Essay on Mathematical Understanding (Otten)

7/26/2019 Essay on Mathematical Understanding (Otten)

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Sam Otten

EDUC 782

Fall 2009Word Count: 1086

Essay on Mathemat!al Understandn"

Why Understanding?

#he $atonal Coun!l o% #ea!hers o% Mathemat!s &$C#M' de(elo)ed ther

Principles and Standards &2000' on a %oundaton o% mathemat!al learnn" *th

understandn"+ What ,ust%es su!h a %oundaton- .nstead o% relyn" solely on

)hloso)h!al ar"uments that !orres)ond *th a !on!e)ton o% mathemat!s as a sense/

man" ds!)lne $C#M !ted a small !olle!ton and n so don" alluded to a lar"er

resear!h ase &3l)atr! Martn 4 S!h%ter 2005' o% em)r!al studes )ontn" to the

(alue o% learnn" mathemat!s *th understandn"+ For eam)le rans%ord ro*n and

Co!n" &1999' n ther tome on learnn" %ound that understandn" s a ey %a!tor *th

res)e!t to learners )ro%!en!y n !om)le domans+ .n )art!ular learners *ho memore

%a!ts or )ro!edures *thout the asso!ated understandn"s may not no* *h!h %a!ts are

rele(ant under !ertan !r!umstan!es or *hat )ro!edure s a))ro)rate n !ertan

stuatons+ .n ths *ay learnn" *th understandn" s (aluale not only %or ts o*n sae

ut n su))ort o% the !on!urrent learnn" o% %a!ts and )ro!edures+

earnn" *th understandn" a!!ordn" to $C#M also su))orts %uture learnn"

o% mathemat!al slls and !on!e)ts+ #he *or o% S!hoen%eld &1988' and Sem) &1976'

)onts to the use%ulness o% understandn" *hen attem)tn" to mae sense o% ne*

stuatons and uldn" ne* meann"s+ Moreo(er $C#M )osts that learners *ho

understand mathemat!s *ll e more sel%/mot(ated to !ontnue learnn" and more

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!on%dent than learners *ho do not understand+ &.t s nterestn" ho*e(er that the

)ara"ra)h statn" ths at the to) o% )a"e 21 n PSSM does not !ontan any re%eren!es+'

; thrd le" o% $C#Ms ar"ument %or learnn" *th understandn" rests on !ontet+

.n our modern !om)uter/ased so!ety *e no lon"er need to rely on humans to )er%orm

routne mathemat!al !om)utatons<!om)uters are usually %aster and less lely to err+

#hus an nd(dual *ho only learns )ro!edures s no etter than a !om)uter+ ;n

nd(dual *ho understands ho*e(er !an e an ntell"ent user o% su!h te!hnolo"es and

!an e!ome ada)t(ely )re)ared to meet the ne* !hallen"es o% the %uture+ ;s =eremy

3l)atr! *rote >the ?ualty o% the l(es our !tens lead de)ends on *hether they are

e?u))ed *th mathemat!al tools %or thnn" aout )rolems that !on%ront them@ &1985

)+ 506'+ #he >tools %or thnn"@ that he re%eren!es are !ertanly !hara!tered y

understandn" not d(or!ed %rom t+

Whence Misunderstanding?

#he ao(e s e(den!e %or some o% the (alues o% learnn" *th understandn"+

#hou"h some may dsa"ree *th the role that understandn" should )lay n s!hool

mathemat!s and may )resent !ounter(aln" e(den!e to ths e%%e!t e(eryone !an

usually a"ree on the detrmental ?ualty o% )ersstent msunderstandn"+ Why s t then

that su!h msunderstandn" ests- For eam)le many students ele(e that

( ) 222

baba +=+ orbaba A

2

5

2

A5

2+=

+ &Mat 1980'+ #hese msunderstandn"s seem to

e related to (ald al"era! denttes su!h as ( ) baba 222 +=+ and2

A

2

5

2

A5 baba+=

+

*h!h may ha(e een o(er"eneraled nto the errors ao(e+ #he d%%eren!e et*een the

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(ald denttes and the student errors has to do *th the !on!e)t o% lnear trans%ormatons+

Mult)l!aton y 2 and d(son y 2 &or mult)l!aton y 1B2' are lnear trans%ormatons

and so thnn" o% them as a %un!ton t s true that ( ) ( ) ( )b f a f ba f +=+ + S?uarn" and

tan" the re!)ro!al o% d(son y 2 ho*e(er are not lnear trans%ormatons so the

same )ro)erty does not a))ly+ Errors o% the ty)e mentoned ao(e may !ome %rom an

e)e!taton that all trans%ormatons eha(e as n!ely as lnear trans%ormatons+

;nother msunderstandn" that has een do!umented n the resear!h lterature s

!on%uson *th he"hts o% tran"les &utrre 4 =ame 1999'+ Many learners ele(e that

the he"ht o% a tran"le should e a sde o% the tran"le and others ele(e that the he"ht

should al*ays run (ert!ally *th res)e!t to ther )a)er+ .t seems )lausle that the he"ht/

as/sde/o%/a/tran"le msunderstandn" s related to the smlar &and (ald' noton o% ase/

as/sde/o%/a/tran"le+ ;lso the he"ht/runs/(ert!ally/on/the/)a"e msunderstandn" s

lely rooted n the ty)!al )ra!t!e o% dra*n" tran"les so that the ase runs horontally

