Strategies of pre-service primary school teachers for solving addition problems with negative numbers

This article was downloaded by: [Anadolu University]On: 21 December 2014, At: 05:53Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Click for updates

International Journal of MathematicalEducation in Science and TechnologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tmes20

Strategies of pre-service primaryschool teachers for solving additionproblems with negative numbersRut Almeidaa & Alicia Brunoa

a Department of Mathematical Analysis, University of La Laguna,La Laguna, SpainPublished online: 20 Jan 2014.

To cite this article: Rut Almeida & Alicia Bruno (2014) Strategies of pre-service primaryschool teachers for solving addition problems with negative numbers, InternationalJournal of Mathematical Education in Science and Technology, 45:5, 719-737, DOI:10.1080/0020739X.2013.877605

To link to this article: http://dx.doi.org/10.1080/0020739X.2013.877605

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

http://crossmark.crossref.org/dialog/?doi=10.1080/0020739X.2013.877605&domain=pdf&date_stamp=2014-01-20

http://www.tandfonline.com/loi/tmes20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/0020739X.2013.877605

http://dx.doi.org/10.1080/0020739X.2013.877605

Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

International Journal of Mathematical Education in Science and Technology, 2014

Vol. 45, No. 5, 719–737, http://dx.doi.org/10.1080/0020739X.2013.877605

Strategies of pre-service primary school teachers for solving additionproblems with negative numbers

Rut Almeida and Alicia Bruno∗

Department of Mathematical Analysis, University of La Laguna, La Laguna, Spain

(Received 16 April 2013)

This paper analyses the strategies used by pre-service primary school teachers forsolving simple addition problems involving negative numbers. The findings reveal sixdifferent strategies that depend on the difficulty of the problem and, in particular, onthe unknown quantity. We note that students use negative numbers in those problemsthey find easy and resort to other strategies in more complex problems. Furthermore,the problem-solving strategies reveal lingering difficulties involving negative numbers,such as mistakes with arithmetic rules and a lack of rigour in the mathematical notationused in the operations.

Keywords: pre-service primary school teachers; negative numbers; addition problems

1. Introduction

Introducing negative numbers to 12- or 13-year-old students requires them to changestrongly rooted conceptions about numbers that are built up all throughout primary school.When negative numbers are presented, conceptual and procedural changes involving aknowledge of numbers take place: the existence of numbers less than zero; expanding themeanings and uses of numbers into new contexts; having the addition and multiplicationof two numbers potentially yields a result that is less than either operand (in contrast tosubtraction and division); the identification of the addition and subtraction operations (a +(−b) = a − b; a − (−b) = a + b); new arithmetic rules (such as the signs for a product);and changes to the symbology and notation ( + a = a or the need for greater numbers ofparentheses).

The concern over improving the teaching–learning of negative numbers has been madeevident by the publication of numerous works starting in about 1970, a trend that has grownto the current day.[1–5]

Most of the research has involved secondary education, although the intuitive ideas ofprimary school students on negative numbers have also been studied.[6–8] It has been welldocumented in research that students have difficulty with integer operations.[2,3,9,10]

Research with pre-service teachers about mathematical knowledge and teaching ofnegative numbers are less numerous. Those that have been made deal with very differentaspects of negative numbers: representation on the number line, models of teaching andproblem solving. Research shows difficulties in pre-service teachers’ conceptual under-standing of negative numbers [4] and their representation on the number line.[11] It isnecessary to further deepen in how pre-service teachers address this topic since their math-ematical knowledge influences in how they teach contents later. The fact that pre-service

∗Corresponding author. Email: [email protected]

C© 2014 Taylor & Francis

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14

http://dx.doi.org/10.1080/0020739X.2013.877605

mailto:[email protected]

720 R. Almeida and A. Bruno

teachers’ misconceptions might be passed on to their future students provides a furtherstrong impetus for researching this issue.

In this paper, we present a study on how pre-service primary school teachers solve simpleaddition problems involving negative number. We analyse the strategies and reasoning theyfollow in solving questions in this topic considering the structure of the problems (change,compare, change–compare and equalize) and the position of the unknown. The results revealgaps in the numerical knowledge of those pre-service primary school teachers analysed,gaps that are filled in by their pre-existing knowledge of positive numbers.

2. Review of the literature

Research has shown that primary school students can use negative numbers in simpleaddition and subtraction operations before the age prescribed by the official curriculum(the end of primary education in some countries and the beginning of secondary educationin others). Students do this by resorting to simple strategies and building mental modelsbased on their knowledge of positive numbers.[6,7,12]

In their research, Mukhopadhyay, Resnick and Schauble [8] and Carraher [13] analysedhow students who had received no previous instruction on negative numbers solved additionproblems involving debts and noticed how they could formulate them verbally, but had greatdifficulty when trying to write the operations formally.

The students’ initial ideas reflect a significant gap between their understanding of thespecific significance of negative numbers and how they represent and handle these numbersusing signs and symbols.

In keeping with the terminology in Janvier,[14] most of the models that we find forteaching negative numbers resort to the equilibrium model or to the number line model.

In the equilibrium model, positive and negative whole numbers are represented usingchips of two different colours, such as black and white. If the black chips represent negativenumbers and the white chips positive numbers, the number −1 can be represented usingany combination of chips in which the black chips outnumber the white chips by one. Thebasic idea of this model is that one white and one black chip cancel each other out, givingzero. The various operations are defined based on this rule: a sum is defined as ‘combining’or ‘joining’ chips, subtraction as ‘removing’ chips and multiplication as ‘repeated sums’ ofgroups of chips.

