i s a r a t io ? What Is a p ro p o rtio n statem ent?

O ther lead in g questions are suggested in the w ri tte n experiment

in the appendix.

42

Through th is p re -lab th e .stage has been s e t fo r a c la ss r e c i ta ­

t i o n on ra tio s and p ro p o rtio n s as they apply to ra te problems o f v a r i­

o u s k in d s .

For the same u n it o f work a second c la s s o f the three was as­

s ig n e d a post-lab approach. This laboratory type was designed to fo l­

low th e discussion o f the concep t in c lass r e c i ta t io n . Here the ma­

t e r i a l s fo r the equipment w ere provided but s p e c if ic d irec tio n s fo r

t h e i r u ti l iz a tio n were m inim ized. Exploration and discovery in an ap-

p l i c a to r y setting was th e o b je c tiv e . Im provization with o th e r read ily

a v a i la b le m aterials was encouraged. Thus, s p e c if ic app lica tio n s from

th e generalized concepts a lre a d y discussed a s w e ll as extension of

th e s e generalizations r e s u l t e d from student p a r tic ip a tio n in the post­

l a b .

For a comparison o f th e p re -lab and p o s t- la b approaches, again

c o n s id e r Experiment I) in a p o s t- la b s e tt in g . These students have a l ­

re a d y discussed ra tio s and p ropo rtions and have solved problems using

p ro p o rtio n s and ra te p a irs i n reg u la r periods o f class r e c i ta t io n .

Each committee i s s u p p lie d with an in c l in e d plane, a b a l l , and a

s to p watch. The objective o f th e post-lab i s to promote innovation and

e x p lo ra tio n through the a p p l ic a t io n of the p rev iously discussed con­

c e p t s . Teacher guidance i s e v e r present, b u t sp e c if ic in s tru c tio n s fo r

im plem enting the basic co n cep ts are minimized to allow as much freedom

a s p o ss ib le .

Y/hat can we do w ith th e s e item s, the board , the b a ll , and the

w a tc h , to get an ap p lic a tio n o f a tim e-ra te -d is tan ce problem? Not a l l* >

com m ittees need do the same ty p e problem. I s th e board o r in c lin e d

plane r e a l ly necessary? With these r e l a t i v e l y simple le ad in g ques­

t io n s , the committees a re in an atmosphere o f exploration, se a rc h in g

fo r some application o f what they already know in theory. The teacher

should, move among committees to determine w hat ideas they h av e end to

ask o th e r leading q u es tio n s to promote new id e a s . I f one g roup cannot

th in k o f a good a p p lic a tio n , the teacher can often explain t h e methods

of a working committee w ith slight a l t e r a t io n s suggested. 2?or example,

i f Committee A is r o l l in g the hall from a dead s ta r t on one m ark to a

second mark on the in c l in e d plane, suggest to Committee B t h a t th ree

marks be placed on the in c lin e d plane a t X, Y, Z, that th e b a l l be re­

le a s e d a t X, but th a t th e clock be used to tim e the descent o f the ball

from p o in t Y to point Z . Committee C m igh t simply drop th e b a l l from

a f ix e d height to the f l o o r . Committee D m ight use two b a l l s , d if fe r ­

en t i n s ize and weight, and compare the tim es of descent. However,

su g g estio n s by the te a c h e r should be k e p t a t a minimum to a l lo w fo r

freedom of thought and a c t io n on the p a r t o f the students.

Thus, for a l l f i f t e e n experiments w hich were used in t h e eighteen

weeks o f th is research study the p re-lab and post-lab d i f f e r e d by the

approach to the experim ent. The m athem atical concepts being emphasized

were th e same for each laboratory type.

The experiments a s they are p resen ted i n the appendix a r e not

w r i t t e n specifically f o r e ith e r labora to ry approach. The re q u ire d

m a te r ia ls are l is te d and a set of suggested questions are s t a t e d which

the te ach e rs may o r may n o t use according to the type of la b o ra to ry ap­

p ro ach desired. The te a c h e rs of the e ig h th grade students i n th i s

s tu d y soon found th a t g en e ra l plans f o r a n experiment are e s s e n t i a l ,

44

b u t f l e x ib i l i ty i s a m ost important p ro p e rty i f the s tuden ts a re to

w i l l in g ly p a rtic ip a te i n th e experim ents. No ideas are ignored. Every

su g g e s tio n is considered f o r i t s value before i t i s e i th e r used,

ad ap ted fo r use, o r d isc a rd e d . The r o le o f the teacher i s to g e t the

experim ent s ta rted by p la n tin g some g en e ra l ideas and then l e t the s tu ­

d e n ts nu rtu re , c u l t iv a te , weed, seed, and hopefully h arv est a worth­

w h ile crop of m athem atical concepts from th e i r experim ental endeavors.

The remaining c l a s s from each sch o o l was assigned a no-lab ap­

p ro ach during th is s i x weeks on f r a c t io n s , r a t io s , and p rop o rtio n s.

The t r a d i t io n a l c la ss r e c i ta t io n s were employed. Some models and de­

v ic e s were involved, a s they normally a re when concepts a re explained /

by th e teacher, but d i r e c t student p a r t ic ip a t io n in experiments in a

la b o ra to ry se ttin g was n o t perm itted.

N ature of the U n its Covered

During the f i r s t s ix weeks o f th e school year, the t r i a l and re ­

fin em en t period, both te a c h e rs agreed to include the study of s e ts , and

s e t n o ta tio n even though one tex t did n o t begin w ith th i s inform ation .

The knowledge of s e ts prov ided a u n ify in g concept used throughout the

y e a r i n the study o f m athem atics. Also included in th is t r i a l period

were th e following to p ic s :

One-to-one correspondences, c a rd in a l number, number l in e s , names

f o r numbers, place v a lu e , fa c to rs , exponents, decimal system, poly­

nom ia l expressions, b ase 7 and base 12 system s, and the commutative,

a s s o c ia t iv e , and d i s t r ib u t iv e p ro p e r tie s .

45

U nit X, the f i r s t u n i t of work involved in the s t a t i s t i c a l study,

included the following m a te ria l: f ra c tio n s as quo tien ts, equ ivalen t

f ra c tio n s , common denominators, comparing f ra c tio n s , com putational

m anipulation with f ra c tio n s , rec ip ro ca ls , complex f ra c tio n s , decim als,

repeating and term inating decimals, r a t io s , equivalent r a t io s , r a te s

and proportions, scale drawings, ra tio and p e r cent, and in te r e s t prob­

lems.

U nit I I included: p o in ts , lin e s and p lanes, angles, congruency,

construc tions, perpendiculars, angle b is e c to rs , p a ra l le l l i n e s , simple

closed f ig u re s , tr ia n g le s , q u a d rila te ra ls , polygons, perim eters, c i r ­

c le s , circumference, and th e number P i.

U nit I I I included: diagonals o f polygons, sums of in t e r i o r and

e x te r io r angles in polygons, areas of sp e c ia l polygons and c i r c le s ,

in sc rib ed and circumscribed polygons, space fig u re s , and su rface areas

and volumes of prisms, cy linders and spheres.

Por the most p a r t , the teachers covered the p e r tin e n t m a te ria l

contained in the 1961 (50) and 1964 (31) ed itio n s o f e ig h th grade

mathematics (Book Two) published by S ilv e r Burdett Company. They re ­

organized and supplemented the re la ted chap ters where they deemed i t

necessary to produce e s s e n tia l ly the same mathematics courses i n the

two schools.

Treatment V aria tions with A b ility Levels and U nits

The in s tru c tio n a l and te s tin g procedures were repeated f o r two

more u n i ts o f eighth grade mathematics, c a lle d Geometry I and Geometry

I I . The study of each o f these un its began w ith a d iagnostic t e s t and

46

concluded w ith an analogous achievement t e s t . The exact number o f days

o r weeks in which th e u n its were to be taught was n o t specified. Some

c la s s e s ea s ily completed the u n i ts in s ix weeks, w hile the lower

ach iev ers requ ired s lig h tly more tim e. Enrichment work took up th e

s la c k and te s ts were adm inistered a t the same tim e fo r a l l c la sse s .

To assure th a t a l l students from the sample had the same oppor­

tu n i t i e s to study mathematics during the school y e a r , a ro ta tio n o f

la b o ra to ry assignm ents was made a t the end of each o f the three u n i t s .

The lab o ra to ry type assignments in each school fo r the th ree

u n i t s included in th is study appear in Table 2 and Table 3. Other con­

v e n ien t views o f those same assignments are p resen ted in Tables 4 and

5 where a b i l i ty grouped c lasses in the schools a re mapped onto u n i t s of

work.

TABLE 2

ASSIGNMENT OP ABILITY LEVELS TO LABORATORY TYPES PER UNIT AT MERGER

Lab - iyp® Unit I Unit I I U nit I I I

P re - la b High Medium Low

P o st- la b Medium Low High

N o-lab Low High Medium

47

TAHOE! 3

ASSIGNMENT OP ABILITY LEVELS TO LABORATORY TYPES HER UNIT AT LAKEVIEW

Lab - Type Unit I U n it I I U n it I I I

P re-lab Low High Medium

P ost-lab High Medium Low

No-lab Medium Low High

TABLE 4

ASSIGNMENT OP LABORATORY TYPES TO ABILITY LEVELS PER UNIT AT MERGER

A b ility Levels Unit I Unit II U nit I I I

High

Medium

Low

P re-lab

P o st-lab

No-lab

No-lab

Pre-lab

Post-lab

Post-lab

No-lab

P re-lab

48

• TABLE 5

ASSIGNMENT OP LABORATORY TYPES TO ABILITY LEVELS PER UNIT AT LAKEVIEW

A b ility Levels Unit I Unit I I U nit I I I

High Post-lab Pre-lab No-lab

Medium No-lab Post-lab P re-lab

Low Pre-lab No-lab P ost-lab

Testing

Before any c lasses o r lab o ra to ries were held fo r a p a r t ic u la r

u n it of work, a teacher-made diagnostic t e s t was adm inistered to d e te r­

mine the pre-knowledge o f th e included mathematical concepts (see Ap­

pendix B)* These te s ts were constructed jo in t ly by th e teach ers and

the author and the same t e s t was given a t each school a t th e appro­

p r ia te tim es. Tables o f these te s t sco re s appear in C hapter IV.

At the conclusion o f th is un it a teacher-made achievement te3 t

was adm inistered (see Appendix c). The item s on th is t e s t were anal­

ogous to those of the diagnostic te s t p rev io u sly given. These se ts o f

te s t scores along with the students' in te l l ig e n c e q u o tien ts comprised

the e s se n tia l data fo r the s t a t i s t i c a l a n a ly s is of th is u n i t .

49

Questionnaire

An attempt was made to determine the immediate reactions of the

studen t3 and the te a c h e rs to each o f the fifteen experiments through

eva luation check sh e e ts (see Appendix D). Unfortunately, these forms

were n o t always re tu rn e d due to a la c k of su ffic ien t time during the

lab o ra to ry period to s a tis fa c to r i ly check them. However, an evaluation

of those responses received by the au tho r w ill be g iven in Chapter IV.

V/hen junior h ig h school s tuden ts are asked to re a c t to statem ents

regarding th e ir op in ions of a school ac tiv ity , there a re many in fluenc­

ing f a c to r s . This i s especially tru e when they are requested to r e c a l l

events which occurred over a long perio d of time and to make comparisons

concerning these e v e n ts .

Students in t h i s age group are whimsical to th e extent th a t the

most re c e n t experiences often become the chief c r i t e r i a by which a l l

o ther experiences a re evaluated. These experiences may be good o r bad

as seen from the s tu d e n ts ' viewpoint. In either case , the emotional

s ta te o f the students and the degree o f teacher-student rapport are

o ften re f le c te d in th e a ttitu d es of th e students toward th e ir assigned

work.

There i s no way o f knowing the r e a l reasons f o r the instantaneous

opinions of jun ior h ig h school s tu d e n ts . In sp ite o f th is known sh o rt­

coming in evaluating students' op in ions, another S tudent In te res t Ques­

tio n n a ire was construc ted and d is tr ib u te d one week a f t e r the olose of

the th i r d u n it (see Appendix E). I t was hoped th a t an in te re s t trend

would be evident when the entire s e t o f responses was viewed. The re ­

su lts o f th is endeavor are also summarized in Chapter IV.

50

Features Limiting the Scope of the Study

Although the s t a t i s t i c a l ana lysis of data in Chapter IV t r e a t s

each school separately , the fa c t th a t the two school d i s t r i c t s from

which th e samples we re se lec ted possess many s im ila r c h a ra c te r i s t ic s

has a l im itin g effec t on the app lica tio n of the re s u lts o f t h i s study

fo r the teacher in another type d i s t r i c t . The ru ra l s e t t in g , th e homo-

geniety o f the liv in g standards and c u ltu ra l experiences, and th e ra ther

conservative approach toward untested ideas a re c h a ra c te r is t ic s which

many school d is t r ic ts do n o t have. Consequently, the s tu d en ts have

sim ila r environmental backgrounds which influence th e ir a d a p ta b i l i ty to

a d if fe re n t type of le a rn in g experience. This in fluence i s d i f f i c u l t ,

i f not im possible, to measure.

Who i s to say what makes a good c lass o r a bad c lass? Y et, in

one of the schools the teachers of a l l e ighth grade su b jec ts agreed

th a t th is group of students on the whole was one of the p o o re s t in

recent y ears ; poor in the sense o f th e i r la ck ad a is ic a l a t t i t u d e toward

school in general. Perhaps the na tio n a l c r is e s and the u n c e r ta in ! ty of

the tim es contribute to th is l is t le s s n e s s in many geographical a re a s .

Only two teachers were involved in th is study . Both o f them were

raised and educated in northw estern Pennsylvania; though th ey b o th have

many years of teaching experience, th e ir teaching loads a re n o t l ig h t .

There was no released time from usual school r e s p o n s ib i l i t ie s f o r these

teachers. One is a s s is ta n t coach of a major sp o rt whose team became

involved in tournament play fo r the S tate Championships. There was no

bard evidence that th is caused any d is tra c tio n from, nor d e r e l ic t io n in

51

f u l f i l l i n g th e classroom re s p o n s ib il i t ie s , bu t undoubtedly the con­

fusion made th e task more d i f f i c u l t . The s tu d en ts , too, must have been

a ffec ted by th e pressures o f tournament involvement a t th i s le v e l .

I l l h e a l th n ecess ita ted th a t e ith e r a su b s titu te teach e r or the

assigned s tu d e n t teacher rep lace one of the reg u la r teachers fo r p a rt

of th e tim e during th is study . Again, there i s no evidence th a t th is

in fluenced th e re su lts of the Btudy, but s tuden ts may have reac ted d if ­

fe re n tly under these cond itions.

The la rg e class enrollm ents re la tiv e to the s ise o f th e c la ss­

rooms dim inished percep tive ly the amount of in d iv idualised a tte n tio n

given the s tu d en ts during labo ra to ry periods. Both teachers were as­

signed s tu d e n t teachers during p a r t of the p ro je c t , and t h e i r a s s i s t ­

ance probably helped to a l le v ia te th is s i tu a t io n .

