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1Volume 116| Number 9/10

September/October 2020

Research Article

https://doi.org/10.17159/sajs.2020/8125

© 2020. The Author(s). Published

under a Creative Commons

Attribution Licence.

AUTHOR:

Craig Pournara1

AFFILIATION:

1Wits School of Education, University

of the Witwatersrand, Johannesburg,

South Africa

CORRESPONDENCE TO:

Craig Pournara

EMAIL:

craig.pournara@wits.ac.za

DATES:

Received: 04 Apr.2020

Revised: 15 May 2020

Accepted: 18 May 2020

Published: 29 Sep. 2020

HOW TO CITE:

Pournara C. Grade 9 learners’ algebra

performance: Comparisons across

quintiles, insights from errors and

curriculum implications. S Afr J

Sci. 2020;116(9/10), Art. #8125,

7 pages. https://doi.org/10.17159/

sajs.2020/8125

ARTICLE INCLUDES:

☒ Peer review

☐ Supplementary material

DATA AVAILABILITY:

☐ Open data set

☐ All data included

☒ On request from author(s)

☐ Not available

☐ Not applicable

EDITOR:

Jenni Case

KEYWORDS:

learner error, linear equations,

didactic cut, integers, quintile 5

maths performance

FUNDING:

South African National Research

Foundation (grant numbers 105901,

115261, 71218); First Rand

Foundation

It is well known that learners’ performance in mathematics in under-resourced secondary schools in

South Africa is poor. However, little is known about the performance of learners in top-performing well-

resourced secondary schools apart from their good results in Grade 12. In this study, the performance

of Grade 9 learners in top-performing quintile 5 (i.e. well-resourced) schools was compared with that of

learners in quintile 1–3 (i.e. poorly resourced) schools using a 45-item test. While the quintile 5 learners

obtained higher test scores, the performance pattern across the test items was very similar for both

quintile groupings. A detailed error analysis of 89 quintile 5 learners’ responses to equation items revealed

difficulties in applying the standard equation-solving procedure, and in operating with negatives and

subtraction, particularly on like terms. These problems may be related to a poorly conceived curriculum in

the areas of integers and equations. It is recommended that Grade 8 and 9 teachers pay regular attention

to all four arithmetic operations on integers. Furthermore, the teaching of equations should give greater

attention to the cognitive shifts required in solving equations with letters on both sides of the equal sign.

Significance:

• Learner performance patterns on a test were similar for learners from top-performing quintile 5 schools

and learners from lower quintile schools, although the quintile 5 learners obtained higher test scores.

• Quintile 5 learners’ ability to solve linear equations correctly is substantially impacted by their difficulties in

simplifying two algebraic terms to a single term, par ticularly when negatives and/or subtraction are involved.

• Particular aspects of the curriculum may par tly be responsible for the difficulties learners experience

with integers and equations.

Introduction

Research suggests that Grade 9 learners in quintile 5 schools, which are well resourced, are approximately 4 years

ahead of their counterparts in poorly resourced quintile 1–3 schools.1 This tells us about the relative per formance

of the two quintile groupings in South African schools and we know only too well about the poor mathematics

performance of learners in under-resourced schools2 but we still do not know much about what is happening at

Grade 9 level in well-resourced quintile 5 schools. While many quintile 5 schools produce excellent results at Grade

12 level, research conducted on the mathematics performance of high per formers writing the National Benchmark

Tests3 has shown that these learners have difficulties with apparently basic ideas such as percentage and

inequalities. Many of these learners will come from quintile 5 and high-fee independent schools. Another indication

that all is not well comes from informal discussions with heads of mathematics departments and teachers in

quintile 5 schools who despair that many learners in Grades 8 and 9 are not performing at desired levels.

