An Analysis of Addition and Subtraction Word Problems In

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An Analysis of Addition and Subtraction Word Problems in

Mathematics Textbooks Used in Malaysian Primary School Classrooms

Parmjit Singh, University of Technology MARA Malaysia ! [email protected]"

Teoh Sian Hoon University of Technology MARA Malaysia

Mathematics textbooks are integral in most classroom#based teaching and learning as both teachers and $u$ils use them as a source of

mathematical learning% This study embarked on a mission to uncover the ty$es of &ord $roblems associated &ith addition and subtractionavailable in the text books used in Malaysian Primary Schools% 'irstly it examines the distribution of (( categories of addition and subtraction

&ord $roblems based on )an de Walle*s +(,,-. model and secondly it analyses $u$ils* achievement in accordance &ith these categories% Thefindings revealed that the textbooks did not ade/uately distribute or re$resent all the (( categories of &ord $roblems and analysis of the $u$ils*scores based on tests made u$ of /uestions re$resenting the various categories suggested a relationshi$ &ith the distributions of the ty$es of

$roblems%

Background

0n almost every sub1ect area at nearly every grade level students and teachers of mathematics are ex$ected to

use a textbook as a resource +Amit 2 'reid 34435 6luth 3447.%Textbooks $lay a very im$ortant $art in the

teaching and learning $rocess in schools as the textbooks $rovide an im$ortant foundation for teachers in

assisting $u$ils to learn mathematics +Ministry of 8ducation 3449.% Currently in Malaysia all textbooks are

$rovided free for $u$ils in government schools% These textbooks are &ritten by a team of &riters &orking for anumber of $ublishers and can be considered the best according to the s$ecifications outlined by the Ministry of

8ducation% 0n 3449 and 344: the Ministry of 8ducation $ublished ne& mathematics textbooks for Primary ( and

Primary 3 &hile the $ublication of the ne& textbooks for Primary 9 Primary : Primary 7 and Primary ;

occurred from 3447 to 344-% These textbooks are the basis of school instruction and the $rimary source of

information for schools and teachers% This is because the $roblems set as exercises in these textbooks are almost

al&ays assigned as home&ork for the $u$ils and act as a $latform for discussion of mathematical conce$ts

among $u$ils and teachers% The assignments of these exercises as home&ork for $u$ils* conce$tual develo$ment

are sourced from these textbooks by most teachers +Porter 'loden 'reeman Schmidt 2 Sch&ille (,--.%

Schmidt Mcknight and Rai<en +(,,=. have described in general terms the role of textbooks as >bridges bet&een

the &orlds of $lan and intentions and of classroom activities sha$ed in $art by those $lans and intentions? +$%

79.% 0n short one could say that textbooks may $lay a significant role in the attem$t to achieve an intended

learning outcome for classroom teaching and for most $u$ils they $rovide the ground&ork for the content to be

learned as &ell as the conce$tual understanding that $u$ils construct during class activities +Amit 2 'reid 34435

Porter et al% (,--.%)arious researchers +e%g% 'an 2 6aeley 34445 'an 2 @an 344=5 'reeman 2 Porter (,-,5 Stodolsky

(,-,5 @an 2 'an 344;. have investigated from different $ers$ectives the &ays mathematics teachers use

textbooks in their classroom settings% 'reeman and Porter*s +(,-,. study focused on textbook usage by

elementary teachers based on the content taught and textbook content &hile other studies +@an 2 'an 344;5 and

'an 2@an 344=. com$ared American Chinese and Singa$orean school textbooks% Schmidt Mcknight and

Rai<en +(,,=. analysed textbooks based on de$th#of#content#coverage in the textbooks used in the United States

and in other countries% Stodolsky +(,-,. studied the use and influence of textbooks in classroom learning and

teaching% She $ro$osed that the use and influence of textbooks should be analy<ed &ith res$ect to to$ics content

and its com$arison &ith literature on a similar to$ic% lkun and Toluk +3449. analy<ed the content of school

mailto:[email protected]

mailto:[email protected]



mathematics textbooks in Turkey% Their study found that textbooks did not ade/uately re$resent all ty$es of

addition and subtraction $roblems and students &ere less successful on the $roblem ty$es underre$resented in

textbooks% 0n echoing this line of investigation this current research investigated the distribution of &ord

$roblems in the to$ical context of addition and subtraction in Malaysian Primary School textbooks% 0t is

significant to highlight that one of the main thrusts of the Primary Mathematics curriculum in Malaysia is to

develo$ basic com$utation skills com$rising the four o$erators of addition subtraction multi$lication and

division% The usage of the o$erators of addition and subtraction are introduced as early as Primary one in

Malaysia schools &ith the ob1ective to develo$ com$utation skills and the ability to use these skills in solving&ord $roblems% 'urthermore the curriculum also $laces em$hasis on $roblem solving communication

mathematical reasoning and mathematical connections and re$resentations +Curriculum Bevelo$ment Centre

3449.%

The investigation re$orted in this $a$er studied the distribution of &ord $roblems related to the o$erations

of addition and subtraction in Malaysian $rimary school mathematics textbooks% Solving &ord $roblems

embodying additive structures is an im$ortant as$ect of the ne& curriculum for $rimary and secondary Schools

in Malaysia% The term &ord $roblem* is often used to refer to any mathematical exercise for $u$ils stated in a

