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A CLINICAL STUDY OF THE MATHEMATICAL INCOMPETENCE OF SOME UNIVERSITY
STUDENTS
A THESIS PRESENTED IN PARTIAL
FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF P HILOSOPHY
IN EDUCATI ON AT MASSEY UNIVERSITY
GORDON HENRY KNIGHT
1982
., . , .
f • .. •
( ii) ABSTRACT
The obj ectives of the study are to contribute to the c l e ar ident if ication and unders tand ing of the factors which l ie behind the severe mathe ma t ical d if f icul t ies exper ienced by so me otherwise able un ivers ity student s .
A care ful descr ipt ion of the pheno menon , wh ich might l e ad to an explanation , is dependant on an understand ing o f the cognit ive proce s ses of the ind ividua l s concerned . Consequent
ly a research method and theoret ical perspect ive were chosen which would enable 'a study of these proce s ses to be made as they were used in solving mathe ma t ical problems . The method was based on the P iaget ian c l in ica l interv iew a nd the theoret ical background wa s e s sent ia l ly that of S k e mp ' s
( 1 9 7 9 ) mode l of intel l igent behaviour . The pr inc ipa l ad vant
ages of this mode l were it s structural rather than g loba l features and the close relat ions h ip imp l ied between the cogn it ive and a f fective deter minants of behaviour .
Twenty s ix subj ects were of mathema t ical abil it ies presented w ith the s a me pr imary -secondary school
interv iewed having ·a wide r ange and intere s t . Each subj ect wa s sequence o f tasks taken from the arith me t ic - a lgebra syllabu s . The responses were probed in an unstructured manner .
The analys is of the interv iew data had two stages . F irstly , in arder to prov ide an overv iew , a for mal cod ing was under
taken in which the response to each i tern wa s c la s s if ied a ccord ing to the level of unders tand ing ind icated . The res u l t ing data wa s ana lysed in it ia l ly in an ent ire l y descr ipt ive manner and then wa s subj ected t o Latent Response An alys is . F o l l owing th is statis t ical analy s is a c loser c l in ical ana l y s is wa s made u sing a mul t iple-cod ing approach to build up a mosaic of evidence concern ing the conceptua l structures used by the subj ects .
The pr inc ipa l conc lu s ions of the s tudy relate f ir s t l y to the v ita l importance of the a v ailabil ity of appropriate in it ia l f ra meworks for the succe s s ful hand ling of mathe ma t ic s .
, ... ,
I t is argued th at the absence of such f r ameworks , or s che ma, interpreted in the l ight of · Skemp ' s theory , expl ains both the affect ive react ion of subj ects hav ing d if f icu lty w ith mathe mat ic s , and the development of inte l l igent behaviour w ithin one for m of knowledge but not in another .
Secondly , the ev idence of the study ind icated that it was unl ikely that the d if f icult ies wh ich the students were h av ing with mathema t ics were due e ither to the abstract nature of the concepts involved or to the logica l nature of the subj ect matter .
Th irdly , the top ic of fra c t ions e merged c l �arly as the mos t l ike ly source o f rea l d if f icu lty . It is suggested that genera t ions of curr iculum de s igners have s �r iou s l y under estimated the dif f iculties a s soc iated with learning in this area .
F ina l l y a t tent ion is drawn to the neces s ity · for teachers to con s t antly mon itor the development of the cogn it ive structures of the ir students and to be sen s it ive to s ignal s in the af fec t ive doma in which might ind ic ate developing proble ms in the cognit ive area . I n this way the v ic ious intera c t ion of cognit ive and affec t ive reactions to math
ematic s , which is the most d istre s s ing feature ·of the problem, might be avoided .
(iv)
PREFACE AND ACKNOWLEDGEMENTS
My i n tere s t in the topic of this thes is a r ises f rom s ome twenty f ive years of teaching mathema t ics in schoo l , technical col lege , teachers ' college , and univers ity . I t wou ld be impos s ible t o acknowledge a l l of the for ma t ive inf luences dur ing that t ime . However I w ish to express my spec ia l debt to the fo l lowing for the ir particular support over the per iod in wh ic h this the s is ha s taken shape .
To my supervisor , Dr D . M . McAlpine for h i s help and encour age
ment throughout the s ix years of th is proj ect .
To Mr E . L . Archer for h is part icular a s s i s t ance with the f in al stages of the s tudy .
To Dr M . R . Carter for h is construct ive c r it ic ism and intere s t .
T o Mrs F ield send for her eff ic ient typing .
Fina l ly , I owe a spec ia l debt of gra t itude to the stu dents who were prepare d to hare their math e mat ical sou l s hopi ng th at by doing so they would contribute to the unders t and ing o f the complexity of mathe mat ics learning .