A ll d iag n o stic te s ts and achievement t e s t s were constructed

through th e combined e f fo r ts o f the two teachers and the au th o r. The

r e l i a b i l i t y co e ff ic ie n ts f o r these te s ts were calcu lated by a Kuder-

Richardson form ula discussed by Harry A, Greene e t a l . , in Measurement

and E valua tion in the Secondary School (13:74). The c o e ff ic ie n ts

c a lcu la ted were a l l s u f f ic ie n tly high to show reasonable r e l i a b i l i t y

(see Table 6 ) . The formula used provides a "footnote c o e ff ic ie n t" -

which may underestim ate, bu t never overestim ates, the r e l i a b i l i t y coef­

f i c ie n t .

Because of i t s s im p lic ity and because i t fu rn ishes a r e s u l t of s u f f ic ie n t accuracy f o r many purposes, t h i s method i s recommended fo r use by teachers in estim ating the r e l i a b i l i t y of th e i r informal ob­je c tiv e exam inations.

52

TABLE 6

RELIABILITY COEFFICIENTS FOR DIAGNOSTIC AND ACHIEVEMENT TESTS

D iagnostic Test Achievement Test

Unit I

E'­en• .95

Unit I I .94 • 90 .

Unit I I I .89 .92

T his completes th e ch ap ter on th e organization of the study.

Chapter IV presents the methods of c o lle c tin g and analyzing the data

and th e re s u lts of th is a n a ly s is .

CHAPTER IV

DATA AlfD ANALYSIS OP DATA

S eparation of Schools

The common fac to rs of s im ila r textbooks, u n i ts , and evaluative

devices used in the two schools and discussed in Chapter V were no t

considered s u f f ic ie n t grounds fo r an an a ly sis o f combined d a ta . Two

teachers were involved. T heir knowledge of su b jec t m a tte r, th e i r

teaching methods, and th e ir lab o ra to ry f a c i l i t i e s and equipment were in

a l l p ro b ab ility n o t equivalent. The d a ta from the th ree c lasses in

each school d i s t r i c t were tr e a te d sep a ra te ly in the an a ly s is . That i s ,

the data are rep o rted , recorded, and tre a te d as r e s u l t s of two separate

experiments w ith mathematics la b o ra to r ie s . No attem pt i s made to draw

s ta t i s t i c a l comparisons across schoo ls, Mercer d a ta are reported and

analyzed f i r s t .

Control V ariab les—Mercer

There were two sets o f c o n tro l v a r ia b le s , the s e t of in te llig en ce

quotients and th e se t of d ia g n o s tic t e s t scores fo r each c la s s . Table

7 reports the means of these v a r ia b le s f o r each c la ss fo r each u n i t of

work a t Mercer,

53

54

TABLE 7

CONTROL VARIABLE MEANS BY CLASS AND UNIT AT MERCER

C lass IQUnit I

Diagnostic TestUnit I I

Diagnostic T estU nit I I I

D iagnostic T est

1 111.27 66.73 86.23 66 .42

2 107.56 65.04 81.30 73. 4S

3 96.63 36.67 38.37 52 .44

I t i s in te re s tin g to note th a t the t e s t means are ranked i n th e

same order as the IQ means except fo r Unit I I I where class 2 h a s th e

h ig h e s t mean sco re . Further discussion of t h i s f a c t appears i n th e

a n a ly s is of d a ta .

Dependent Variable—-Mercer

The v ariab le being te s ted fo r effect upon achievement and m o tiv a ­

t i o n i s the type o f laboratory treatm ent. Table 4 in Chapter 1X1 p re ­

se n te d the lab o ra to ry assignments fo r the se v e ra l classes a t M ero e r fo r

th e three u n its o f study. R ecall tha t a r o ta t io n of labora to ry A ssign ­

m ents a t the conclusion of each u n it made i t p o ss ib le for each c l a s s to

experience each laboratory type once during th e study. A com parison of

th e laboratory types may be found in Chapter I I I .

55

Independent Variable—Mercer

The percentage scores of the achievement te s ts fo r each u n i t w ere

v i t a l to the an a ly s is . Table 8 reports the class means fo r these

sco res, although more important c r i te r ia fo r the analysis of data a re

the adjusted mean scores of the achievement te s ts in Table 9. Table 10

records the same s e t o f adjusted mean scores as Table 9» but maps u n i t s

in to laboratory type assignments. The reasons fo r and th e methods o f

moving these adjustments in the achievement scores are now d iscu ssed .

TABLE 8

MEAN ACHIEVEMENT TEST SCORES FOR EACH CLASS BY UNITS AT MERCER

Class U nit I Unit I I U n it I I I

1 75.38 70.69 7 8 .77

2 73.63 68.22 69 .41

5 52.59 60.00 63 .00

TABLE 9

ADJUSTED MEAN ACHIEVEMENT TEST SCORES FOR EACH CLASS BY UNITS AT MERCER

Class U nit I Unit I I U n it I I I

1 68.0 65.6 7 7 .1

2 70.3 65.0 6 4 .3

56

TABLE 10ADJUSTED MEAN ACHIEVEMENT TEST SCONES EON

EACH LABORATORY TYPE AT MERCER

lab Type U nit I • Unit I I Unit I I I

Pre-lab 68.0 65.0 69.7

Post-lab 70.3 68.1 77.1

No-lab 61.2 65.6 74.3

Analysis o f Covariance--M ercer

Three E s t a t i s t i c s , one fo r each of the th ree s e ts of v a riab le s

in a u n i t , were obtained by running the scores fo r each of the e igh ty

students through two consecutive computer programs on the IBM 360/40

computer a t Slippery Rock State C o llege , Pennsylvania (6 ). The f i r s t

program w hich ac ts upon the co n tro l v a riab les and th e dependent v a ria b le

is a m u lt iv a r ia te ana ly sis of v a ria n ce ca lled MANOVA. The output from

th is program becomes the input f o r a MANOVA with covaria tes ca lled

COVAR. The output of COVAR i s th e E s t a t i s t i c which re la te s the pos­

sib ly s ig n i f ic a n t e ffe c t of the la b o ra to ry treatm ents on the achieve­

ment s c o re s fo r each u n i t a f te r an attem pt i s made to p a r t ia l out the

e ffec t o f th e two contro l v a r ia b le s . Tables 11, 12, and 13 rep o rt the

re su lts o f COVAR fo r the Mercer c la s s e s .

57

TABLE 11

analysis op covariance for achievement differences amongTHREE EXPERIMENTAL CLASSES WITH ASSIGNED LABORATORY

TYPES CONTROLLING ON PRIOR MATHEMATICS ACHIEVEMENT AND INTELLIGENCE ON MERCER UNIT I

Source o f Variation d f Sum of Squares Mean Square f

Between 2 594.04 297.02

W ithin 75 15493.50 206.58 1.44

T o ta l 77

_ TABLE 12

ANALYSIS OP COVARIANCE FOR ACHIEVEMENT DIFFERENCES AMONG THREE EXPERIMENTAL CLASSES WITH ASSIGNED LABORATORY

TYPES CONTROLLING ON PRIOR MATHEMATICS ACHIEVEMENT AND INTELLIGENCE ON MERCER UNIT I I

Source o f Variation d f Sum of Squares Mean Square f

Between 2 38.84 19.420.24

W ithin 75 5993.25 79.91

T ota l 77

5 3

TABLE 13

ANALYSIS OP COVARIANCE POR ACHIEVEMENT DIFFERENCES AMONG THREE EXPERIMENTAL CLASSES WITH ASSIGNED LABORATORY

TYPES CONTROLLING ON PRIOR MATHEMATICS ACHIEVEMENT AND INTELLIGENCE ON MERCER UNIT I I I

Source of V ariation df Sum of Squares Mean Square f

Between 2 1940.88 970.441 0 .1 9

W ithin 75 7145.00 95.26

T otal 77

Por the degrees of freedom ind icated , based on. the number of

treatm ents minus one and the number of students in the sample minus

f iv e , an P - r a t io of a t le a s t 3*12 i s required befo re sign ifican t d i f ­

ferences in achievement due to laboratory treatm ent i s indicated beyond,

the 0.05 le v e l . An P -ra tio o f a t le a s t 4*90 in d ic a te s significance b e­

yond the 0.01 le v e l (29*402).

In a d d itio n to the P -scores, COVAR p r in ts ou t the adjusted mean

achievement scores fo r each c lass in every u n i t . These were recorded

in two ways in Tables 9 and 10.

The P - ra t io s of 1.44 and 0.24 fo r Unit I and U nit I I re sp e c tiv e ly

a t Mercer in d ic a te th a t in n e ith e r case i s the re ju s t if ic a t io n fo r r e ­

je c tin g the n u l l hypothesis. However, the P -score o f 10.19 fo r U n it

I I I shows a s ig n if ic a n t e f fe c t upon the achievement scores due to th e

types o f lab o ra to ry assignm ents. This s ig n ifican ce i s beyond the 0 .0 1

le v e l.

59

W ith th e assumption t h a t COVAR, as i t was applied to the avail­

able d a ta , was able to e q u a l iz e the control variab les, a comparison of

the a d ju s te d mean achievement scores w ill s im ila rly compare the e ffec ts

of the la b o ra to ry treatm ents* No s ig n ific an t differences a re reported

fo r U nit I o r Unit XX. P o r U n it I II in M ercer the p o s t- la b ranks

h ighest, follow ed by p re - la b and no-lab in th a t order* I t i s in te rs

esting to n o te that though t h e differences o f adjusted means were not

s ig n if ic a n t fo r Unit I o r U n i t I I they d id share the same d irec tio n as

those f o r U n it I I I . In a l l c a se s , the c la s s with the p o s t- la b assign­

ment c o n s is te n tly had the h ig h e s t mean sc o re . Because o f th e ro tation

of la b o ra to ry assignments e v e ry class had i t s highest achievement when

assigned th e post-lab approach* The tem ptation is to a r r iv e h as tily a t

the co n c lu sio n that the p o s t - l a b approach y ie ld s superior achievement

fo r s tu d e n ts of a l l a b i l i ty le v e ls . However, only the U n it I I I F-score

i s s ig n i f ic a n t ly large. S in c e only high a b i l i ty students received the

p o st- lab treatm ent for U n it I I I , we can conclude no more th an that

p o st- lab was superior fo r h i g h ab ility s tu d en ts for the to p ic s in­

cluded i n U n it I I I .

To fu r th e r te s t th is conclusion and to possibly determ ine other

s ig n if ic a n t differences i n achievement due to treatm ents, a t - t e s t was

run on th e adjusted mean s c o r e s to make a pair-w ise comparison of the

trea tm en ts f o r Unit I I I . The resu lts of th e se te s ts are shown in Table

14.

►

60

TABLE 14

t-SCOHES PROM PAIR-WISE COMPARISONS OP TREATMENTS POR UNIT I I I AT MERGER

Pairs df Required t- s c o re * t-s c o re Superior Treatment

Pre-Post 51 2.68 2.80 Post

Pre-No 51 2.68 2.07 —

Post-No 52 2.66 15.35 Post

*Required fo r sig n ifican ce a t the 0 ,01 le v e l; 2.01 i s required a t the 0.05 l e v e l .

This ta b le clearly shows th a t there a re s ig n if ic a n t d ifferences

in achievement f o r the p o s t- la b , p re-lab comparison and esp ec ia lly fo r

the p o st-lab , no -lab comparison. Post-lab s tu d en ts achieved s ig ­

n if ic a n tly h ig h e r on the average than e i th e r o f the o ther two classes.

The t-sco re o f 2.07 fo r the p r e - la b , no-lab p a ir in g was not su ff ic ie n tly

high to w arran t any decision ab o u t superior labo ra to ry treatm ents a t

the 0.01 le v e l , but i t did in d ic a te s ig n ific an ce beyond the 0.05 lev e l

in favor o f th e pre-lab .

Another Analysis—Mercer

Prom an o th er point of v iew , u t i l iz in g standard deviations and

standard sc o re s , the to ta l e f f e c t of the lab o ra to ry types across a l l

three u n i ts may be observed. Yftthin each u n i t the standard deviation

fo r the d is t r ib u t io n of ad ju s ted achievement scores fo r the pooled

sample i s p a r t o f the COVAR p r i n t out. The d iffe rences between each

61

c la ss ad justed mean and the adjusted mean f o r the e n tire sample i s

e a s ily ca lcu la ted . Dividing these d iffe ren ces by the standard devia­

t io n fo r the B am p le y ie ld s the v aria tio n s of the c lass ad ju s ted means

from the sample adjusted means in te rns of standard sco res . Table 15

tab u la tes th i s information f o r the Mercer sample.

TABLE 15

VARIATION OP THE ADJUSTED MEAN ACHIEVEMENT SCORES PROM THE ADJUSTED MEAN OP THE POOLED SAMPLE IN

TERMS OP STANDARD SCORES AT MERCER

Lab-Type

Unit I V ariation Class

Unit I I V ariation Class

U nit I I I V ariation Class

Sum of V aria tions

P re-Lab 0.11 1 -0 .14 2 -0 .06 3 -0 .09

P o st-Lab 0.27 . 2 0.22 3 0.71 1 1.20

o’ 1

-0 .30 3 -0 .07 1 -0 .63 2 -1 .00

T otals 0.08 0.01 0.02 0.11

In a l l cases, the r e la t iv e e ffe c ts o f the three la b o ra to ry t r e a t ­

ments across a l l three u n i ts may be d iscerned by comparing the sums of

the resp ec tiv e v aria tio n s.

Por the Mercer sample the expected occurs. The p o s t- la b t r e a t ­

ment has a la rg e r v a ria tio n sum across the th ree u n its th an e i th e r the

p re -lab o r no-lab treatm ents. The p re -lab sum ranks second and the no­

la b sum i s sm allest. Regardless of the u n i t studied o r th e general

62

a b i l i ty le v e l o f the claases talcing the post-lab experiments, the ad­

ju s te d mean sco re s fo r the p o s t- la b treatment a re g re a te r than those

f o r the pooled sample mean in every case. For U n it I I I , where the

COVAR F -ra tio in d ica ted s ig n if ic a n t differences i n achievement, th e

mean v a ria tio n f o r the p o st-lab i s considerably l a r g e r than the sample

mean*

Table 16 rep o rts the f a c ts and figures of T ab le 15, but maps

c la sse s onto lab o ra to ry trea tm ents a t Mercer, I n t h i s table one can

view the sums o f v a ria tio n s by c lasse s across th e th re e treatments.

For the e n tire study , the h ig h es t a b il i ty class ra n k s f i r s t , the lo w e s t

a b i l i ty c lass ranks second, and the middle a b i l i t y c la ss ranks th i r d .

The comparatively poor showing f o r Class 2 during th e no-lab treatm ent

seems to account fo r the drop to th ird place in t o t a l a ffect. The

read e r i s reminded th a t Class 2 had the highest mean diagnostic sco re

f o r Unit I I I , b u t th e ir achievement was far below expectation.