&ay that enhances a&areness of the semantic structure of the $roblem in con1unction &ith the numerical

re$resentation% According to Car$enter Moses and Debout +(,--. it is essential for the $u$il to think about and

analy<e a &ord $roblem before making an attem$t to solve it% This is because as stated by 6rulik and Rudrick

+(,-3. the &ord $roblem is a situation &hich demands resolution and that there is no easy a$$roach to solving it%

0n this sense for exam$le carefully chosen &ord#$roblems can $rovide a rich context for learning addition and

subtraction conce$ts +Ereer (,,=.%

The term >additive structures? covers $roblems involving addition and subtraction o$erations andkno&ledge of addition and subtraction conce$ts and skills is a $rere/uisite for almost all $rimary school

mathematics to$ics% Substantial research +Car$enter Moser 2 Debout (,--5 Clements (,,,5 Peterson 'ennema

2 Car$enter (,-,. has investigated and found that $u$ils* conce$tions of &ord $roblems demanding addition

and subtraction &ere often vague% 'or young $u$ils it is not easy to model $roblem situations mathematically%

Pu$ils &ho have difficulties &ith reading com$utation or both are likely to encounter difficulties &hen

attem$ting to solve &ord $roblems +Fitendra 2 Gin (,,=.% They are unable to com$rehend the semantics of the

&ord $roblems and this affects the translation into mathematical symbolism% The cure for the >0 can*t do &ord

$roblems? syndrome &ould a$$ear to be ade/uate instruction in using mathematics as a language for $roblem

solving in the curriculum +Parm1it 344;.%

Table (

Categorizing Additive and Subtractive Word Problems Using Van De Walles !"##$% &odel

SHoCategor y 0nformation Problem

(% FRU 'oin (esult

Un)no*n

Iani has (3 flo&ers in the basket% Sarah gave her = more% Io& many flo&ers

does Iani have altogetherJ

3% FCU 'oin C+ange

Un)no*n

Had<irah had - mangoes% 'arah gave her some more% Ho& Had<irah has (7

mangoes% Io& many did 'arah give herJ

9% F0U 'oin ,nitial

Un)no*n

Tasha had some s&eets% Aisha gave her , more% Ho& Tasha has 34 s&eets% Io&

many s&eets did Tasha have at firstJ

:% SRU Separate (esult

Un)no*n

A<har bought (3 $encils% Ie gave 7 $encils to Ran1it% Io& many $encils does

A<har have no&J

7% SCU Separate C+ange

Un)no*n

Ialim catches (- fishes% Ie gave some to Ali% Ho& Ialim has = fishes left% Io&

many did he give to AliJ

;% S0U Separate ,nitial

Un)no*n

Anis baked some cookies% She gave ; to Chong% Ho& Anis has (3 cookies left%

Io& many cookies did Anis bake at firstJ=% CBU Compare

Di--erence

Un)no*n

Binesh has (9 balloons and Kina has : balloons% Io& many more balloons does

Binesh have than KinaJ

-% CKU Compare arger

Un)no*n

Mira read ; storybooks% Alya read (3 storybooks more than Mira% Io& many

storybooks did Alya readJ

,% CSU Compare Smaller

Un)no*n

A<man has : stam$s fe&er than Kim% Kim has (= stam$s% Io& many stam$s does

A<man haveJ

(4% PWU Part/*+ole W+ole

Un)no*n

Siti has (9 small teddy bears and ; big teddy bears% Io& many teddy bears does

she have altogetherJ



((% PPU Part/*+ole Part

Un)no*n

Mimi bought (- a$$les from the su$ermarket% (9 of them are red and the rest are

green% Io& many green a$$les did Mimi buyJ

Some &riters have argued that from a structural $ers$ective there are (( different categories of /uestions

in the form of &ord $roblems for addition and subtraction o$erations +Peterson 'ennema 2 Car$enter (,-,5

)an de Walle (,,-.% Although out&ardly similar /uestions in the (( categories +see Table (. can vary greatly in

difficulty for $u$ils +lkun 2 Toluk 34495 Peterson 'ennema 2 Car$enter (,-,.% While solving different &ord $roblems $u$ils are not only challenged to com$rehend relationshi$s bet&een language and mathematical

$rocesses but also to ex$erience sense making and mathemati<ation of realities +Ereer (,,=5 Reusser 2 Stebler

(,,=5 Wyndhamn 2 Sal1o (,,=.%

Parm1it*s +344;. study of $u$ils* achievement in addition and subtraction &ord $roblems used )an de

Walle*s model% Ie re$orted that many $u$ils found /uestions in the CBU CSU and PPU categories +refer to

Table (. as difficult based on their lo& scores% Although one might ex$ect $u$ils to be able to contextuali<e

$roblems that relate to real#&orld settings often they are unable to move bet&een the semantic structures to the

associated mathematical symbolisms because of the mismatch bet&een their theoretical kno&ledge and &hat

they have ex$erienced in the mathematics classroom%

A study by lkun and Toluk +3449. utili<ing )an Be Walle*s model found that the textbooks used in

