bauersfeld hidden dimensions in the reality of mathematics classroom.pdf

8/9/2019 BAUERSFELD HIDDEN DIMENSIONS IN THE REALITY OF MATHEMATICS CLASSROOM.pdf

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H E I N R I C H B A U E R SFE L D

H I D D E N D I M E N SI O N S I N T H E SO - C A L L E D R E A L I T Y

OF A MATH EMATI CS CLAS SROO M 1

What is a rite? asked the tittle prince.

Those also are actions too often neglected ,

said the fox. They are what make one day

different from other days, one hour from

other hours.

Antoine de Saint Exupgry (1943, p. 84).

Abstract. Teaching and learning of mathematics in classrooms is interpreted as a situation

of human interaction in an institutionalized setting. Using theories and categories from

different disciplines - e.g., ethnomethodology, linguistics, cross-cultural studies - a docu-

mented mathematics classroom episode is re-analyzed. The analysis of the example is used

to identify four hidden dimensions in the classroom process and thus deficient areas of

research: The constitution of meaning through human interaction, the impact of institu-

tional settings, the development of personality, and the process of reducing chssroom

complexity. Consequences for teacher training towards more reflected classroom exper-

ience are drawn.

1. A NEARLY TRUE STORY

In old Russia two men meet in a train somewhere between Moscow and

Warsaw. Since the beaver collars indicate they are both merc hant s one of

the m asks: Where are you going? .. . To Moscow, the other replies. Hey,

says the first one, if you say you go to Moscow you mus t really want me to

believe that you go to Warsaw. But this train is headed for Moscow and this

makes certain that you travel to Moscow. So, why are you lying to me ?

These two men are talking not only a bout directions, but more, they are

concerned with their mutual expectations and with their subjective inter-

preta tion of what they 'really' do. Though the replying man tells the 't ruth '

from our contextual view, the asking protagonist understands the utterance

as a lie.

Let us t ry an explanation. (This is not to kill a joke by explaining, but

rather to use the explanatio n to illustrate a more imp orta nt problem.) Com-

peting merchants will hardly disclose good sources and addresses to each

other. The questioner therefore expects a non-d estin ation as answer. Knowing

these rules of the interact ion and hearing the obviously true destinat ion he

EducationalSmdiesinMathematies

11 (1980) 23-41. 0013-1954/80/0111-0023501.90

Copyright 9 1980 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.


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2 4

H E I N R I C H B A U E R S F E L D

m u s t c o n s t r u e a l ie . T h u s w e l a u g h a b o u t a m a n , w h o s e e m s t o b e t h e c a p t iv e

o f h i s e x p e c t a t i o n s . H e b e c a m e a c c u s t o m e d t o t h i s g a m e a n d f o r h i m i t i s

r e a l it y , h is r e a l it y . S e e n f r o m a m o r e g e n e r al p o i n t o f v i ew o u r ' t r u t h ' a b o u t

t h e c a se is n o b e t t e r , n o r m o r e v a l id , t h a n is h is ' t r u t h ' - a l t h o u g h w e e n j o y a

l a rg e r m a j o r i t y s u p p o r t in g o u r i n t e r p r e t a t i o n .

T h i s s t o r y a b o u t ' s i t u a t i o n s ' , ' r u l e s ' , ' e x p e c t a t i o n s a n d i n t e r p r e t a t i o n s ' , a n d

' s u b je c t iv e r e a l it ie s ' b r in g s m e d i r e c tl y t o m y t h e m e : h i d d e n d i m e n s i o n s in th e

s o -c a ll ed r e a l i ty o f a m a t h e m a t i c s c l a s sr o o m . A f t e r a s h o r t o v e r v i ew o f m a t h e -

m at i c s l ea rn ing as a soc i a l ac t i v i t y and t he ro l e o f r e l a t ed t heo ry , t he con-

s t it u t iv e p o w e r o f h u m a n i n t e r a c t i o n w il l b e c o n c r e t e l y d e m o n s t r a t e d w i t h a

d o c u m e n t e d c l a s s ro o m s i tu a t i o n . F o l l o w i n g t h i s , f o u r d e f i c ie n t a r e a s o f r e-

sea rch i n ma themat i c s educa t i on wi l l be i den t i f i ed and d i scussed wi th a v i ew to

c h a n g in g p a r a d i g m s o f r e s ea r c h . F i n a l ly , I w i ll c o m e b a c k t o m y m a i n c o n c e r n

o f p r e - se r v ic e a n d i n -s e rv i ce t e a c h e r t r a in i n g a n d m a k e s o m e p r e l im i n a r y

conc lus ion s fo r i t .

2. T H E C O N T R I B U T I O N O F S O C I A L S C I E N C E S

T o v ie w t h e l ea r ni n g a n d te a c h in g o f m a t h e m a t i c s a s a s oc ia l p r o c e s s - a

jo in t l y p rod uce d soc i a l s e t t l e m en t a s Lee S . Shu lm an pu t s i t ( s ee No te 3 ; a l so

B a u e r s fe l d , 1 9 7 8 ) - s e e m s t o b e a fa i rl y r e c e n t i ss u e. A l t h o u g h t h e a n c i e n t

G r e e k s p r o v id e d u s w i t h f a m o u s e x a m p l e s o f m a t h e m a t i c s i n s tr u c t io n t h r o u g h

dialogue (e .g . , P la to 's Menon w e s till d o n o t h a v e m u c h i n f o r m a t i o n a b o u t t h e

s o c ia l d i m e n s i o n s o f g e n e r a t in g m a t h e m a t i c a l k n o w l e d g e a n d o f d e v e l o p i n g

ind iv idua l ma themat i ca l power wi th in t he c l a s s room. Pa r t i cu l a r l y r e sea rcher s i n

m a t h e m a t i c s e d u c a t io n h a v e n o t s p e n t m u c h t i m e o n t h e s e d i m e n s i o n s o f

human in t e r ac t i on . O the r d i sc ip l i nes l ong s ince have p roduced r e l evan t r e sea rch

s p e c if ic t o t h e i r d i s c ip l in e s a l t h o u g h n o t t o t h e l e a r n in g o f m a t h e m a t i c s .

S p e a k i n g o f h i d d e n o r n e g l e c te d d i m e n s i o n s w i t h i n m a t h e m a t i c s e d u c a t i o n is

o n l y r e la t iv e l y t r u e i n t h e s e ns e t h a t r e s e a rc h e r s h a v e n o t m a d e u s e o f re l e v a n t

d e v e l o p m e n t s i n t h e o t h e r d i sc i pl in e s .

E x a m p l e s o f s u c h c o n t r i b u ti o n s c o m e f r o m s y m b o l ic i n te r a c ti o n i sm

( B l u m e r , 1 9 6 9 ; G o f f m a n , 1 9 6 9 ) , l in g u is ti c s ( G u m p e r z , 1 9 7 2 ; H e r r l it z , 1 9 7 7 ) ,

e t h n o m e t h o d o l o g y ( C ic o u re l, 1 9 7 4 ; M e h a n , 1 9 7 5 , 1 9 7 9 ) . T h e d e m a r c a t i o n

among these d i sc ip l i nes i s d i f f i cu l t , because o f t he i r i nc reas ing i n t egra t i on

t h r o u g h i n t e rd i s c ip l in a r y p r o c e d u r e s - p r o c e d u r e s w h i c h m i g h t al so b e n e f i t

m a t h e m a t i c s e d u c a t i o n . T o p i c s s u c h a s t h e g e n e r a t i o n o f m e a n i n g a n d t h e

fun c t i on o f language i n soc i a l s i t ua t ions , t he ac tua l shap ing o f beh av ior and

c o g n i ti v e p e r f o r m a n c e t h r o u g h h u m a n i n t e r a c t io n , th e s p e c i f i ty o f c o m m u n i -

ca t i on i n i ns t i t u t i ona l i zed se t t i ngs , e t c . , apparen t l y fo rce i n t e rd i sc ip l i na ry

a p p r o a c h e s a n d h a v e fo r m e d a n ew t y p e o f h u m a n s c ie n ce .


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S O - C A L LE D R E A L I T Y O F A M A T H E M A T I C S C L A S S R O O M 25

Th ere i s a f ina l po in t to mak e in th i s in i t ia l ove rv iew. F ro m the ve ry beg in-

n ing m y con cern i s bo th pragm at ic , a s we l l a s h igh ly theore t i ca l . I t i s p ragm at ic

s i n c e m y g o a l i s t o i m p ro v e m a t h e m a t i c s t e a c h i n g a n d l e a rn i n g t h ro u g h b o t h

t e a c h e r ' s a n d s t u d e n t ' s ac t io n s . It is t h e o re t i c a l b e c a u s e t h e i m p ro v e m e n t

a n d t h e d i f f e r e n t i a t e d o r i e n t a t i o n r e q u i r e t h e m o s t so p h i s t ic a t e d , r e p ro -

duc ib le the ore t i ca l f ram ew ork avai lab le . Bo th aspec t s , the p ragm at ic and the

theor e t i ca l , ac t ion and re f lec t ion , a re deep ly in te rwov en . Albe r t E ins te in has

pu t i t sha rp ly : I t i s a lways the theo ry wh ich dec ides wh a t can be obse rv ed

(M e h ra , 1 9 7 3 , p . 2 6 9 ) . T h o u g h f ro m a p h y s i ci st o n e m i g h t e x p e c t t o h e a r t h e

c o m p l e m e n t a ry s t a t e m e n t : I t i s a lw a y s t h e o b s e rv a t io n (o r t h e ' r e a l it y ' ) w h i c h

d e c i d e s t h e t h e o ry . I n t h e h u m a n s c i e n c e s d i f f e r e n t a c t i o n s a n d d i f f e r e n t

c o n c e rn s o f t e n p ro d u c e d i f f e r e n t t h e o r i e s , a n d d i f f e r e n t t h e o r i e s i n t u rn

pro du ce d i f fe ren t rea li t ie s .