In the number line model, the numbers are both positions on the line and movementsalong the line. Addition in this model can be a combination of two movements or a movementfrom one position to another. Subtraction means ‘adding the opposite’, that is, ‘moving inthe opposite direction’, or as ‘the difference between two positions’. Multiplication isdefined as the ‘repeated sum’ of movements.

The research that supports the equilibrium model does so based on the fact that additionmaintains the same meaning as with positive numbers, that is, ‘to add’, and subtractionmaintains the meaning of ‘take away’. The findings show it to be an intuitive model for thenotion of opposite, as well as for addition and simple subtractions (a − ( + b)); however,subtractions of the type a − (−b) do not fare as well.[15,16]

Authors who propose the number line model highlight how this model differentiatesbetween the signs of the numbers and the signs of the operations, and the ease withwhich subtractions can be seen as the addition of the inverse.[7,17] Cunningham [5] wassuccessful in using the number line to teach negative numbers in remedial mathematicsclasses designed to prepare community college students for the high stakes exit examinationnecessary to graduation and college-level work. Hativa and Cohen [18] and Altiparmak and

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14

International Journal of Mathematical Education in Science and Technology 721

Ozdogan [3] taught negative numbers in a computing environment using the number lineas an intuitive model.

Faced with this research, different authors have noted that the number line combinedwith traditional subtraction rules is ineffective for making sense of subtractions involvingnegative numbers.[14,19] Janvier [19] criticizes the ambiguity of this model, since numbersare states (points on the line) or changes (arrows or movements along the line). He alsonotes that many students who do well with the model pay more attention to the operationthan to the model itself.

In short, there is no consensus on the best model, although all researchers agree thatthe greatest challenge is posed by the subtraction of negative numbers, such as a − (−b),and by explaining the multiplication rule that ‘negative times negative equals positive’.

As we noted, the knowledge that pre-service primary and secondary school teachershave of negative numbers has not been the subject of much research. Widjaja, Stacey,and Stente [11] studied number line representations of negative decimals used by 94pre-service primary school teachers. The authors note that understanding the orderingof negative decimals and placing them on a number line are a challenging topic for asignificant proportion of the pre-service teachers. Steiner [4] conducted a study with pre-service elementary teachers on how their conceptual understanding of integer addition andsubtraction understanding is impacted by the use of a novel teaching model that combinesthe features of the number line model with those of the equilibrium model. This studyshowed that the equilibrium model aided the participants’ understanding of the algorithmsfor addition and subtraction of integers.

3. Solving addition problems

In the same way as positive numbers, efforts to associate negative numbers to realityare evident, particularly noteworthy is the research conducted into simple addition wordproblems. Simple addition word problems with positive numbers solved using x + y = z orx − y = z have been the subject of intensive studies. Interested readers can find an extensivebibliography in the overview of research conducted by Fuson [20] and Verschaffel and DeCorte.[21]

The most common classification for addition problems distinguishes between change,compare, equalize and combine problems.[20] In this paper, we consider change, compareand equalize problems and a variant of problem that can be considered as a hybrid betweenchange and compare and that we call change–compare. This problem is described below.

3.1. Compare and equalize

Certain numerical situations give rise to two states that are compared: state a (‘John has 2Euros’) and state b (‘Peter has 5 Euros’). We will use the form a + d = b, where d is thedifference. There are two basic ways to express the difference.

In compare problems, the difference is expressed through the use of ‘more than’ (‘Peterhas 3 Euros more than John’) or ‘less than’ (‘John has 3 Euros less than Peter’).

In equalize problems, we state the amount by which the smaller number must beincreased to give the larger number (‘If John makes 3 Euros, he will have as much as Peter’)or by which the larger number must be decreased to give the smaller one (‘If Peter loses 3Euros he will have as much as John’).

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


3.2. Change and change–compare

Other situations feature an initial state i (‘John had 2 Euros’), a variation v (‘and then made3 euros’) and a final state f (‘now he has 5 Euros’). These problems are of the form i +v = f. There are two ways to express the variation.

In change problems, the variation is expressed simply (‘John made 2 Euros’ or ‘Johnlost 2 Euros’).

In change–compare problems, the variation is expressed with ‘more than’ or ‘less than’,as in compare problems (‘John now has 3 Euros more than before’).

The above distinction between form and expression is not typical. In research on ad-dition problems, change–compare problems are regarded as compare problems; however,we thought it interesting to distinguish between them, since these problems were studied inBruno et al.,[22] who concluded that the expression of the variation in change and change–compare problems was important to the success of primary school students. Moreover, im-portant research has proven the importance of the expressions used in word problems.[23,24]Fuson and Willis [25] also noted that compare problems are, in general, more difficult thanequalize problems; in other words, the difficulty depends on how the difference is expressed.

3.3. Unknown quantity

When solving addition problems, not only is the difference between the structures relevantbut also is the position of the unknown variable.

In this paper, we will call unknown 3 (U3) problems those in which the unknownquantity is the third in the form a + d = b; that is, the state b, or the third in the formi + v = f, that is, the final state f.

Unknown 2 (U2) problems are those in which the unknown is the second term in theabove forms; that is, the d and v in the expressions a + d = b and i + v = f, respectively.