$rimary schools in Turkey did not ade/uately re$resent all ty$es of addition and subtraction $roblems% The FCU

F0U SCU S0U CBU CKU and CSU categories &ere under#re$resented% They further argued that this

unsystematic distribution of &ord $roblems categories may $revent $u$ils from develo$ing a rich re$ertoire of

the addition and subtraction conce$ts in the categories that are under#re$resented%

The Purpose of the Study

Doth textbooks and &ord $roblems occu$y an im$ortant $osition in the teaching and learning $rocess and as

Dall and Cohen +(,,;. $ointed out >curriculum materials could contribute to $rofessional $ractice if they &ere

created &ith closer attention to $rocesses of curriculum enactment? +$% =.% Taking into consideration the

im$ortance of mathematics textbooks used in classrooms cou$led &ith the difficulty $u$ils face in solving &ord

$roblems es$ecially in the conce$ts of additive structures this research &as designed to analy<e the content of

additive structures in textbooks used in Malaysian $rimary schools%

Researchers +Riley Ereeno and Ieller (,-95 )an de Walle (,,-. have modeled addition and subtraction

$roblems into categories based on the kind of relationshi$s involved% The classification by Riley et% al% +(,-9.

model &as based on the classification of Change +3 ty$es. Combine +; ty$es. and Com$are +; ty$es.

com$rising (: categories% While )an de Walle*s +(,,-. model &as classified into Foin $roblems +9 ty$es.Se$arate Problems +9 ty$es. Part # Part # Whole ProblemsLcom$are $roblems +3 ty$es. and Com$are or 8/uali<e

Problems +9 ty$es. com$rising (( categories% 'rom these t&o models Riley et% al% +(,-9. (: categories model

seemed more extensive than )an Be Walle*s (( categories% This &as because $roblems such as the follo&ing

&ere not addressed in )an Be Walle*s model% Doth these models com$rise similar categories using different

names% Io&ever )an de Walle*s model seemed not able to re$resent the follo&ing three ty$es in his category

(% There &ere : a$$les in the basket% T&o more a$$les &ere added% Ho& there is the same number of

a$$les as oranges in the basket% Io& many oranges are in the basketJ

3% There &ere (3 a$$les in the basket% 7 of them &ere removed so there &ould be the same number of

a$$les as oranges in the basket% Io& many oranges &ere in the basketJ

9% There &ere some boys in the team% 'our of them sat do&n so each girl &ould have a $artner% There are =

girls in the team% Io& many boys are in the teamJ

0n general both )an de Walle*s model and Riley et% al*s% model &ere similar besides the absence of the three

ty$es of $roblems sho&n above from the latter% Io&ever this study ado$ted the former model because it &as

easier to analy<e the content of a textbook based on (( categories com$ared to the (: categories of the latter%

Secondly as this study focused on Primary ( and Primary 3 textbooks the ty$es of $roblems as sho&n above

&ere not included at this level +$rimary ( and Primary 3. based on the Malaysian mathematics syllabus%

According to )an de Walle*s model there are (( different categories of $roblems in addition and

subtraction5 out of &hich four re/uire addition &hile seven re/uire subtraction% Therefore the intention of this

t&o#fold study &as to analy<e



(% The distribution of the ty$es of &ord $roblems &ith regards to addition and subtraction conce$ts

available in textbooks used in Malaysian schools in Primary ( and Primary 3 using )an de Walle*s +(,,-.

model%

3% Pu$ils* relative achievement on &ord $roblems in these (( different categories%

0f textbooks $lay an im$ortant role in the teaching and learning of mathematics is it $ossible that $u$ils*conce$tual understanding of addition and subtraction is inhibited due to inade/uate o$$ortunities $rovided to

ex$erience these different ty$es of $roblems as a result of imbalance in their school textbooksJ 'urthermore is it

$ossible that $u$ils are facing these difficulties because they are not ex$osed to certain ty$es of $roblems in their

classroom learningJ Nuestions such as these dra& attention to the need to investigate &hich of the categories

cause most difficulty for $u$ils and &hy%

Together &ith these textbooks there &ere also activity books &hich &ere to su$$lement the textbooks% 0n

other &ords at each level a textbook and t&o activity books &ere su$$osed to be used in the teaching and

learning $rocess in the Malaysian $rimary mathematics classroom%

To be noted that these textbooks used in this study &ere $ublished in the 8nglish language adhering to the

$olicy of teaching Mathematics and Science in 8nglish since early 3449% Io&ever this $olicy has been reversed

in August 344, &here effective from 34(3 the teaching of mathematics and science &ill be reverted back to

Dahasa Malaysia the national language% The 8ducation Ministry is in the $rocess of translating all the math

books into Dahasa Malaysia% Although the $olicy has been reversed the content of the books remains the same%

0n vie& of this this $a$er is not affected &ith the reversal of the $olicy%

Method

T&o modes of methodological analysis &ere utili<ed for this research% 'irst document analysis &as used in

analy<ing the distribution of the ty$e of &ord $roblem categories involving addition and subtraction o$erations in