TABLE 16VARIATION OF THE ADJUSTED MEAN ACHIEVEMENT SCORES

FROM THE ADJUSTED MEAN OF THE FOOLED SAMPLE MAPPING MERGER CLASSES ONTO TREATMENTS

C lass Pre-Lab Post-Lab No-Lab T o ta ls

1 0.11 0.71 - 0 .0 7 0.75

2 -0 .1 4 0.27 - 0 .6 3 -0 .5 0

3 -0 .0 6 0.22 - 0 .3 0 -0 .1 4

T o ta ls -0 .0 9 1.20 - 1 .0 0 0 .1 1

Mercer Q uestionnaire Analysis

The qu estio n s end the percentages of s tu d en ts in each c la ss who

checked corresponding responses a re l is te d in Table 17* The f i r s t s tu ­

dent opinion expressed on the questionnaire in d ic a ted th e ir p references

fo r p re-lab , p o s t- la b , and n o -lab in th a t order. E'er a l l th ree c la sse s

a t Mercer th e re were no s ig n if ic a n t d ifferences i n the percentages o f

students s e le c tin g p re-lab o r p o s t- la b treatm ents. In each of th e two

upper a b i l i ty c la sse s whose IQ means are more n e a r ly the same, only one

student chose n o -lab as the p re fe rre d treatm ent. Such was not th e case

fo r the lower a b i l i t y c la ss . Approximately o n e - th ird of the s tu d en ts

in th is c la s s , w ith mean IQ le s s than 100, chose no -lab as th e i r favor­

i t e .type.

Responses f o r item two in d ic a te th a t more th a n two-thirds o f the

students in th e upper a b i l i ty c la sse s found mathematics la b o ra to rie s

made th is school year more in te re s t in g . The same student in each c lay s ,

who se lec ted n o -lab in item one, found the year more boring, and a few

studen ts could sense l i t t l e d iffe ren ce due to th e changed approach.

The responses to th is item fo r th e lower a b i l i ty c la s s are n early

evenly div ided , w ith a s lig h t margin favoring th e 'foiore in te re s tin g "

choice.

The medium c la ss responded to item three a s one might p re d ic t in

l ig h t of th e i r responses to item s one and two. However, a t l e a s t two

studen ts from th e high c lass p re fe rre d some la b to no-lab and were n o t

bored by the approaches, but suggest no-labs a t a l l fo r next y e a r 's

e igh th g rad ers . S ix ty per cen t o f the low c lass suggest no-labs a t

64

TABLE 17

EESPONSES TO THE STULENT INTEREST QUESTIONNAIRE BY PERCENTAGES POR EACH CLASS AT MERCER

ClassesHigh Me d im Low

111.3* 107.6 96.6

1 . The kind of la b I l ik e b es t wasa} before c lass d iscussion o f the to p ic bJ a f te r c la ss d iscussion of th e to p ic c) no lab a t a l l

2. Mathematics lab s have made th is year compared with l a s t yeara) more in te re s tin gb) more boringc) no d if fe re n t

3. I suggest th a t next y e a r 's e ig h th gradersa) continue having a t l e a s t the same

number of lab s as we hadb) have more la b s than we hadc) have no la b s a t a l l

4* These lab experiments on the wholea) helped me to understand mathematics

b e tte rb) confused me more than they helped mec) made no d iffe rence in my a b i l i ty to

understand mathematics

5. When my teacher announced th a t we were going to have another labo ra to ry experimenta} I looked forward to i tbJ I sa id to m yself, "So w hat."c) I sa id to myself, 'Wo, no ag a in ."

52 46 3044 50 37

4 4 33

69 82 414 4 26

27 14 33

44 30 1644 66 2412 4 60

52 82 608 4 21

40 14 19

69 72 4551 21 33

0 7 22

*Mean IQ to n ea re s t ten th

65

a l l . This change also in d ica te s an hypothetical a t titu d e of "Well, i t

was a l l r ig h t , but don 't do i t ag a in ."

B e tte r than one-half o f th e students in e a ch class a t Mercer f e l t

an Increase in understanding concepts resu lted from laboratory e x p e ri­

ments. S ixty per cent fo r the low class i s uncommonly high in l i g h t of

th e ir responses to the f i r s t th re e items. The f a c t th a t about o n e - f i f th

of the students in th is c lass were confused i s co n s is te n t with the p re ­

viously recorded a ttitu d e s in item s one and two, but inconsistent w ith

th e ir responses to item th ree , looking ahead to item five, one can see

th a t once again about o n e -f if th o f the lower c l a s s reg istered d isg u s t

w ith the e n tire procedure. Nearly h a lf of t h i s c la s s looked forward to

experiments possibly because they f e l t i t a ided t h e i r understanding o f

mathematics. This i s e s se n tia lly the same group who found mathematics

la b o ra to rie s more in te re s tin g (item 2). C le a r ly , the percentages o f

responses to items two and f iv e a re consisten t f o r a l l three c la s se s a t

th is school.

In genera l, the responses to a l l items r e f le c t in g p ositive i n t e r ­

e s t and a sense of increased understanding a re encouraging. However,

the personal reasons fo r these responses are n o t to ta lly known. Two

contrary statem ents w ritten by students who fav o red laboratories

emphasize the complexity o f analyzing the resp o n ses to th is q uestion ­

na ire ; " I 'd l ik e to have more o f them because th e y made me understand

math a l o t b e t t e r , " and "They go t us out o f p a r t o f the olass—they

were o .k ."

66

The studen t evaluation forms (see Appendix D), which were sup­

posed to be checked by studen ts and co llec ted by the teachers following

each experim ent, were no t used with enough re g u la rity to w arran t much

d iscu ss io n . Any attempt to draw conclusions from the sm all number of

ev a lu a tio n s and comments received could no t be ju s t i f ie d .

Before any conclusions based on th is study a t Mercer a re pre­

m ature ly s ta te d , an examination of the data from the lakeview school

i s i n o rder. Bata, completely analogous to th a t from Mercer, were col­

le c te d and recorded. The content of each u n it fo r the two schools was

b a s ic a lly the same throughout the study.

Control V ariables—Lakeview

Id e n tic a l d iagnostic te s ts were adm inistered to the Mercer and

th e Lakeview students during the study. Table 18 rep o rts th e means f o r

the s e t of In te llig en ce quo tien ts and the s e t of d iag n o stic t e s t scores

f o r each c la ss per u n it a t Lakeview.

TABLE 18• CONTROL VARIABLE MEANS BY CLASS AND UNIT

AT LAKEVIEW

C lass IQU nit I

D iagnostic TestU nit I I

D iagnostic TestU nit I I I

D iagnostic Test

1 111.89 68.22 72.93 72.19

2 102.44 54.39 45.44 51.61

3 102.95 54.38 48.14 63.71

67

. C lass 2 and Class 3 were ranked by the Lakeview teachers as me­

dium and low a b i l i ty c la s se s re sp e c tiv e ly . Throughout th is rep o rt

these rankings w ill be used even though th e i r mean IQ scores are es­

s e n t ia l ly equal. Except f o r a s l ig h t v a r ia t io n in Unit I I I , the ranges

o f the means of the d iag n o stic scores fo r these two c lasse s are re la ­

tiv e ly sm all. Class 1 c o n s is te n tly scored h igher on the diagnostic

t e s t s .

Dependent V ariab le—Lakeview

Table 6 in Chapter I I I shows the o rd e r o f labo ra to ry assignments

and th e i r ro ta tio n s during the s tu d y . The correspondence between types

of lab o ra to ry approaches i n i t i a l l y assigned fo r Unit I and the a b i l i ty

le v e ls o f the c lasse s d i f f e r s from th a t o f Mercer by one ro ta tio n .

This was not a r e s u l t o f design sin ce assignm ents were randomly made by

the resp ec tiv e te a ch e rs . The p o in t i s , no attem pt was made to dupli­

ca te th e treatm ent assignm ents a t the two schools.

Independent V ariable—Lakeview

Table 19 ta b u la te s the c la s s means f o r achievement t e s t scores

fo r each u n it a t Lakeview. The more rev ea lin g ad justed mean achieve­

ment scores appear in Tables 20 and 21. These ta b le s contain the same

f ig u re s , but Table 20 maps c la s se s in to u n i ts w hile Table 21 maps

labo ra to ry types in to u n i t s .

68

TABLE 19

MEAN ACHIEVEMENT TEST SCORES POE EACH CLASS BY UNITS AT LAKEVIEW

Class U nit I Unit I I Unit I I I

1 70.56 90.22 70.19

2 41.39 45.67 48.06

3 39.29 64.43 56.67

TABLE 20

ADJUSTED MEAN ACHIEVEMENT TEST SCORES POE EACH CLASS BY UNITS AT LAKEVIEW

Class Unit I U nit I I Unit I I I

1 61.5 87*1 62.9

2 47.8 48.1 55.8

3 45.5 66.4 59.3

69

TABLE 21

ADJUSTED MEAN ACHIEVEMENT TEST SCORES FOR EACH LABORATORY TYPE AT LAKE V LEM

Lab-Type Unit I U nit I I U nit I I I

Pre-Lab 45.5 8 7 .1 55.8

Post-Lab 61.5 4 8 .1 59.3

No-Lab 47.8 66 .4 62.9

Analysis o f Covariance—Lakeview

MANOVA and COVAR were consecutively ru n on the Lakeview data.

The number of students in th i s sample was 66 . Thus, the degrees of

freedom fo r w ith in v a r ia tio n was 61. An F - r a t i o of a t l e a s t 3.14 in ­

d ic a te s sign ificance of treatm ent beyond the 0 .0 5 le v e l w hile an F-

r a t lo o f 4*95 o r g rea te r means sig n ifican t d if fe re n c e s in achievement

are a t tr ib u ta b le to lab o ra to ry treatments a t th e 0.01 le v e l of prob­

a b i l i ty (24:402). COVAR r e s u l t s fo r Lakeview were p rin ted in Tables

22, 23* and 24.

70

TABLE 22ANALYSIS OP COVARIANCE POR ACHIEVEMENT DIFFERENCES AMONG

THREE EXPERIMENTAL CLASSES WITH ASSIGNED LABORATORY TYPES CONTROLLING ON PRIOR MATHEMATICS ACHIEVEMENT

AND INTELLIGENCE AT LAKEVIEW UNIT I

Source of V aria tio n df Sum of Squares Mean Square f

Between 2 2711.54 1355.776.89

Within 61 12007.85 196.85

Total 63

TABLE 23ANALYSIS OP COVARIANCE POR ACHIEVEMENT DIFFERENCES AMONG

THREE EXPERIMENTAL CLASSES WITH ASSIGNED LABORATORY TYPES CONTROLLING ON PRIOR MATHEMATICS ACHIEVEMENT

AND INTELLIGENCE AT LAKEVIEW UNIT I I

Source of V aria tion df Sum of Squares Mean Square f

Between 2 10972.32 5486.1626.57

Within 61 12594.06 206.46

Total 63

TABLE 24ANALYSIS OP COVARIANCE POR ACHIEVEMENT DIFFERENCES AMONG

THREE EXPERIMENTAL CLASSES WITH ASSIGNED LABORATORY TYPES CONTROLLING ON PRIOR MATHEMATICS ACHIEVEMENT

AND INTELLIGENCE AT LAKEVIEvV UNIT I I I

Source o f V aria tio n d f Sum of Squares Mean Square f

Between 2 362.66 181.330.88

W ithin 61 12520.86 205.26

Total 63

71/

For Units I and I I a t lakeview, the F -ra tio s o f 6*89 and 26.59

were both s u f f ic ie n tly la rge to r e g i s t e r sign ificance beyond the 0.01

le v e l. R eferring back to Table 21, and comparing ad justed mean

achievement sco res, i t was found th a t f o r Unit I the post-lab treatm ent

was superior. Since only high a b i l i t y studentB received the post-lab

treatm ent fo r U nit I , we can only conclude th a t p o st- lab was superior

fo r high a b i l i ty studen ts fo r the to p ic s included in Unit I .

An in spec tion o f the ad justed means fo r Unit I I a t lakeview shows

a d iffe re n t p a t te rn . As a m atter o f f a c t , the p o st- lab has the

sm allest ad justed mean among the th re e types. The p re -lab c lass had an

extremely high ad ju sted mean of 87*1 and the no-lab studen ts ranked

second w ith an ad ju sted mean of 6 6 .4 . Only high a b i l i ty students re ­

ceived the p re -lab treatm ent fo r U n it I I , and so we can only conclude

th a t p re-lab was superio r fo r h igh a b i l i t y students fo r the top ics in ­

cluded in Unit I I .

The F -score fo r Unit I I I i s so low th a t no comparison of the

equal adjusted means i s warranted.

A pair-w ise comparison of th e ad ju sted mean achievement scores

fo r each treatm ent f o r Unit I and U n it I I presents a c le a r view of the

treatm ent re la tio n s h ip s . The t - s c o re s re su ltin g from these comparisons

are presented in Tables 25 and 26.

72

TAHOE 25

t-SCORES PROM PAIR-Y/ISE COMPARISONS OP TREATMENTS POR UNIT I AT LAKEVIEW

P airs df Required t-sco re* t - s c o r e Superior Treatment

P re-P ost 46 2 .40 3 .7 4 Post

Pre-No 37 2.43 0 . 0 5 ' —

Post-No 43 2.42 3 .2 6 Post

*Required t-score f o r sign ificance a t the 0.01 le v e l .

TABIE 26

t-SCOEES PROM PAIR-Y/ISE COMPARISONS OP TREATMENTS POR UNIT I I AT LAKEVIEW

Pairs df Required t-sco re* t - s c o r e Superior Treatment

Pre-Post 43 2 .4 2 9 .0 7 Pre

Pre-No 46 2 .40 5 .0 1 Pre

Post-No 37 2 .43 4 .0 4 No

^Required t-score f o r sign ificance a t the 0*01 le v e l .

The superio rity of th e post-lab tre a tm e n t over bo th p re-lab and

no -lab fo r Unit I is q u ite ev ident from, th e respective t- s c o re s of 3*74

and 3*26* Class 3 vdth a mean IQ of 102*93 end a d iagnostic te s t mean

o f 54*38 was assigned p re -lab * Class 2 w ith a mean IQ o f 102*44 and a

diagnostic t e s t mean of 54*39 was assigned no-lab . The equivalent

73

co n tro l variab les and the n ea rly equivalent ad justed mean achievement

sco re s of 45#5 and 47.8 in d ica te no s ig n ific an t d iffe rence between pre-

l a h and no-lab treatm ents fo r Unit I . A t-sc o re of 0.05 v e r i f i e s th is

conclusion.

For Unit I I both the p re -lab class and the no-lab c la ss achieved

more s a tis fa c to r i ly than the p o st-lab studen ts. The resp ec tiv e t -

sco re s for these comparisons o f 9.07 and 4*04 are s u f f ic ie n tly h igh to

in d ic a te sign ificance beyond the 0.01 le v e l. I t i s worthwhile to note

t h a t pre-lab stu d en ts achieved s ig n ific an tly b e t te r than no-lab s tu ­

d en ts as in d ica ted by a t-sco re of 5.01. Thus, even though p o s t- la b

treatm ent appeared le a s t e ffec tiv e fo r Unit I I , p re -lab v/as more

b e n e fic ia l than no-lab fo r th i s u n i t . Again, the high a b i l i ty c la ss

was assigned the laboratory treatm ent which proved most e f fe c t iv e fo r

U n it I I .I

Another A nalysis—Lakeview

Tables 27 and 28 fo r Lakeview are analagous to Tables 15 and 16

f o r Mercer. The sums of the v a ria tio n s of ad justed means fo r each

trea tm en t from th e adjusted mean fo r the pooled sample are found across

a l l three u n its i n Table 27.