Finally, unknown 1 (U1) problems are those where the unknown quantity correspondsto state a or to the initial state i (problems of this type are not considered in this paper).

The position of the unknown is relevant because problems that are solved using sub-traction pose two meanings for this operation: In U3 problems, the subtraction means totake away; while in U1 and U2 problems, the subtraction involves the distance betweentwo amounts. As noted by Selter, Prediger, Nurenborger and Hußmann,[26] the model ofsubtraction as a distance is used less than that of subtraction as ‘taking away’, as it isreflected in the strategies adopted by students.

3.4. Solving addition problems with negative numbers

Researches done about additive problems with negative numbers have shown they are diffi-cult for many secondary students. The difficulties posed by problems with negative numbersdepend primarily on the aforementioned variables: structure, position of the unknown, typesof numbers or context. Not all structures are equally simple; for example, change problemsare simpler than those that combine two changes.[1] However, the unknown variable in theproblem has more effect on the difficulty than the structure of the problem.[27] The signsof the numbers also play a role in both the difficulty and in the strategy used to solve theproblems. Marthe [2] noted that problems involving numbers of opposite signs are morecomplex than those with the same sign. Although the context of the word problem does nothave as much of an effect on the difficulty as the structure,[9] it can, in some cases, affect

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


the outcome. The debt context is easier to understand and more often leads to a success-ful resolution, which indicates its suitability for introducing the study of these numbers,whereas the chronology context (BC and AD) leads to higher failure rates than others.[27]

This makes it necessary, therefore, to look for teaching methods that help to establishrelations between abstract knowledge and contextualized situations.

4. Objectives and methodology of the study

Negative numbers are introduced in Spain at the last year of primary school (12 year old) andcontinuous in secondary school (12–14 year old). It is of interest to know how pre-serviceprimary school teachers solve additive tasks that provide an introduction to operations withnegative numbers. This paper focuses on how pre-service primary school teachers solveaddition problems involving negative numbers.

4.1. Objectives of the research

The general purpose of this research is to expand our knowledge of how certain types ofaddition problems involving negative numbers are solved by pre-service primary schoolteachers, with a focus on:

(1) studying the strategies used by pre-service primary school teachers to solve additionchange, compare, change–compare and equalize problems based on the structureand the unknown;

(2) analysing the reasoning behind the use of negative numbers when solving problems.

4.2. Methodology

We administered a written test with eight addition problems, four of the U3 type andthe other four of the U2 type. The contexts used were such that they implied a verticalrepresentation: elevator, sea level and temperature.

For the case of negative numbers, different studies have shown that the sign of thenumbers (positive and/or negative) affect the way the problems are solved and their levelof difficulty. In our study with negative numbers, the problems are of the form:

positive number + negative number = negative number (in the four U3 problems),negative number − positive number = negative number (in the four U2 problems).

From all possibilities, we chose problems with these signs in numbers because theyimply situations from positive to negative ones going through zero.

Table 1 shows the complete wording for the eight problems used in this study, notingthe structure, the unknown, the sign of the numbers, the operation and the context.

Note that the U3 problems employed in the study can be solved by using a sum, themeaning of which is to add a negative movement, although they can also be solved with asubtraction, the meaning of which is to ‘lower’: 6 − 7 = −1; 4 − 6 = −2; 2 − 6 = −4; 3− 5 = −2. The U2 problems are all solved using subtractions, the meaning of which is thedifference between two states and which allow for a simplified notation: −2 −3 = −5; −5−4 = −9; −1 −4 = −5; −5 − 3 = −8.

The test thus described was taken by 137 pre-service primary education second-yeareducation majors at the University of La Laguna (Tenerife, Spain) who had taken courses

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


Table 1. Statements and types for the eight problems used in the study.

Unknown 3

Luis recorded the temperature in the morning and at night.The temperature, which was 6◦ above zero in the morning,dropped 7◦ over the course of the day. What was thetemperature at night?

Change (Ch3)Initial state + variation = final stateUnknown: final state( + 6) + (−7) = −1; 6 − 7 = −1

Before moving, the elevator in a building was on the fourthfloor. Now it is six floors below its initial position. Whatfloor is it on now?

Change–compare (Ch-Cp3)Initial state + variation = final stateUnknown: final state+ 4 + (−6) = −2; 4 − 6 = −2

A bird flies 2 metres above sea level and a fish is 6 metresbelow the bird. At what position is the fish?

Compare (Cp3)State 1 + difference = state 2Unknown: state 2+ 2 + (−6) = −4; 2 − 6 = −4

The temperature in Madrid is 3◦ above zero. If thetemperature in Madrid drops 5◦, it will be the same as inBilbao. What is the temperature in Bilbao?

Equalize (Eq3)State 1 + difference = state 2Unknown: state 2+ 3 + (−5) = −2; 3 − 5 = −2

Unknown 2An elevator was on the third floor and then moved to thesecond underground floor. What was the elevator’s motion?

Change (Ch2)Initial state + variation = final stateUnknown: variation−2 − ( + 3) = −5; −2 − 3 = −5

The temperature in Paris in the morning was 4◦ above zeroand fell over the course of the day to 5◦ below zero. Howmany degrees lower was the temperature in the evening thanin the morning?

Change–compare (Ch-Cp2)Initial state + variation = final stateUnknown: variation−5 − ( + 4) = −9; −5 − 4 = −9

A building has two elevators, A and B. Elevator A is on thefourth floor and elevator B in the first underground floor.How many floors lower is elevator B than A?