Primary ( and Primary 3 textbooks together &ith the accom$anying activity books for each grade% 'or $ur$oses

of com$arative analysis commercial texts +or &orkbooks. from an established $ublisher &ere also analy<ed%

These texts &ere analy<ed according to the benchmark of the eleven ty$es of standard &ord $roblems as sho&n

in Table ( modeled by )an de Walle +(,,-.% The researcher inde$endently categori<ed each $roblem in these

textbooks in accordance &ith the given categories% 0n ensuring the validity of the analysis inter#rater member

check &as also undertaken to determine the accuracy of the analysis% 0nter#rater member check is a $rocess

&hereby another rater other than the researcher is asked to verify the $roblem in accordance to the category% All

&ord $roblems that could be solved using addition and subtraction of natural numbers &ere included &hile

symbolic ex$ressions such as >- O = J? and $hrases such as >9 less than (4J? &ere excluded%

Secondly an achievement test &as administered to 943 $u$ils from Primary ( and Primary 3 in order to

/uantify the $u$ils* achievements according to the eleven categories% The sam$les com$rised ((; and (-; $u$ils

from Primary ( and Primary 3 classes res$ectively &ith ages ranging from seven to nine years% The actual

schools in &hich the students &ere located &ere selected randomly from five urban schools in a district in the

state of Selangor Malaysia% nce the schools had been selected the $u$ils &ere selected from the >to$? classes

for each of the res$ective grades the aim being to measure their understanding of &ord $roblems in each of the

(( )an de Wall categories% These $u$ils &ere selected from the to$ classes because the researchers &ant the

content kno&ledge to be the main issue in this study instead of the language if the &eaker $u$ils &ere to be

selected% An instrument &as constructed in &hich the $roblems &ere ada$ted from the (( categories +see Table

(.% There &ere (( $roblems in this instrument &ith one re$resentative $roblem for each category% Pu$ils*

res$onses &ere categori<ed based on the follo&ing :#$oint scale 9 Q All correct5 3 Q MinorLcarelessLsilly

error+s.5 ( Q Some attem$t but unlikely to lead to a solution5 4 Q Ho attem$t

Table 3

Distribution o- Word Problems in t+e &alaysian Primary " 0e1tboo) and t+e Commercially/Publis+ed

Wor)boo) FR

U

FC

U

F0

U

SR

U

SC

U

S0

U

CB

U

CK

U

CS

U

PW

U

PP

U

Primary (School Textbook +. 37%-

7%3 ;%3 9,%3 9%( 9%( 7%3 3%( 4 -%9 3%(

Primary ( Commercial &orkbook +. 37 3 3 3: : 7 7 = (( (( 7



Results

Distribution o- Word Problems in t+e Primary " 0e1tboo) and in t+e Commercially/Publis+ed Wor)boo)2 across

t+e "" Categories

This section details the distribution of &ord $roblems in textbooks used in schools across the (( categories for

each of the res$ective levels% Table 3 sho&s the distribution of &ord $roblems according to categories found in

the Malaysian Primary ( mathematics textbook +together &ith activity books.% 0t indicates that in the Primary (

textbook the SRU category had the highest re$resentation +9,%3 of all $roblems. follo&ed by the FRU

category +37%- of all $roblems.% The other categories &ere relatively s$eaking infre/uently re$resented

+ranging from 4 to -%9. Q a $roblem in the CSU category &as not to be found in the Primary ( text%

3igure "a% Percentage of distribution of &ord

$roblems in the Primary ( School textbook

3igure "b% Percentage of distribution of &ord $roblems

in the Primary ( commercial &orkbook%

As sho&n in 'igure (b the commercially#$ublished text had reasonably similar re$resentations &ith FRU

+37. having the highest re$resentation and SRU +3:. the next highest% Io&ever categories such as CKU+=. CSU +((. and PWU +((. have a much higher re$resentation than in the text used in the schools%

Clearly in both texts the em$hasis is on SRU and FRU /uestions% Io&ever the more abstract categories such as

CKU CSU and PWU receive greater attention in the commercially#$ublished text% 'igures (a and (b sho& the

distributions in a bar chart%

Figure 1c. Percentage of correct answers in the test according to the eleven categories

among Primary 1 pupils

Primary " Pupils Ac+ievements on 4uestions in t+e "" Categories

'igure (c indicates the $ercentage of correct res$onses obtained by Primary ( $u$ils for the /uestions in the ((

categories% The highest scores &ere obtained for categories FRU +,:%- correct. follo&ed by PWU +-(. CKU

+=9%9. and S0U +7;.% The categories &hich $osed the greatest difficulty for the $u$ils &ere F0U +,%7. CSU+(:%;. FCU +33%:. and CBU +39%3.% 0t is to be noted that the four categories for &hich $u$ils gave the most

correct res$onses &ere FRU S0U CKU and PWU and each of these involved the o$eration of addition% The

/uestions for other seven categories &hich involved the o$eration of subtraction &ere not as &ell ans&ered%

Table 9

Distribution o- Word Problems in t+e &alaysian Primary 5 Sc+ool 0e1tboo) !P5S0% and Commercially/