The Wits Maths Connect Secondary (WMCS) project is a research and development project at the University of the

Witwatersrand. While our mandate is to focus on teacher professional development in lower quintile schools, we

were curious to compare the performance of learners taught by teachers with whom we work, with the performance

of Grade 9 learners in quintile 5 schools. We knew the quintile 5 learners would obtain higher marks but we wanted

to compare performance patterns over the entire test, i.e. the trends in which items had a higher/lower number

of correct responses. We also wanted to investigate learner errors and to compare these with previous findings

of learners’ performance in algebra in lower quintile schools.4 As the ability to solve equations is fundamental for

future success in mathematics, quintile 5 learners’ responses to three linear equation items were investigated to

gain insight into their fluency in solving equations and also into their fluency in algebraic manipulation. The research

was framed by two questions:

• What similarities exist in the test performance patterns of quintile 5 and quintile 1–3 learners?

• What are the most common errors made by quintile 5 learners on linear equation items?

Literature review

Research on learners’ approaches to solving linear equations and the errors they make goes back to the late 1980s.5

There has been a resurgence in research on learner performance on linear equations and notions of equality in

recent years.6-8 One of the key findings of this accumulated research is that learners must be taught formal methods

to solve equations of the form a

χ

+b=c

χ

+d because their informal methods, which are adequate for equations of

the form a

χ

+b=c break down for equations when there are letters on both sides and/or where there are two terms

with letters on one side, e.g. 2

χ

+5–

χ

=4. This breakdown (or discontinuity) has been referred to as the didactic

cut5 and the cognitive gap.9 An equation such as 3

χ

–4=11, can be solved arithmetically by saying ‘what multiplied

by 3 and then subtract 4 gives me 11’ or 3×☐–4=11? Clearly the solution is 5. However, this approach cannot

Grade 9 learners’ algebra performance:

Comparisons across quintiles, insights from errors

and curriculum implications

2Volume 116| Number 9/10

September/October 2020

Research Article

https://doi.org/10.17159/sajs.2020/8125

be applied to equations of the form a

χ

+b=c

χ

+d and so learners must

be taught to operate on the letters using inverse operations. The initial

research on the didactic cut and cognitive gap involved learners who had

not yet been taught formal procedures for solving equations. Research

conducted with older learners who had already learned equation-solving

procedures has challenged the existence of the didactic cut.7

Given that the research presented here also involves learners who

have been taught procedures for solving equations, I shall rather use

the notion of epistemological obstacle10 in speaking about learners’

difficulties in making the transition to formal methods for solving

equations. An epistemological obstacle involves ‘knowledge which

functions well in a certain domain of activity and therefore becomes

well-established, but then fails to work satisfactorily in another context

where it malfunctions’11. Thus this notion of obstacle is concerned with a

presence rather than an absence of knowledge. With reference to solving

equations, the knowledge which has previously worked well refers to

arithmetic approaches for solving equations. These methods need to be

replaced with new knowledge for solving equations that have letters on

both sides (or two terms with letters on one side).

The remainder of the literature review provides an overview of existing

research on common errors in solving linear equations. This will provide

the reader with the necessary background for the analysis which follows.

Approaches to and errors in solving equations

Kieran12 identified seven approaches to solving equations, five of which

are informal, including undoing or working backwards and trial-and-error

substitution. She also distinguished two formal methods: transposing of

terms (change side, change sign) and performing the same operation on

both sides. The informal or arithmetic methods can be used for equations

with letters on one side only while the formal or algebraic methods are

necessary to solve efficiently equations with letters on both sides.

Four common errors have been identified in solving linear equations.

Two of these are the redistribution error and switching addends error.12

A redistribution error involves adding a term to one side but subtracting it

from the other side. A switching addends error involves ‘moving’ a term

across the equal sign without changing its sign. In this study, I refer to

this as a moving error and I distinguish between moving constants and

moving a letter-term. The other inverse error13 occurs when learners use

the incorrect inverse operation, e.g. given 5

χ

=2, a learner may subtract

5 from both sides instead of dividing by 5, giving

χ

=–3 as the solution.

Learners making the familiar structure error14 ‘force’ their answer to fit

the form

χ

=k by eliminating additional letters as necessary. For example,

a learner who manipulates an equation to obtain 3

χ

=12

χ

, might first

divide by 3 to get

χ

=4

χ

and then drop the letter on the right side and

write

χ

=4. I refer to this error as familiar form because it appears to be

driven by learners’ desire to produce a final answer of form

χ

=k.