Compare (Cp2)State 1 + difference = state 2Unknown: difference−1 − ( + 4) = −5; −1 − 4 = −5

A hang glider is flying 3 metres above sea level and a diveris swimming 5 metres below sea level. How many metresdoes the hang glider have to drop to be at the same altitudeas the diver?

Equalize (Eq2)State 1 + difference = state 2Unknown: difference−5 − ( + 3) = −8; −5 − 3 = −8

on mathematics and on mathematics education. The test was given during a normal 1-hourclass session.

5. Results

The analysis of the results takes into account whether the answer was right/wrong andclassifies the solving strategy. On many occasions the students used different strategies forthe same problem.

5.1. Problem-solving strategies

The students solved the problem using six basic strategies, some of them simultaneously,both in a correct or incorrect way. The strategies are as follows.

(1) Use an operation involving positive numbers.(2) Use an operation involving negative numbers.

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


4 + 5 = 9 the temperature was 9º lower. Justification: The temperaturedropped by the positive 4º it was during the day plus the 5º negative at night

Figure 1. Student 67’s answer (problem Ch-Cp2).

(3) Use a number line.(4) Count sequentially.(5) Give a verbal explanation.(6) Make a drawing.

An example of how each strategy was used by different students to solve the sameproblem, change–compare Ch-Cp2, is given below:

Problem Ch-Cp2

The temperature in Paris in the morning was 4◦ above zero, and fell over the course ofthe day to 5◦ below zero. How many degrees lower was the temperature in the evening thanin the morning? [Answer with negative numbers: −5 − ( + 4) = −9; −5 − 4 = −9]

5.1.1. Use an operation involving positive numbers

Students who use this strategy solve a problem by adding or subtracting positive numbers,using the numbers given in the problem statement. Using positive numbers means that inorder to obtain the right answer, the result must be contextualized to the negative situation.Example: Figure 1, student 67.

5.1.2. Use an operation involving negative numbers

Solving problems with negative numbers means using the two numbers in the problemstatement (one positive and one negative) and adding or subtracting them. Student 112(Figure 2) gave an answer resulting from an operation with negative numbers, as shownbelow.

5.1.3. Use a number line

This strategy consists of displaying the numbers on a number line (horizontal or vertical)and comparing the positions of the two numbers on the line or moving from one to the other(see student 2’s answer, Figure 3).

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


4 – (-5) = 9

The temperature was 9 degrees lower at night

than in the morning.


The temperature dropped 9º, since to get from 4º to -5º thetemperature would have to fall by 9º. Answer: 9º


Many students drew a number line, with most of them being vertical, due to the contextof the problem.

5.1.4. Count sequentially

Another frequently used strategy is to write part of the number sequence, starting andfinishing with the two numbers given in the problem statement. The number sequence waswritten horizontally or vertically (though most were vertical). The vertical sequence maybe regarded as a simplified version of the number line, though it can also be thought of asa way of counting. This is more apparent when the numbers are written horizontally. Anexample of this is given in student 36’s answer (Figure 4).

5.1.5. Give a verbal explanation

Give a verbal explanation consists of writing the positions of the numbers given in theproblem statement in detail, along with the movements or comparisons made betweenthem, so as to obtain a contextualized solution verbally. A student utilizing this strategy

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


Morning Night


9 degrees. Because if you count from 4 to -5,you get 9 degrees less.


does not write down any operations or use visual aids, but rather describes the situation andthe actions (or comparisons) that take place. This strategy implies either a mental countingprocess to go from one position to the other or an estimate of the distance between thenumbers. A mental calculation may also be performed. In any case, these parallel strategiesare not reflected in the students’ answers. One example of this strategy is shown in theanswer from student 76 (Figure 5).

5.1.6. Make a drawing

In this strategy, a picture or a diagram is drawn to describe the problem using the numbersgiven in the problem, and the answer deduced from this picture or diagram. The picturedrawn is sometimes very detailed and accurately portrays the situation given; other timesthe picture is a crude diagram featuring the numbers and/or the actions or comparisons, asin the case of student 57’s answer (Figure 6).

The pictures or diagrams accompany other strategies, with the pictures being used tobetter explain another predominant strategy.

Although it is clear that the number line is a graphic representation, we have not includedit in the strategy ‘make a drawing’ because of the importance the number line has in themathematical knowledge, specially in the introduction of negative numbers. While makinga drawing implies doing a diagram or a picture related with the context, number line is theusual numerical representation in mathematics.

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


If the temperature fell at night, the lower thetemperature the colder it is (9º).


5.2. Problem-specific strategies

In analysing the results, we considered as correct all the answers that give the negativenumber that answers the problem, or the opposite positive number within the context of theproblem. For example, for the change problem Ch3, whose answer using negative numbersis ( + 6) + (−7) = −1, or 6 − 7 = −1, the answers ‘−1’ or ‘1 degree below zero’ wereboth marked as correct, regardless of the strategy used to obtain the answer.

Table 2 shows the percentage of right answers for the eight problems in the test. Allof the U3-type problems exhibited high percentages of right answers (between 88% and92%), except for the comparison problem, which tested lowest of all (68%), a finding weanalyse later. The U2 problems featured a slightly lower percentage (76%−77%) than theU3 problems, except for the change problem, which was answered correctly by 88% of thestudents, a figure that is closer to the percentages for most of the U3-type problems.