G�H. Knight November 1 9 82
TABLE OF CONTENTS
ABSTRACT
PREFACE AND ACKNOWLEDGEMENTS TABLE OF CO NTENTS
LIST OF FIGURES LIST OF TABLES
CHAPTER 1
CHAPTER 2
CHAPTER 3
CHAPTER 4
THE NATURE OF THE PROBLEM I n troduc tion
The Symptoms
Ma themati c a l Hierarch ies
THEORETICAL PERS PECTI VE AND CHOICE OF METHOD
Research Me thods Fac tor ial S tud ie s C l in ical S tudie s
Theore tica l Background Indiv idua l D if f erence s Piaget' s Cl inica l Me thod
C l a s s if ica tion of I n terv iew Responses Re l iabil ity and Va l id ity
THE CHOICE OF SUBJECTS A ND DE VELOPMENT OF INTER VIEW TASKS
Selection of Subj ects The Ma thema tida l Ta sks
The Conduc t of the Interviews
THE CLINICAL A NALYS I S - A CASE STUD Y The Me thods of Ana l y s is
The Interv iew
The Ma thema tical Tasks : Na tura l Numbers
{V J
Page ii
V
v iii
X
1 1 3 9
1 2 1 2 1 3 1 5 1 9 22 2 8
32 3 6
4 1 4 1 42 5 0
5 7 5 7 5 9 6 1 6 1
\ VJ.}
T AB LE OF CONTENTS CONTINUED
CHAPTER 4
CHAPTER 5
CHAPTER 6
CHAPTER 7
CONTINUED Fract ions
Var iable s Problems Log ic Conc l u s ions
GENERA L CLINICA L ANA LYS I S Background Infor ma t ion The Cognit ive S y mptoms
Fra mewor ks
Equiva lent fra c t ions Addit ion of fractions
Mu ltip l ic Rt ion of a fra ct ion by ·� whole nu mber
D ivis jon of a whol e nu mber by a
6 6 7 0 7 7 7 9 8 1
8 3 8 3 8 8 8 8 9 0 9 4
9 6
fraction 9 8
Mu ltipl ication of f ractions 9 9 D ivis ion of one fract ion by anothe r 1 0 0
Log ic 1 0 1
Abstract ion 1 0 3
STATISTICAL ANALY S I S
The As ses s ment of Understand ing Method of Ana ly s is and Results Latent Response Ana l y s is by I te m Compar ison between Content Area s
SUMMARY , CON CLU S I O NS , EDU CAT JON1'1L
I MP L ICATIONS AND SUGGESTIONS FOR FURTHER RESEARCH
1 0 7 1 0 7 1 0 9 1 1 5 1 3 0
1 3 6
Su mmary of the Research 1 3 6
The Limitat ions of the Study 1 3 9
Conc lusions 1 4 1
The Character is t ic s o f the Les s Able
Students 1 4 1
The Character · o f the Mathe mat ica l
Dif f icu l t j es 1 4 6
The C ontrad iction 1 5 0
TAB LE OF CONTE NTS CO NTI NUED
APPE NDI X A LATE NT RESPONSE ANALYSIS
APPE NDI X B NEW ZEALAND PRIMARY SCHOOL SYL LABUS RE VS ION OF FRACTIONAL NUMBERS
BIBLIOGRAPHY
' • .
(vii)
1 5 3
1 5 7
1 6 6
LIST OF FIGURES
Figu re
1 Affective symptoms
2 The relationship between content are as in the school ari thmetic-algebra sequence
3 H i e r achy of topics i n the school a ri th
met i c - a lgebra sequence
4 Conceptual hierarchy for fractions
5 A hierarchy of selected subconcepts .of the f raction concept
6
7
D i s tribution of pos i t i ve re sponses - Instumental under standing
D i s tribution of pos i t ive responses - Relationa l unders tandi ng
8 Compari son of s cores accordi ng to i n s t rum
enta l understanding and relationa l under- · s tanding c r i teria
9 Compari son of rankings by ins t rument al unders tand i ng o f items 1 4 and 2 5 . Numbe r of pos i t i ve responses by subj ects at rank � x
1 0 Compari son o f rank i ngs by instrumental
understand ing of i tme s 1 4 and 2 5 . Propo rt ion of pos i t ive responses by s ubj ects a t rank � x
1 1· Latent respons e ana ly s i s . Forms o f t rac.e l ines
( vi i i )
Page
8
43
4 4
45
4 6
1 1 1
1 1 2
1 1 3
1 1 4
1 1 4
1 1 6
LIST OF FIGURES CONTINUED
F ig ).lre ·
1 2
1 3
1 4
1 5
Compar ison of rank ings by instrument al un derstanding of items 14 and 25. Latent response ana l y s is trace l ines
Proport ion of subj ects at rank
pos i t ive responses · by :::; x for items 13 -20.
Rel a t iona l under s tan ding
Latent response ana l y s is trace l ines for items 1 3-20. Re lationa l underst an ding
Latent response for items 13-20.
ing
ana l y s is trace · l ines Instrumenta l un de rs tand-
1 6 La tent response ana l y s is trace l ines
for items on fract ions . Relat iona l under-
(ix)
Page
1 20
1 22
1 23
1 24
s tanding 1 26
1 7
1 8
Latent response ana l y s is for items on f ra c t ions . under stan ding
Proport ion of pos it ive subj ects at r ank � x . - relat iona l un derstan ding
trace l ines Instrumenta l
responses by Content area s
1 9 Latent re sponse ana ly s is trace l ines .
1 27
1 3 1
Content areas - rel a t iona l un ders t an ding 1 3 2
20
21
Propo rt ion of pos it ive subj ects at r ank :::; x .
responses by Content are as - instrumenta l understan ding
L atent response ana l y s is trace l ine s . Content areas - ins trumental understan ding
1 3 3
1 3 4
LIST OF TABLES
T able
1 The content are as , que s t ions , and source s
'
of the 38 interview cards
2 Groups of s ubj ects accord ing to tot al s core
3 Analysis by ite m. Re lationa l u nder s t anding Location par a meter M i and d iscr imi n at ion para meter a.
1
4
5
6
Ana l y s is s tand ing .
by ite m. I nstru mental Loca t ion para meter d iscr imin at ion parameter a .
1
M .
1
under
and
Ranking of ite ms by f rom least ·d if f icult
locat ion p ar ameter to most d if f icu l t . Relat iona l understand ing .
Ranking o f from lea s t
items by d iff icult
locat ion para meter to most d if f icul t . Instrumental under stand ing .
Ana l y s is by content a rea . d iscr i minat ion para meter s
Loc at i on an d
(X)
Page
52
1 1 7
.1 1 8
1 1 9.
1 28 . ·
1 29
1 3 0