Meaning of the equal sign

Learners’ conceptions of equality are clearly important in solving

equations. Seminal research identified two different views of the equals

sign: as a do something signal and as an indication of equivalence.15,16

The former operational view is typically associated with unidirectional

reasoning about equations and is frequently drawn on to solve equations

of form a

χ

+b=c. For example, as noted above in the case of 3

χ

–4=11,

the learner reasons ‘what multiplied by 3 and then subtract 4 gives me

11?’ Here the learner treats the right side as the result of operations

performed on the left side. The latter relational view is associated with

solving equations of the form a

χ

+b=c

χ

+d. Research in the USA found

that, across grade levels, learners who demonstrated a relational view of

the equal sign, were better able to solve linear equations.6 However, the

authors note that despite learners’ inadequate conceptions of equality,

attention to the equal sign is typically not addressed in secondary school

curricula in the USA. The same is true in the South African secondary

curriculum.

Errors in operating on algebraic symbols

A fundamental component of early algebra involves making sense of

new symbols and notation. In arithmetic 4+½=4 ½ but in algebra one

Grade 9 learners’ algebra performance

Page 2 of 7

cannot simply juxtapose the two symbols, i.e. 4+a≠4a. In algebra, a+b

can be seen as the process of adding b to a as well as the resulting

object.17 The difficulties in making sense of the new notation explain,

to some extent, why learners make errors when working with like and

unlike terms, usually conjoining them to produce closure. I work with an

expanded notion of conjoining which distinguishes additive conjoining

from subtractive conjoining as two categories of errors that may involve

like or unlike terms. While additive conjoining involves addition of positive

terms, subtractive conjoining involves negatives and/or subtraction, e.g.

7–

χ

=7

χ

; 3

χ

–

χ

=3 and –

χ

+

χ

=

χ

.

Errors with negatives

The minus symbol can be viewed as an operator (subtraction) or as a

sign (negative). Hence when learners encounter an equation such as

2+3

χ

=5–2

χ

, the –2

χ

can be seen as subtracting 2

χ

or as negative 2

χ

.

This duality of the minus symbol poses significant difficulty for learners.

Equations involving negatives are more difficult because they are not

easily modelled using a balance model.18 Local research found that

learners had greater difficulty in dealing with algebraic expressions when

they involve negatives, either as sign or as operation.4

Research on subtraction and negatives has identified a range of errors

associated with the minus symbol. For example, right-to-left reasoning

involves subtracting a larger number from a smaller one18, e.g. 4–7=3 or

5

χ

–7

χ

=2

χ

. Overgeneralised integer rules may also lead to errors, e.g. if

the multiplication rule is expressed as ‘a minus and a minus gives a plus’,

this may lead to the expression –2

χ

–3

χ

being simplified to +5

χ

because

explicit attention is not given to the operation. The error of detaching

the minus sign may occur when learners add numbers or terms with

a leading negative19, e.g. –2+5=–7 and –3

χ

+

χ

=–4

χ

. In both cases

learners detach the minus symbol and isolate it from the expression. They

then perform the addition and re-attach the minus symbol to the answer.

Research design and methods

The research reported here stems from a larger study of learner

performance. A one-hour test was administered to Grade 9 learners in late

September/early October 2018. The first part of the analysis involved a

comparison of the performance of these learners with that of pre-existing

data from quintile 1–3 schools on the same test. The second part of the

study involved further qualitative analysis of the quintile 5 data only with

particular focus on learner errors in three test items.

Test instrument

The test consisted of 45 items dealing with number, algebra and function.

Most items were typical curriculum items that learners would encounter

in text books and tests, and they spanned Grades 7–9 content. The test

had been piloted in 2016 with lower quintile schools but we were unsure

how the items would perform with a quintile 5 sample. The comparative

analysis of both quintile groupings deals with all 45 items. For reference,

the broad content area of each question is given in Table 1.