This po in t i s o f ten exp res sed in e duc a t ion b y say ing tha t re sea rch f ind ings ,

l ike a th eo ry on ce r ta in c la s s room events , need spec ia l t rans fo rm at ion in to

t e a c h i n g p r a c t i c e ; o r , t h a t t h e r e is l i tt l e d i r e c t c o n n e c t i o n b e t w e e n r e se a r ch

a n d e d u c a t i o n a l p r a c t i c e (F re d K e r l in g e r , 1 9 7 7 , p . 5 ) ; o r , . . . w h a t is g o o d

t h e o ry fo r o n e p u rp o s e is n o t a g o o d t h e o ry fo r a n o t h e r (E rn e s t H i lg a rd ,

see Note 2 ) . Al l these s ta tements a re on ly d i f fe ren t expres s ions in an edu-

ca t iona l s e t t ing o f the general po in t m ade b y E ins te in .

3. T H E C O N S T I T U T I V E P O W E R O F H U M A N IN T E R A C T I O N

T w o d i s se r t a ti o n s m a rk c o rn e r s t o n e s fo r t h e d i sc u ss io n o f h u m a n i n t e r a c t i o n

i n t h e m a t h e m a t i c s c l a s sro o m : G e o rg e B e rn a rd S h irk 's 'A n E x a m i n a t i o n o f

C o n c e p t u a l F r a m e w o rk o f B e g in n in g M a t h e m a t i c s T e a c h e r s ' ( 1 9 7 2 ) , a n d

S tan ley Er lwanger ' s 'Case S tud ies o f C hi ld ren 's Co ncep t ions o f M athemat ics '

(1974) . Both were d i rec ted by Jack A. Eas ley , Unive rs i ty o f IHinois , Urbana /

Champaign .

Er lwanger ' s case s tud ies a re re la ted to p rograms f rom Ind iv idua l ly P re -

s c r ib e d In s t ru c t i o n . H i s d o c u m e n t a t i o n o f s t u d e n t s ' m a t h e m a t i c a l m i s c o n -

c e p t i o n s a n d d e f i c ie n c i e s d e m o n s t r a t e h o w m a t h e m a t i c s l e a rn i n g c an b e

d a m a ge d b y r e st ri c te d t e a c h e r - s t u d e n t c o m m u n i c a t i o n - a r e s tr i ct io n w h i c h

l e a d s t o t h e n e a r l y t o t a l a b s e n ce o f n e g o t i a ti o n s o v e r m e a n i n g s. I t s h o u l d b e

c l e a r t h a t t h e f a d in g f a s c i n a ti o n o f p ro g ra m m e d i n s t ru c t io n d o e s n o t o f f e r a

s a t i s f a c t o ry e x p l a n a t i o n fo r t h e n o n -a p p e a ra n c e o f f u r t h e r r e s e a r c h w i t h

such case s tudies .

Sh i rk ' s work wi th beg inn ing teachers g ives a s t r ik ing example of the in f lu -

e n c e o f s u b je c t iv e t h e o r i e s a b o u t m a t h e m a t i c s t e a c h in g , t h e s t u d e n t ' s r o l e ,

a n d t e a c h e r ' s r o l e . M o re o v e r , h i s d o c u m e n t s g iv e an i d e a o f t h e f r a g il i ty o f


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2 6


c l a s s r o o m d i s c o u r se a n d o f t h e i m p a c t o f th e s e s o c i al si tu a t i o n s o n m a t h e -

m a t i c s l ea r n in g . T h e r e f o r e , 1 w i ll t a k e a b r i e f e x a m p l e f r o m S h i r k ' s t r a n s c r ip t s

a n d u s e i t f o r c o m m e n t s b a s ed o n t h e o r i e s f r o m o t h e r h u m a n s c ie n c es .

T h e e p i s o d e p r e s e n t s a n e a r ly p a r t o f a b e g i n n in g t e a c h e r ' s le s s o n w i t h

e igh th g rader s a t Urbana Jun ior High School . The top i c i s about s l i des , f l i ps ,

a n d t u r n s f r o m Motion eometry w r i t t e n b y R u s s e l Z w o y e r a n d J o M c K e e b y

P h i l l i p s ( a p r o d u c t o f M a x B e b e r m a n ' s U I C S M ) . I n t h e p r e c e d i n g l e s s o n T o m ,

the t eache r , has d e f ined pa ra l le l l ines i n t e rm s o f s li des. Th e le s son und er

d i s c u ss i o n o p e n s w i t h s t u d e n t ' s w o r k i n g o n p o s i t iv e e x a m p l e s . T h e e p i s o d e

w h i c h I a m g o i n g t o r e a n a ly z e s t a rt s w i t h l in e 3 9 o f t h e t r a n s c i p t. T h e t e a c h e r

presen t s a counte r - exa m ple , two in t e r sec t i ng l i nes ( see F igure 1 ).

J

f

i

Fig. 1

" T o m , ta p e o f 4 / 4 / 7 2 " ( f r o m G . B. S h ir k , 1 9 7 2, p p . 1 7 3 - 1 7 4 ) .

T - t e a c h e r , T o m K - s t u d e n t , K e v i n R - s t u d e n t , R e g g ie

3 9 T :

4 0

41 K:

4 2 T :

4 3 K :

4 4 T :

4 5

4 6 K :

4 7 T :

4 8 K :

4 9 T :

5 0 R :

51 K:

5 2 T :

53

5 4 K :

55 T :

56

. . . . . . l o o k a t t h e n e x t f ig u r e , r ig h t b e l o w

i t . Ya . . . . now , a re t hose tw o l i nes pa ra ll e l?

N o p e .

W h y n o t ?

T h e y c r o ss e a c h o t h e r .

O K , b u t , u h . . . . a c c o r d i n g t o w h a t I ' v e s a id a b o u t p a ra l le l

l in e s, w h a t c a n ' t t h e y d o ?

C a n ' t c ro ss t h e m . . . ( ?)

What?

C a n ' t . . . t h e y w o n ' t , t h ey w o n ' t . . . I 'd r at he r n o t . . . ( ID )

What Reggie?

U m .

9 T h e y w o n ' t c om e t o g e t h e r . . .

OK , bu t w ha t d id I say , wh a t d id I s ay on the f ir s t fi gure?

w h a t c o u l d y o u d o t o g e t f r o m o n e l in e t o t h e o t h e r ?

Slide.

O K . T h e r e ' s a s li de a r r o w t h a t ' l l g o . . . . t h a t ' l l t a k e o n e

l ine i n to t he o the r .


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SO-CALLED REALITY OF A MATHE MATIC S CLASSROOM 27

57 K:

58 T:

59

60

61 R:

62 T:

63

64 R:

65 T:

66 R:

67 T:

68

69 R:

70 T:

71 R:

72 T:

73 R:

74 T:

75 R:

76 T:

77 R:

78 T:

79 R:

80 T:

81 R:

82 T:

83

84

85

86

9 Arrow.

Right? Is there a slide arrow . . . . on the second figure?

9 Reggie? On the second figure, can you draw a slide

arrow that'll go from one of these lines to the other?

(?) Not any more.

Like th e. . , a slide ar ro w, .. , will that take the,

9 will that go from one to the other?

I don't know.

Well, you remem ber what a slide arrow did?

Hm?

It . . . . it moved a figure along that slide. Can you draw

one that ' ll do that?

Oh.

OK, is that a slide arrow?

? )

What is that ? What . . . . what did you draw?

A circle.

Well, you remember 9

(laughter)

9 What that was called?

? )

Does anybody re me mbe r . . .

A r o t a t i n g . . .

OK, we called it a wh a t ? .. . A turn?

A tu rn arrow 9

OK, so what you were starting to draw in was a turn arrow,

right? But I'm just talking about slides9 Can you draw

a slide arrow? Just a straight line, a straight arrow,

that'll go from one, that'l l take one line to the other?

(pause for students to wor k) .. .

First I shall follow Shirk's own interpre tation and then add my c omment s

later:

In this episode, Tom was working for a compound goal which he wanted the students to

reach. . , that they recognize the lines illustrated were not parallel and realize there was a

reason for it through the definition of slides. The students did accomplish the first part

of this goal; but when they invoked a reason other than that which Tom sought, it was

rejected by him as being inappropriate. The students weren't connecting the lessons

together. Tom could only see this as a completely unexpected deficiency in the student's

understanding of a lesson which he believed had been learned earlier, so he turned the

lesson toward this deficiency in an effort to correct for it. (Shirk, 1972, p. 46.)


