mathematics as the art of seeing the invisible

8/13/2019 MATHEMATICS AS THE ART OF SEEING THE INVISIBLE

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MATHEMATICS AS THE ART OF SEEING THE INVISIBLE

Jorge Soto-Andrade 1

Department of Mathematics, Faculty of ScienceCentre for Advanced Research in ducation !CAR"

#niversity of Chile

The role of metaphors and the switch in cognitive modes in relation to visualization

in learning and teaching mathematics is discussed, based on examples and case

studies with students and teachers. We present some preliminary evidence

supporting our claims that visualization requires the activation of various metaphors,

that it is rather hampered than facilitated by traditional teaching in mathematics, but

it is however a trainable capacity in teachers and students.

INTRODUCTION

$hen teaching mathematics to first year undergraduates %ho intend to ma&or in'umanities or Social Sciences at the #niversity of Chile, %e usually as( them toconcoct aphorisms to descri)e their ongoing e*perience of mathematics, as the courseunfolds+

An often emerging aphorism is Mathematics is the art of seeing the invisi)le., aclose (in to /isuali0ation offers a method of seeing the unseen. !McCormic(,DeFanti, 2ro%n, 1345, p+ 6", ta(en up as leitmotiv in Arcavi !7886" %hendiscussing the role of visual representations in the learning of mathematics+

At the other end of the spectrum, in a recent paper !Fau* 9ates, 788:" onsupersymmetric representation theory, in theoretical physics, %e read

;he use of sym)ols to connote ideas %hich defy simple ver)ali0ation is perhaps one ofthe oldest of human traditions+ ;he Asante people of $est Africa have long )eenaccustomed to using simple yet elegant motifs (no%n as Adin(ra sym)ols, to serve &ust

this purpose+ $ith a nod to this tradition, %e christen our graphical sym)ols asAdin(ras+.


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these pictures %hich %e have called adinkras. Adin(ra is an Ashanti !$est African" %ordmeaning sym)ol %ith hidden meaning+ Bnce you create these pictures from theeuations, there are things you can see more clearly than you can from a page of figuresand num)ers

?n Fig+ 1 %e see some e*amples of classical Adin(ras and in Fig+ 7, their analogues designed)y 9ates and colla)orators+

?n the latter, %hich sho%s the folding of a cu)e, %hite nodes stand for )osons, )lac(nodes for fermions+

2etter late than never, the (ey role played )y visuali0ation in the teaching andlearning of mathematics has received slo% )ut increasing recognition in the lastdecades+ See =resmeg !788>" for a comprehensive revie%+ o%adays its rather lo%status in mathematics and mathematics education is )eing progressively upgraded+

?n this paper %e intend to e*plore, through e*amples, the didactical role ofvisuali0ation in mathematics together %ith its interplay %ith metaphors and transitingamongst cognitive modes+ $e )uild for that on our previous research ande*perimental findings !Soto-Andrade, 788>, 7885" concerning the role of metaphorsand cognitive modes in the teaching-learning of mathematics+

After setting up our tentative theoretical frame%or(, %e put for%ard our main%or(ing hypotheses related to necessity of metaphor activation in visuali0ation and%e proceed to report on some specific e*amples of the role of metaphors andcognitive modes in the process of visuali0ation+ ?n this %ay %e get some preliminarye*perimental evidence to support our %or(ing hypotheses and also suggestions forfurther research along these lines, that %e discuss in the final section+

7

Figure 1

Figure 7


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THEORETICAL FRAMEWORK

Nature and Role o Meta!"or#

?t has )een progressively recogni0ed during the last decade !Araya, 7888 nglish,1335 ,7885 and many others" that metaphors are not &ust rhetorical devices, )ut also

po%erful cognitive tools that help us to )uild or to grasp ne% concepts, as %ell as tosolve pro)lems in an efficient and friendly %ay+ ;hose are indeed conceptualmetaphors. !