Table 1: Test content areas and question numbers

Content area Question numbers

Number 1, 2, 3, 4

Algebra 5, 6, 7, 9, 10

Pattern and function 8, 11, 12

Learner responses to each item were coded as correct, incorrect or

missing, with no provision for partially correct responses. A learners’

test score is simply a count of the number of fully correct responses.

The response coding was led by members of the project team, with a

group of research assistants. Coding and capturing of responses were

moderated – approximately 15% and 20%, respectively. No errors were

found in either process.

3Volume 116| Number 9/10

September/October 2020

Research Article

https://doi.org/10.17159/sajs.2020/8125

The error analysis and coding were conducted by the author alone. The

focus was on three linear equation items, as shown in Table 2.

Table 2: Linear equation items

Item number Item

Q9a 3ꭓ-2=10

Q9b 3ꭓ-2=4+ꭓ

Q9c 2-3ꭓ=7-ꭓ

Q9a contains a letter on the left side only and can easily be answered

using arithmetic approaches. Q9b and Q9c contain letters on both

sides but Q9c involves subtraction on both sides of the equation and

is therefore more cognitively demanding than Q9b. These items and

the associated learners’ responses have potential to reveal evidence of

learner difficulties in working with equations of the different forms. They

also reveal learner errors when the combining of algebraic terms was

not the main goal of the manipulations. Consequently, they have potential

to reveal errors in solving equations as well as errors in simplifying

algebraic expressions. This algebraic work would have been completed

in the first half of the year and thus been examined by mid-year.

Sample

The sample from quintile 1–3 schools consisted of 1139 learners from

19 schools, taught by 25 teachers. Schools were selected because

their mathematics teachers had completed a professional development

course offered by the WMCS project in 2016 or 2017. The selected

learners were taught by these teachers in 2018. The quintile 5 sample

of 824 learners, taught by 22 teachers, was drawn from four quintile 5

secondary schools, all of which had an existing relationship with the

WMCS project. They are top-performing schools in their respective

districts and/or top feeder schools to the University of the Witwatersrand.

The University’s rankings for feeder schools are determined as a ratio

of the number of applications to the number of enrolments from that

school in a particular year. The large number of teachers is worth noting

because it reduces the impact of individual teacher effects on the results.

The sub-sample for the error analysis consisted of 89 learners, across

the four quintile 5 schools, who got Q9a correct but Q9b and Q9c

incorrect. These criteria suggest possible evidence of an epistemological

obstacle in solving more complex linear equations.

Ethical clearance was obtained from the University of the Witwatersrand

ethics committee (H17/01/01) and the Gauteng Department of Education

(M2017/400AA). All schools were assured that their identity would

remain confidential and that no comparisons would be made between

schools. Parents and learners were assured that the testing would not

impact learners’ marks and that they could withdraw at any point. They

were also assured of the confidentiality of individual results.

Coding for error analysis

Learner responses were coded according to the approach used and the

errors made. Because there are no interview data, it is difficult to infer the

underlying reasoning informing learners’ written responses. Coding was

based on interpretations of what had been written, looking at changes

between successive lines of a response together with individualised

annotations which learners may have provided such as arrows indicating

the moving of a term across the equal sign. While the approaches and

errors are reported per item, I also compared each learner’s responses

to all three items, looking for similarities and differences that might assist

in coding their errors.

I distinguished between algebraic and arithmetic approaches to solving

the equations. For the purposes of this article, an algebraic approach

involves manipulating expressions and operating on the letters. An

Grade 9 learners’ algebra performance

Page 3 of 7

arithmetic approach involves substitution of a possible solution or an

undoing approach. For example, solving 3

χ

–2=10 by substitution might

look as follows: 3(4)–2=10. An undoing approach might be written as:

10+2→12÷3→4

The error analysis was conducted on Q9b and Q9c. Drawing from other

analyses of similar data14,20, I distinguished three broad categories:

1. Equation errors – errors in applying inverse operations, collecting like

terms and constants on opposites sides of the equal sign, and isolating

the letter to determine the solution.