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H E I N R I C H B U E R S F E L D

The critical aspect centers aro un d To m 's expectation that the students wo uld know all

of the consequences of the definition of 'parallel'. Not seeing this, T orn interpreted their

problem as having to do w ith slides; and this bothered him for he b elieved that slides

had been adequately c ov ered an d, therefore, the students should know them. H e was

also assuming that the students wo uld appreciate everything that he said and therefore,

the problem would have to lie elsewhere, i .e. , in their more b asic preparations wh ich

he thought had been covered earlier.

ibid,

p. 46.)

Fo r t he r e - ana lys i s i t is use fu l t o no t e t he m a jor sh i f ts i n t he s t ud en t -

t e a c h e r i n t e r p r e t a t i o n o f t h e s i tu a t io n . T h e e p i s o d e t h e n s p l it s i n t o f o u r p a r t s .

P ar t 1 , l i nes 39-51: T h e t e a c h e r d o e s n o t s u c c e e d i n u s i n g t h e c o u n t e r -

example t o i n f e r t ha t i n t e r sec t i ng l i nes cannot be pa ra l l e l ( t he re i s no s l i de

a r r o w w h i c h w o u l d m o v e t h e l in e s t o g e t h e r ) . U n e x p e c t e d l y h e r ec e iv e s a m u c h

s i m p l e r a n s w e r , n o t i n v o k i n g m o t i o n g e o m e t r y c o n c e p t s , t h e y c r o s s e a c h

o th e r ( li ne 43) . A lbe i t cor r ec t , t he t eache r r e j ec t s t he answer a s i nad equ a t e

o n t h e b a s i s o n l y t h a t t h e y s h o u l d r e m e m b e r w h a t h e s a i d i n t h e p r e v i o u s

e x a m p l e . i b M , p . 47 . )

T h e s t u d e n t s b e c o m e c o n f u s e d a n d u n c e r t a i n a s e v i d e n c ed b y K ' s s t a m m e r -

ing and br i e f w i thdraw al ( l ine 48) .

Part 11, l ines 52-64: T o m i s n o w d i r e c t i n g t h e s t u d e n t ' s a t t e n t i o n t o w a r d

t h e d r a w i n g a g a i n in a n e f f o r t t o g e t t h e m t o s e e t h e c o n n e c t i o n b e t w e e n i t

and t he s l i de . i bM, p . 4 8 . ) R e p e a t e d l y h e u se s t h e k e y w o r d s l id e a r r o w .

The s tuden t s t ry t o guess t he t ea che r ' s i n t en t i ons . The i r answer s a r e shor t

a n d c a u t i o u s : S l i d e ( li ne 5 4 ) a n d . . . a r r o w ( li n e 5 7 ) . T h e i r u n c e r t a i n t y

i n cr e a se s . T h u s u n d e r t h e t e a c h e r ' s p r e s si n g R eg g ie m o d i f i e s h i s a n s w e r s f r o m

N o t a n y m o r e ( li ne 6 1 ) t o I d o n ' t k n o w ( li ne 6 4 ) . W i t h t h e i r i n it ia l

e f f o r t s r e je c t e d , a n d T o m e m p h a s i n g s lid es , t h e s t u d e n t s b e g i n t o l o o k a r o u n d

for wa ys t o s li de t he tw o l ines t og e the r fo r t he re w as no th ing i n t he ea r l ie r

p o r t i o n s o f th e l e ss o n s a b o u t ' n o s li d e ' o r ' n o t p a r a l l e l' .

ib id ,

p . 4 8 . )

P ar t I I I , l i nes 65-86:

St il l t he t eacher has n o t g iven up h i s i n i ti a l a im.

H i s i m p u l s e W e l l , y o u r e m e m b e r ( li n e 7 4 ) is a n a t t e m p t t o ge t R e g g ie

t o p u t t o g e t h e r t h e f o r m a l p r in c ip l es b y p o i n ti n g o u t t o h i m t h a t w h a t h e h a d

d r a w n w e r e t u r n s r a t h e r t h a n s l i d e s ib id , p . 49) . Reggie ' s ' f a i l u r e ' ( a s seen

f r o m t h e t e a c h e r ' s e y e s ) j u s t if i e s t h e c a u s a l a s c r i p ti o n t h a t t h e s t u d e n t s h a v e

f o r g o t t e n a ll a b o u t s lid es . T h e r e f o r e t h e t e a c h e r b e g i n s t o ' r e t e a c h ' th e c o n c e p t

t o w a r d s t h e e n d o f t h i s s e c ti o n .

T h e s t u d e n t s , h o w e v e r , i n a n e f f o r t t o c o m e u p w i t h th e a n sw e r t h e y

tho ug ht T om was l ook ing fo r , ( nam ely , a slide a r ro w) i nven t ed s lide l ines

b e t w e e n t h e t w o i n t er s e c ti n g l i n e s

ib id ,

p. 49) , so Reggie a t l ine 69 (see

F i g u r e 2 ). R e g g i e 's m i s i n t e r p r e t in g t h e t e a c h e r ' s q u e s t io n ( l in e s 7 4 - 7 6 ) W h a t

tha t was ca l l ed? i s com ple t e ly i n l ine wi th t h i s l ook in g fo r s lides. Th e d rawings

o f t h e s t u d e n t s i n t h i s p a r t ( s e e F i g u r e 3 ) e x p o s e t h e e x t e n t t o w h i c h t h e


7/19

S O - C A L L E D R E A L I T Y O F A M A T H E M A T I C S C L A S S R O O M 2 9

Fig. 2 . Reggie a t l ine 69 . Fig . 3 . Stud ents a t l ine 85

t e a c h e r ' s p re s s in g h a s c o n t r i b u t e d t o ' s p o i li n g ' t h e s t u d e n t ' s c o n c e p t o f ' s l i d e '

a n d o f ' s li d e a r r o w ' . N e v e r th e l e s s t h e t e a c h e r ' s i n t e r p r e t a t i o n t h a t t h e s t u d e n t s

h a v e n o t l e a r n e d t h e i r s l id e l e s s o n is n o t d i m i n i s h e d b u t r a t h e r r e i n f o r c e d .

Par t IV , l ines 87-155: ( T h is c o n c l u d i n g p o r t i o n o f t h e e p i s o d e is n o t

i n c l u d e d i n t h e a b o v e q u o t e . ) T h e r e m a i n d e r o f t h i s f i rs t e p i s o d e i n v ol v e s

T o m ' s a t t e m p t s t o r e t e a c h t h e c o n c e p t o f sl id e s t o t h e s t u d e n t s . i bM, p . 5 1 . )

A n d h e w o r k s o n t h a t r e h e a rs a l u n t i l h e g et s t h e c o n v i c t io n t h a t h e h a s

r e c o n n e c t e d t h e s t u d e n t s w i t h s l id e s a n d p a r a l l el l i n e s .

i bM,

p . 5 2 . )

O n a m o r e g e n e r a l l e v e l S h i r k e x p l a i n s t h e e p i s o d e u s i n g t h e t e r m s 's p l i t

p e r s o n a l i t y ' a n d ' g u e ss i ng a h e a d ' ( th e l a t t e r f r o m J o h n H o l t , 1 9 6 4 ).

A ' sp l it pe rs on a l i ty ' . . , occurs when the teacher is t eaching one lesson and the s tuden ts ,

in an e f fo r t in ' p sych ou t ' the teacher , a re ac tua l ly lea rn ing an o th e r . . . The ' sp l i t pe rson-

a l i ty ' in the le sson occur red as a resu l t o f the concep tua l f rameworks which governed

To m 's act ions, ra the r than being a part o f those frame work s themselves. Tom, act ing in

accordance with his f rameworks, in terpreted the s tudent ' s behavior in a cer ta in way and

then acted in a ma nner consis ten t with his conceptua l f rameworks. Fo r their par t , the

studen ts a t tem pte d to guess ahead and therefo re acted differently . These act ions resulted

in one pa th taken by th e s tuden ts whi le Tom was t ry ing to lead them a long ano ther .

(Shirk, 1972, p. 43.)

I n t h e a b o v e a n a l y s i s S h i r k u s e s t h e c l a ss i c a l r e l a t i o n o f c a u s e a n d e f f e c t a s

h e m a t c h e s c a u s e s w i t h p e r s o n a l a t t r ib u t e s . T h a t i s, h e t r a c e s t h e o u t c o m e s

b a c k t o p r o p e r t i e s a n d a c t i o n s o f s i n g le in d i v i d u a l s . A s a r e s u l t h e i s l e d t o

s o m e w h a t d i s c o u r ag i n g c o n c lu s i o n s a n d r e c o m m e n d a t i o n s . F r o m h i s f i n d in g

t h a t t h e r e w a s n o c h a n ge d i s c e r n ib l e w i t h i n t h e c o n c e p t u a l f r a m e w o r k s

h e c o n c l u d e s t h a t t h e f u t u r e t e a c h e r e d u c a t i o n p r o g r a m s m u s t b e s o d e s i g ne d

s o a s t o b e a s si m i l a bl e t o t h e p r e e x i s ti n g c o n c e p t u a l f r a m e w o r k s o f th e

s t u d e n t t e a c h e r ( S h i r k , 1 9 7 2 , p . 1 6 5 ) . I s h a l l t r y a n a l t e r n a t e a n s w e r t o t h e

t e a c h e r t r a in i n g p r o b l e m l a t e r , b u t f i rs t le t m e g iv e a n i n t e r p r e t a t i o n o f t h e

e p i s o d e f r o m a d i f fe r e n t p a r a d i g m o f s o c i al a c ti o n .