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and reflection upon pictures, images, diagrams, in our minds, on paper or %ithtechnological tools, %ith the purpose of depicting and communicating information,thin(ing a)out and developing previously un(no%n ideas and advancing understandings+

?n a systemic approach to the classroom, it may )e argued that learning itself appears

as an emergent process, that may encompass visuali0ation to a greater or lesserdegree+

o%adays there is %idespread interest in understanding ho% visuali0ation interacts%ith the didactics of mathematics, and ho% %e could enhance its usefulness inmathematics education !$oolner, 788: =resmeg, 788>"

?n %hat follo%s %e adhere mainly to the frame%or( laid )y 7884, of E classes of 68 primaryschool teachers each, enrolled in a 1E months program aiming at upgrading theircommand of elementary mathematics+ Bn the other hand these primary schoolteachers have made the recurrent o)servation in their classrooms that from Eth gradeon%ards the children are uite una)le to ta(e advantage of visuali0ation %hentac(ling a pro)lem+ ;heir feeling is that this is a result of the instruction the childrenhave received from previous teachers, %hich tend to repress spontaneousvisuali0ation in favour of rote learning of algorithms+

;his pro)lematics leads naturally to some (ey uestions regarding visuali0ation !cf+=resmeg, 788>"

:


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$hat conversion processes are involved in a smooth transit amongstdifferent cognitive modes, specially from seuential-ver)al to simultaneous-nonver)al, as an antidote to unimodal cognitionL

$hat is the role of metaphors in the s%itching from one cognitive mode to

another, necessary to visuali0ationL 'o% can teachers help learners to transit )et%een visual and ver)al

cognitive modes, i+e+ )et%een the iconic and the sym)olic domain, forspecific mathematical notionsIL

Re#ear+" ",!ot"e#e#

Bur main %or(ing hypothesis is that metaphors and cognitive modes are (eyingredients in a meaningful teaching-learning process+ ?n particular, %e claim that

visuali0ation is elicited )y the activation of previous metaphors that entail a s%itch inthe cognitive mode of the learner+

;his applies specifically to visuali0ation understood as the a)ility to see theinvisi)le., typically in the case of arithmetical or alge)raic o)&ects and processes+;he situation may )e less clear cut in cases li(e the rope around the earth pro)lem.!Arcavi, 7886, p+ 73 -68", for instance, %here %e imagine a rope laid around thecircumference of the earth and as( ho% much longer should it )e if %e %ant to placeit on si*-foot-high poles all the %ay+ ou could argue that %hen you visuali0e )othropes around the earth and you deform their shape, from circular to suare, to see

)etter %hat is going on and to solve the pro)lem, you are not activating anymetaphor, )ecause you remain in the geometric realm+ 'o%ever, %e can also pointout, that usually %e transit first from the concrete. setting of the pro)lem to asym)olic one, )y %riting the formula for )oth perimeters involved+ Bne calculatesthe ans%er )ut one is not really convinced unless one really sees it+ ;hen a metaphoractivates that allo%s you to transit from the alge)raic formulation to the suare %ithrounded-off corners. visuali0ation+

?t is also one of our %or(ing hypothesis that the competences regarding multimodalcognition, use and creation of metaphors and visuali0ation are traina)le and that

measura)le progress can )e achieved in one semester course or even in a one %ee(%or(shop+ ;his, in spite of the strongly unimodal !seuential-ver)al" training mostteachers have )een e*posed to+

Regarding students, our %or(ing hypothesis is that they %ould significantly improvethe uality and ro)ustness of their learning if they can dra% from a suita)le spectrumof metaphors and approach pro)lems in a multimodal %ay, emphasi0ingvisuali0ation+

$e present )elo% some e*amples, tested in teaching mathematics courses for severalyears, to various audiences of students, to illustrate the didactical use of differentcognitive modes, in particular visual ones, for approaching mathematical o)&ects and

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their interplay %ith the use of metaphors+

RESEARCH BACKGROUND AND METHODOLOG3

;he )ac(ground for our e*perimental research consisted in several courses, to %it

4 Mat"e'at%+# 5 A one semester general mathematics course, given since 7888 tofirst year students of the 2achelor =rogram in 'umanities and Social Sciences at the#niversity of Chile+ ;his course aims at introducing the students to themathematical %ay of thin(ing and cognitive attitudes of mathematics+ ;he class has68 to :8 students+ *periments are carried out during the lessons !t%ice 38 minutes%ee(ly" and during e*ams !7 hours each, : in a semester"+ and :8 in 7885+


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Figure >

Bf course, you could use your calculator, to get 8+6678 %ith four digits !seuential-ver)al mode"+

2ut do yo really see. that the sum is very close to 1N6L

'o% could the students tac(le this pro)lem in a nonver)al - seuential mode firstL

An implicit or underlying metaphor %ould )e =roduct is area.+

So, they could try to represent, or visuali0e, 1N: as a su)-region orportion of a region that stands for the %hole, the unit+ Moststudents, and in-service teachers, try this visuali0ation !Fig+ 1"+