74

•TABUS 27

VARIATION OP THE ADJUSTED MEAN ACHIEVEMENT SCORES PROM THE ADJUSTED MEAN OP THE POOLED SAMPLE IN TERMS OP

STANDARD SCORES AT LAKEVIEW

U nit I Unit I I Unit I I I Sum ofType V ariation Class V ariation Class Variation Class V ariations

Pre-Lab -0 .52 3 1.20 1 -0.28 2 0.40

P ost-Lab 0.64 1 -1 .54 2 -0.03 - 3 -0.94

No-Lab -0 .36 2 -0 .24 3 0.22 1 -0.38

T otals -0 .24 -0 .58 —0.10 -0.92

According to the sums of v a ria tio n s in th is ta b le , th e ranking of

the labo ra to ry types in descending o rder of contributing e f f e c t on

achievement was p re - la b , no -lab , and p o s t- la b . Even though the post­

lab ra te d h ighest in U nit I and nearly equal to the average o f the

o thers in Unit I I I , th i s treatm ent su ffe red a p a rtic u la r ly hard setback

during Unit I I . One can only speculate on the reasons f o r th is unex­

pected decline in performance (see Chapter V).

Table 28, mapping th e varia tio n s by classes onto trea tm ents, in ­

d ica ted th a t the h ig h est a b i l i ty c lass performed best reg a rd le ss of the

labo ra to ry type assigned . Class 3 ra te d second in o v e ra ll achievement,

and Class 2 was l a s t i n th e ranking. C lass 2 had a low v a r ia tio n score

of -1 .54 fo r p o st-lab treatm ent which la rg e ly accounts f o r the low

75

o v e ra ll showing of th e s e students and fo r the negative sum of varia ­

tio n s fo r post-lab a c ro s s u n its .

TABLE 28

valuation of the adjusted mean achievement scores promTHE ADJUSTED MEAN OP THE POOLED SAMPLE MAPPING

LAKEVI E.7 CLASSES ONTO treatments

Class Pre-Lab Post-Lab No-Lab Totals

1 1.20 0.64 0.22 2.06

2 -0.28 -1 .54 -0 .36 -2.18

3 -0.52 -0.04 -0 .2 4 -0.80

Totals 0.40 -0 .94 -0 .3 8 -0.92

Lakeview Questionnaire Analysis

Table 29 p resen ts th e re s u lts of the in te re s t questionnaire fo r

the Lakeview students. The responses f o r Classes 2 and 3 with compar­

able IQ scores are s im i la r fo r a l l questions with C lass 3 reg is te rin g

s l ig h t ly higher on an a t t i t u d e sca le . A considerable number of s tu ­

dents from these c la s s e s were bored by the laboratory experiments (item

2) , and perhaps th is i s why they p re fe rred no-lab (item l ) . These same

studen ts recommended n o - la b s fo r next y e a rfs c lasses (item 3) perhaps

due to th e i r general f e e l i n g of d isgust fo r labora to ry experiments

(item 5)* Confusion seemed, to play a major ro le fo r th e o n e-fifth of

Class 2 with this r e l a t i v e l y negative reac tio n to the program (item 4 ) .

76

TABES 29

RESPONSES TO THE STUDENT INTEREST QUESTIONNAIRE BY PERCENTAGES POR EACH CLASS AT LAKEVIEtY

High111.9*

ClassesMedium102.4

Low103.0

1 . The k ind of lab I lik e b e s t wasbefore class d iscussion o f the top ic 74 19 40

b) a f t e r c lass discussion o f the top ic 23 38 44c) no la b a t a l l 3 43 16

2. Mathematics labs have made th i s year compared with la s t year, a) more in te re s tin g 80 46 36b) more boring 3 16 28c) no d iffe re n t 17 38 36

3* I suggest th a t next year*s e igh th graders a) continue having a t l e a s t the same number

of la b s as we had 51 34 52bl have more labs than we had 46 45 28c) have no labs a t a l l 3 21 20

4. These la b experiments on th e whole a) helped me to understand mathematics

b e t te r 71 65 64b) confused me more than th ey helped me 0 19 8c) made no difference i n my a b i l i ty to

understand mathematics 29 16 28

5. Y/hen my teacher announced th a t we were going to have another laborato ry experiment a) I looked forward to i t 51 50 28bJ I sa id to myself, "So w h at." 40 34 52c) I sa id to myself, '*No, n o t again ." 9 16 20

*Mean IQ to nearest te n th .

77

Perhaps a more p o sitiv e approach to the percentages in Sable 29

w ill be more revealing and encouraging. For Class 1, l i t t l e needs to

be said} the record speaks fo r i t s e l f . Seventy-four p e r cent of Class

1 chose pre-lab f i r s t , and th is c lass had th e ir h ig h est mean achieve­

ment scores with p re -lab , Unit I I . Even though Class 2 perfonaed so

poorly Y/ith post-lab treatm ent, 38 per cent of them p re fe rred th is type

of approach. Prom se lec tio n s (a) and (b) of item th ree we see th a t

n early 80 per cent of th is c lass recommends a t le a s t as many experi­

ments fo r next y e a r 's c lasses. S ix ty-five per cent were o f the opinion

th a t th e ir understanding of mathematics was improved by th e laboratory

experiments (item 4 ), and h a lf of th is group looked forward to the next

laboratory experience during the study (item 3).

A summary of the study and fu rth e r in te rp re ta tio n s of the data

are included in Chapter V.

CHAPTER V

SUMMARY AND INTERPRETATIONS

The Problem

(The schoo l year which the academically m aturing , jun io r h ig h

school s tu d e n ts spend in th e eighth grade i s a very important y e a r .

Their ex p erien ces during t h i s year d irec tly in f lu e n c e the d e c is io n s fo r

fu tu re endeavors which th ey are called upon t o make before e n te r in g the

n in th g rad e . S tudents i n t h i s age group a r e s u b je c t to s p li t- s e c o n d

decisions and spontaneous ac tio n s largely b a s e d on recent ex p e rie n ce s .

I f t h e i r d e c is io n s fo r th e fu tu re are to l e a d s a t is fa c to r i ly to m ental

and s o c ia l p ro d u c tiv ity , th e se students m ust b e effectively ch a llen g ed ,

m otivated, and taugh t by t h e i r eighth grade te a c h e r s . The te a c h e rs

must be p ro p e rly prepared f o r th is task, and t h e i r search fo r im proved

pedagagical techniques must never cease.

In th e i r , search f o r b e t te r methods, t h e mathematics te a c h e rs o f

th ree ju n io r - s e n io r high schools in Mercer C o u n ty , Pennsylvania de­

cided during a seminar to weigh the merits o f mathematics la b o ra to r ie s .

Two e ig h th grade teachers i n particu lar from d i s t i n c t school d i s t r i c t s

were in te r e s te d enough in th e further study o f th e effectiveness o f

mathematics la b o ra to r ie s to volunteer th e i r s e r v ic e s . The a u th o r , who

was in v i te d to p resid e a t th e seminar, and t h e tv/o volunteers proceeded

to e s ta b l i s h a program o f in q u iry and ex p erim en ta tio n .

78

79

Belated l i t e r a tu r e

Before say a ttem p ts were made to Include the use of mathematics

la b o ra to r ie s in th e i r classrooms, some p ertin en t l i t e r a tu r e was re ­

viewed. The perusual o f available l i t e r a tu r e provided u se fu l guide­

l in e s f o r the implementation of labora to ry experiments in mathematics

classroom s.

I t was discovered th a t most o f the published m ateria l p erta in in g

to th e sub jec t was d i r e c te d toward teachers and students in the*ele­

mentary grades. However, comprehension of the ab s tra c t n a tu re of

mathem atics i s often d i f f i c u l t fo r o lder students a lso . Mathematical

models o f a l l kinds may provide an avenue of re tu rn from th e concrete

to th e a b s tra c t.

Some equipment f o r mathematics lab o ra to rie s has been commerciallyt

prepared and tested . Examples of th i s type are desk-top computors,

lo g ic puzz les, games, geoboards, and Cuisenaire rods. G enerally the

b ridges from the co n cre te to the a b s tra c t have been constructed and

tra v e rse d with some d eg ree of success using these and s im ila r models.

The equipment which o th e r teachers have successfu lly used to motivate

s tu d en ts to actively p a r t ic ip a te in the process of learn ing mathe­

m a tic a l concepts c o n s is ts simply o f those physical items which are

e i th e r a lready in o r a r e brought by the students in to the ordinary

classroom .

Those teachers who have been able to ask the proper questions of

th e i r s tu d en ts lead ing them to discover basic concepts, reg a rd le ss of

the type o f equipment employed, teachers who have allowed th e students

to use th e i r ingenuity to expand and extend these i n i t i a l id e a s , have

80

indicated th a t labo ra to ry techniques do enhance th e learning process a t

the elementary school level*

A few teachers o f ju n io r h igh school ch ild re n have attem pted to

te s t the e ffec tiv en ess of mathematics la b o ra to r ie s . She r e s u l t s of

these informal evaluations are s im ila r to those involving elementary

students. Thus, the conclusion th a t properly used laboratory tech­

niques w ill probably produce p o s itiv e r e s u lts i s supported by nearly

a l l of the reviewed, p e rtin en t l i t e r a tu r e . But th i s l i t e r a tu r e p r i­

marily rep o rts inform al s tu d ies .

I t was also discovered th a t the id ea of u s in g laboratory methods

in a mathematics c la ss i s n e ith e r r e s t r ic te d to schools in the United

S tates nor i s i t a recen tly developed id ea . European textbooks, some

w ritten over 200 years ago, include adaptable id e a s fo r contemporary

experiments. ' .

O rganization o f the Study

A fter approximately f iv e months o f search ing , planning, a t ­

tempting, and re f in in g , an organized p lan to t e s t the effec tiveness of

laboratory techniques was put in to operation by th e two e igh th grade

teachers a t the two Mercer County school d i s t r i c t s .

Three a r b i t r a r i ly chosen c la sse s from th e two eighth grades were

randomly assigned d is t in c t labo ra to ry approaches f o r a six weeks u n it

of work on f ra c tio n s , r a t io s , and p ro p o rtio n s. These classes were

homogeneously grouped according to o v e ra ll academic achievement. This

grouping i s standard procedure in both schools and n ecess ita ted th a t

81in ta c t c lasses w ith d if fe r in g in te ll ig e n c e averages be involved i n the

study .

Two more un it3 of work on geometry, each approximately s i x weeks

in len g th , were completed using lab o ra to ry techn iques. The c la s s a s­

signments to d is t in c t lab o ra to ry approaches were ro ta te d fo r e a ch u n it

o f work. Thus, every c la s s had the opportun ity to study mathem atics

fo r s ix weeks under each labora to ry approach*

The th ree lab o ra to ry approaches were c a lle d p re-lab , p o s t - la h ,

and n o -lab . For p re -lab and p o s t- lab sev e ra l s im ila r experim ents p e r

u n it were incorporated in to the c la ss se ss io n s . The labora to ry was the

classroom, and the equipment fo r the experim ents consisted of en v iro n ­

mental gadgets. The d iffe ren ces between p re - la b and post-lab w ere ac­

centuated by the types of leading qu estio n s asked by the te a c h e rs . In

p re - la b , predesigned experiments were used in the hope of g u id in g the

studen ts to conjecture about the concepts to be discussed in fo llow ing

r e c i ta t io n sessio n s. In p o s t- lab , l e s s te ach e r guidance was in vo lved

while m a te r ia ls fo r studen t innovated a p p lic a tio n s o f p rev iously d is ­

cussed concepts were sought. The teach e rs played an extremely im­

p o rtan t p a r t in making the lab o ra to ry approaches d is tin c tiv e a s they

in i t i a t e d the use of th e to o ls and m otivated th e students th rough

th e i r questioning to d iscover more id e a s and /o r ap p lica tio n s . The no­

la b approach was the standard classroom procedure o f five c la s s r e c i t a -

tio n s p e r week with no experiments invo lv ing s tu d e n t p a r t ic ip a t io n in ­

cluded.

The n u ll hypothesis th a t was te s te d s t a t i s t i c a l l y was t h a t the re

i s no s ig n if ic a n t d iffe ren ce in achievement between eighth grade 1

82

classes -taught e i th e r without labora to ry techniques o r with one o f the

two d ire c t labora to ry approaches.

S ta t i s t ic a l Analysis of Data

The data co llec ted to te s t th i s n u ll hypo thesis, fo r the 146

students in the sample, included recen tly obtained in te llig en ce quo­

t ie n ts , d iagnostic t e s t scores f o r each u n it recorded from teacher-made

te s ts adm inistered before each u n it began, and achievement te s t sc o re s

fo r each u n it derived from teacher-made te s ts adm inistered a t the con­

clusion o f each u n i t . The class averages from each o f these se ts o f

scores provided the inpu t fo r a computer ca lcu lated an a ly sis of co -

variance ca lled COVAR.

The two contro l v ariab les , which the COVAR attem pted to eq u a liz e

fo r each c lass per u n i t , were the IQ scores and th e scores on th e u n i t

d iagnostic te s t s . The independent variab le was th e laboratory ap­

proach and the dependent (c r ite r io n ) variable was th e u n it achievement

te s t sco res . An F -ra tio v/as p rin ted out fo r each u n i t of work a t each

school d i s t r i c t . Of the six re su lt in g F-scores, th r e e were of s u f ­

f ic ie n t magnitude to warrant a re je c tio n of the n u l l hypothesis beyond

the 0.01 le v e l.

For the lakeview students s ig n ific an t d iffe re n c e s in achievement

due to labora to ry treatm ent were ind icated fo r U n it I and Unit I I ,

while f o r the Mercer students the s ig n ifican t F -sco re occurred f o r

Unit I I I .

F air-w ise comparisons of the adjusted mean achievement sco res were

made fo r the c lasses i n the th ree cases with la rg e F - ra tio s . The

83

p o st-lab treatm ent was most e ffec tiv e fo r Unit XII a t Mercer and fo r

Unit I a t Lakeview. The p re -lab treatm ent was most e ffec tiv e fo r U nit

II a t Lakeview. I t i s in te re s tin g th a t the c lass w ith the p o st-lab ap­

proach f o r th is Unit I I had the lowest adjusted mean achievement score

of a l l th re e c lasses . N atu rally , th is generates a question regarding

the reaso n s fo r th is complete rev ersa l o f the achievement rankings.

The rev ersa l o f the e ffec tiveness of laboratory patterns fo r U nit

I I a t Lakeview should cau tio n us th a t changes in o ther variab les can

change th e laboratory treatm ent e f f e c t . These o ther variab les inc lude

the degree of d i f f ic u l ty o f the u n it being studied and the teacher-

student rap p o rt which d ir e c t ly in fluences a t ti tu d e s of the studen ts.