A d e s c r i p t i o n o f t h e s i t u a t i o n a s c o n s t i t u t e d t h r o u g h t h e i n t e r a c t i o n o f

t h e p a r t i c i p a n t s c a n c h a l le n g e t h e u s u a l c a u sa l m o d e l , cast d o u b t o n p r e d i c ti v e

c o n c l u s i o n s a n d p o s s i b l y s he d l i g h t o n t h e u s e o f l a n g u ag e in t h e m a t h e m a t i c s

c l a s s r o o m .

The cons t i tut ion of the social s i tuat ion. P r i n c i p a l l y , a n d t a k e n a s a p i e c e o f

a n o n g o i n g p r o c e s s , t h e e p i s o d e c a n n o t b e r e c o n s t r u c t e d s u f f i c i e n t l y - n e i t h e r


8/19

3 0


fro m p ersona l va riab les , f rom cha rac te r i s t ic s o f the s ing le pa r t i c ipan t , o r f rom

t h e d o c u m e n t e d s p e e c h p ro d u c t i o n . H e n c e , f r o m a d d i t i o n a l i n te rv i ew s w i t h

t h e t e a c h e r , a n d f ro m e s sa y s a n d ' c o m m e n t c a rd s' w h i c h t h e t e a c h e r h a d t o

w r i t e , S h i rk h a s d i s t il le d t h e t e a c h e r ' s c o n c e p t u a l i z a t io n o f m a t h e m a t i c s

e d u c a t i o n , o f h i s r o l e a s a t e a c h e r , a n d o f t h e s t u d e n t ' s r o le . S h k k u s e s t h i s

s e t o f s t a t e m e n t s o n l y t o e x p l a i n t h e t e a c h e r ' s m o v e s .

W e d o n o t h a v e a n y c o m p a ra b l e i n fo rm a t i o n a b o u t t h e s t u d e n t s . H o w e v e r ,

t h e i n t er a c ti v e n a t u r e o f t h e p ro c e ss a n d t h e m u t u a l r e la t e d n e ss o f e x p e c t a -

t i o n s a n d i n t e rp r e t a t i o n s c a n b e p a r t i a l l y r e c o n s t ru c t e d f ro m t h e t r a n s c r i p t .

Fo r the purp ose of th i s ana lys i s cons ide r Table I . Th e f i rs t tw o l ines in Table I

r e c o n s t ru c t t h e t e a c h e r ' s i n t e rp r e t a t i o n a n d t h e s t u d e n t ' s i n t e rp r e t a t i o n o f

t h e fo u r p a r t s. (A m o re d e t a il e d a n a ly s is c an b e d e v e l o p e d t h ro u g h fo l lo w i n g

t h e d i sc u s si o n s t e p -b y - st e p .) C o m p a r i n g t h e m u t u a l i n t e rp r e t a t i o n s in c o l u m n s

gives a ro ugh b u t suf f i c ien t ly c lea r idea .

The teacher ' s immedia te ob jec t ives change fo l lowing h i s changing in te r -

p r e t a t i o n s o f t h e p ro c e s s . ' G u e s s i n g a h e a d ' t h e s t u d e n t ' s i n t e rp r e t a t i o n o f t h e

t e a c h e r ' s i n t e n t i o n c h a ng e s a s w e l l. B y n o m e a n s a r e t h e a c t i o n s o f th e t w o

s id e s, te a c h e r a n d s t u d e n t s , r e a c ti o n s o n l y t o t h e p r e c e d in g m o v e o f t h e o t h e r

s ide . I t i s commonly be l i eved tha t ind iv idua l s reac t to the ac t ions o f ano the r

w h e n i n f a c t t h e y r e a c t t o t h e i r s e l f- c o n s t ru c t e d i n t e rp r e t a t i o n . Y e t , 'r e a c t i o n '

i s mis lead ing . Fa r f rom the s imple mode l o f s t imulus - response the pa r t i c ipan t ' s

ac t ions in th i s soc ial s itua t ion a re genera ted th rou gh com pl ica ted , in te rna l

re f lec t ive ac t iv i ty . Th is sub jec t ive re f lec t ive ac t iv i ty t akes in to account no t

o n l y t h e a c t u al a n d p e rc ei v a b le m o v e s o f t h e o t h e r s b u t a ls o t h e m o re g e n e ral

i n t e rp r e t a t i o n s o f t h e s i tu a t i o n , a n d o n e ' s o w n ro l e in t h a t s i tu a t io n . F u r t h e r -

m o re , a c t u a l i n t e rp r e ta t i o n s o f r e la t e d fo rm e r e x p e r i e n c e s e x e rc is e an i n f lu e n c e

o n t h e c u r r e n t o n g o i n g i n t e rp r e t a t i o n . E a c h p a r t i c i p a n t ' s a c t i o n s c o n t r i b u t e

t o t h e c h a n g e o f t h e o t h e r ' s , o f th e i r i n t e rp r e t a t i o n a n d t h e i r a c ti o n s . A n d

t h ro u g h t h is p ro c e ss t h e y c o n t r i b u t e t o t h e c h a n g e o f t h e p a r t i c i p a n t 's o w n

i n t e rp r e t a t i o n a n d a c t i o n . T h u s i t b e c o m e s r e a s o n a b l e t o s p e a k o f t h e ' c o n -

s t i tu t ion ' o f the soc ia l s i tua t ion (Mehan , 1975) . More prec i se ly : The soc ia l

s i tu a t i o n is c o n s t i t u t e d a t e v er y m o m e n t t h ro u g h t h e i n te r a c t io n o f r e fl e c ti v e

sub jec t s E t h n o m e t h o d o l o g i s t s t h e r e fo re d e s cr i be r e a l i t y as a r e f le x i v e a c t i v i t y

(M e h a n a n d W o o d , 1 9 7 5 , p . 8 ) .

T h e e p i s o d e u n d e r d i sc u s si o n is a n e x a m p l e o f t h e c o n s t i tu t i v e p o w e r o f

hu m an in te rac t ion , i .e . , the in te rac t ion co ns t ruc t s the sub jec t s ' va r ious rea l it i es .

Bo th t ea cher and s tudents ac t accord ing to the i r ac tua l sub jec t ive rea l it i es . The

s tuden ts d raw turn a r rows fo r s lides; the t eacher d iagnoses lea rn ing de f ic ienc ies;

a n d t e a c h e r a n d s t u d e n t s w o rk c l e ar l y a t c ro s s p u rp o s e s co n v i n c e d t h e y u n d e r -

s tand the s i tua t ion , c lea r ly .


9/19


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10/19

3 2


Every mom ent is mysterious, as the u nderstood horizon of th e mom ent is inexhaustible.

Every interpretive act ind exe s this mystery in an unpredictable w ay . A person's every

action is thus creative; it reflexively alters the w orld. The person begin s with certain

m aterials that set limits, and the n ac ts and in acting alters those lim its, (Me han and Wood,

1975, p. 203.)

Forms of life are alw ays forms o f life forming. Realities are alway s realities becoming.

(Melvin PoUner, in M ehan, 197 5, p. 32.)

The constitution of meaning

N o t o n l y s u b j e c t i v e i n t e r p r e t a t i o n a n d

assessment ch ange dur ing t h e p roc ess bu t a l so t he a ims , t he ac tua l t a sks , and

e v e n t h e c o n c e p t s . A c o m p a r i s o n o f t h e t e a c h e r ' s a n d th e s t u d e n t ' s v ie w d u r i n g

t h e e p i s o d e o f t h e t a s k (s e e T a b l e I , t h i r d a n d f o u r t h l in e ) e m p h a s i z e s t h e

s e m a n t i c c h a n g e o f t h e p r o b l e m s i tu a t io n . C l e ar ly , e a c h p a r t i c i p a n t ' s v i ew o f

t h e a c t u a l t a s k t o b e d o n e is d i f fe r e n t a n d v a r y i n g d u r in g t h e c o u r s e o f t h e

e p i s o d e . T h e t a s k m u s t b e u n d e r s t o o d a s a f u n c t i o n o f th e s i tu a t io n .

For t he s t uden t s t he concep t o f ' s l i de a r row ' a l so va r i e s ac ross t he ep i sode

( see Tab l e I , fi f t h l i ne) . I n t he beg inn ing t he p rev ious ex per i enc e wi th pa ra l l e l

l in e s ( a n d s lid e a r r o w s m o v i n g t h e m t o g e t h e r ) i s d o m i n a n t . T h e i n t e r v e n t i o n

o f t h e c o u n t e r - e x a m p l e a n d t h e f o l l o w i n g d i s c u s s i o n w i t h t h e t e a c h e r p r o d u c e

d o u b t s a b o u t w h e r e t o l o c a t e t h e b o r d e r l i n e s o f t h e c o n c e p t . W h a t is a n d w h a t

i s n o t t o b e i n c lu d e d ? T h e s t u d e n t ' s i n t e r p r e t a t i o n o f t h e t e a c h e r ' s i n s is t en t

ques t i ons i nc reas ing ly spo i l s t he concep t and l eads t o an a rb i t r a ry guess a s

t o t h e t r u e m e a n i n g . A n y t y p e o f a r r o w , c u r v ed o r u p - a n d - d o w n , i s u s e d b y

the s t uden t s ( see F igures 2 and 3) .