Although the percentages are not low, the problems were more difficult than expectedfor college students since students arriving at the right answer using operations involvingnegative numbers did not have to express it in its ‘canonical’ form. The difficulties in somecases were contextual or stemmed from improper interpretations of the problem statement.In other cases, however, they did reveal a deficiency in the students’ knowledge of negativenumbers.

Table 3 shows the percentages for the strategies used in each of the problems. Most ofthe times only a single strategy was used, and at other times two or three strategies werecombined. Up to 20 strategy combinations were found. One example of such a combinationwas found in student 39’s answer (Figure 7) to problem Ch3, in which he used the strategiesof: use an operation involving negative numbers, use a number line and make a drawing.

Figure 8 shows the percentages for the number of times each strategy is used in the fourU3 problems. The percentages for combined strategies are given in the ‘Others’ column.

The results for the U3 problems, Ch3, Ch-Cp3 and Eq3, are similar both in terms ofthe right answer percentage and solution strategies (see Tables 2 and 3 and Figure 8). The

Table 2. Percentage of right answers.

Unknown 3 problems Unknown 2 problems

Ch3 Ch-Cp3 Cp3 Eq3 Ch2 Ch-Cp2 Cp2 Eq292 90 68 88 88 76 77 77

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


Table 3. Strategy percentages by problem.

Problems

Unknown 3 Unknown 2

Strategy Ch3 Ch-Cp3 Cp3 Eq3 Ch2 Ch-Cp2 Cp2 Eq2

Operation n◦ positives 3 2 24 6 11 23 21 36Operation n◦ negatives 54 37 16 35 5 7 5 3Number line 10 11 9 17 12 22 15 8Count 7 17 18 21 20 20 22 15Explanation 9 15 12 6 39 17 18 8Drawing – 1 5 2 4 1 6 2Operation n◦ positives/number line – – – – 1 2 – 1Operation n◦ positives/count – – – – – – – 2Operation positives/explanation – – – – 1 – 1 2Operation positives/drawing – – 2 – – – 2 11Operation negatives/number line 10 6 1 6 – 2 1 –Operation negatives/count 4 2 2 2 – 1 – –Operation negatives/explanation 2 2 – 2 – – – –Operation negatives/drawing 1 2 2 2 – 2 1 1Number line/count – – – – – 1 – –Number line/drawing – 1 3 – 2 – 1 2Count/drawing – 1 4 – 4 2 5 6Operation positives/number line/drawing – – – – – – 1 2Operation negatives/number line/drawing – 1 1 1 – – – 1Operation negatives/count/drawing – 2 1 – 1 – 1 –

Figure 7. Student 39’s answer (problem Ch3).

Figure 8. Strategy percentages for unknown 3 problems.

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


The fish is -8 meters away from the bird. So going back to the sequence, that’s the numberof steps we have to go down to reach the fish.

Figure 9. Student 75’s answer (problem Cp3).

predominant strategy is Use an operation involving negative numbers followed to a muchlesser extent by Use an operation involving positive numbers and then Make a drawing.The other three strategies have similar percentages, with no single strategy standing out.Change problem Ch3 exhibits the greatest use of operations with negative numbers, mostlyin its simplified form, that is, 6 − 7 = −1.

The results for U3 problem Cp3 exhibit a different trend from the three earlier problems.Of note is that 39 out of the 124 students (32%) answered the problem incorrectly. Theproblem statement is as follows:

Problem Cp3

A bird flies 2 metres above sea level and a fish is 6 metres below the bird. At whatposition is the fish?

Many students interpreted the statement to mean that the bird is at position + 2 and thefish as position −6, and asks for the distance between them, or a comparison between theirpositions. In other words, they restructured the problem into a U2 type of problem. Suchwas the case with student 75 (Figure 9).

The misinterpretation of the problem statement by the students could result from readingthe problem too quickly after having already answered five similar word problems (this wasthe sixth problem in the test), since we believe the wording of the problem to be standardand correct.

The fact that it was interpreted as a U2-type problem results in a diversification ofstrategies of between 12% and 26%, similar to that observed in the actual U2 problems.

U2-type problems also exhibit different results from those shown by U3-type problems(Table 3 and Figure 10).

Without question most notable of all is the considerable drop in the strategy to usean operation involving negative numbers, which was the least used in practically all fourproblems.

To prove the significant difference in the use of operation with negative numbers betweenU2 and U3 we made a chi-square contrast test for independence (α = 0.05) with the SPSS

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


Figure 10. Strategy percentages for unknown 2 problems.

Table 4. Difference in the use of operation with negative numbers between U2 and U3. Results ofchi-square independence test.

Ch Ch-Cp Cp Eq

Negative Other Negative Other Negative Other Negative Other

U2 % 6.5 93.5 11.3 88.7 8.1 91.9 4.8 95.2Rα −10.4 10.4 −6.8 6.8 −3.2 3.2 −7.7 7.7

U3 % 71 29 51.6 48.4 22.6 77.4 47.6 52.4Rα 10.4 −10.4 6.8 −6.8 3.2 −3.2 7.7 −7.7

χ 2 = 108.8 χ 2 = 46.8 χ 2 = 10.1 χ 2 = 58.6df = 1 df = 1 df = 1 df = 1

p = .000 p = .000 p = .002 p = .000

21 software (Table 4). This test showed us that the use of negative numbers in operations isstrongly dependent to the type of item. We found the use of negative numbers is associatedto U3 (there are evidences analysing adjusted residuals Rα), whereas U2 is related with theuse of other strategies.