Publis+ed Wor)boo) !P5CPW%

FRUFCUF0USRUSCUS0UCBUCKUCSUPWUPPU

P3ST 9(%

9

9%

(

4 37%

4

9%

(

:%

=

=%

-

4 (%; (7%

;

=%

-

P3CPW (7%

3

9 4 3:%

3

;%

(

4 4 ;%

(

(3%

(

3=%

9

;%

(

Distribution o- Word Problems in t+e Primary 5 0e1tboo) and t+e Commercially/Publis+ed Wor)boo) across

t+e "" Categories

Table 9 sho&s the distribution of the categories for /uestions in the Primary 3 mathematics textbook used in

schools% 0t indicates that in the Primary 3 textbook the FRU category had the highest re$resentation +9(%9 of all

$roblems. follo&ed by the SRU category +37%4 of all $roblems. and the PWU category +(7%;.% The other

categories &ere relatively s$eaking infre/uently re$resented +ranging from 4 to =%- re$resentation there

being no $roblem in either the F0U or the CKU categories.%



3igure 5a . of distribution of &ord $roblems in

the Primary 3 school textbook%

3igure 3b% of distribution of &ord $roblems in the

Primary 3 commercially#$ublished book .

'igure 3b sho&s the distribution of the categories in the commercially#$roduced book% The highestre$resentations are for the PWU +3=%9. SRU +3:%3. FRU +(7%3. and CSU +(3%(. categories% There &ere no

$roblems in the F0U S0U and CBU categories% Clearly 'igures 3a and 3b the em$hasis in both of these books is

on $roblems in the FRU SRU and PWU categories%

Primary 5 Pupils Ac+ievements According to t+e "" Categories

'igure 3c sho&s the $ercentage of correct res$onses obtained by Primary 3 $u$ils for the /uestions in each of the

(( categories% Nuestions that &ere ans&ered correctly most fre/uently &ere in the FRU +-=%;%- correct. PWU

+=7%9. SCU +=4%7. CKU +;=%=9. and SRU +;;%=. categories% The most difficult /uestions &ere in the

CBU +37%3 correct. CSU +94%(. FCU +9(%-. F0U +97. and S0U +9;%7. categories% Three of the four

categories for &hich correct res$onses &ere most fre/uently &ere in the FRU CKU and PWU categories &hich

involved the o$eration of addition% The /uestions in the seven >subtraction? categories &ere less likely to be

ans&ered correctly% Com$arison across the categories for Primary ( and Primary 3 indicated that results tended

to be similar es$ecially for /uestions in the FRU PWU and CKU categories%

3igure 5c. Percentage of correct ans&ers according to the (( categories among Primary 3 $u$ils%

Analyses o- Arte-acts6 Pupils Wor) 7ased on Computations in Wor)s+eets

This section analyses $u$ils* +Primary ( and Primary 3. incorrect res$onses% The data from the &orksheets based

on $u$ils* com$utations suggested that $u$ils tended to utili<e the o$eration of addition instead of subtraction

&hen they faced obstacles in modeling the situation% The follo&ing exam$les from $u$ils* &ork exem$lify this

reasoning%

0tem 9

0as+a +ad some s*eets. Ais+a gave +er # more. 8o* 0as+a +as 59 s*eets.

:o* many s*eets did 0as+a +ave at -irst;

Pupils Computation6

0tem 7

:alim catc+es "$ -is+es. :e gave some to Ali. :alim +as < -is+es le-t.

:o* many did +e give to Ali;

Pupils Computation6

0tem =

Dines+ +as "= balloons and ina +as > balloons. :o* many more balloons does Dines+ +ave t+an ina;

Pupils Computation6

0tem ,

Azman +as > stamps -e*er t+an im. im +as "< stamps. :o* many stamps does Azman +ave;

Pupils Computation6

0tem ((

&imi boug+t "$ apples -rom t+e supermar)et. "= o- t+em are red and t+e rest are green. :o* many green apples did &imi buy;

Pupils Computation6

These five items indicate that both Malaysian Primary ( and Primary 3 $u$ils used the addition o$eration instead

of subtraction to solve the given $roblems% 'or exam$le the res$onse sho&n for 0tem 9 seems to indicate that the

$u$il did not com$rehend the relationshi$ bet&een the /uantities , and 34% The &ord $roblem stated that > Ais+a

gave +er one more?% Pu$ils might have inter$reted the &ord >more? as addition% The recognition and a$$lication



of key &ords during the com$utational $rocess might have $layed a role in $u$ils* faulty reasoning thus leading

to incorrect res$onses%

Similarly for 0tems 7 and = the key &ords > gave? and >more? $robably influenced $u$ils &hen modeling

the situation% The &ord > gave? in 0tem 7 might have been inter$reted as becoming more as &hich might have

misled them into using the addition o$eration% The ma1ority of the students &ho obtained an incorrect res$onse

gave the ans&er 37 &hich is the sum of (- and = or = and (-% Again in item = the &ord > more? again seems to

have triggered the $u$ils to add (9 to : &hich e/uals (=%

The $robable reason that young children are confused is $robably because during their first year at schoolchildren are told by their teachers that you get more by 1oining or adding or getting or finding etc% and

you get less by taking a&ay or se$arating or giving a&ay or losing or finding the difference bet&een