W i t h o u t f u r t h e r i n f o r m a t i o n o n t h e s t u d e n t ' s t h i n k i n g t h e e f f e c t o f t h e

re t each ing i s d i f f i cu l t to ev a lua t e . Sure ly t he r e s idua l s t a tus o f t he c on cep t

' s l i de a r row ' w i l l d i f f e r f rom i t s i n i t i a l s t a tus bu t i t may no t be improved .

D u e t o t h e h i g h a ff e c t iv e l o a d d u r i n g P a rt I I , t h e c o n c e p t n o w m i g h t b e v u ln e r -

ab l e t o fu tu re mi sunder s t and ing i n s imi l a r s i t ua t i ons .

Thus , t he l og i ca l p r i nc ip l e o f i den t i t y i s no t app l i cab l e . The word ' s l i de

a r r o w ' d o e s n o t m e a n t h e s a m e t o e v e r y p a r t i c i p a n t . M o r e o v e r , t h e m e a n i n g

c h a n g e s d u r i n g t h e e p i s o d e r e p e a t e d l y a n d r e m a r k a b l y . B u t , i f p r o b l e m s a n d

c o n c e p t s b e c o m e f u n c t i o n s o f t h e s i t u a t io n i n s t e ad o f b e i n g c o n s t a n t a n d

s t a b l e , i t t h e n b e c o m e s n e c e s s a r y t o c o n s i d e r t h e s o c i a l c o n s t i t u t i o n o f m e a n -

ing, i .e. ,

the constitution of meaning through human interaction

H e r b e r t B l u m e r m a k e s t h e s a m e p o i n t : S y m b o l i c i n t e r a c t io n i s m s ee s

m e a n i n g s a s s o c i a l p r o d u c t s , a s c r e a t i o n s t h a t a r e f o r m e d i n a n d t h r o u g h t h e

d e f in i n g a c t iv i ti e s o f p e o p l e a s t h e y i n t e r a c t . ( 1 9 6 9 , p . 5 .) H o w e v e r , a s a

m a t t e r o f p r i n c i p l e , t h e r e i s s m a l l c h a n c e o f p r e d i c t i n g t h e o u t c o m e s o f s u c h

e p i s o d e s a t t h e i r b e g i n n i n g . N o r i s th e r e m u c h c h a n c e o f m a k i n g p r e d i c ti o n s

a b o u t a l a te r s ta g e f r o m t h e b a s i s o f p r e c e d i n g o n e . S i n ce w e c a n n o t a s c r ib e

t h e c o n s t i t u t i o n o f m e a n i n g t o o n e s in gle p a r t i c i p a n t ( e . g ., t h e t e a c h e r ) , w e


11/19


are n o t in a po s i t ion to use causa l mode ls a s adequ a te desc r ip tors o f ind iv idua l

s o ci a l i n t e r a c t i o n -p a r t i c u l a r l y n o t o f m a t h e m a t i c s t e a c h i n g a n d l ea rni n g .

A s e v e ry d a y m e a n i n g s d o n o t m e e t t h e c a n o n s o f l og i c, t h e y a re tr a n s fo rm e d

by l i t e ra l desc r ip t ion . These t rans formed meanings a re amenable to causa l

mod e ls . Eve ry day l i fe i s no t . (Mehan , 1975 , p . 66 . )

At th i s po in t the ana lysi s l eads in to a revo lu t ion of fundam enta l pa rad igm

(fo l lowing T hom as S . Ku hn , 1962) . In the hu ma n sc iences ru les a re used d i f fe r -

e n t l y f ro m i n t h e n a t u r a l s c ie n c es . F o r e x a m p l e , t h e y a re ru l es a b o u t t h e c o n -

s t i tu t ing of s i tua t ions and meanings ra the r than ru les about the s i tua t ions and

meanings themse lves . Th ey a re ru les abo ut s t ruc tur ing the p rocess ra the r tha n

a b o u t s t r u c t u r e o f t h e p ro c e s s. (S e e H u g h M e h a n 's t it l e ' S t ru c t u r i n g th e S c h o o l

S t ru c t u r e ' , 1 9 7 8 ) . In t h e h u m a n s c ie n c es t h e i n t e rp r e t a t i o n -a s s e s s m e n t p a r a d ig m

wi l l rep lace the ca use -e f fe c t pa rad igm bor row ed f rom the n a tura l s c iences.

The role of language

T h e a b o v e e p i s o d e a ls o p ro m p t s a n e w l o o k a t t h e

role o f language in math em atics teaching. Y ears ago linguis t ic research wou ld

have used the u t t e rances in Sh i rk ' s ep i sode for a syn tac t i ca l and a semant ic

ana lys i s o f the ma te r ia l . For example , the ana lys i s might have po in ted to a

lack of ad jec t ives . Or , i t m ight have no ted tha t the un i t s o f speech cons i s t

o n l y o f s h o r t o r b ro k e n s e n t e n c e s , t h a t p a r a t a c t i c a l s t r u c t u r e s d o m i n a t e , o r

t h a t m a n y d e i c t i c w o rd s a p p e a r (w o rd s w h i c h d e m o n s t r a t e o r p o i n t t o s o m e -

t h i n g - ' h e r e ' , ' t h i s ', ' t h e r e ' , e tc . ) . W i t h o u t a n y a d d i t io n a l i n fo rm a t i o n th e

ana lys t w ould speak of a ' r e s t r i c ted code ' and pe rhaps a re su l ting poo r ly

d e v e l o p e d m e a n i n g .

I t i s t h e fu n d a m e n t a l i d e a o f C h o m s k y ' s t h e o ry (1 9 5 7 ) t h a t w e c a n n o t

r e c o n s t ru c t t h e c o n s t i t u t i o n a l p ro c e s s o f c o m m u n i c a t i o n f ro m t h e s u r f a c e ,

tha t i s, f rom record ed speech . Ne i the r 'd i scovery-m ode ls ' ( fo r the d i scovery

o f n e w g ra m m a rs ) n o r ' d e c i s i o n -m o d e l s ' ( f o r t h e d e c i s i o n a b o u t t h e a d e q u a c y

o f a g ra m m a r ) w ill w o rk . C h o m s k y m a i n t a in s t h a t t h e se m o d e l s o n l y m a k e

use o f the l ingu is ti c da ta . S ince we have to ana lyze the ru les o f s truc tur ing

c o m m u n i c a t i o n w e m u s t a n a l y z e e v a l ua t iv e s t r u c t u r e s o f in d i v id u a l i n t e r-

pre ta t ion and as ses sment and th i s ana lys i s requ i res an *eva lua t ion-mode l '

( fo r the eva lua t ion among ex i s t ing grammars ) . At th i s po in t the re i s a change

of pa rad igm f rom objec t ive ( l ingu is t i c ) da ta to in te rpre ta t ive s t ruc tures a s the

objec t o f ana lys i s . S ince Chomsky ' s work soc io l inguis t s have inc reas ing ly

s tud ied the use o f l anguage in soc ia l in te rac t ion .

An in te res t ing and he lpfu l i s sue i s the concept o f

indexicality

I f t h e r e

i s n o fu r t h e r i n fo rm a t i o n , t h e n m o s t p e o p l e w i l l n o t u n d e r s t a n d t h e d i s c o u r s e

i n l in e s 4 1 -5 1 o f t h e e p i so d e :

4 1 K : N o p e .


12/19

3 4


4 2 T :

4 3 K :

4 4 T :

45

4 6 K :

4 7 T :

4 8 K :

4 9 T :

W h y n o t ?

T h e y c r o s s e a c h o t h e r .

O K , b u t , u h , . . , a c c o rd i n g t o w h a t I ' v e sa id a b o u t

pa ra l le l l i nes, w ha t can ' t t hey do?

C a n ' t c ro ss t h e m . . . ( ?)

What?

C a n ' t . . . t h ey w o n ' t , t h ey w o n ' t . . . I 'd r at he r n o t . . . ( ID )

What Reggie?

5 0 R : U m .

5 1 K : . . . T h e y w o n ' t c o m e t o g e t h e r . . . ( S h ir k , 1 9 7 2 , p . 1 7 3) .

M a t h e m a t i c s e d u c a t o r s m a y e v e n b e c o m e d o u b t f u l a b o u t t h e t o p i c a n d

m e a n i n g o f t h e d i s cu s s io n i f t h e y d o n ' t k n o w t h a t t h e d o c u m e n t i s r e la t e d

t o m a t h e m a t i c s i n s t ru c t i o n . B a r- H il le l h a s d e f i n e d s u ch u t t e r a n c e s t h a t

r e q u i re c o n t e x t u a l i n f o r m a t i o n t o b e u n d e r s t o o d a s i n d e x i c a l e x p r e s s i o n s .