P o ss ib ly due to th e re la tiv e ease of re la tin g the basic geom etric

concepts in Unit I I to th e previously stud ied and in tu itiv e ly under­

stood geometry, and perhaps, due to more en th u s ia s tic teacher involve­

ment because a student teach er began a s s is t in g during th is u n it, the

achievement average f o r the p re-lab c la ss was considerably higher than

for e i t h e r of the o th e r two c lasses . Enthusiasm fo r a d iffe ren t sub­

jec t a r e a , which e igh th grade students u su a lly ex h ib it, may have been

u n s a tis fa c to r i ly enhanced because o f the delayed physical app lica tions

for th e p o s t- la b c la s s . By the time experiments were scheduled a pas­

sive a t t i t u d e leading to poor achievement may have re su lted . When th is

r e la t iv e ly low a b i l i ty c la s s was asked to s e le c t the laboratory type

they m ost p referred , 19 p e r cent o f them chose p re -lab , 38 per cent

se lec ted p o s t- la b , and 43 per cent o f them preferred no-lab. This gen­

e ra l disenchantment w ith experiments in mathematics i s consisten t w ith

84

the r e la t iv e ly poor showing c la ss 2 made throughout the study ( r e f e r to

Tables 20 and 28)*

limitations of COVAR f o r thiB Study

E lash o ff (10:383) c a l ls a n a ly s is o f covariance a d e l ic a te in ­

strument. Several assumptions are l i s t e d in th i s revealing a r t i c l e

which must be met before the r e s u l t s of COVAR can be accepted w ithout

re se rv a tio n s . The f i r s t of these assum ptions deals w ith random se­

le c tio n s . Covariance assumes a random s e le c tio n of studen ts f o r the

classes and random assignments of c la s se s to trea tm en ts. However,

since th is s i tu a t io n i s r a re ly p o ss ib le in educational research ,

E lashoff s ta te s th a t th i s an a ly s is may be used w ith caution i f in ta c t

classes a re used , but treatm ents a re randomly assigned . This i s the

s itu a tio n f o r th i s study as n e c e s s ita te d by the a c tu a l schoo ls ' env iron -i

ments.

There i s a p o s s ib i l i ty th a t COVAR could no t remove a l l of th e•<

bias caused by d if fe r in g a b i l i ty levelB among the th ree c la sse s a t each

school. Of course, the range of a b i l i t y le v e ls from the low est pos­

s ib le to th e h ig h est possib le in each school could have caused un­

a lte ra b le b ia s . This was n o t the ca se . The two e ig h th grade c la sse s

with h ig h est a b i l i ty a t Mercer were no t considered n o r was the h ig h e st

ranked c la ss a t lakeview. Whether o r no t the re s u lt in g ranges o f

a b i l i ty a re too g rea t to be overcome i s a m atte r o f judgment.

As can be ea s ily discerned from Table 20, the h ighest a b i l i t y

class a t Lakeview had the h ig h est a d ju s ted mean achievement reg a rd le ss

of th e la b o ra to ry treatm ent employed. I t was because o f th i s

occurrence th a t a second view o f the e f f e c ts o f labo ra to ry treatm ents

was made using the amounts o f v a r ia tio n o f th e adjusted mean achieve­

ment scores from the ad ju sted mean fo r th e t o t a l sample in terms of

standard sco res . A fter th e se v a r ia tio n s were ca lcu la ted f o r each c la ss

per u n i t , they were summed across a l l th re e u n i t s , These sums were

then ranked to compare e f f e c t s of lab o ra to ry trea tm en ts. Ag might be

expected, these rankings d id n o t com pletely c la r i f y the an a ly s is . The

Mercer scores (see Table 15) show a decided advantage f o r those stu­

dents tak ing the p o s t- lab trea tm en t. This confirm s the an a ly sis by

COVAR. lakeview sums o f v a r ia tio n s (see Table 27) ranked treatm ents in

descending o rder o f e f f e c t a s p re - lab , n o - la b , p o st- lab .

Apparently th is r e s u l te d from a com bination of two fa c to rs . One,

the so -ca lled medium c la ss made an extrem ely poor showing fo r Unit I I

under the p o st- lab trea tm en t, and two, th e b ia s of in te llig e n c e could

no t be completely overcome by COVAR re s u l t in g in a r e la t iv e ly good

score fo r the h ighest a b i l i t y c la ss in U n it I I I under a no -lab tr e a t­

ment.

Another assumption which, according to E lashoff, must be met i s

th a t w ith in each trea tm en t, c r i te r io n sco re s have a l in e a r regression

on the con tro l v a riab le sc o re s . Also, th e s lo p e of the regression

l i n e s should be the same f o r each trea tm en t. To te s t f o r these re ­

quirem ents, scattergram s p lo t t in g c r i te r io n scores ag a in st control

v a riab le scores were made f o r a l l th ree tre a tm e n ts . In th e judgment o f

the au tho r, th e re appears to be no se rio u s v io la t io n o f these

86

assumptions regarding lin earity * The basis fo r assuming a t l e a s t some

r e l ia b i l i ty on the resu lts of COVAR i s Elashoff*s statem ent,

Generally, v io la tion of the assumptions of l in ­e a r i ty , homogeneity o f regressions, norm ality , o r homogeneity of variances w ill be le s s se rious i f ind iv iduals have been assigned to the treatm ents a t random and the x v a riab le has a normal d is­tr ib u tio n (10:398).

The labo ra to ry treatm ents were in i t i a l ly assigned a t random and

then assignments were ro tated fo r each new unit* The two co n tro l

variab les, IQ scores and diagnostic t e s t scores, were p lo tte d on dis­

trib u tio n curves to check fo r norm ality . The only se riously skewed

d is trib u tio n i s fo r the Mercer Class 2, Unit I I I d iagnostic t e s t which

was decidedly skewed negatively . In o ther words, th is c la ss performed

exceptionally w ell on th is t e s t . This c lass was assigned th e no-lab

approach fo r U nit I I I , and th e i r raw achievement t e s t mean score was«

ranked second as one might expect fo r th is medium a b i l i ty group (see

Table 8). However, when COVAR adjusted the achievement means, th is

c la ss f e l l to th e lowest ranked c la s s . Apparently the good per­

formance on th e diagnostic t e s t followed by average or below average

achievement influenced the adjustm ent. Expectations were n o t met by

th i s class. Even the low a b i l i ty c lass re la tiv e ly overachieved the

medium a b il i ty c lass fo r th is u n i t . This tends to su b s tan tia te tha t

including some laboratory experiments, regard less of th e ir type , moti­

v a tes the students to higher achievement than does to ta l ly excluding

experiments. The question o f COVAR*s a b i l i ty to make the proper ad­

justment in s p i te of the skewed d is tr ib u tio n i s s t i l l an open one.

87

P in a l In te rp re ta tio n s

1 . la b o ra to ry methods o f teaching m athem atical concepts to

e ig h th grade s tu d e n ts , where they are encouraged to a c tiv e ly p a r t ic ­

ip a te in the m anipulation of devices fo r discovery purposes, can p ro ­

duce h igher achievement te s t scores than does a t r a d i t io n a l , no la b ­

o ra to ry method o f teaching.

2. Of th e two lab o ra to ry approaches, p re - la b before c lass r e c i ­

ta t io n s and p o s t- la b a f te r c la s s d iscussions, the p o s t- la b approach

more o ften th an n o t produces h ig h er achievement t e s t sco res . This con­

c lu s io n i s based on the COVAR an a ly s is and a lso on the summation o f

e f f e c ts using standard score evaluations.

3* Due to hard to measure variab les such as students* p erso n a l

problems, excessive e x tra -c u rr ic u la r a c t iv i t i e s , s tu d en t-teach er r a p -t

p o r t , and th e degree of challenge in the lab o ra to ry experim ents, th e

re a c tio n s o f e ig h th graders to any labora to ry approach are somewhat ‘un­

p re d ic ta b le . The s k i l l of the teacher to guide, but n o t ovezvguide,

and the amount o f contagious enthusiasm exh ib ited by th e in s tru c to r

seem to be very im portant elem ents in the classroom procedures.

4 . S tuden ts w ith average o r above average a b i l i t y almost unan­

imously p re fe r some type of mathematics lab o ra to ry a c t iv i ty . Nearly

o n e -th ird o f th e lower a b i l i ty studen ts , w ith IQ scores le s s than 100,

p re fe r no la b o ra to ry experiments of the discovery type. Perhaps t h i s

i s because they become confused by the absence of s p e c if ic in s t ru c t io n s .

The hypothesis th a t the m ajo rity of eighth grade s tu d e n ts , re g a rd le ss

o f th e i r a b i l i t y le v e ls , p re fe r some experiments in th e classroom i s

accepted as v a l id .

83

5. Approximately tw o-thirds of the students in the sample a t

each school were o f the opinion th a t laboratory experiments were help­

fu l in explaining mathematical concepts. Nearly one-fourth of the

higher a b i l i ty s tu d en ts were no t aware of any appreciable improvement

in th e ir a b i l i t i e s to comprehend the concepts through experiments, but

they prefer la b o ra to ry methods to trad itio n a l class re c i ta t io n s .

6. The m a jo rity of the students in the sample, regard less of

th e ir a b ili ty l e v e ls , s ta ted th a t they looked forward to the next ex­

periment. Thus, th e pre-lab or post-lab approach to discovering mathe­

m atical concepts through discovery experiments acted, a t le a s t , as moti­

vating fac to rs f o r most eighth grade students in th is study.

7. G eneralizing from a sp ec ific s e t of experiments i s r isk y .

When judging th e7 e f fe c ts of mathematics labo ra to ries on students a t

Mercer and lakeview schools, one must judge the in te rn a l v a l id ity of

th is study. When one attempts to decide to what o ther populations

these re su lts can be applied in general, ex ternal v a l id ity i s involved.

Campbell and S tan ley warn th a t th is type of guessing a t laws involving

situa tions s im ila r to those stud ied must be done with caution .

The experiments we do today, i f successful, w ill need re p lic a tio n and cross valida tion , a t o ther times under other conditions before they can be­come an estab lished p a r t of science, before they can be th e o re tic a lly in te rp re ted with confidence (4*3).

89

Recommendations f o r F a rth e r Study

With th e cautionary words o f Campbell and Stanley uppermost in

the author1 s m ind, the fo llow ing recommendations a re made f o r r e la te d

s tu d ie s .

1. A re s e a rc h study, s im i la r to the one described in th i s paper,

could be conducted w ith ju n io r h ig h school c la s s e s whose s tu d en ts are

se lec ted com pletely a t random from a la rg e s tu d e n t population and whose

labo ra to ry tre a tm e n ts are a lso random ly ass ig n ed , fhe r e s u l t s o f an

an a ly sis o f covariance on such a study would be more r e l ia b le , and could

be used to e i t h e r confirm o r deny th e conclusions of th is study .

2* S p ec ia l e f fo r t could be made to d es ig n e ffe c tiv e experim ents

f o r a mathem atics labora to ry le a d in g to a d isco v ery of basic concepts

f o r c lasses v /ith above average a b i l i t y . A pparently these s tu d en ts can

be highly m otivated to eagerly e x p lo re m athem atics i f the experim ents

a re challenging enough and e f f e c t iv e ly designed to make them worth the

time and e f f o r t .

3. A s tu d y could be made to improve th e standard experiments f o r

low achievers to encourage them to i n i t i a t e t r i a l s of th e i r own making

f o r the sake o f conjecturing a b o u t g e n e ra liz a tio n s , fhe e f f e c t of

th ese refined experim ents may be compared th rough s t a t i s t i c a l a n a ly s is

w ith the r e s u l t s o f the cookbook v a r ie ty commonly used w ith lower

a b i l i ty students#

4. Elem entary and ju n io r h ig h school s tu d e n ts who have been re ­

peated ly exposed to mathematics la b o ra to r ie s o v e r a period o f th ree o r

more years may become so accustom ed to th is m ethod th a t they p assiv e ly

90

p a r tic ip a te . I s an approach a l te rn a tin g la b o ra to r ie s and class r e c i ­

ta t io n s within each y e a r 's study b e t te r than a l te r n a t in g methods an­

n u a lly ? The r e s u l t s of a c a re fu lly planned study to answer th is ques­

t io n s could prove very valuable to the coordinators o f mathematics cur­

r ic u la .

BIBLIOGRAPHY

1 . Armstrong, J . R . Representation modes a s they in te ra c t w ith cog­nitive and m e n ta l development of the re ta rded to promote mathe­matical l e a r n in g . A research rep o rt presented a t the annual meet­ing of the AERA, Los Angeles, 1969*

2 . Bruner, Jerom e S. Towards a Theory o f In s tru c tio n . Cambridge: Harvard U n iv e rs ity Press, 1966, 10-11.

3 . Callahan, Jo h n J . and Jacobson, Ruth S. "An Experiment Y/ith Re­tarded C h ild re n and Cuisenaire Rods," The Arithmetic Teacher. XIV (January, 1 9 6 7 ) , 10-13,

4 . Campbell, D onald Thomas and S tanley, J u lia n C. Experim ental and Quasi-experimental Designs fo r Research. Chicago: Rand McNally, 1963.

5 . Clarkson, D avis M. 'Mathematical A c t iv i ty .11 The A rithm etic Teacher, XV (October, 1968), 493-498.

6 . Cooley, YftJLliam Y/. and Lohnes, Paul R. Cooley-Lohnes Programs.New York:. Jo h n Wiley and Sons, 1967.

7 . Cuisenaire Company of America, In c ., 12 Church S tre e t , New Rochelle, New York 10805 (1971 C ata log).

8 . Dale, Edgar. Audio Visual Methods in Teaching. New York: The Dcyden P re s s , In c . , 1946, 317*

9 . Davidson, P a t r i c i a A, "An Annotated Bibliography o f Suggested Manipulative D evices,11 The A rithm etic Teacher, XV (O ctober,1968), 509-524.

10 . Elashoff, J a n e t D. "Analysis' of Covariance: A D elicate In s tru ­ment, " Am erican Educational Research Jo u rn a l, VI (May, 1969), 383- 401.

11 . Pennema, E . H. A study of the r e la t iv e e ffec tiv en ess o f a mean­ingful c o n c re te and a meaningful symbolic model in le a rn in g a selected m athem atical p rin c ip le . A research rep o rt p resen ted a t the annual m eeting of the AERA, M inneapolis, 1970.

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1 2 .

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23.

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F itzgera ld , V/., Bellamy, D., Boonstra, P ., Oosse, W., and Jones,J . labo ra to ry Manual fo r Elementary Teachers. Boston: P rind le ,Weber, and Schmidt, In c ., 1969.

Greene, Harry A ., Jorgenson, A lbert N., and Gerberich, J . Raymond. Measurement and Evaluation in the Secondary School. New York: Longmans, Green and Co., 1954, 74.

Higgins, Jon L. "The mathematics through science study* a tti tu d e changes in a mathematics la b o ra to ry ," SMSG Reports No. 8 , Stan­fo rd U n iv ersity , 1969*

Hi.13.man, Thomas P. "A Current L isting of Mathematics Laboratory M a te ria ls .11 School Science and Mathematics. LXVIII (June, 1968), 488-490.

Hoffman, Ruth Iren e . "The Slow Learner - Changing His View of M ath," The B u lle t in of the N ational A ssociation of Secondary School P r in c ip a ls (April. 1968).' 86-89.

H o llis , Loye Y. "A Study to Compare the E ffec t of Teaching F ir s t and Second Grade Mathematics by the Cuisenaire-Gattegno Method w ith a T ra d itio n a l Method," School Science and Mathematics. LXV (November, 1965), 683-687.

Kieren, Thomas E. '(Manipulative A ctivity in Mathematics Learn­in g , " Journal f o r Research in Mathematics Education, Volume 2 (May, 1971 )V 228-229.

Leeb-Lundberg, K ristina . 'K indergarten Mathematics Laboratory- N ineteenth-Century Fashion," The Arithmetic Teacher. XVII (May, 1970), 372.

Lick. Dole W. "Why Not M athematics," The Mathematics Teacher.LXIV (January, 1971), 86.