+Coombs and Iarcourt (,-; $% xxvii.% According to Kean Clements and Bel Cam$o +(,,4. analysis the

confusion &hich young children ex$erience &ith /uestions &hich s$ecifically contain the &ords more and

less often carries over to /uestions involving giving taking finding losing etc% and /uestions in &hich t&o

sets are combined or com$ared% They tend to a$$ly to such /uestions the same kind of strategies that they a$$ly

to /uestions &hich contain more and less%

(elations+ip bet*een t+e Category o- Word Problems and Problem Di--iculty

According to Tables 3 and 9 the FRU and SRU categories had the highest re$resentation in the Primary ( and

Primary 3 textbooks% Hot sur$risingly $u$ils in both these levels obtained a high $ercentage of success rates of

,:%- and -=%; in both these categories &ith Primary ( $u$ils doing better than their counter$arts in Primary3%

Although the FRU category is highly re$resented in textbooks and activity books it &as found that both

Primary ( and Primary 3 $u$ils failed to res$ond correctly in this category% nly :=%: and ;;%= of Primary (

and Primary 3 $u$ils res$ectably ans&ered this /uestion correctly% Category had the lo&est re$resentation in the

$rimary mathematics textbooks% According to 'igures (a and (b this category &as unre$resented +4. for

Primary ( and under#re$resented +(%; . for Primary 3% The results from $u$ils* achievement &ere also

unsatisfactory &ith only (:%; and 94%( of Primary ( and Primary 3 $u$ils res$onding correctly to the

/uestion in this category%

Table : sho&s a strong statistically significant correlation +r 4%-7 p ! 4%4(. bet&een corres$onding item

facilities for Primary ( and Primary 3 $u$ils% This means that if a /uestion in a certain category &as ans&ered

correctly by Primary ( students then it &as also tended to be ans&ered correctly by Primary 3 $u$ils% Kike&ise

a /uestion ans&ered $oorly by Primary ( $u$ils tended to be ans&ered $oorly by Primary 3 $u$ils%

There &as also a strong statistically significant correlation + r 4%-; p ! 4%4(. bet&een the distribution

categories of &ord $roblems bet&een Primary ( and Primary 3 school textbooks% This indicated for exam$lethat &here there &as a high $ercentage for a $articular category in the Primary ( text the same &ould be likely to

be true in the Primary 3 text% A similar statement &ould be true if >high? &ere to be re$laced &ith >lo&%? A

moderately statistically significant correlation +r 4%;: p ! 4%47. existed bet&een scores item facilities for

Primary 3 $u$ils and the distribution of &ord $roblem categories in the Primary 3 school text% This suggested

that if a certain category &as &ell re$resented in the Primary 3 textbook then the $u$ils tended to ans&er the

/uestion from that category correctly%% Similarly if a certain category &as unre$resented or $oorly re$resented

in the Primary 3 textbook then the $u$ils tended to ans&er the /uestion from that category incorrectly%

0nterestingly though there &as no statistically significant correlation +r 4%9; p " 4%47. bet&een scores item

facilities for Primary ( $u$ils and the distribution of &ord $roblem categories in the Primary ( school text%

Table :

Correlations &atri1 bet*een Pupils Scores and distribution o- Word Problems in Primary " and Primary 5

Categories

Primary ( Primary 3 Text Primary ( Text Primary 3

Primary ( score Pearson Correlation ( %-:=+. %97- %;(9+.

Sig% +3#tailed. %44( %3-4 %4:7

H (( (( ((

Primary 3 score Pearson Correlation ( %74- %;:(+.

Sig% +3#tailed. %((( %49:

H (( ((

Bist%Text Primary (

Pearson Correlation( %-;9+.

Sig% +3#tailed. %44(



H ((

Bist%Text Primary 3

Pearson Correlation(

Sig% +3#tailed.

H ((

Correlation is significant at the 4%4( level +3#tailed.%

Correlation is significant at the 4%47 level +3#tailed.%

Discussion and Conclusions

Textbooks $lay a crucial role in the Malaysian mathematics teaching and learning $rocess in schools% The t&o

im$ortant $edagogic functions as succinctly $ut by )an Bormolen +(,-; cited in Fohnson 3447. are the

curricular and conce$tual $ers$ectives% The former relates to the $rogression of learning materials &hile the latter

relates to the conce$tual develo$ment of learners% The role of textbooks in the t&in $rocesses of teaching and

learning mathematics has received increasing attention over the last t&o decades at least%

As mentioned earlier the study that has been summari<ed &as a t&o#fold study% The distribution of &ord

$roblem categories found in Malaysian $rimary schools mathematics textbooks based on )an de Walle*s +(,,-.

(( model &as identified and $u$ils* achievement on /uestions re$resenting the various categories &ere

com$ared% Also a com$arison of the school textbooks &ith similar commercially#$ublished &orkbooks &as

carried out%8ntries in Table 7 indicate that the &ord $roblem categories based on )an de Walle*s model &ere not

systematically distributed throughout the Primary ( and Primary 3 mathematics textbooks% Three of the

categories namely FRU +3-%;. SRU +93%(. and PWU +(3. &ere &ell re$resented but other categories &ere

either under#re$resented or not re$resented at all 0n terms of $u$ils* achievements across the categories the

findings of this study &ere similar to that obtained by Parm1it +344;.% Analyses indicated that $u$ils in Primary (

and Primary 3 ex$erienced difficulties &ith com$rehending /uestions in categories FCU +3=%(. F0U +33%9.