;hen, as o)served )y our teachers in their classes, E th and >th

graders tend to place !1N:"7 side )y side %ith the mar(ed fourth ofthe suare and so on !Figs+ : and E"+ Bnly roughly one out of tenteachers see. spontaneously in this %ay that you are ta(ing oneout of three all the time.+ Some of themhave a hard time seeing. this, even after

)eing told so+

Blder students and in-service teachers aremore prone to place the smaller suares

along the diagonal, as inFig+ >+ ;hat %ay, theysuddenly see.+ oticethat this last figure affords a truly nonseuential !simultaneous"nonver)al approach to our sum+ ?n fact, if you %ant to couch it in%ords you could say that this affords a transparent %ay for Itrisecting the suare using only suares, including an efficientestimate for the remainder., in case you stop short of infinity+++

?n our Random $al(s. course some studentsconcocted various visuali0ations, li(e the oneillustrated in Fig+ 5, %here succesive transversal

partitions reveal a &oint region that can )e seen to

comprise almost 1N6 of the total area, if you alsomar( its symmetrical t%in as in Fig+ 4+

?n Arcavi !7886" a different visuali0ation %hose stage is theeuilateral triangle - is recalled !Fig+ 3"+ ?n our case studies %ith firstyear university students, in-service

5

Figure 6

Figure E Figure :

Figure 5 Figure 4

Figure 18 Figure 3


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primary school teachers and their students !from Eth to 4th grade", this visuali0ationdidnOt appear spontaneously as an alternative to the suare partition+ ?t only did afterprompting+ Any%ay, once students and teachers got the idea of the euilateraltriangle, the %hole procedure unfolded very smoothly !see Fig+ 18"+

E)a'!le @( Su' o !o*er# o =>

;hen %e suggested the case of the sum of po%ers of 1N6 instead of 1N:+

Students and teachers uic(ly visuali0ed step%ise the triangles in Fig+ 11+

$hich are the ingredients of these visuali0ation processes, up to no%L

Bur vie%point is that these visuali0ations emerge than(s to the activation of severalmetaphors+ ;he crucial one is the =roduct is Area., or Fractions are =ortions.metaphor, %hich develops into the =o%ers of a Fraction arise from ?terated=artitions. metaphor+ ;hen the Adding is putting together. metaphor comes into

play+ $e argue that visuali0ation is grounded on these metaphors, %hich entail as%itch in cognitive mode, from ver)al to nonver)al !visual" and finally, fromseuential to simultaneous+

?ndeed, %hen the metaphor is activated, or runs., a s%itch in cognitive mode arises

first from seuential-ver)al to seuential-nonver)al %hen %e represent our po%ers of1N: one )y one, step%ise, and then to simultaneous-nonver)al, %hen %e see. thatour sum is almost 1N6+ otice that %hen %e compute it, %e come to this conclusionstep%ise, )ut %hen %e visuali0e it, %e see it at a glance, truly in a simultaneous-nonver)al mode+ ?t is a ahaP. e*perience, that is unli(ely to occur %hen %e computestep%ise %ith the calculator+

E)a'!le @( Su' o !o*er# o =>

How much is 1N E Q !1N E"7 Q !1N E"6 Q !1N E": Q !1N E"E L

*an you calculate, or estimate the value of this sum)*an you +see how much it should be)

4

Figure 11


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=ractically minded students %ould fetch their calculator and get

+7 Q +77 Q +76 Q +7: Q +7E 8+7:337,

%hich is appro*imately +7E %ith an error less than +8881+

;hose %ho li(e fractional yoga !a minority" %ould get1NE Q 1N7E Q 1N17E Q 1N>7E Q 1N617E !>7E Q 17E Q 7E Q E Q 1"N617E 541N617E,

%hich loo(s rather opaue as a result+ After some pondering, they %ould reali0eho%ever that -! times # is (!'#, so that our unyielding fraction is very close to1N : 541 N 617:+Br the other %ay around, since 617EN: 541T , they can get