The fact that many students solved the U3 problems with negative numbers and thenabandoned this strategy in favour of others with the U2 problems indicates a lack ofknowledge regarding the meaning of subtraction as the difference between two states.

Of the four U2 problems, Cp2 and Ch-Cp2 present comparable strategies. It appears thatthe similarity in the problem statements, in terms of how the comparison and the variationare expressed, led the students to employ the same strategies.

In change problem Ch2, the strategy of giving a verbal explanation stands out abovethe rest (40%). Most of the answers relying on this strategy stem from the way in whichthe problem statement asks the question. The question ‘What was the elevator’s motion?’led many of the pre-service teachers to provide an answer that only indicated the directionof motion, such as ‘the elevator moved down’. In other words, the students did not give anumerical answer or write a sentence with a numerical value of the type ‘the elevator wentdown five floors’.

In equalize problem Eq2, 54% of the students wrote the operation with positive numbers,3 + 5 = 8. This is explained by the fact that a considerable number of students (at least25%) exhibited what [7] called the divided number line model. When a student uses this

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


model, he describes the distance from a positive number to 0, and then adds the distancefrom 0 to the negative number given in the problem statement.

Problem Eq2

A hang glider is flying 3 metres above sea level and a diver is swimming 5 metres belowsea level. How many metres does the hang glider have to drop to be at the same altitude asthe diver?

Many of the students did not consider a continuous motion from a + 3 to −5 position,but rather split up the motion in two and added the absolute values. An example of thereasoning behind the divided number line method is shown in the answer given by student 35:

8 meters. I added 3 to 5, since he would have to drop 3 meters to get to sea level, plus another5 to drop to the diver’s level.

Student 17’s answer to the same problem shows fairly complex reasoning. This studentpictured the divided number line in two parts, with the zero as the meeting point of the twomotions and then he wrote: ‘He has to go down 8 metres. I took away the 3 metres abovesea level to get to zero. Then I added the −5 from under the water, 5, to get to zero. Then,I added the 3, I subtracted and the 5 I added to get 8’.

In contrast to these forms of reasoning, students who did not use this model saw thedistance or movement as continuous from + 3 to −5. For example, student 33 replied: ‘Thehang glider has to go down 8 metres. I count from 3 down to −5: 3, 2, 1, 0, −1, −2, −3,−4, −5’.

5.3. Reasoning used to solve problems

In this section, we present other reasoning methods in addition to those shown above.

5.3.1. Negativity

We analysed if the students used negative numbers at any time while solving the problems,regardless of whether or not they actually resorted to operations with negative numbers.For example, student 67 (Figure 1) wrote down an operation with positive numbers, butacknowledged the negativity of the temperature by writing −5. The results of this analysisare shown in Table 5.

The use of negative numbers was not uniform for the eight problems. Their use wasgreater in the U3 problems (except for the compare problem) since, as we have seen, thepredominant strategy was to rely on operations with negative numbers. The U3 compareproblem was viewed in many cases as a U2 problem, which is why their answers moreclosely approximated those for this type of problem.

Table 5. Use of negative numbers in the eight problems.

U3 problems U2 problems

Ch3 Ch-Cp3 Cp3 Eq3 Ch2 Ch-Cp2 Cp2 Eq292 94 55 88 58 67 63 32

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


The two problems that saw the lowest use of negative numbers were those set in thecontext of sea level. Although not an objective of our research, it would be interestingto analyse the influence of the context on the strategies used by pre-service teachers inaddition problems with negative numbers, as has already been done with secondary schoolstudents.[27]

We also noted that some students did not correctly identify the problem statementsas involving negatives, and solved the problem as if the two numbers were positive. Thisleads to wrong answers since every problem involves a positive and a negative number. Anexample of this is the answer of student 91 to the U2 change problem, which is solved usingthe operation −2 − ( + 3) = −5.

Student 91’s answer: ‘3 − 2 = 1. I thought that if it was on the 3rd floor and it wentdown to the 2nd, that I subtract the two amounts’.

The student interpreted the second underground floor as being above ground level; inother words, he fails to understand the numbering scheme of the floors in the context of anelevator.

5.3.2. Failure to identify subtraction as a difference

Table 1 shows the operations that must be performed to solve the problems with negativenumbers. Note that the U3 problems can be solved with an addition if a signed number isused (6 + (−7) = −1), although it also allows for a subtraction if the simplified expression(6 − 7 = −1) is used. Moreover, the way the problem is stated, the order of the numbersin the statement coincides with the order in which they appear in the operation. In a way,the problem can be read linearly (that is, the numbers can be written in the same order theyappear in the statement by placing whichever sign, positive or negative, corresponds to thenumbers).

The U2 problems, in contrast, are solved with a subtraction in which the numbers arewritten in the opposite order to that given in the statement. We note this fact because veryfew students wrote the operation using negative numbers (see Table 3). Many of those whogave wrong answers to these problems set up an operation with negative numbers in theorder given in the statement. See, for example, student 72’s answer to problem Ch-Cp2, thecorrect operation for which, using negative numbers, is −5 − ( + 4) = −9 or 4 − (−5) =9, if the 9 is interpreted in a negative context. Student 72’s answer: ‘ + 4◦ in the morning.−5◦ at night. + 4 − 5 = −9. Add the two and use the sign of the larger number. It fell9 degrees’.