( q u o t e d b y M e h a n , 1 9 7 5 , p . 9 3 ) . T h u s a n ' i n f o r m e d ' o u t s i d e r c a n e v e n h a v e

d i f f i cu l ty under s t and ing wha t i s go ing on in a d i scuss ion . I t i s d i f f i cu l t t o

r e a li z e w h a t t h e p a r t i c ip a n t s i n t en d t o s a y a n d to i d e n t i f y th e m e a n i n g th e y

c r e a t e i n t h e g i v e n s i t u a t i o n . A s a p r e r e q u i s i t e o f c o m m u n i c a t i o n p a r t i c i p a n t s

h a v e t o s h a re c o m m o n u n d e r s t a n d i n g s w h i c h t h e y t a k e a s a n i m p l i ci t b a s is o f

r e f e r e n c e w h e n s p e a k i n g t o e a c h o t h e r . W h i l e s p e a k i n g , e a c h p a r t i c i p a n t

an t i c ipa t es t h e u nder s t an d ing and th e i n t e res t s o f t he spec i f ic addressee . The

s p e e c h g e t s o rg a n i z e d t h r o u g h t h e e x p e c t a t i o n o f w h a t th e a d d r e s se d p e r s o n

a l r e a d y k n o w s . E a c h s p e a k e r u se s h is i n t e r p r e t a t i o n o f t h e g iv e n s i tu a t i o n a n d

o f t h e a d d r e ss e e as a n i n d e x f r o m w h i c h h e f o r m s h is u t t e r a n c e s a n d f r o m

w h i c h h e d e c i d e s h i s ' c h o i c e o f g r a m m a r ' .

I n t h e c l a s s r o o m t h e t e a c h e r ' s i n s t ru c t i o n s a re i n d e x ic a l e x p r e s s io n s

w h i c h r e q u i re s t e a c h e r s a n d c h i ld r e n t o e m p l o y c o n t e x t u a l l y b o u n d i n te r -

p r e t a t i v e p r a c ti c e s t o m a k e s e n se o f t h e se in s t r u c t i o n s . ( C i c o u r e l, 1 9 7 4 ,

p . t 2 9 . ) W h a t a p a r t i c i p a n t s a y s n o t o n l y t r a n s p o r t s th e i n t e n d e d m e s s a g e ,

b u t o v e r a n d a b o v e th e m e s sa g e t h e u t t e r a n c e c o n t a i n s i n f o r m a t i o n a b o u t h i s

u n d e r s t a n d i n g o f t h e t o p i c , h i s i n t e r p r e t a t i o n o f t h e s i t u a ti o n , h i s e x p e c t a t i o n s

o f w h a t t h e o t h e r s m i g h t k n o w , a s w e l l a s h i s p r e s e n t e m o t i o n a l c o n c e r n s .

He nce , i nd ex ica l i t y i s ano the r l abe l fo r t he t hes i s t ha t t h e s i t u a t i o n i n f l u

e n c e s h o w l a n g ua g e is u s e d . N o t o n l y c o n t e n t a n d m e a n i n g s a r e n e g o t i a t e d

and con s t i t u t ed i n the soc i a l s i t ua t i on , bu t a l so the use o f l anguage and the

p e r f o r m a n c e o f t h e s p e a k e r a re c o - d e t e r m i n e d . T h i s is t r u e n o t o n l y f o r t h e

s y n t a c t i c a l s t r u c t u r e o f t h e u t t e r a n c e s b u t a ls o f o r t h e c h o i c e o f w o r d s . ( F r o m

t h i s p o i n t o f v i e w B e r n s t e i n ' s d i s t in c t i o n ( 1 9 6 5 ) b e t w e e n ' e l a b o r a t e d ' a n d

' r e s t r i c t e d ' c o d e s m i g h t b e m o r e a n i s s u e o f t h e i n d e x i c a l a n d r e f l e x i v e


13/19

S O -C A L L E D R E A L I T Y OF A M A T H E M A T I C S C L A S S R O O M 35

c o n s t i t u t i o n o f t h e s i t u a t io n r a t h e r th a n o f t h e c o m p e t e n c e o f t h e s p e a k e r s .)

M a t h e m a t i c i a n s , i n p a r t i c u l a r , h a v e i n v e s t e d m u c h e f f o r t i n p r o d u c i n g

u n i v e rs a l s t a te m e n t s , a n d m o s t s c h o o l m a t h e m a t i c i a n s w o u l d c l a im a n y m a t h e -

mat i ca l s t a t ement a s non- index i ca l , i . e . , a s un ive r sa l and ob j ec t i ve . However ,

t h i s convic t i on b locks i ns igh t i n to t he

irreparable incompleteness

o f u t t e r a n c e s ,

a n d m o r e g e n e r a l , of any symbolic action. E a c h u t t e r a n c e , j u s t a s e a c h s y m b o l i c

form, i s necessa r i l y i ncomple t e , because i t has t o be f i l l ed i n wi th meaning

v ia c o n t e x t u a l i n t e r p r e t a t i o n . T h r o u g h i ts g e n e si s a n d c h a i n o f d e f i n i ti o n a

c o n c e p t i n e v i ta b l y g e t s i n f il t ra t e d w i t h c o n t e x t u a l i n f o r m a t i o n . A n d e v e r y

a t t e m p t a t re p a i r in c r e as e s t h e n u m b e r o f s y m b o l s t h a t n e e d t o b e r e p a i r e d .

( M e h a n a n d W o o d , 1 9 7 5 , p . 9 3 .) T h e r e f o r e , u n d e r s t a n d i n g m a t h e m a t i c s is n o t

on ly a case o f l og i c , o r o f d ive rgen t t h ink ing , o r o f p rop er de f in i t i ons . As f a r

a s u n d e r s t a n d i n g is r ea l iz e d i n so c ia l i n t e r a c t io n ( o r t h r o u g h c o m m u n i c a t i o n ,

w h i c h is t h e v e r y s a m e t h i n g ) i t i n e s c a p a b l y b e c o m e s d e p e n d e n t u p o n t h e

i n t e r p r e t a ti v e , i n d e x i c a l, a n d r e fl e x iv e c o n s t i t u t i o n o f m e a n i n g .

4. F O U R D E F I C I E N T A R E A S O F R E S E A R C H -

M A T H E M A T I C S E D U C A T I O N ' S H I D D E N D I M E N S I O N S

T h e p r o c e s s o f t e a c h in g a n d l e a r n in g m a t h e m a t i c s c a n b e v i ew e d m o s t a p t l y

as a h igh ly com plex hu m an in t e r ac t i on i n an i ns t it u t i ona l i zed se t t ing - an in t e r -

a c t i o n w h i c h f o r m s a d i s ti n c ti v e p a r t o f t h e p a r t i c i p a n t ' s l if e . F o u r a s p e c t s i n

th i s is sue dese rve m ore d e t a i led d i scuss ion s ince t he y r epresen t w eak a reas o f

r esea rch .

1 . Teach ing and l ea rn ing mathemat i c s i s r ea l i zed t h rough

human inter

action. I t i s a k i n d o f m u t u a l i n f lu e n c i n g , a n in t e r d e p e n d e n c e o f t h e a c t i o n s

o f b o t h t e a c h e r a n d s t u d e n t o n m a n y le v el s. I t is n o t a u n i l a te r a l s e n d e r -

r e c e i v e r r e l a t i o n . I n e v i t a b l y t h e s t u d e n t ' s i n i t i a l m e e t i n g w i t h m a t h e m a t i c s

is m e d i a t e d t h r o u g h p a r e n t s , p l a y m a t e s , t e a c h e r s. T h e s t u d e n t ' s r e c o n s t r u c t i o n

o f m a t h e m a t i c a l m e a n i n g i s a c o n s t r u c t i o n v i a s o c i a l n e g o t i a t i o n a b o u t w h a t i s

m e a n t a n d a b o u t w h i c h p e r f o r m a n c e o f m e a n i n g g e ts t h e t e a c h e r 's ( o r t h e

p e e r ' s ) s a n c t i o n . H o w c a n w e e x p e c t t o f i n d a d e q u a t e i n f o r m a t i o n a b o u t

t each ing and l ea rn ing wh en we neg l ec t t he i n t e r ac ti ve con s t i t u t i on o f i nd i-

v idua l meanings?

2 . Teach ing and l ea rn ing m athe m at i c s i s r ea li zed i n institutions w h i c h t h e

s o c i e t y h a s s e t u p e x p l i c i t ly t o p r o d u c e s h a r ed m e a n i n g s a m o n g t h e i r m e m b e r s .

I n s t i t u t io n s a r e re p r e s e n t e d a n d r e p r o d u c e d t h r o u g h t h e i r m e m b e r s a n d t h a t i s

w h y t h e y h a v e c h a r a c t e r i s t i c i m p a c t s o n h u m a n i n t e r a c t i o n s i n s i d e o f t h e

ins t i t u t iona l . Th ey con s t i t u t e no rm s and ro les ; t hey dev e lop r it ua l s i n ac t i ons

and i n mea nings ; t he y t end t o sec lus ion and se l f - suf fi c i ency ; and t h ey eve n


14/19

3 6


p r o d u c e t h e ir o w n c o n t e n t - i n th i s ca s e, s c ho o l m a t h e m a t i c s . H o w r el ia b le

a r e s t u d i e s o n t h e e f f e c t s o f m a t h e m a t i c s e d u c a t i o n i f t h e y d o n o t t a k e i n t o

a c c o u n t t h e i n s t i t u ti o n a l i m p a c t o n t e a c h e r a n d s t u d e n t ? T h e q u e s t i o n

b e c o m e s c r u ci al w h e n o n e t h i n k s a b o u t a n y a p p l ic a t io n o f k n o w l e d g e l ea r ne d

a t s chool t o s i t ua t i ons ou t s ide o f school .