CBU +3:%3. CSU +33%:. and PPU +:9%;.% There are at least t&o reasons that might have been res$onsible

for this%

Table 7

Distribution o- Word Problems and Pupils Ac+ievement across t+e "" Categories

Categories Pr ( Pr 3 Mean Pr ( Pr 3 Mean(%

FRU 37%- 9(%9 3-%; ,:%- -=%; ,(%3

3%

FCU 7%3 9%( :%3 33%: 9(%- 3=%(

9%

F0U ;%3 4 9%( ,%7 97%4 33%9

:%

SRU 9,%3 37%4 93%( :=%: ;;%= 7=%(

7%



SCU 9%( 9%( 9%( :3%3 =4%7 7;%:

;%

S0U 9%( :%= 9%, 7(%3 7;%7 79%,

=%

CBU 7%3 =%- ;%7 39%3 37%3 3:%3

-%

CKU 3%( 4 (%( =9%9 ;=%= =4%7

,%

CSU 4 (%; 4%- (:%; 94%( 33%:

(4%

PWU -%9 (7%; (3%4 -(%4 =7%9 =-%3

((%

PPU 3%( =%- 7%4 9-%- :-%: :9%;

Category ( : and (4 &ith highest distribution5 Category ( : - and (4 utili<ing addition o$erations

'irst the com$arison bet&een the textbook analysis and $u$ils* achievements indicated that that $u$ils

$erformed least &ell on /uestions in categories that &ere under#re$resented or not re$resented at all in the school

textbooks% n the other hand the students tended to do &ell on /uestions for categories +e%g% FRU SRU and

PWU. that &ere &ell re$resented in the text%

The assum$tions here is that /uestions in those categories that &ere &ell#re$resented in the textbooks

tended to be correctly ans&ered because $u$ils had considerable ex$erience in solving such $roblems in their

schools* mathematics classrooms%

The second $oint is that four of these eleven categories in the )an de Walle model involve the o$eration of

addition but the other seven categories involve the o$eration of subtraction% The /uestions for the four categories+&ith the success rate in $arenthesis. involving the o$erations of addition &ere as follo&s

FRU +,(%3.

:ani +as "5 -lo*ers in t+e bas)et. Sara+ gave +er < more. :o* many -lo*ers does :ani +ave altoget+er;

8umber sentence6 "5 < BBB

The o$eration needed to solve this $roblem is addition%

S0U +79%,.



Anis ba)ed some coo)ies. S+e gave to C+ong. 8o* Anis +as "5 coo)ies le-t. :o* many coo)ies did Anis ba)e

at -irst;

8umber sentence6 BBB "5


CKU +=4%7.

&ira read storyboo)s. Alya read "5 storyboo)s more t+an &ira. :o* many storyboo)s did Alya read; 8umber sentence6 "5 BBB


PWU +=-%3.

Siti +as "= small teddy bears and big teddy bears. :o* many teddy bears does s+e +ave altoget+er;

8umber sentence6 "= BBB


The success rate obtained by $u$ils in the test &ere higher in these four categories +,(%3 79%, =4%7 and

=-%3. than the scores in the other categories +e%g% 3=%( 33%97=%( 7;%: 79%, 3:%3 33%: and

:9%;.% There is a strong $ossibility that the high scores obtained for these four categories involving addition

might not be linked to $u$ils* ability in modeling the situation but rather through guessing% This is because from

these four categories it can be said that S0U +79%,. and CKU +=4%7. involve a higher order $roblem#solving

skill as com$ared to that re/uired for the com$utation of SRU or F0U and yet the $u$ils obtained a higher

$ercentage success rate in the former categories &hen com$ared to the latter% These $u$ils might have com$uted

the numbers utili<ing addition o$erations in deriving the ans&ers% 0t may not have been based on their conce$tual

understanding in modeling the situations% This may be because the first arithmetic o$eration that $u$ils learn in

early mathematics after counting is addition and $u$ils tend to fall back on their $rior ex$erience of construction

&hich they are familiar &ith in modeling the situation% This study suggests that &hen $u$ils face difficulties in

com$rehending $roblems they fall back on the o$eration &ith &hich they are most familiar namely addition%

They tend to commit to the o$eration of addition if there is a barrier that $revents them from com$rehending or

contextuali<ing the given $roblem% This &as /uite $revalent &hen these $u$ils obtained a much higher success

rate in CKU +=4%7. as com$ared to CSU +33%:. though it involves similar contexts &ith differences in terms of

>more? and > -e*er?2 &here the former category involves the o$eration of addition and the latter subtraction%

Another category that concurs &ith this line of reasoning is the analysis of S0U as com$ared to FCU or F0U% 'rom

the $roblems given S0U re/uires a higher thinking mode of contextuali<ation as com$ared to FCU or F0U% @et

the success rate obtained by $u$ils on the former is much higher +79%,. as com$ared to the latter +3=%( and