2. Letter errors – inappropriate or incorrect execution of operations on

terms with letters.

3. Numeric errors – operations on constants where the outcome of the

operation is incorrect. These are not reported here.

Each category was then sub-divided and errors were allocated specific

codes, as described below. I included sub-codes for subtraction/

negatives in each of the three categories. I assigned codes to each

response based on the three broad categories as well as an ‘other’

category. It was possible for a single response to have multiple codes.

I then dealt systematically with each category identifying sub-codes

based both on the literature discussed above and on the data.

Equation error codes

Six sub-codes were identified for equation errors:

1. Move term with letter – a term involving a letter is moved unchanged

across the equal sign

2. Move constant – a constant is moved unchanged across the equal sign

3. Incorrect inverse – additive inverse is applied when multiplicative

inverse is required, or vice versa

4. Divide binomial by monomial – binomial is incorrectly divided by

monomial to isolate letter on one side (see Figure 1a)

5. Familiar format – inappropriate manipulation of one/both sides of

equation to force the form

χ

=k as the final line of the response.

This code was only applied when comparing the last two lines of a

response, as shown in Figure 1b.

In addition to the five codes above, I included an incomplete code for

responses where the learner had not produced an answer in the form

χ

=k (see Figure 1c).

a c

b

Figure 1: Equation errors: (a) binomial divided by monomial, (b) familiar

format and (c) incomplete response.

Letter error codes

Letter errors were distinguished on two dimensions: those involving

addition and/or positive terms, and those involving subtraction and/or

negatives. The matrix in Table 3 provides examples of typical errors. The

examples of subtraction/negatives with like terms include instances of

right-to-left reasoning and detaching the minus sign.

4Volume 116| Number 9/10

September/October 2020

Research Article

https://doi.org/10.17159/sajs.2020/8125

Grade 9 learners’ algebra performance

Page 4 of 7

Table 3: Examples of letter operation errors

Like terms Unlike terms

Addition/positives 3ꭓ+ꭓ=3ꭓ6+ꭓ=6ꭓ

4+ꭓ=5ꭓ

Subtraction/negatives 3ꭓ-ꭓ=3ꭓ

3ꭓ-ꭓ=3

-3ꭓ+ꭓ=-4ꭓ

ꭓ-3ꭓ=2ꭓ

3ꭓ-4=-ꭓ

7-ꭓ=7ꭓ

3-2ꭓ=1

There were several instances of learners over-generalising exponential

laws, e.g.

χ

+

χ

=

χ

2. These errors were separated from the addition-of-

like-term errors shown above in order to determine the extent to which

learners were still making typical conjoining errors that do not involve

exponents. The errors involving exponents were coded as ‘other’.

Analysis and results

The analysis is reported in two sections. I begin with the comparison

of the overall performance and per formance patterns of the quintile 5

group and the quintile 1–3 group. This is followed by the analysis of

the responses of the quintile 5 sub-sample to the three equation items.

Overall performance and performance patterns

Table 4 shows that the mean score for the quintile 5 group (24.67) is

more than 2.5 times the mean score of the quintile 1–3 group (9.34).

This is to be expected and does not merit further discussion. However,

a comparison of the performance patterns across the 45 items is of

interest (see Figure 2). An obvious difference in the two graphs is that

the quintile 5 group performed better than the other group on every item.

Again, this is to be expected. More interesting is that the graphs have

very similar shapes with peaks and dips in similar places. This suggests

that learners across the quintiles find the same questions ‘easy’ and

‘difficult’.

Table 4: Mean scores and standard deviations

Learner group NMean Standard deviation

Quintile 1–3 1139 9.34 6.64

Quintile 5 824 24.67 9.03

Given that the quintile 5 sample is drawn from top-performing schools,

it may seem surprising that there were 21 items which fewer than 50%

of learners answered correctly. There is a noticeable downward trend

in the quintile 5 performance, interspersed with a few peaks. Better

performance is generally associated with the numeric items in questions

1 to 4 which focus mainly on integers. Thereafter, most items involve

algebra and this is where the downward trend becomes most noticeable.