This answer is interesting because the student writes the numbers in the order given inthe statement (using the sign of the number as the sign of the operation), and yet he obtainsthe correct result, despite not solving the operation correctly. This indicates that the studentknew the answer to the problem and wrote it as the result of the operation without applyingthe rules of arithmetic. This same student also wrote operations with negative numbers andsolved them correctly in the U3 problems with the following operations: 6 − 7 = −1 and4 − 6 = −2.

We conclude that many students recognized the situations as ‘negative’, but did notidentify the structure of the U2 problems as subtraction problems. This same procedurehas also been identified in secondary school students. In [27] the authors concluded thatU2 problems with negative numbers are more complex than U3 problems, and they tend toemploy procedures that are not always valid, like that of writing the numbers in the sameorder as in the problem statement.

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


Figure 11. Student 6’s answer (problem Eq2, Ch3 and ChCp3).

Something else that we noticed is that some students place little importance on howthey write operations with negative numbers. This is reflected in several cases in whicha mathematically incorrect operation is written vertically (see Figure 7, student 39). Thisis also seen in the answer by student 6 (Figure 11), who used the same notation in threedifferent problems and who, in the last one, claims to be subtracting when in fact he isadding because he knows the answer to the problem must be 8.

It seems as though the students forget the notation they learned for negative numbers,opting instead to write positive numbers in a vertical format. Despite the extensive mathe-matical training they received over the years, they do not question the contradictions thatmay arise from their notation. It is as if what matters is obtaining the right answer, and notthe mathematical notation used to express it.

5.3.3. The step from positive to negative

The fact that the problems featured a positive and a negative number highlighted somethingthat had not been considered beforehand: the difficulty with zero faced by some studentsand that in some cases led to incorrect answers.

The zero in the number line drawings resulted in two types of problems, one involvingthe counting process, in which it was skipped when counting spaces, and another in whichthe zero was not shown, as was the case with student 4 (Figure 12) and equalize problemU2.

When the strategy involved counting, we also find that the zero was sometimes omittedfrom the count, as in student 73’s answer to problem Ch2: ‘The movement was down

Figure 12. Student 4’s answer (problem Eq2).

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


4 floors. I looked for the difference between the two floors counting backwards, for example,2, 1, −1, −2’.

In this case, we are faced with a contextual difficulty in that the student did not takeinto account the building’s ground floor, a mistake the student also makes with the otherproblem involving an elevator.

Another surprising example of problems with zero is found in student 50’s answer toproblem Ch-Cp2: ‘It dropped 10◦, since you have to add the 4◦ above zero, the 5◦ belowand the 0◦, which has to be taken into account. The temperature thus dropped by 10◦’.

6. Conclusions

In this paper, we analyse the strategies employed by pre-service primary school teachersto solve addition problems involving negative numbers. We analysed eight problems usingfour different structures (change, change–comparison, comparison and equalize) and twopositions for the unknown in each structure (which we call unknown 2 and unknown 3).The problems were set in contexts that rely on a vertical frame of reference (temperature,sea level and elevators).

We encountered a variety of strategies in the answers given by pre-service primary schoolteachers to solve the addition problems. These strategies reflect different ways of thinkingand degrees of knowledge regarding negative numbers. The students in question identifiedthe problems with situations involving negative numbers, since at some point during theprocess they wrote numbers with a negative sign. And yet, they did not express the operationwith the same ease in every case and resorted to other strategies, resulting in a wide range ofanswers. Finding the proper answer to the problems was not exceedingly difficult (the errorrate was between 8% and 32%), but using an operation with negative numbers did provecomplex. We found significant differences in the solving strategies based on the positionof the unknown quantity. The most notable finding in our research is that pre-serviceteachers incorrectly formulate those problems in which the unknown quantity is a variationor difference involving negative numbers, resorting instead to ‘common sense’ to expressa contextualized or graphical solution. The long teaching–learning process of pre-serviceteachers has not led them to distinguish between subtraction problems associated with‘taking away’ and those involving the ‘difference between two states’. However, as notedin [26] situations with subtraction as a difference are easier for explaining the significanceof subtracting negative numbers, since subtraction as ‘taking away’ (or ‘lowering’ in ourproblems) implies a more sophisticated concept.

The variety of solutions encountered in our research also reflected other aspects relatedto the pre-service teachers’ knowledge of negative numbers, such as a lack of formality intheir mathematical notation and a difficulty in representing the number line (not includingthe zero or counting incorrectly along the line).

The number of problems studied was limited in this research, and certain variables suchas the number types, unknowns and vertical contexts were fixed. We believe that expandingthis research to other types of problems involving negative numbers would expand thefindings of the study on pre-service primary school teachers.

AcknowledgementsThis work is part of the EDU2011-29324: Models of Formal and Cognitive Competence in Numer-ical and Algebraic Thinking of Primary and Secondary School Pupils and Primary School TraineeTeachers, Ministerio de Ciencias e Innovacion, Madrid, Spain.

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14


References[1] Vergnaud G, Durand C. Structures additives et complexite psychogenetique [Additive struc-

tures and psychogenetic complexity]. La Revue Francaise de Pedagogie. 1976;36:28–43.French.