3 . M a t h e m a t i c s e d u c a t i o n c o n s t i t u t e s a distinctive part of the student s

l if e a s we l l a s o f t he t each er s . An yo ne w ho i s ac t i ve i n m a the m at i c s , w i ll

l e a rn s o m e t h i n g a b o u t h i m s e l f , e s p e c i al ly s in c e t h e a c t i v i ty h a p p e n s i n i n te r -

a c t iv e s i t u a ti o n s . O n t h e o t h e r h a n d o n e c a n l e a rn m a t h e m a t i c s o n l y b y

ac t i ve ly engag ing h i s p rev ious knowledge o f r e l a t ed sub j ec t s and ac t i ons .

T h e r e f o r e m a t h e m a t i c s e d u c a t i o n is d e e p l y r e l a t e d t o th e m a n - m a d e w o r l d

o f s y m b o l s a n d m e a n i n g s , t o c o m m o n s e ns e , a n d t o e v e r y d a y l if e . M a t h e -

m a t i c s e d u c a t i o n d e p e n d s o n o u r s o c i al a n d h i s to r i c c o n d i t i o n s . H o w c a n w e

d a r e t o m a k e a n y p r e d i c t i o n a b o u t t h e m a t h e m a t i c a l a b i li ti e s o f a s t u d e n t

and ab ou t h i s chances t o d eve lop t hese ab i l it i e s i f no t b y ca re fu l l y re l a t i ng

s u c h s t a t e m e n t s t o h is p e r s o n a l i ty a n d b a c k g r o u n d ?

4 . Sc i en ti s ts a r e n o t t he on ly on es wh o have d i f f i cu l t ie s dea l ing w i th

h igh ly complex i s sues . The o r i en t a t i on fo r ac t i ons and dec i s ions i n t he c l a s s -

r o o m c o n t i n u o u s l y r e q u i r e s t h e

reduction of complexity.

O n t h e o t h e r h a n d

t h e u n d e r s t a n d i n g a n d t h e e f f e c ti v e r e d u c t i o n o f c o m p l e x i t ie s d e m a n d s t h e i r

t o t a l u n f o l d i n g a n d c o m p l e t e r e c o n s t r u c t i o n . T o d a t e s c i e n ti f ic a n a ly s i s h a s

b e e n i n c a p a b l e o f r e d u c i n g t h e c o m p l e x i t i e s o f a n a c t u a l m a t h e m a t i c s cla ss -

r o o m s u f f i ci e n t ly f o r g u i d in g a t e a c h e r s d e c i s io n s . Y e t w i t h o u t s u c h g u id a n c e

i t is imposs ib l e t o p l an e f f ec t i ve t eache r t r a in ing p rogram .

5 . F U N D A M E N T A L P R O B L E M S O F

R E S E A R C H A ND D E V E L O P M E N T

W i t h i n t h e l a s t y e a r s m y v i ew o f t h e s t r u c t u r e o f t h e c l a s sr o o m p r o c e ss h a s

be en change d as a re su l t o f co l l ec t ing i n form at ion f rom several d i sc ip l i nes,

p a r t i c i p a t i n g i n m a t h e m a t i c s l e s s o n s , a n d a n a l y z i n g v i d e o - t a p e d m a t h e m a t i c s

l e ss o n s. M y s u b j ec t iv e c h a n g e i n cl u d e s t h e a i m s o f m y w o r k , t h e s u b j e c t t o b e

s t u d i e d , t h e m e t h o d s o f r e s e a r c h , a s w e l l a s t h e u n d e r l y i n g p a r a d i g m s o f m y

t h i n k in g . T h i s p e r s o n a l e v e n t is w o r t h o f m e n t i o n s in c e d i sc u s si o n s w i t h

co l l eagues l eads me to be l i eve t ha t my sub j ec t i ve d i f f i cu l t i e s on ly mi r ro r

m u c h m o r e f u n d a m e n t a l d i f f i c u l t i e s w i t h i n o u r p r o f e s s i o n . P h i l o s o p h e r s

of sc i ence agree t ha t such d i f f i cu l t i e s w i th in a p rofes s ion a r e s t rong i nd i ca to r s

o f f u n d a m e n t a l c h a n g e o f p a ra d i g m s .

I n t h e p r e s e n t t r a n s i ti o n s ta g e t h r e e t h e s e s s e e m t o b e o f im p o r t a n c e :

1 . M a t h e m a t i c s e d u c a t i o n i s d e e p l y i n n e e d o f

theoretical orientation.

W e


15/19


h a v e t o o m u c h r e s e a r c h o n t o o s m a l l a t h e o r e t i c a l b a s i s . M a n y o p e n i n g a d -

d r e ss e s o f A P A , A E R A , a n d S I G / R M E , a s w e l l a s j o u r n a l a r ti c le s f r o m w i t h i n

t h e l a s t y e a r s h a v e c o m p l a i n e d a b o u t t h i s p r o b l e m , e .g ., L e e C r o n b a c h ( 1 9 7 5 ) ,

L e e S . S h u l m a n ( 1 9 7 9 ) , E r n e s t H i lg a r d ( 1 9 7 6 ) . P e r h a p s t h e r e a r e t o o m a n y

s h o r t - t e r m e d r e s e a r c h c o n t r a c t s . P e r h a p s t h e r e i s t o o m u c h p r e s c r i p t i o n f o r

' a c c e p t a b l e ' r e s e a r c h p r o g r a m s . O r , p e r h a p s , t h e r e i s n o s u p p o r t f o r m e t h o d o -

l o g ic a l h e r e t i c s a n d t h u s n o e n c o u r a g e m e n t f o r y o u n g r e s e a r ch e r s t o t r y u n u s u a l

a p p r o a c h e s . F o r s u r e , t h e r e h a s y e t t o a p p e a r a n a d e q u a t e f o r u m f o r th e o r e t i c a l

d i scuss ion . W ha tever t he cause , t do no t be l i eve t ha t t he re a r e no t en oug h new

ideas .

2 . R e s e a r c h a n d p r a c t i c a l d e v e l o p m e n t s f o l l o w dif ferent paradigms T h e

m ain s t r eam of r e sea rch s til l f o l l ows t he p a rad igm o f t he na tu ra l s c i ences ,

s t a t ing an ob j ec t i ve educa t i on a l r ea l i t y , us ing we l l de f ined an d quan t i f i ab l e

c o n c e p t s , a n d a n a l y zi n g t h e r e l a ti o n s h i p a m o n g t h e m t h r o u g h s t at is t ic a l

m e a n s . F o r a l o n g t i m e w e h a v e h e a r d a n d a c c e p t e d c o m p l a i n t s a b o u t t h e

comple t e l ack o f c l a s s room appl i cab i l i t y o f r e sea rch r esu l t s . (See Ker l i nge r

( 1 9 7 7 ) f o r a s u m m a r y o f a r e s e a r c h e r' s v i ew . )

O n t h e c o n t r a r y , t h e m a j o r i t y o f a t e a c h e r ' s c la s s r o o m d e c is io n s ar e m a d e

v ia c o m m o n s e n se a n d i n t u i t i o n r a t h e r t h a n t h r o u g h r a t i o n a l an a l y se s b y

sc i en t i f i c means . I f she /he i s a good t eacher t hen he r /h i s ac t i ons a r e based

o n a m o r e d i f f e r e n t i a t e d p e r c e p t i o n o f t h e c l a s s r o o m e v e n t s t h a t r e s e a r c h

r e c o g n i ze s . S h e / h e is m o r e o p e n t o c o n t e x t u a l c h a n g es , ' k n o w i n g ' a s t u d e n t ,

u s i n g t a c i t k n o w l e d g e (M . P o l a n y i ) , a n d i n f o r m a l r e f l e c te d e x p e r i e n c e s.

C o m p a r e d w i t h t h e s e ' h a r d ' s o c i a l f a c t s , c u r r e n t r e s e a r c h a p p e a r s a s ' s o f t -

w a r e ' . I t i s n e c e s s a r y t h a t r e s e a r c h i n m a t h e m a t i c s e d u c a t i o n t a k e s n o t i c e

o f t h i s gap i f a c l a im for p rac t i ca l r e l evance i s t o be e s t ab l ished .

3 Interdisciplinary approaches

a re p romis ing , i f no t necessa ry , fo r c los ing

m a t h e m a t i c s e d u c a t i o n ' s ' c r e d i b i li t y g a p ' . W i th i n t h e b r o a d a r e a o f s o c ia l

sc i ences t he d i sc repanc i es be twe en r a t i ona l i s ti c and he rm ene ut i c desc r ip t i ons ,

b e t w e e n n a i v e a n d s c i en t if i c c o n s t r u c t s h a v e b e e n r e a li z ed a n d i n v e s t ig a t e d

much ea r l i e r . I t i s t ime t o i n t egra t e t hese f i nd ings i n to our p rofes s ion and

t o t r a n s f o r m t h is k n o w l e d g e i n t o s p e c i fi c c o n d i t i o n s o f l e a rn i n g a n d t e a c h i n g

m a t h e m a t i c s .