The three highest peaks from question 8 onwards are associated with

numeric work: intercepts of the graph of a linear function (Q8a, Q8b);

simple linear equation that can be solved without algebraic manipulation

(Q9a); and a function machine with numeric inputs and outputs (Q11i,

Q11ii). Thus the overall picture of quintile 5 performance is that learners

have difficulty with algebraic work and functions. The downward trend

is less obvious for the quintile 1–3 group because performance flattens

from question 5 onwards, apart from the peaks which occur at similar

places to those of the quintile 5 graph.

Three factors must be borne in mind when interpreting the lower than

expected performance of the quintile 5 learners. Firstly, learners did

not prepare for the test and so the scores merely provide a once-off

measure on a particular day. Secondly, the data were collected in late

September to early October and, in at least one school, teachers were

still completing a section that was included in the test. Consequently,

many learners may not yet have consolidated some of the test content.

Thirdly, a learner’s score indicates completely correct responses and so

scores likely will under-estimate learners’ knowledge of the test content.

For example, a blank response and a response containing a minor slip in

a calculation both count as zero.

Given the similarities in the performance pattern of the two groups, there

is value in studying the errors made by the quintile 5 group because

these will likely provide useful insights for both groups which may in turn

lead to recommendations relevant to the teaching of algebra across all

quintile schools.

Error analysis of responses to equations items

Table 5 provides a summary of learner performance on the equation

items for both groups. In both groups there is a general decrease in

performance from Q9a to Q9c with the percentage drop from Q9b to Q9c

being larger. The drop in performance between Q9a and Q9b suggests

that some learners experience some kind of obstacle in working with

equations with letters on both sides. This appears to be exacerbated

by the presence of additional negatives in Q9c, particularly given that

the letters are being subtracted on both sides of the equation. It is also

possible that performance on Q9c was lower because the solution is

rational whereas the other two solutions are integers.

Figure 2: Learners’ performance on all test items.

5Volume 116| Number 9/10

September/October 2020

Research Article

https://doi.org/10.17159/sajs.2020/8125

Table 5: Learner performance on linear equation items

Item number Item Quintile 5

(N = 824)

Quintile 1–3

(N = 1139)

Q9a 3ꭓ-2=10 87.6 33.80

Q9b 3ꭓ-2=4+ꭓ73.4 15.28

Q9c 2-3ꭓ=7-ꭓ40.4 5.00

Learners’ approaches to solving equations

The analysis from here focuses on the quintile 5 sub-sample. Learners’

approaches to Q9a in contrast to their approaches to Q9b and Q9c

were of particular interest because this might reveal the extent to which

they have overcome the epistemological obstacle described earlier.

Table 6 shows that the vast majority used algebraic approaches for

Q9a. Of greater interest, however, are the learners who used arithmetic

approaches.

Table 6: Approaches to Q9a

Approach Number of responses Percentage

Algebraic 63 73.3

Arithmetic Substitution 15 17.4

Undoing 3 3.5

No evidence 5 5.8

Total 86 100

For those using arithmetic approaches to correctly answer Q9a, it was

important to investigate their approaches to Q9b and Q9c: did they shift

to an algebraic approach or did they still attempt an arithmetic approach?

Eighteen learners successfully used arithmetic approaches for Q9a,

either substitution or an undoing approach. Only two of these learners

attempted an algebraic approach for Q9b and Q9c.

Of the 15 learners who used a substitution approach for Q9a, 11 used

the same approach for Q9b and 9 used it for Q9c. Of the three learners

who used an undoing approach for Q9a, two attempted this approach for

Q9b and Q9c while the third learner approached both items algebraically.

There were five learners who only provided answers and so their

responses were coded as ‘no evidence’.

In summary, most learners either used all arithmetic approaches or all

algebraic approaches. Those adopting arithmetic approaches for all

three items constituted 23.8% of the sub-sample. It appears that these

learners have not yet recognised the need to reject an inadequate method

and replace it with a procedure that makes use of inverse operations.