Low Achievement Motivation P ro je c t, 1164 26th S tre e t, Des Moines, Iowa 50311.

Iiucow, W illiam H. "An Experiment with the Cuisenaire Method in Grade T hree," American Educational Research Journal, I (May,1964), 159-167.

Marx, Robert. '(Mathematics Can Be Fun," School Science and Mathematics. LXVTII (February, 1968), 123-129.

May, Lola J . "Learning Laboratories in Elementary Schools in V/innetka." The Arithmetic Teacher. XV (October, 1968), 501-503.

93

25. Nasca, Donald. "Comparative M erita o f a Manipulative Approach to Second-Grade A rith m etic ," The A rithm etic Teacher. X III (March, 1966), 221-226.

26. National Education A ssociation. "Secondary School Math in 5 P it ie s ." Today*s Education (May. 1970), 50-51.

27. Passy, Robert A. "The E ffect o f Cuisenaire M ateria ls on Reasoning and Computation." The Arithmetic Teacher. X (November, 1963), 439- 440.

28* P h illip s , Joe. ’'Pu tting the Tic in Arithme," School Science and Mathematics. LXVI (March, 1966), 216.

29. Popham, W. James* Educational S t a t i s t i c s . New Yorks Harper & Row, P ub lishers, 1967.

30* Rosskopf, Myron P . , Morton, Robert L ., Hooten, Joseph R ., and Sitomer, Harry. Modem Mathematics: fo r Junior High School.Book 2. New Je rsey : S ilv er B urdett Company, 1961.

31. Rosskopf, Myron R ., Morton, Robert Lee, More dock, H. Stewart and G ilbert, Glenn A. Modem Mathematics: Through Discovery. BookTwo. New Jersey : S ilv e r B urdett Company, 1964.

32. Spencer, P e te r L. and Byrdegaard, M arguerite. B uilding Mathe­matical Concepts in the Elementary ‘School. New York: Holt, Rinehart & Winston, In c ., 1952.

33* T in ti, Robert. 'Mathematics Through Cardboard C arpen try ," The Arithmetic Teacher. XVIII (March, 1970), 209*

34. Trueblood, C. R. "A Comparison o f Two Techniques f o r Using Visual Tactual Devices to Teach Exponents and Non-decimal Bases in Ele­mentary School M athematics." Unpublished doctoral d is se r ta tio n , Pennsylvania S ta te U niversity , 1967.

35. Vance, J . H. "The E ffec ts o f a Mathematics Laboratory Program in Grades 7 and 8—An Experimental S tudy ." Unpublished doctoral d isse r ta tio n , U niversity o f A lberta , 1969.

36. Wilkinson, Jack. (Doctoral d is s e r ta t io n in process o f being com­p leted), Iowa S ta te U niversity , Ames, c i te d by F itz g e ra ld , Wil­liam M., About Mathematics L ab o ra to rie s . Michigan S ta te Univer­s ity , 19701 (Mimeographed.)

57. Woodby, Lauren. Mathematics Laboratory , MOREL Annual Report Volume VI, Michigan-Ohio Regional Educational Laboratory,D etroit, 1967.

APPENDIX A

EXPERIMENTS WITH SUGGESTED PROCEDURES

AND LEADING QUESTIONS

I

.. n.

94

95

• UNIT I

EXPERIMENT A .

The only m ateria l needed for th is simple experiment i s a quantity

o f paper, p referab ly colored construction paper.

Each student s e le c ts one sheet which represen ts one whole p a r t.

The paper i s to m in h a l f , using a fo ld in g process to f in d a dividing

l in e . The phrase "dividing lin e" i s emphasized to equate the ideas of

halving the paper and d iv id ing i t in to -two equal p a r ts .

One of the halves i s again folded and halved. Y/hat p a r t of the

whole i s represen ted by one of the sm alle st pieces of paper? How do

you know th a t i t i s one-fourth? Are you dividing one by fo u r or are

you tak ing one-half tim es one-half? i s th e re any d iffe rence in these

two operations?

Now, divide one-fourth by two or halve a fourth* Y/hat fra c tio n a l

p a r t o f the whole do you have now in the sm allest po rtion?

I f we were to continue taking o ne-half of the sm allest pieces of

paper on our desks, would the pieces ev e r become la rg e r? Y/hat i s 1/2

x 1 /2? I f we m ultip ly any number by o n e-h a lf, w ill the product ever

be a number la rg e r than one m ultiplicand?

Y/hat would happen i f we had s ta r te d by tearing our paper in to

th ree equal p a rts? How would we accomplish th is? How can we obtain a

p iece of paper rep resen ting 1/9 of the whole? 1/27?

Can you fig u re out how to take tw o-th ird s of a whole by fo ld ing and

tea rin g ? V/hat about 4 /9? Compare the products with the m ultip licands

in each case. Try another frac tiona l value such as th ree -e ig h th s .

UNIT I

EXPERIMENT B.

Using s t r i p s o f co n s tru c tio n paper and the methods of Experiment

A, fasten p ieces o f paper to g e th e r to form two s t r ip s measuring 1 1/4

and 2 3/8 u n i t s . Now, by fo ld in g these s t r i p s , divide them in to

eighths o f a u n i t . How many e ig h th s a re th e re in 1 1/4? In 2 3/8?

I f Y/e combine a l l of these e ig h th s to g e th e r by p lacing the s t r ip s end

to end, how many eigh ths do we have? How many u n its do we have? Y/hat

i s the sum of 1 1 /4 + 2 3 /8? Can you t e l l me why we chose to divide

the u n its in to e igh ths?

S ta r t again v/ith new s t r i p s rep resen tin g 1 1/3 and 2 5 /6 u n i ts .

V/hat should we use as a common f ra c t io n a l p a r t? V/hy use s ix th s ? Could

we use tw elfth s? Y/hy d o n 't we?

V/hat would you use i f you wished to f in d the sum 1 1/3 + 1 1/4?

1 1 / 3 + 2 3 /8? 2 5/6 + 2 3 /8?

UNIT I

EXPERIMENT C.

F ractional p a r ts of a u n i t measure can be related to f ra c t io n a l

p a rts of one hundred u n its . The requ ired apparatus c o n s is ts o f any

s tick divided in to equal p a r ts such as a one foo t ru ler, a m e te r stick ,

and several p ieces of s tr in g th re e o r four f e e t in length.

On a f l a t su rface , the f lo o r i f necessary , form a t r i a n g l e using

the meter s t ic k as one side and two pieces o f s tring as th e o th e r two

sid es . The s tr in g s may or may n o t be the same length. C a ll the point

where the s tr in g s meet, po in t P. The meter s t ic k and p o in t P are to

remain fixed throughout th is experim ent.

/co

Select a denominator fo r your fru c tio n s such as e ig h t. Place the

ru le r p a ra l le l to the meter s t ic k w ith i t s l e f t end on th e one string

side of the tr ia n g le and move i t up o r down so tha t the number eight

f a l l s on the o th e r s trin g . See th e p ic tu re above.

98

From P s tr e tc h s tr in g s cu tting the r u le r a t points 1, 2 , 3> 4, 5,

6 , 7* Each of these s tr in g s w ill cut the m eter s t ic k a t a d i s t i n c t

numeral. 1/8 = 7/100, 3/8 = 7/100, e tc .

Select o th e r denominators such as f iv e , s ix , seven o r e lev en . I f

a measuring s t ic k longer than a ru le r i s a v a i la b le , try 16 o r 20 as a

denominator.

Check your r e s u l t s by long d iv ision .

t

99

UNIT I

EXPERIMENT D.

A sp h erica l o b je c t, such as a te n n is b a l l , is to be r o l l e d down an

in c l in e d plane. I f p o ss ib le use a trough made by folding m a te r ia l such

as aluminum flash ing . Measure a d e f in ite leng th , perhaps s ix f e e t ,

along th e trough c le a r ly marking both endpo in ts. With a s to p watch,

time th e r o l l of the b a l l from one mark to the next. Do n o t push the

b a l l , r e le a s e i t from a s ta n d s til l a t th e top maik. Perform t h i s part

of th e experiment fo u r o r five times w ith o u t changing the an g le of the

in c lin e d plane. Average the times recorded and use th is average in the

fo llo w in g ca lcu la tio n s.

I f th e ball r o l l s s fe e t in t seconds, how many fe e t does i t aver­

age p e r second? How many inches per second, yards per second, fe e t per

m inu tes, f e e t per hour, miles per hour?' Use the re su ltin g form ulas to

c a lc u la te the speed o f your b a ll.

In c rease the slope o f the inc lined trough and see what happens to

the average speed of th e b a ll. I f we r a i s e the one end tw ice a s high ,

as i t was o rig in a lly , do we double the average speed of the b a l l? Try

i t th r e e times higher th an the f i r s t p o s i t io n . Can you determ ine the

average speed of the b a l l i f i t i s dropped from a sp ec ific h e ig h t to

the f lo o r ? Why do we use the phrase, average speed? I s th e a c tu a l

speed o f the b a ll ever le s s than the average speed? Is i t e v e r more

than th e average speed? When?

100

UNIT I

EXPERIMENT E.

Everyone knows how to play t ic - ta c - to e using 0 's and X 's . This

i s a sim ilar game u sin g fra c tio n s . The idea i s to take tu rn s placing

f ra c t io n s whose values a re le ss than one in the boxes o f a th re e by

th re e square a ttem pting to get a sum o f one e ith e r v e r t ic a l ly , h o ri­

z o n ta lly , or d iag o n ally . Your opponent w ill attempt to block your ef­

f o r t s by c a re fu lly se le c tin g h is f ra c tio n s and a t the same time try to

g e t a sum of one f o r h im self.

Other r e s t r ic t io n s to make the game more in te re s tin g are to use

only frac tio n s l e s s than one-half, o r to change the sought a f t e r sum

to 5 /8 , or 5 /4 , o r 3 /2 .

Decimal f ra c t io n s may also be used, e ffec tiv e ly in th i s game. I t

i s in te re s tin g to mix fra c tio n s , decim als, and percen ts in the same

game.

Computational s k i l l s decidedly improve as various v ers io n s of this

experiment are rep ea ted ly employed#

Can you th in k o f o ther games l ik e these?

101

UNIT I I

EXPERIMENT P .

P ro trac to rs and s t r a ig h t edges are commonly used too ls f o r mathe­

m atical ex e rc ise s in an experim ental s e t t in g . Some useful experiences

to promote discovery a re ,

1 . Measure the s iz e o f a given angle.

2. Construct an angle of sp e c if ic s iz e .

3 . Construct an angle whose measure i s the sum of two given ang les.

4 . Construot two in te rs e c t in g l in e s and measure the fo u r re ­s u lt in g angles.

5. Construct a t r ia n g le and measure the th ree angles. Find the sum.

6. Measure an e x te r io r angle o f the same tr ian g le . Compare i t s s iz e w ith the sum o f the two non-adjacent in te r io r ang les.

7. Sketch two l in e s which appear to be p a ra l le l . Cut them w ith a tra n sv e rsa l. Measure the re su lt in g angles. Pind equal measures.

8 . Construct p a r a l le l l in e s using equal corresponding an g les.

9. Construct a r ig h t angle. Construct a r ig h t tr ia n g le .Measure the two sm aller ang les, f in d the sum of th ese two measures.

10. Use your p ro tra c to r to draw a c i r c le . Draw two r a d i iforming an angle o f 80°• Draw two chords from a f ix ed poin t on the c irc le to th e endpoints of these ra d ii . Measure the angle formed by these two chords. Try i t again w ith o ther chords.

102

UNIT I I

EXPERIMENT 0.

Gather numerous c irc u la r o b je c ts such as j a r l i d s , p la s t ic covers,

paper p la te s , paper o r p la s t ic cups, sm all o r la rg e wheels, e tc .

Using a f le x ib le measuring ta p e , such as one used in sewing, to

estim ate the circum ferences of th e c i r c u la r o b jec ts co lle c ted . Care­

f u l ly measure th e la r g e s t chord p o ss ib le (the d iam eter) of each o b jec t.

Record these in a ta b le w ith columns f o r Circumference and Diameter.

Using long d iv is io n , f in d the r a t io o f th e circum ference to the diam­

e te r f o r each of th e o b je c ts .

Should your answers be the same f o r a l l cases?

Should they a l l be 22/7, 3*14, o r something e ls e ?

M e or f a ls e ? 22 s 7 : * 314 : 100.

Why do we use the Greek l e t t e r tt to rep re sen t the r a t io of the

circumference to th e diameter?

What ra t io was used in B ib lic a l tim es? See I Kings 7:23*

UNIT I I

EXPERIMENT H.

The construction of p a r a l le l ru le rs u sin g s t r ip s of l i g h t card­

board connected with brass fa s te n e rs to allow fo r f lex ib le j o i n t s , can

be in te re s t in g and educational.

On two s tr ip s e ig h t inches long and one inch wide, c e n te r two

p o in ts s ix inches ap a rt . Cut two o ther s t r i p s f iv e inches i n le n g th

and cen te r two points each th re e inches apart* Jo in these s t r i p s with

the fa s te n e rs a t the po in ts marked with the two equal length s t r i p s op­

p o s ite each o ther.

What fig u re i s formed? I f the angle s iz e s are changed by pushing

sideways on the f ig u re , are th e opposite s id e s s t i l l p a r a l le l? Measure

the angles of a figu re formed a f t e r marking on a sheet of p a p e r alongt

■the edges o f your p a ra l le l r u le r . Are th e corresponding a n g le s equal

in measure?

Given a lin e I and a p o in t P not on L, can you use your r u l e r to

co n stru c t a l in e M through P which i s p a r a l le l to l in e L? How many

such l in e s do you suppose can be thus drawn?

Sketch an angle o f any convenient s iz e on a sheet of p a p e r . On

the same paper draw two l in e s w ith your p a r a l l e l ru le r each o f which

i s p a r a l l e l to one o f the ray s o f the given an g le . Extend th e s e lin e s

u n t i l they in te rs e c t to form fou r angles. Measure these a n g le s . How .

do they r e la te to the o r ig in a l angle you sketched? Which p a i r s o f rays

could you se le c t to make the newly constructed angle congruent to the

o r ig in a l? How do th e i r d irec tio n s compare w ith those of th e o r ig in a l?

I s th e re ju s t one answer to the preceding question?

104

UNIT I I

EXPERIMENT I .

The s p e c ia l m aterials needed fo r th e following e x e rc is e s are

sc isso rs and sheets of paper. Heavy wax paper i s most s u i ta b le since

the creases become white l in e s and i t s transparency makes i t e a s ie r to

superimpose p o in ts and l in e s .

C onstruc t, by fo ld ing the wax paper, each of the fo llow ing geo­

m etric f ig u re s .

1 . s t r a ig h t lin e s

2. perpendicular l in e s

3. perpendicular b ise c to r of a l in e segment

4 . p a r a l le l lin e s

3* an g les '

6. ang le b isectors

7 . tr ia n g le s with each angle b ise c te d

8 . tr ia n g le s w ith th ree a lt i tu d e s

9 . tr ia n g le s w ith th ree perpendicu lar b isectors o f th e

resp ec tiv e side3

10. tr ia n g le s with, th ree medians

11. rec tang les and squares

12. e q u ila te ra l tr ia n g le .