U n l i k e t h e n a t u r a l s c ie n c es t h e h u m a n s c ie n c e s m u s t d e a l w i t h a n o b j e c ti v e

s o c ia l r e a l it y o n t h e o n e h a n d , y e t o n t h e o t h e r h a n d m u s t d e a l w i t h a s m a n y

rea l i t i e s a s t he re a r e r e f l ec t i ve sub j ec t s . Pa radoxica l l y , modern phys i c i s t s

h a v e a h i g h l y d e v e l o p e d u n d e r s t a n d i n g o f e x p l a n a t o r y m o d e l s l o s i n g t h e i r

m e a n i n g i n t h e l i g h t o f m o r e c o m p r e h e n s i v e th e o r i e s. F o r e x a m p l e , t h e

q u e s t i o n o f t h e ' d i v is i b i li t y ' o r t h e ' c o n s i s t e n c y ' o f a l ig h t q u a n t u m ( p h o t o n )

m a k e s n o s e ns e in a g e n e ra l t h e o r y o f e l e m e n t a r y p a r t i cl e s b e c a u s e t h e t h e o r y


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3 8 H E I N R I C H B A U E R S F E L D

d e s c r ib e s t h e r e l a ti o n s a m o n g e l e m e n t s b u t i s n o t c o n c e r n e d w i t h th e n a t u r e

o f t h e e l e m e n t s t h e m s e l v e s . T h i s i s v e r y n e a r t o a n i m p o r t a n t i s s u e o f c o n -

s t it u t iv e e t h n o m e t h o d o l o g y . T h e s t r u c t u r in g a c t iv i ti e s o f t h e p a r t i c i p a n t s

f o r m ( ' c o n s t i t u t e ' ) t h e s o c i a l s i t u a t i o n a m o n g t h e m s e l v e s , a n d t h e p r o c e s s

a n d r u le s o f t h e s e ' s t ru c t u r i n g s ' ( a s M e h a n c a ll s t h e m ) b u i l d a c o r e th e o r y

o f s o c ia l a c ti o n . T h e t h e o r y is r e la t e d t o s t r u c tu r i n g s r a t h e r t h a n t o s t r u c t u r e s

o f t h e s i t ua t i on i n usua l soc ia l sc i ences . Th a t i s, t he re is g rea t e r genera l i zab i li t y

w i t h i n t h e p r o c e s s o f s t r u c tu r i n g s t h a n w i t h i n t h e s t r u c t u r e s t h e m s e lv e s .

6. I M P L I C A T I O N S F O R T E A C H E R T R A I N I N G

Tho se wh o f ind t he d i scussed theor i es and in t e rp re t a t i on s m ore o r l e ss accep t -

a b l e m i g h t fi n d th e m s e l v e s f o r c e d to t h i n k a b o u t c o n s e q u e n c e s . I t s e e m s

l ike ly , t ha t i nnov a t ion i n sch ool s wi ll no t be o f a ve ry r ad i ca l k ind un l ess

t h e c a t e g o r ie s t e a c h e r s u s e t o o r g a n iz e w h a t t h e y k n o w a b o u t p u p i l s a n d to

d e t e r m i n e w h a t c o u n t s as k n o w l e d g e u n d er g o a f u n d a m e n t a l c h a n g e . ( K e d d i e ,

1 9 7 1 , p . 1 5 6 .) S h i r k 's f i n d in g s a b o u t t h e s t a b i li t y o f f u n d a m e n t a l c o n c e p t u a l -

i z a t io n s a c r o ss te a c h e r t r a i n in g a p p a r e n t l y d o n o t l e a v e m u c h c h a n c e f o r th a t

change . How i s i t poss ib l e fo r a beg inn ing t eacher t o overcome the s ix t een o r

m o r e y e a r s o f h is o w n e x p e r i e n c e s a s a s t u d e n t ? P a r t ic u l a r ly , si nc e t h e c o n -

t e x t u a l f o r c e o f t h e s e e x p e r i e n c e s o f t e n d o m i n a t e s a n y l a t e r v e r b a l i n f o r m a t i o n

a b o u t e d u c a t i o n a n d l e a d s t h e b e g i n n e r i n t o a n a l m o s t u n c o n s c i o u s re p r o -

d u c t i o n o f t h e s c h o o l s y s t e m ' s c h a r a c te r is t ic s .

B u t , i f w e f o r m o u r c o g n i t i o n a n d b e h a v i o r a b o u t t e a c h i n g t h r o u g h s o c i a l

s i tu a t i o n s , t h e n w e c a n a ls o c h a n ge t h is f o r m e d c o g n i t io n a n d b e h a v i o r th r o u g h

soc i al s i t ua t ions . We l ea rn t o beh ave in soc ia l se tt i ngs on ly t h roug h the

re f l ec t ed pa r t i c ipa t i on and ac t i on i n soc i a l se t t i ngs . S imi l a r ly , a t eacher wi l l

l e a rn t o t e a c h o r t o c h a n g e h is te a c h i n g p a t t e r n o n l y t h r o u g h r e f l e c t e d t e a c h in g .

Y e t , t h i s i s no t t he ru l ing m od e l o f p resen t p re - se rv i ce t eache r t r a in ing .

U s u a l l y t h e t e a c h e r s t u d e n t l e a r n s a b o u t te a c h i n g in c o n t e x t s v e r y d i f f e r e n t

f r o m c l a s s r o o m s i t u a ti o n s . T h e o r g a n iz i n g i n te r e s t f o r p i c k in g u p k n o w l e d g e

in l ec tu res i s more a long wi th pass ing examina t ions t han r e l a t ed t o l a t e r c l a s s -

r o o m a p p l i c a ti o n . T h r o u g h v a r i o u s l e c t u re s a n d s e m i n a r s t h e s t u d e n t t e a c h e r

co l l ec t s i ncom ple t e ec l ec t i c kno wled ge and she /he i s l e f t w i th t he unass i s t ed

t ask o f i n t egra t i ng t h is know ledge in to an app l i cab l e sys t e m for t he l i v ing c lass -

r o o m .

I f th e c o n s t i tu t i v e p o w e r o f s o c ia l s i tu a t io n s o n b e h a v i o r , m e a n i n g a n d

language i s a s s trong as a s sumed he re , t he n the s tuden t t eache r wi ll have t o

s p e n d m u c h m o r e t i m e p l a n n i n g , a c c o m p l i s h i n g a n d r e f l e c t i n g u p o n r e a l

c l a s s r o o m t e a c h i n g e x p e r i e n c e . F r o m t h e v e r y b e g in n i n g th e t e a c h e r t o b e m u s t


17/19


e n c o u n t e r a n a d e q u a t e c o m p l e x i t y o f so c ia l c l a s s r o o m e x c h a n g e s . ' A d e q u a t e '

m e a n s t h a t t h e c o m p l e x i t y o f t h e t e a c h i n g - le a r n i n g s i tu a t io n m i g h t b e r e d u c ed

i n q u a n t i t y , e .g ., v i a a r e d u c e d n u m b e r o f s t u d e n t s t o t e a c h o r a r e d u c e d

a m o u n t o f l e s s o n t i m e , b u t n o t r e d u c e d i n q u a l i t y ( as s im u l a t i o n g a m e s o r

v i d e o - t a p e a n a l y s e s , e .g . , w o u l d c a u s e ) . I f w e c l a i m t o e d u c a t e h u m a n b e i n g s ,

t h e n a t e a c h e r w i l l h a v e t o r e c e iv e a m u c h m o r e c a r e f u l , h o l i s t i c p r e p a r a t i o n .

T h i s , o f c o u r s e , w i ll re q u i r e s u p p o r t a n d d e v e l o p m e n t o n t h e si de o f

r e s e a r c h a s w e l l . A n d t h i s r e s e a r c h , a t l e a s t a r e a s o n a b l e p a r t o f i t , w i l l h a v e

t o f o l l o w t h e i n t e r p r e t a t i v e p a r a d i g m . " S c i e n c e a t i t s b e s t is t h u s l i k e a f i r m

b u t g e n t l e h a n d t h a t h o l d s a b u t t e r f l y w i t h o u t c r a s h in g i t . " ( K e n n e t h S . B o w e r s ,

1 9 7 3 , p . 3 3 2 . )

University o f ielefeld

From Semp~, Wie sag ich ' s me inen K indern?

9 1960 by Diogenes Verlag, Ziirich

N O T E S

This pa per was develope d for an invited address a t the pre-session of th e Special In terest

Grou p fo r Resea rch in Mathemat ics Educ a t ion (S IG/RME) fo r the annu a l mee t ing o f the

Na tional Council of Teachers in Mathe m atics (NCTM ), Boston, Ap ri l 18, 1979.


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4 0 H E I N R I C H B A U E R S F E L D

2 H i l g ar d , E . R . , P r e s e n t a t io n a t t h e S y m p o s i u m o n 7 5 t h Y e a r b o o k o f N S S E , A E R A

annua l m ee t ing , A pr i l, 1976 .

3 Shu lma n , L . S ., P res en ta t ion a t the S IG/R ME pre -s es sion o f the NCTM annua l mee t ing ,

Bos ton , Apr i l 18 , 1979 .

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8 0 , 3 0 7 - 3 3 6 .

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posium on Motivation 1975,

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UTB 642 , W. F ink Ver lag ,

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G o f f m a n , E . : 1 9 6 9 ,

Strategic Interaction,

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/ekts,

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of Communication,

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Niem eyer , T i ib ingen .

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cational Research Journal 4 , 2 , 145-167 .

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