33%9.% 0n other &ords the more difficult category +S0U. &hich re/uires the o$eration of addition $roduced a

higher success rate as com$ared to the less difficult categories +FCU and F0U. &hich $roduced a lo&er success

rate% 0n this res$ect a /ualitative study ought to be undertaken in assessing $u$ils* construction of kno&ledge

based on their conce$tual advances in Malaysian settings%

The three categories +FRU SRU and PWU. that are most fre/uently re$resented in the $rimary school

textbooks seem to re$resent the lo&est cognitive $rocess needed for the solution as com$ared to the other

categories +es$ecially CBU CKU and CSU. &hich are under#re$resented or not#re$resented at all% The lexicon in

the semantic structure such as E-e*er? and >more? in the latter categories +CKU and CSU. re/uire a higher#order

cognitive o$eration in order to com$rehend and contextuali<e the $roblems as com$ared &ith the other

categories% The inability of $u$ils to com$rehend and contextuali<e the $roblems com$ounded &ith the lexicon

used might be the root of the difficulties that the $u$ils ex$erienced% The failure of many $u$ils to ans&er

correctly some of the $roblems es$ecially those for categories CBU CSU and PPU a$$ears to have been due to

their lack of ex$erience &ith certain situations rather than &ith any difficulty &ith o$erating on numbers

+Parm1it 344;.%Word $roblems &hich start &ith unkno&ns are relatively difficult for young $u$ils +Peterson 'ennema 2

Car$enter (,-,. since they find it difficult to model the $roblem situations mathematically% Textbooks often

include only a fe& of these categories of &ord $roblems thus limiting children*s ability to learn the o$erations

meaningfully +Ereer (,,=5 Peterson 'ennema 2 Car$enter (,-,.% That kind of conclusion concurs &ith the

findings of this study &ith Primary ( and Primary 3 $u$ils and their textbooks in Malaysian schools%

Doth textbooks and &ord $roblems occu$y an im$ortant $osition in the teaching and learning $rocess and

as $ointed earlier by Dall and Cohen +(,,;. >curriculum materials could contribute to $rofessional $ractice if

they &ere created &ith closer attention to $rocesses of curriculum enactment? +$% =.% This sim$ly means that

there is a $ossibility that the $u$ils* inability to solve difficult $roblems may be due to the fact that they have not



obtained ade/uate ex$osure to or ex$erience &orking &ith these $roblems in their classroom learning% The

$resum$tion here is that an analysis of textbooks is a necessary but not sufficient condition to understand &hat

really ha$$ens in actual classroom teaching +'an 2 hu 3444.% When categories of &ord $roblems have been

rarely ex$erienced by $u$ils then the $u$ils can be forgiven for having difficulty >translating? the $roblems into

mathematically correct statements if all of a sudden they are asked to solve them% This can result in the $u$ils*

conce$tual develo$ment in addition and subtraction being inhibited%

!ducational "mplications

The first im$lication is that $u$ils* ability to link addition and subtraction &ith additive and subtractive &ord

$roblem situations develo$s slo&ly throughout $u$ils* early stages of mathematics learning +Ereer (,,=5

Munirah 34475 Parm1it 344: 344;5 Wyndhamn (,,=.% 0t is likely that this shortcoming might be directly

linked to the inade/uate balance of content in the mathematics textbooks used in classroom teaching% The result

of this research indicates that Primary ( and Primary 3 textbooks $ublished by the Ministry of 8ducation since

3449 +in stages. used in Malaysian classroom teaching do not ade/uately re$resent the totality of different

categories for addition and subtraction &ord $roblem /uestions &hich have been &ell#researched in the literature%

Kess ex$osure to these ty$es of $roblems might be one reason that many $u$ils* faced difficulties in these

conce$ts% 0n vie& of this textbooks develo$ers should consider including more of the different ty$es of $roblems

in textbooks%

Secondly the distribution of the &ord $roblem categories in the textbooks has not been systematically

based on research findings% 0n the Malaysian Primary ( and Primary 3 textbooks there is an over em$hasis on

certain categories &hile other categories are hardly re$resented or indeed not re$resented at all% The 8ducationMinistry should em$hasi<e the need for textbook &riters to take into account the findings of research out$uts%

0t is recommended that in#de$th research study be carried out by the Ministry of 8ducation on the content

ade/uacy of current mathematics textbooks used in classrooms es$ecially for the early $rimary grades% This is

necessary because at this stage $u$ils are being ex$osed and becoming familiar to &ord $roblems% Carefully

chosen &ord#$roblems can $rovide a rich context for learning mathematical conce$ts +Ereer (,,=.% Secondly at

these early grades $u$ils &ho have trouble &ith fundamental mathematical conce$ts &ill be likely to continue

having $roblems &ith mathematics throughout their schooling life% Struggling $u$ils can /uickly cultivate a

strong negative attitude to&ards mathematics% 'urther research is clearly essential to ex$lore ho& kno&ledge of

children*s learning of mathematics can be a$$lied to the design of instruction via textbooks to bridge that ga$

that still exists bet&een teachers* teaching and children*s learning in mathematics%

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An Analysis of Addition and Subtraction Word Problems In