These paper folded constructions a re quite simple excep t fo r the

e q u ila te ra l tr ia n g le . F o llo w the steps ou tlin ed below to assu re suc­

cess.

105

! • Construct a p a i r o f lin e s K and 1 perpendicular a t po in t P*

2* C onstruct two l in e s M, N p a ra l le l to 1 cu ttin g K a t Q and R so th a t PQ = QR*

5* Fold the paper so th a t point R f a l l s on l in e M a t R* and th e crease goes through point P. The crease cuts l in e IT a t T.

4* Row, PT iB one side of the e q u ila te ra l tr ia n g le and the o th e r vertex S on l in e 1 can be ea s ily located by fo ld in g T down to l in e It on a crease along PR1 • Triangle STP i s e q u i la te ra l . Can you explain why th is construction works?

106

UBIT I I

EXPERIMENT J .

The m a te ria ls fo r th i s labo ra to ry experiment a re pick-up s t ic k s

and b i ts of p la s t ic clay* The ob jective i s to co n stru c t polygons of

various sizes and shapes, Tinkertoy s tic k s and wheels can also be used

e ffec tiv e ly , but th e i r f l e x ib i l i t y i s more r e s t r ic te d ,

1 , Construct an e q u ila te ra l triang le* Is i t a lso Isosceles?

Construct a second e q u ila te ra l tr ia n g le using one o f the sides of th e

f i r s t tr ian g le as a base. Construct another with th e same vertex

shared by the f i r s t two. Do you see anything sp ec ia l about the p o si­

tio n s of the s id es? Are any of them co llin ear? Continue to construct

tr ia n g le s four and f iv e . Can you make a sixth? How many sticks are

needed to complete the s ix th tr ia n g le ? V/hat i s a hexagon? What i s a

reg u la r hexagon?

2, Construct a parallelogram . How do you make i t in to a rec ­

tangle? Are the d iagonals of a rec tan g le equal? Check those of a gen­

e ra l rhombus. Are th e diagonals of a rhombus perpendicular?

3, Construct a tr ia n g le by p u ttin g together i n a lin e an even

number of s tic k s on each s id e . Use a d iffe re n t number fo r each sid e

such as 2, 4, and 6 , Using as many s t ic k s or p a rts o f s tick s as needed,

jo in the midpoints o f the sides forming a smaller t r ia n g le . Do you

need any f ra c tio n a l p a r ts o f the s tic k s? How do the perimeters of the

two tr ian g le s compare? Are there any p a ra l le l l in e s i n the figure?

How do the angles o f the two tr ia n g le s compare?

107

UNIT I I I

EXPERIMENT K .

On s h e e ts of cardboard cu t from sid es of cardboard cartons,

fasten a s h e e t o r sheets of graph paper* By using thumb tacks and

elastic s t r i n g s , foim various polygons such as tr ia n g le s , squares, rec­

tangles, para lle log ram s, rhombuses, reg u la r hexagons, e tc . Estim ate

the area o f each figure by counting the squares enclosed by i t s s id e s .

D iagonals of polygons w ith more than three s id e s may be stre tch ed

to divide th e fig u re in to tr ia n g le s . Estim ate the sums of a l l in te r io r

angles. Can you guess what the sum o f a l l the e x te r io r angles might

be?

Using a piece of regular s t r in g as a compass, construct a c irc le

with a ra d iu s o f ten u n its . E stim ate the area by counting the squares

enclosed.

In s c r ib e a regular polygon in th e c i rc le . I s i t s area more or

less than th e c irc le ? Double the number of s id e s . How do the areas of

polygon and c i r c le compare now?

Circum scribe a regu lar polygon about the c i r c le . Is i t s a rea

more o r l e s s than the c irc le ? V/ill doubling the number of sides make

i ts area l a r g e r o r smaller?

I f we continued to increase th e number of s id es of both the in ­

scribed and circumscribed polygons, what happens to th e i r enclosed area

as they r e l a t e to the area of the c i r c le ?

Can th e a rea of a c irc le and th e area o f an inscribed or circum­

scribed polygon ever be exactly the same?

unit h i

EXPERIMENT 1 .

The area of a curved surface i s more d i f f i c u l t to measure than,

th a t of a f l a t surface. Therefore, the purpose o f t h i s experiment i s

to r e la te the la te r a l a reas of cy linders to areas o f rec tan g les and to

re la te the area of a hemisphere to the a rea of a c i r c le by in tu i t iv e

th inking.

l e t ' s f i r s t review the formulas fo r the circum ference and the2area of a c irc le since these w ill be req u ired . C = tt d and A = t r r ,

I f we take a rec tan g u la r piece of paper and bend i t s one edge in

a c irc u la r manner, a cy lin d er i s formed. Thus, the l a t e r a l area o f a

cylinder i s exactly the same as the a rea of th is re c ta n g le . What i s

the formula fo r the area o f a rectang le?

How can we determine the dimensions required to ca lcu la te the

la te r a l a rea of a cy linder? The height i s e a s ily measured and the

circumference can be ca lcu la ted (estim ated) a f te r th e diam eter i s de­

termined. How can we use these values to ge t the l a t e r a l surface a rea

of the cylinder?

The m aterials requ ired fo r the n ex t p a r t of our experiment are

h a lf a sphere, made by cu ttin g e i th e r a rubber b a l l o r a wooden croquet

b a ll in h a l f , and a len g th o f re la t iv e ly heavy chord s tr in g .

On the c ircu la r cross section o f the hemisphere t ig h t ly c o il the

heavy chord, s ta r tin g a t th e cen ter, u n t i l the e n t i r e su rface i s

covered by one and only one layer of s t r in g . C arefu lly mark the end

of the chord where i t runs o ff the edge o f the c i r c le . Now, unwind the

chord and measure i t s length* How can the area of the c irc le be c a l­

culated now th a t we lmow the leng th of the coiled chord? Does the un ­

co iled chord in any way resemble a rectangle?

Using chord from the same b a ll , co ll the chord over the curved

surface of the hemisphere. Try to wind the chord w ith the same com­

pactness as you used on the c irc u la r cross sec tio n . Completely cover

the su rface w ith one and only one lay er of twine.

What can we do now to compare these two areas? Can you "g u e s ti-

mate" the formula fo r the a rea o f a sphere when th e rad iu s i s known?

UNIT I I I

110

EXPERIMENT H.

The fo llow ing experiment req u ires a su ffic ie n t number of wooden

cubes one in ch on an edge to co n stru c t s o lid fig u res .

V/hat u n i ts of measure a re used to measure volumes? Of course,

q u a rts and g a llo n s are used, bu t what u n i ts can we use which are re ­

la te d to l in e a r and area measure?

Build the follow ing and "guestim ate" th e ir volumes.

1 . R ight rec tan g u la r prism .

2. Oblique rec tan g u la r prism .

3* R ight tr ia n g u la r prism .

4 . C ylinderical tow er (estim ate fo r blank a r e a s ) .

5 . Pyramid. '

6 . C ircu la r cone.

Check your estim ated answers a g a in s t those c a lc u la te d through th e

form ulas in our te x t .

Ill

UNIT I I I

EXPERIMENT N.

Rather sp e c ia l equipment i s necessary fo r the experiment to de-

termine o r su b s ta n tia te formulas f o r the volumes o f pyramids, cones,

and spheres.

Our desire i s to re la te the volumes o f these o b jec ts to volumes

of prisms and c y lin d e rs , the form ulas fo r which we already know.

V *= lwh : V = TTr2 h .P c

Thus, we need a cone and a c y lin d e r w ith the same base size and

the same h e ig h t, and a pyramid and a prism w ith th e same base s ize and

heigh t.

Can you guess what we in tend to do w ith these? W ill you venture

a guess as to how many cones w ill f i l l the cy lin d er? How many pyramids

w ill f i l l the prism ? >

Sand, o r some o th e r g ranular substance , i s b e t te r than a l iq u id

since i t cleans up more ea s ily .

Check your "guestim ates" by f i l l i n g the la rg e r volumes with the

sm aller co n ta iners .

Take a cy lin d e r whose heigh t i s th e same as i t s radius and com­

p le te ly f i l l i t w ith w ater. Place th e con ta in er in a la rg e , empty pan.

Make sure the pan i s empty. Now, ta k e a b a ll w ith th e same diam eter as

the cy linder and com pletely immerse i t in the w ater allowing the excess

water to overflow in to the pan. Remove the c y lin d e r from the pan and

completely empty i t . Now, pour th e w ater from th e pan in to the empty

cy linder. This rep re sen ts the volume o f the sphere . Y/hat f ra c tio n a l

p a rt o f the cy lin d e r i s f i l le d ?

2Since V = 2 /3 n r h fo r the sphere, and since

2/5 Trr2 . 2r = 4/3 " r 5.

Do you agree?

UNIT I I I

EXPERIMENT 0.

Three dimensional figures a r e o ften d iff icu lt t o sketch on a tv/o

dimensional p lane. To a s s is t the m inds eye in th ese In te rp re ta t io n s ,

th e pick-up s t ic k s and p la s tic c lay can he very h e l p f u l .

Construct a cube, a re c tan g u la r prism, a r e c ta n g u la r pyramid, a

tr ia n g u la r prism , and a tr ian g u la r pyramid using th e s t i c k s and c lay .

P lace them c a re fu lly on the tab le i n fro n t of you.

On a sheet o f paper sketch e x a c tly what you s e e . Try not to move

y o u r head as you ske tch so th a t y o u r perspective w i l l be the same.

Another method which may be u se d involves the u s e o f a l ig h t be­

h in d the object and a tran e lu sc ien t paper secured by p la c in g i t in a

fram e of cardboard on wood. The shadow, of the o b je c t can be traced on

th e paper producing a two dimensional look at a th re e dimensional

o b je c t .

Can you ske tch a sphere? V/hat special markings a re needed to

g iv e the i l lu s io n o f th ree dimensions?

Try sketching a cone and show cross sections fo rm ed when the cone

i s s lic e d a t various angles.

/

APPENDIX B

DIAGNOSTIC TESTS

J

114

115

EIGHTH GRAHB DIAGNOSTIC TEST ON UNIT I .Name

P art I

1* I s 12 a common denominator o f the f ra c tio n s 1 /2 and 5/6?

2* What i s the le a s t common denom inator of 2/3 and 1/9?

3 . Express 1 /3 , 2 /5 , and 5/6 w ith th e same denominator,

4* 3 3/8 i s equal to 2 and how many eigh ths?

5. What i s the rec ip ro ca l o f 7, 1 /2 , 4 /3 , 2 1 /4 , .80?

6 . Write out how you would read th e follovdng:

a . .3985 b . 36.384

7. Write in polynominal foim: 35.793*

8 . Write 1/3 as a decimal.Perform the in d ica ted o p era tio n s and express your answers in sim plest form.

10. | - f =

11 JL _ —2.J “ L * 6 10

12 . 3 3 /8- 7/8

13 a s .13* 2 5 x 7 -

14. 12.5 x .06 =

1, X 7,15. £ • 5 =

16. .0048 7 .16

116

Part XI

Simplify the following r a t io s .

________ 1 . 96 * 27

2. 6 : 26

________ 4. 1.1 to .01

_________ 5. In the proportion 3*4 = 30 : 40 name the means.

Solve the following proportions f o r the m issing numbers.

_________ 6 . 1 : 2 k ? : 12

_________7. 7 * 3 = 4 : ?

Write the following ra tio s as p e r cents.

8 . 40 * 100

.________ 9. 1 * 1

10. 283 : 1000

________11. 25$ of 484 = ?

12 . Y/rite 5/8 as a p e r cen t.

________13. Y/rite 172.4$ as a decimal.

14. V/rite 100$ as a decim al,

________15. 128 i s 2$ of ?

117

EIGHTH GRADE DIAGNOSTIC TEST ON UNIT I I .Name

On the space provided before each statem ent place the l e t t e r o f the answer th a t best completes the statem ent.

_____ 1 . A s tr a ig h t l in e has

a. no beginning and no end po in tb. a d e f in ite beginning and a d e fin ite end po in tc. a beginning poin t and no end point d* no beginning point but an end point

______ 2. A l in e segment i s

a . a d e f in ite number o f po in tsb. sometimes curvedc. a subset o f a s tr a ig h t l in e d* the same as a s t r a ig h t l in e

______ 3. Two p o in ts on a l in e and a l l the points between i s

a . an inchb. a l in e segmentc. a rayd. an angle i

4 . I f we have two lin e s in the same plane th a t d o n 't meet, wec a ll them

a . p a ra l le l l in e sb. perpendicular l in e sc. oblique l in e sd. skew l in e s

_____ 5. Perpendicular lin e s meet to form a

a . s tr a ig h t angle b* 45° anglec. 90° angled. none o f these because they don 't meet

_____ 6. A simple closed fig u re d iv ides a plane in to how many se ts?

a . 1,b. 2c. 3d. 4

118_ 7- A co m er o f a polygon ia called

a . an angleb. a v e r te xc. a diagonald. two rays

_ 8 . A diagonal i s a l in e segment th a t jo in s two non-adjacent v e r t ic e s of a polygon.

a . Trueb. F a lsec. Sometimes, hard to t e l l

_ 9. A square I s a (an)

a . octagonb. pentagone. q u a d r ila te ra ld. hexagon

10. An angle i s formed by two in te rse c tin g

a. ray sb. l in e sc. segmentsd. d iagonals '

11. A ray has

a . no beginning and no end pointb. a d e f in i te beginning p o in t and a d e f in i te end pointc. a beginning po in t and no end poin td. no beginning po in t bu t an end poin t

12. (The in strum en t used to measure angles i s c a lle d a

a . compassb. r u le rc. m eter s t ic kd. p ro tra c to r

119

Approximate the measure o f th e follow ing angles using your p ro trac to r.

13.C

14.

E

1 5 .

BATE- CLASS

EIGHTH GRABE BIAGNOSTIC TEST OH UNIT I I I .

I .

O c

1 . I n the figures above, which are closed fig u res?

Aj B only (B) B and E (c) A, B, and C (d) A, C, and B E) a l l of them

2. F igure C i s ca lled a (A) pentagon (b) hexagon (c) heptagon(d) octagon (e ) none o f these

3* A polygon with th re e s id e s i s ca lled a (A square (B) t r i ­an g le (C) q u a d r ila te ra l (b) decagon (e ) none of these

4. The sum of the m easures of the angles o f a q u ad rila te ra l i s (A) 90° (B) 180° (C) 270° (B) 360° (E) 400°

5. The sum of the le n g th s of the s id es of a polygon i s called th e (A) boundary (b) a rea (c) perim eter (b) circumfer­ence (E) size

6. A c i rc le separates th e plane i n t o s e ts of p o in ts .(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

7 . The lin e segment connecting the cen ter of a c irc le w ith a p o in t on the c i r c le i s ca lled the (A) a rc (b) diameter(c ) radius (b) se m ic irc le (e ) a l l o f th e se .