541N617E !541T - T "N617E 1N : 1N617E 1N : - 1N17E88

so that the value of our sum is 1N:, up to an error smaller than 1N18888+

So, either the decimal %ay or the fractional %ay, %e cancalculate our sum+ 2oth %ays are ho%ever relentlessly ver)al-seuential and they definitely disgust many students,

particularly those motivated )y the humanities or the socialsciences+

'o% could you see. %hat is not visi)le in the precedingcalculationL

$hen so prompted, most students try to visuali0e 1NE in a

pentagon, as in Fig+ 17+ 2ut then, they are stuc(P ;hey see no transparent %ay topartition an isosceles triangle into fifthsP $hat to doL

;o )e a)le to visuali0e %hat is less visi)le, %e need to change our underlyingmetaphorP

A good idea is to revisit the more favoura)le case of the sum of po%ers of 1N:

How could we visualize our sum with the help of a different metaphor)

After some pondering, Uarel, a student in our Random %al(s. course, said that the%hole geometric visualisation in the suare reminded him of a fernV

From this, after some discussion, the hydraulic tree in Fig+ 16 emerged+

3

Figure 17

Figure 16


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?ndeed, the root of the tree stands for the unit suare, the coloured son., for the coloured1N: of the suare, the coloured grandson, for the coloured !1N:"7, etc+ $e imagine that 1litre of %ater is poured into the root, and that it flo%s do%n%ards, splitting %ith euanimity

into four fourths at each node+ $e collect the %ater coming out the leftmost duct, at allstages+ ;here are many possi)le %ays of representing !1N:"7 in this hydraulic tree, the

pro)lem is choosing the most transparent one for visuali0ing the value of our sum of po%ersof 1N:+ ;he one sho%n in Fig+ 16 %as preferred )y most students+

?t is uite clear no%, that this visuali0ation carries over to the case of 1NE, or 1N6, orany 1Nn, and even furtherV

$hat does this e*ample sho% usL

otice that our =roduct is area . metaphor forsa(es in the case of 1NE )ecause it

doesnOt help us much to see that our sum is almost 1N :+;o visuali0e further %e need to s%itch to another metaphor for the product, one that%e have met )efore.

=roduct is concatenation., i+ e+ ;o multiply is to concatenate )ranchings+.

otice that %hat %e have met )efore is for instance - successive enlargements orreductions of a photographic print or image+

Specially %hen dealing %ith multiplication of fractions, this metaphor is very helpful+?t is closely related to the hydraulic metaphor or the pedestrian metaphor !Soto-

Andrade, 788>, 7885"So eventually, visuali0ation of closely related mathematical o)&ects or processes, mayreuire the activation of different (inds of metaphors, %ith different scopes+

E)a'!le ?( T"e nu'.er #euen+e( a Far Ea#t &%#ual%0at%on

?n Soto-Andrade !7885" %e reported on some preliminary results concerning thefollo%ing challenge, posited to students and teachers

/s it possible to represent the usual numerical sequence, up to some number, in a

non verbal and non sequential way) 0et us say the sequence 1, !, ', (, 2,3(.

$e suggested first to try to represent this num)er seuence in a less ver)al %ay thanusual+ As a preliminary, %e proposed the )inaryactivity suggested in Soto-Andrade !788>", i+ e+%e as( the students to get the )inary descriptionof their num)er in the classroom, %ithoutcounting themselves first+ ;o do this, they &uststand up, try to match up in pairs, chec( %hetherthere is or not one odd man out.+ ;hen the pairs

do the same, and so on+$hen the pairing game is over, %e &ust as(

18

QuickTime and aTIFF (LZW) decom presso r

are needed to see this pi cture.

Figure 1:


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%hether there is an unmatched person, an unmatched pair, an unmatched uadruplet,and so on+ ?f the ans%ers are S B B S B S, for instance, %eencourage the students to codify them in a less ver)al %ay+ ventually they re-discover the ? Ching !i Jing" codification a )ro(en line for B, a continuous line

for S, or something euivalent+ ?f they dra% the lines from top to )ottom, theyget the he*agram in the second column !from the left", > th ro%, in the suarearrangement sho%n in Fig+ 1:+ ;his arrangement, due to the Chinese philosopher andmathematician Shao ong !1811-1855" represents e*actly the )inary seuence ofnum)ers 8 to >6+ Students and teachers successfully completed this activity andmost of them reported later having understood for the first time the )inary descriptionof a num)er+

$e are then a)le to visuali0e our num)er seuence as a seuence of diagramsho%ever %e %ould li(e to push this visuali0ation even further

*ould we draw a picture that describes the whole situation in a non verbal andsimultaneous way) 4o that 5ust a glance of it should be enough to reconstruct it

correctly, and to get back from it the whole binary 3# hexagram sequence)