%or!n" the he"ht to !ommonly run (ert!ally+ &; smlar %ndn" *th youn" !hldren s

that they do not re!o"ne tran"les that st ase* on ther )a)er+'

One )ossle road e)lanaton %or these msunderstandn"s les n e*tts

dstn!ton et*een the necessary and arbitrary n mathemat!s &e*tt 1999'+

$e!essary truths are "rounded n the aoms o% mathemat!s *h!h themsel(es are

"rounded n the realty o% our un(erse and so n ths sense are determned+ Eam)les o%

ne!essary truths are the !ommon )ro)ertes o% the *hole numers and the ytha"orean

#heorem+ ;rtrary as)e!ts o% mathemat!s on the other hand !ould e other*se<*e

ha(e sm)ly a"reed on them as !on(entons+ For eam)le *e use the >G@ symol %or

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addton and *e tend to dra* tran"les *th ther ase at the ottom+ oth o% these thn"s

!ould e d%%erent *thout a%%e!tn" the underlyn" mathemat!al truths the ne!essary+

Wth res)e!t to the msunderstandn"s dent%ed ao(e )erha)s learners ha(e

!on%lated the ne!essary and the artrary+ #hey may ele(e that rules su!h as the

dstrut(e )ro)erty *ere artrarly de!ded u)on &ndeed the term >rule@ may su""est

ths' rather than determned y the eha(or o% the numer system and so t *ould mae

sense to also a))ly ths rule to smlar stuatons su!h as s?uarn" a nomal+ #hs may

also e ren%or!ed y the %a!t that ths artrary a))roa!h s easer than the ne!essary

a))roa!h so % mathemat!s s artrary *hy not mae t easy- .n the !ase o% tran"le

alttudes t s )ossle that learners ha(e nter)reted the !ommon ut artrary orentaton

o% tran"les *th ase on ottom as a ne!essary as)e!t o% the stuaton thus !on!ludn"

that he"hts should al*ays run (ert!ally+

What role does nstru!ton )lay- .% tea!hers are themsel(es una*are o% the

dstn!ton et*een the mathemat!ally ne!essary and the mathemat!ally artrary then

t *ll e nearly m)ossle %or them to "ude students n man" the dstn!ton+

;ddtonally %o!usn" on rules and )ro!edures may send the messa"e that the ds!)lne

s !om)letely artrary under!uttn" the )osslty o% students seen" the need to sear!h

%or ne!essary truth &e+"+ Erl*an"ers enny &1975''+

Where from here?

. ele(e that there s )romse some e)lanatory )o*er n the noton o% the

ne!essary and the artrary n mathemat!s+ Future *or may de(elo) the %rame*or

%urther and use t to mae sense o% some student msunderstandn"s+ Su!h *or *ould

!ertanly need to n!or)orate nter(e*s *th students %or the )ur)ose o% un!o(ern" ther

H



!on!e)ton o% mathemat!s+ .t may e *orth*hle to eamne tea!hers !on!e)tons o%

mathemat!s alon" these dmensons as *ell to n(est"ate my !lams ao(e aout the

role o% nstru!ton+ .n )art!ular the doman o% al"era! )ro)ertes and

o(er"eneralatons seems )art!ularly r)e %or a))lyn" these deas+

Ie%eren!es

rans%ord =+ D+ ro*n ;+ + 4 Co!n" I+ I+ &1999'+ How people learn: Brain, mind,

eperience, and school + Washn"ton DC: $atonal ;!ademy ress+

Erl*an"er S+ + &1975'+ ennyJs !on!e)ton o% rules and ans*ers n .. mathemat!s+

!ournal of "hildren#s Mathematical Beha$ior, % 7/26+

utrre ;+ 4 =ame ;+ &1999'+ reser(!e tea!hersJ understandn" o% the !on!e)t o%

alttude o% a tran"le+ !ournal of Mathematics &eacher 'ducation, ( 2A5/27A+

e*tt D+ &1999'+ ;rtrary and ne!essary: ; *ay o% (e*n" the mathemat!s

!urr!ulum+ )or the *earning of Mathematics, %+&5' 2/9+

3l)atr! =+ &1985'+ Edtoral+ !ournal for esearch in Mathematics 'ducation, %- 506+

3l)atr! =+ Martn W+ + 4 S!h%ter D+ &Eds+'+ &2005'+ . research companion to

Principles and Standards for School Mathematics+ Ieston K;: $C#M+

Mat M+ &1980'+ #o*ards a !om)utatonal theory o% al"era! !om)eten!e+ !ournal of

Mathematical Beha$ior, / 96/166+

$atonal Coun!l o% #ea!hers o% Mathemat!s+ &2000'+ Principles and standards for

school mathematics+ Ieston K;: ;uthor+

S!hoen%eld ;+ + &1988'+ When "ood tea!hn" leads to ad results: #he dsasters o%

J*ell/tau"htJ mathemat!s !ourses+ 'ducational Psychologist, (/ 1HA/166+

A



Sem) I+ I+ &1976'+ Ielatonal understandn" and nstrumental understandn"+

Mathematics &eaching, 00 20/26+

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Essay on Mathematical Understanding (Otten)

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