121

______ 8 . A tr ia n g le with no two sides equal i s c a lle d a(n)tr ia n g le •(A) Acute (b) Obtuse (o) Scalene (d) Iso sce les(e ) E q u ila te ra l

______ 9 . A tr ia n g le w ith two sides equal i s c a lle d a(n ) ______triang le*(A) Acute (b ) Obtuse (c) Scalene (d) Iso sce les(e ) E q u ila te ra l

10. A tr ia n g le w ith th ree sides equal i s c a lle d a(n) ______tr ia n g le .(A) Acute (b) Obtuse (c) Scalene (d) Iso sce les(E) E q u ila te ra l

11. The sum of the measures of the angles of a tr ia n g le i s(A) 90° (B) 180° (0) 270° (D) 360° (E) 400°

12. In the space a t the r ig h t , draw a q u a d rila te ra l.

13. Shade i t s in te r io r .

14. Hark a point M in the e x te r io r .

15. Mark a po in t N on the ' boundary.

P a r t I I .

S elec t the answer th a t best completes the follow ing:

1. Area measures the (a) E x te rio r o f a closed figurelb 5 In te r io r of a closed figu re ( c j Distance around a closed figu re (d) Hone of the above

2. Units used to measure a rea must be (a) Cubic u n its(b) l in e a r u n its(c) Square u n its(d) Hone of these

3. Volume measures (a) Capacitylb) Surfaceic) Distance around(d) Hone o f these

4. U nits used to measure volume must be (a ) Square u n its(b) l in e a r u n its(c) Cubic u n its(d) Hone o f these

o

122

5. Match the areas of the following fig u res w ith the corresponding form ulas:

formulas!

rectangle a. £ h (bx + b2)

tr ian g le b. s2

trapezoid c. 2 TTrh

parallelogram d. lw

c irc le e. 1 + w

cylinder f . c + rd

g» 2 *Tr

h. ir Wi

i . V W2tt r ^

k. bh

volumes of the following fig u res w ith the4

cube a. 1 /3 tr r2 h

rectangular prism b. 21 + 2w

trian g u la r prism c. lwh

cylinder d. 2 tt h

square pyramid e. tt r 2h

cone f . 4/3 r 2 - 4/3 r 3

sphere g. s5

h. 4/3 TT r 3

i . Bh

d* 1 + w + h

k. 1/3 Bh

123

7. F ind the a rea of a rec tan g le 3 f t . long and 2 f t . wide.

8 . I f the a rea o f the base of a rectangular prism is 24 square fe e t and the volume i s 96 cubic fe e t , what i s i t s height?

i

APPENDIX C

ACHIEVEMENT TESTS

124

EIGHTH GRADE ACHIEVEMENT TEST ON UNIT I .

Name

5 7 11 . Name 3 common denom inators o f ^ and -jr which are le s s than

200 •

2. What i s the le a s t common denominator fo r the f ra c tio n s in number 17 _____

1 3 53 . Express j > q t 811(1 1 2 w ith the same denominator.

4. 11 f = 10 + y + f - 10 + T

5 . Y/hat i s the re c ip ro ca l o f each of the following?

4a* 2 + 5 • b . 1 . c» ~ •——— -------- 3

d. 3 e • . 031

6. W rite ou t how you would read the follow ing.

a . .3985

b. 36.384

7« W rite in polynomial form : 35«703.

28 . Y/rite y as a decim al.

Perform the in d ica ted opera tions and express your answers in s im p lest form.

126

12. 11 ^11

. 7

15* ■S| x | = ------

14. .22 x .09 =

1K JL i 4 15# 10 * 5 ”

16. .0048 f .016 = _____.

17 . Simplify the fo llo w in g ra tio s ,

a . 144 : 18 ________ .

To, 2 i n . V 2 f t . _________ .

- H

d. 1*1 » .01

EIGHTH GRAHE ACHIEVEMENT TEST ON UNIT I I ,

Choose die best answer. Put the l e t t e r corresponding to the question number on th e answer sh e e t, IX) NOT w rite on the te s t copy. Mark only one answer per question. Erase o r b lo t out any answer you wish to change.

1. When two lin es in te r s e c t , th e ir in te r s e c t io n i s (a ) a point (b)a l in e (c) a plane (d) a ray (e) th e n u ll s e t .

2. When two planes in te r s e c t th e ir in te rs e c t io n i s (a) a point (b)a l in e (c) a plane (d) a ray (e) th e n u ll s e t .

3. When two p a ra lle l l in e s in te rs e c t , t h e i r in te rs e c t io n i s (a) a p o in t (b) a lin e (c) a plane (d) a ray (e) the n u ll se t.

4. The in te rsec tio n o f two skew lin e s i s (a) a p o in t (b) a lin e(c) a h a lf- lin e (d) a plane (e) the n u l l s e t .

5. The angle fozmed by one ray and by an o th er ray perpendicular toi t w ith a common endpoint i s called (a) acute (b ; obtuse (c)r ig h t (d) skewed (d) none of the above.

BOR QUESTIONS 6-10. REFER TO DIAGRAM I .

6. Angle AOB divides th e plane of the, paper in to how many regions?(a) 1 (b) 2 (c) 3 (d) 4 (e) none o f th ese .

7. The rays OA and OB of angle AOB foim th e o f th e angle.(a) rays (b) h a l f - l in e s (c) in te r io r (d j e x te r io r (e) s id es .

8. The shaded region i s ca lle d t h e o f angle AOB. (a) vertex(b) boundary (c) in te r io r (d) sides (e) e x te r io r .

9. P o in t 0 i s called th e (a) vertex (b) boundary (c ) in te r io r(d) s id e s (e) e x te r io r .

10. The unshaded portion i s ca lled the (a ) side (b) v e rtex (c) e x te r io r (d) in te r io r (e) h a lf-p lan e .

128

REFER TO DIAGRAM 2 FOR QUESTIONS 11-19* Choose the c lo se s t e s tim a te .

11* The measure of angle AOE i s9 (a) 60° (b) 90° (c) 120° (d) 135° (e) 180°

12. The measure of angle AOC i s(a) 60° (b) 90° (c) 120° (d) 135° (e) 180°

13. The measure of angle DOE i s(a) 45° (b) 90° (c) 120° (d) 135° (e) 180°

14* The measure of angle AOE i s(a) 45° (b) 90° (c) 120° (d) 135° (e) 180°

15. The measure o f angle AOB i s(a) 45° (b) 60° (c) 90° (d) 120° (e) 135°

16. The measure of angle BOE i s(a) 45° (b) 60° (c) 90° (d) 120® (e) 135°

17. Angle AOB i s(a) acute (b) r ig h t (c) obtuse (d) a l l o f thesethese

18. Angle AOD i s(a) acute (b) r ig h t (c) obtuse (d) a l l o f thesethese

19. Angle BOE i s(a) acute (b) r ig h t (c) obtuse (d) a l l o f thesethese

FOR QUESTIONS 20-22 REFER TO DIAGRAM 3.

20. Angles APC and EPB a r e ______ angles.fa) complementary (b) supplem ental (c) v e r t ic a l(e) none o f these

21. Angles APB and BPS a r e ______ angles.(a) complementary (b) supplementary (c) v e r t ic a lthese (e) none of these

(e) none of

(e) none of

(e) none of

(d) ad jacent

(d) a l l of

22. Angles CPA and APB are _____ angles.(a^Tadjacent (b) v e r t ic a l (c) complementary (d) a l l o f these(e) none o f these

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DIAGRAM 1

DIAGRAM 2

D

DIAGRAM 3

130

23. The measure of the angle th a t i s the complement of 35° Is (a) 35°(b) 45° (c) 55° (d) 145° (e) 155°.

24. The measure of th e angle th a t i s the supplement of 80° i s (a) 10°(b) 30° (c) 50° (d) 80° (e) 100°.

25. In geometry, two fig u res th a t are the same in size and shape arecalled (a ) acute fig u re s (b) adjacent figures (c) supplementary angles (d) congruent figures (e) complementary angles.

MATCHING: Choose the d e fin itio n on the r ig h t th a t best f i t s the wordon the l e f t . Record the l e t t e r of the d e fin itio n on theanswer sh ee t beside the number of the word. The lin e s separate each group of words and d efin itio n s.

26. Geometry a . An angle with a degree measure of 90° .27. Right angle b . The study of angles.28. Acute angle c . An angle w ith a degree measure le s s than 90.29. Obtuse angle d. An angle with a degree measure g rea te r than 90.

30. Perpendicular l in e s a. lin e s NOT in the same plane th a t do NOT31. Oblique lin e s in te rse c t.32. P ara lle l lin e s b. lin e s in the same plane th a t do NOT33. Skew lines in te rse c t .

c. Two lin e s tha t in te rsec t to determinerig h t angles.

‘ e. lin e s th a t in te rsec t in more than onepoint.

34. Adjacent angles a. Two angles with degree measure which35. Complementary angles when added together equal 180 degrees.36. Supplementary ang les b. A se t of points consisting of two d if­37. V ertical angles feren t rays with a common endpoint.

c. Two angles with ( l ) a common vertex,(2) a common ray , and (3) d is jo in tin te r io rs .

d. Two non-adjacent angles determined bytwo in te rsec tin g lin e s .

e . Two angles whose sum of th e i r measuresequal 90 degrees.

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38. M idpoint39. Angle b ise c to r40. Congruent

a . A ray th a t d iv id e s an ang le in to two congruent an g le s .

b . Two l in e s th a t in te r s e c t to determine r ig h t a n g le s .

c . A poin t th a t b is e c ts a l in e segment.d . 'l!wo f ig u re s th a t are the same in size

and shape•e . Two an g les whose sum of th e i r measures

equuls 90 d eg rees.

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EIGHTH GRA.IJE ACHIEVEMENT TEST ON UNIT I I I .

P art I . D efin itio n s:

1. A simple closed f ig u re th a t i s the un ion of l in e segments.

A q u a d rila te ra l w ith ex ac tly two p a ir sof opposite s id es p a r a l le l .

A q u a d rila te ra l has how many s id es?A parallelogram w ith a l l of i t s ang les r ig h t angles.A rec tang le w ith a l l of i t s s id e s equal in len g th .

A parallelogram w ith a l l of i t s s id e sequal in leng th .

A tr ia n g le w ith a l l o f i t s s id es equalin len g th .A tr ia n g le w ith one angle equal to 90 degrees.

2.3.4.

5.

6 .7.

8.

Bart I I . Formulas:

1. Rectangle A. V = 1/3 Bh

2. Triangle B. A K lw

3* C ircle C. V = lwh

4. Prism D. V — TT r 2 h

5. Pyramid E. A ss TT r 2

6 . Cylinder P . A = 1 /2 b h

7. Cone G. V S3 1/3 " r

P a rt I I I . Completion:

a. fo u rb. t r a p e z o idc. sq u a red. p a r a l l e l ­

ograme. r e c ta n g lef . po lygong. rhombush. th re ei . t r a p e z o id j . f iv ek. e q u i l a t e r a l m. s c a le n e n. r i g h t

1. The sum of the measures o f the ang les of a tr ian g le is _ degrees.

2. A tr ia n g le may have no more than one _______ degree angle.

3. A tr ia n g le may have t h r e e ______ a n g le s .

4. The _ _ _ _ _ _ _ of any polygon i s the sum o f th e lengths o f i t s s id e s .

5. The circumference o f any c irc le may be found by using the fo rm u laC S3 •

P a r t I I I , Completion continued:

6 . The perpendicular b isec to r of a segment passes through i t s

7 . A lin e segment whose endpoints are non-adjacent v e r tic e s of a polygon i s ca lle d a _______ o f the polygon.

P a r t IV. True o r F a lse :

_____ 1. AL1 tr ia n g le s are r ig h t tr ia n g le s .

______ 2. All squares are re c ta n g le s .

______ 3, Lines th a t n e ith e r in te r s e c t nor are p a r a l l e l are skew, -

______ 4* A tr ia n g le contains no adjacent v e r t ic e s .

______ 5. A septagon has 7 s id es and 7 v e r tic e s .

_____ 6. ^ABC means the same as £ A,

_____ 7* Two angLes are v e r t ic a l i f they have a common vertex and a common ray between them.

_ 8 . There i s exactly one l in e thrbugh two d i f f e r e n t points,

_ 9* A closed so lid fig u re d iv ides space in to th re e sets of p o in ts .

10. The face s o f a l l polyhedra are polygons.

11. A cube i s a rec tangu la r so lid .

12. A square pyramid has one base.

13* A hemisphere i s h a lf a c i rc le .

14. Another name fo r the v e rte x of a cone i s i t s apex.

_15* An oblique prism con tains no r ig h t an g le s .

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Part V* Find. the areas o f the following:

1 . T riangle: b s= 34" h = 17"

2. Rectangle: 1 = 40" w = 7"

5> C irc le : r = 7"

4> Square: a = 8"

Part VI. F ind the volumes of the following:

1. Prism : 1 = 17, w = 2, h = 10

2. pyramid: B ss 60 sq. f t . . h a 91

?• cy linder: nXII 10'

4. Cone: r == 14" h B 4"

?• 1 cu. f t . = cu. in .

6. 1 cu. yd. s CU. f t .

APPENDIX D

TEACHER AND STUDENT EVALUATION

J

FORMS

135

136

Date

TEACHER'S EVALUATION

Yes No

1 . A ll or n ea rly a l l o f the s tu d en ts w illin g lyp a r tic ip a te d . _ _ _ _ ___

2 . The students were in te re s te d to the extento f o ffering suggestions f o r procedures. _____ ____

3 . The LAB was more than "fun and games" in th a t m athem atical concepts seemedc la r if ie d .____________________________________________________ ___

4 . The students* extemporaneous and/or un­s o lic ite d comments were fav o rab le . ______ ___

5 . O verall, th e experience was s a tis fa c to ry . ______ ___

Comments, p a r t ic u la r ly on the "No" checks:

Date

STUDENT'S EVALUATION OP TODAY'S LAB EXPERIENCE

Yes No

1 . I t was in te r e s t in g . ______ ___

2 . I t helped me d iscover some d iffe re n t ideasabout m athem atics. _____ ___

3 . I t gave me a b e t te r understanding of the mathematics involved.

4 . I s t i l l have some questions about what we were try in g to do.

COMMENTS AND SUGGESTIONS:

APPENDIX E

STUDENT INTEREST QUESTIONNAIRE

4

137

136

STUDENT XNTEHEST QUESTIONNAIRE

Class

During th is school year your c la ss has had a number of mathe­m atics la b o ra to rie s . Sometimes these were given before you ta lked about the mathematical ideas in your c lass and sometimes the lab ­o ra to rie s came a f te r your c lass d iscussions.

There i s a r e a l in te re s t in your ideas about these lab o ra to ry experiments. Please answer the follow ing m u ltip le choice questions by ca re fu lly se lec tin g the best answer provided.

1 . The kind o f lab I l ik e d b es t wasbefore c lass d iscussion of the to p ic

b) a f te r c lass d iscussion o f the to p icc) no lab a t a l l .

_2. Mathematics labs have made th is year compared w ith l a s t y earmore in te re s tin g more boring no d if fe re n t.

3. I suggest th a t next y e a r 's eighth g radersv i

a) continue having a t l e a s t the same number o f lab s as we had bJ have more labs than we had c) have no labs a t a l l .

4. These lab experiments on the wholea) helped me to b e t te r understand mathematicsb) confused me more than they helped mec) made no d ifference in my a b i l i ty to understand mathematics.

5. When my teacher announced th a t we were going to have another labora to ry experiment,a} I looked forward to i tb) I sa id to myself, "So What,"c) I sa id to myself, "No, not ag a in ."

You a re requested to w rite more comments in th e space below about the labora to ry experiences. Be sin cere , but f e e l f r e e to w rite both favorable and unfavorable comments.