[2] Marthe P. Additive problems and directed numbers. Third International Conference on thePsychology of Mathematics Education and Mathematic Education; Warwick, United Kingdom;1979. p. 153–157.

[3] Altiparmak K, Ozdogan, E. A study on the teaching of the concept of negative numbers. Int JMath Educ Sci Technol. 2010;41(15):31–47.

[4] Steiner CJ. A study of pre-service elementary teachers’ conceptual understanding of integers.Kent: Kent State University; 2009.

[5] Cunningham AW. Using the number line to teach signed numbers for remedial commu-nity college mathematics. Math Teaching Res J Online. 2009;3(4):1–40. Available from:http://www.hostos.cuny.edu/departments/math/MTRJ/index.htm

[6] Davidson PM. How should non-positive integers be introduced in elementary mathemat-ics? 11th International Conference for the Psychology of Mathematics Education; Montreal,Canada; 1987. p. 430–436.

[7] Peled I, Mukhopadahyay S, Resnick LB. Formal and informal sources of mental models fornegative numbers. 13th International Conference for the Psychology of Mathematics Education;Paris, France; 1989. p. 106–110.

[8] Mukhopadahyay S, Resnick LB, Schauble L. Social sense-making in mathematics; children’sideas of negative numbers. 14th International Conference for the Psychology of MathematicsEducation; Oaxtepex, Mexico; 1990. p. 281–288.

[9] Bell A. Ensenanza por diagnostico. Algunos problemas sobre numeros enteros. [Diagnosticteaching. Some problems about integer numbers] Ensenanza de las Ciencias. 1986;4(3):199–208. Spanish.

[10] Kuchemann DE. Positive and negative number. In: Hart K, editor. Children’s understanding ofmathematics: 11–16. London: John Murray; 1981. p. 82–87.

[11] Widjaja W, Stacey K, Steinle V. Locating negative decimals on the number line: insights intothe thinking of pre-service primary teachers. J Math Behav. 2011;30:80–91.

[12] Murray JC. Children’s informal conceptions of integer arithmetic. Ninth International Confer-ence for the Psychology of Mathematics Education; Jerusalem, Israel; Noordwijkerhout, TheNetherlands; 1985. p. 147–153.

[13] Carraher TN. Negative numbers without the minus sign. Paper presented at: 14th InternationalConference for the Psychology of Mathematics Education; 1990; Mexico. p. 223–229.

[14] Janvier C. The understanding of directed numbers. Seventh International Conference for thePsychology of Mathematics Education; Jerusalem, Israel; 1983. p. 295–301.

[15] Liebeck P. Scores and forfeits – an intuitive model for integer arithmetic. Educ Stud Math.1990;21:221–239.

[16] Lytle P. Investigation of a model on neutralization of opposites to teach integer addition andsubtraction. 18th International Conference for the Psychology of Mathematics Education;Lisbon, Portugal; 1994. p. 192–199.

[17] Thompson P, Dreyfus T. Integers as transformations. J Res Math Educ. 1988;19:115–133.[18] Hativa N, Cohen D. Self-learning of negative number concepts by lower division elemen-

tary students through solving computer-provided numerical problems. Educ Stud Math.1995;28(4):401–431.

[19] Janvier C. Comparison of models aimed at teaching signed numbers. Ninth InternationalConference for the Psychology of Mathematics Education; Noordwijkerhout, The Netherlands;1985. p. 135–139.

[20] Fuson K. Research on whole number addition and subtraction. In: Grouws D, editor. Handbookof research on mathematics teaching and learning. New York (NY): MacMillan PublishingCompany; 1992. p. 243–275.

[21] Verschaffel L, De Corte E. Number and arithmetic. In: Bishop, A, Clements K, Keitel C,Kilpatrick J, Laborde C, editors. International handbook of mathematics education. The Nether-lands: Kluwer Academic Publishers; 1996. p. 99–137.

[22] Bruno A, Martinon A, Velazquez F. The importance of the expression in the additive wordproblems. Focus Learn Probl Math. 2004;26(1):1 22.

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14

http://www.hostos.cuny.edu/departments/math/MTRJ/index.htm


[23] De Corte E, Verschaffel L. Some factors influencing the solution of addition and subtractionproblems. In: Durkin K, Shire, B, editors. Language in mathematical education. Research andpractice. Buckingham: Open University Press; 1991. p. 117–130.

[24] Teubal E, Nesher P. Order of mention vs order of events as determining factors in additive wordproblems: a developmental approach. In: Durkin K, Shire B, editors. Language in mathematicaleducation. Research and practice. Buckingham: Open University Press; 1991. p. 131–139.

[25] Fuson KC, Willis, GB. First and second graders’ performance on compare and equalize wordproblems. International Conference for the Psychology of Mathematics Education; London:University of London Institute of Education; 1986. p. 19–24.

[26] Selter C, Prediger S, Nuhrenborger M, Hußmann S. Taking away and determining the difference– a longitudinal perspective on two models of subtraction and the inverse relation to addition.Educ Stud Math. 2012;79:389–408.

[27] Bruno A, Martinon A. The teaching of numerical extensions: the case of negative numbers. IntJ Math Educ Sci Technol. 1999;30(6):789–809.

Dow

nloa

ded

by [

Ana

dolu

Uni

vers

ity]

at 0

5:53

21

Dec

embe

r 20

14

Documents

Strategies of pre-service primary school teachers for solving addition problems with negative numbers