Bn this challenge, %hich they tac(led in small groups, the primary school teachersdid )etter than the first year 2achelor students ?n a first class of 68 primary schoolteachers !7E female, E male", after 68 min+ %or(ing in groups, %hen prompted tos%itch to non seuential cognitive style to descri)e the )inary he*agram seuence, Eteachers out of 68 came up %ith diagrams euivalent to the famous Shao ongOsWiantian !2efore 'eaven" diagram, or its inverted form, illustrated in Figs+ 1E and 1>,respectively+

2Xr)ara Math 8 Myriam Math 8 Francisca Math 87885

9erardo Math 87885

Figure 15 Dra%ings )y Math+ 8 students


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As the sample in Figs 15 and 14 sho%s, %e loo( at the same image )ut %e see uitedifferent thing+ Apparently %hat %e see, in this visuali0ation, depends heavily on our

previous history and training+ Some case study %ith professional mathematicians andresearchers in didactics gave also disparate reactions and performances+ Amongmathematicians, alge)raists had a harder time than geometers, )ut one researcher indidactics of mathematics instantly sa%. the )inary tree in Wian ;ian+ ?n fact, roughly

one out of 18 of our students did so, sho%ing that this image may )e the oldeste*plicit emergence of the )inary tree in the history of man(ind+++

D%#+u##%on

$e have sho%n, through several activities carried out in the classroom, ho% someclassical mathematical o)&ects and pro)lematic situations may )e une*pectedlyapproached through various cognitive modes, in particular %ith differentvisuali0ations grounded in alternative metaphors+

$e have seen that it is possi)le to facilitate the activation of these less usualcognitive modes and the emergence of visuali0ation, even for in-service teachershaving )een trained in a unimodal !ver)al-seuential" %ay+ ?n fact, a high percentageof students as %ell as teachers %ere a)le, after some prompting, and initial effort, to

)egin to visuali0e in %ays hitherto un(no%n to them+ ?n this %ay, some of them evenrediscovered visuali0ations of familiar mathematical o)&ects, developed in othercultures !li(e the ancient Chinese, for instance", that favoured visuali0ation more thanours Bne image is %orth 18 888 %ords. Vthey said+

Bur teachers noticed freuently that 6 th or :thgraders %ere more prone to visuali0e

spontaneously, than 5thor 4thgraders, %ho seemed to )e much more ro)oti0ed. tocalculate, %ith little drive to visuali0e+ ;his suggests that the a)ility to visuali0e is

17

Figure 14 Dra%ings )y anonymous M+Sc+ students

2Xr)ara Math8


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rather repressed in children than stimulated, )y traditional mathematical education+

Bur o)servations seem to indicate ho%ever that multimodal cognitive approaches,visuali0ing, in particular, and transiting from one cognitive style to another, aretraina)le, in students as %ell as in teachers, specially )y %or(ing in groups+ First

person reports )y students and teachers )ear %itness of the impact andmeaningfulness that this sort of cognitive e*perience has for them+ Moreover, %eshould mention that some recent reports from our primary school teachers point outto dramatic improvements in the scores of their students in the last nationalassessment test !from 741 to 68: points, for instance, national average stagnating at7:> points", after several months %or(ing %ith e*plicit metaphors and visuali0ations,

)ut %ith no specific test oriented training+

?t %ould )e interesting then to test and measure systematically the depth of learningthat students may achieve %hen taught %ith ample use of visuali0ation, %ith the help

of a )road spectrum of metaphors and various cognitive modes+

Reeren+e#

Araya, R+ !7888"+0a inteligencia matem6tica, Santiago, Chile ditorial #niversitaria+

Arcavi, A+ !7886"+ ;he role of visual representations in the learning of mathematics,7ducational 4tudies in 8athematics, 9'!6", 71E-7:+

2rousseau, 9+ !1334"+ Th:orie des situations didactiques, 9reno)le


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7ducation , July 61"+ ;he #niversity of $estern Australia e%s, 7E!18",p+ 4+

Sfard A+ !1335"+ *ommentary An metaphorical roots of conceptual growth.?n idactique 4ciences *ogn+ 11, 176 1:5+

Soto-Andrade, J+ !7885"+ Metaphors and cognitive styles in the teaching-learning ofmathematics+ ?n D+ =itta-=anta0i, J+ =hilippou !ds+"+

mathematics as the art of seeing the invisible

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