These learners are not operating on the terms with letters in order to

solve the equations. It is worth noting that the mean test score for these

15 learners was 14.2 (31.6%) which suggests that they lack algebraic

fluency more generally.

I now shift to a discussion of the errors made by the learners who used

algebraic approaches for Q9b and Q9c.

Errors in equation operations

A summary of the equation errors is presented in Table 7. A total of 112

equation errors were coded across Q9b and Q9c.

Table 7: Equation errors for Q9b and Q9c

Equation errors Q9b Q9c

Move term with letter 14 7

Move constant 6 11

Incorrect inverse 3 2

Divide binomial by monomial 3 3

Force familiar format for solution 13 14

Incomplete 6 13

Other 6 11

Total 51 61

Not surprisingly, the most common error related to the incorrect use

of additive inverses, with 38 errors involving ‘moving’ a letter/constant

across the equal sign without changing sign. It is worth noting that

only four learners made ‘moving errors’ with both numbers and letters.

There were five instances where learners used the incorrect inverse,

typically subtracting a coefficient instead of dividing. In a small number

of instances learners did not collect like terms with letters and so divided

a binomial on one side of the equation by the coefficient of the variable

on the other side. This typically led to other errors: either the learner

performed the division on the numbers only and then conjoined the

remaining number and letter (see Figure 1a), or dropped the letter from

the expression.

There were a surprising number of responses (27) where learners

manipulated the equation to force a familiar format, i.e.

χ

=k, as the

final line of the response. This took different forms, including dropping/

ignoring ‘unwanted’ letters, e.g. the equation 3

χ

=6

χ

became

χ

=2.

The high prevalence of incomplete responses (19) is closely linked to the

familiar format error. In many of these cases, learners did not manipulate

their equations to produce a familiar format. Instead they stopped with

forms such as a

χ

=b,

χ

=a

χ

or a

χ

/a=b/a.

The other equation errors included converting an equation to an

expression, incorrectly combining terms to generate quadratic forms,

and various manipulations that did not maintain equality.

Errors in letter operations

A total of 98 letter errors were coded for Q9b and Q9c. As described

above, I disaggregated conjoining errors to distinguish (1) additive

conjoining (addition/positives) from subtractive conjoining (subtraction/

negatives); and (2) operating on like terms from operating on unlike

terms. This makes it possible to see more clearly where the majority of

errors occurs.

Table 8: Errors in letter operations

Like terms Unlike terms Total

Addition/positives 2 6 8

Subtraction/negatives 42 13 55

Total 44 19 63

Grade 9 learners’ algebra performance

Page 5 of 7

6Volume 116| Number 9/10

September/October 2020

Research Article

https://doi.org/10.17159/sajs.2020/8125

As can be seen in Table 8, only 8 errors involved additive conjoining,

irrespective of whether learners were operating on like or unlike terms.

By contrast, 55 errors involved subtraction/negatives and 42 of these

involved like terms. This may be surprising in the light of prior research

on conjoining. However, assuming most learners in this sample know

we cannot combine unlike terms but can combine like terms, then it is

not surprising that the majority of errors occurs when combining like

terms. Typical errors included detaching the negative, e.g. –3

χ

+

χ

=–4

χ

and dropping the letter, e.g. 3

χ

–

χ

=3. There was a surprisingly high

number of responses giving 3

χ

–

χ

=3

χ

. One interpretation is that learners

consider only the visible coefficients. In effect they are treating x as 0

χ

rather than 1

χ

. If they separate the numbers and letters, they may reason

3–0=3 and then append the

χ

to obtain 3

χ

. However, interviews are

necessary to investigate this further.

In addition to the above errors, there were 16 instances in Q9c where

learners dropped the negative sign from one line to the next. For example,

–3

χ

became 3

χ

in the following line. There were also 19 letter errors

coded as other. These mostly involved over-generalisation of the addition

law of exponents which typically led to further errors in attempting to

solve an equation that was no longer linear.

Discussion and implications

The overall Grade 9 learner performance on a test of number, algebra

and function covering selected Grade 7–9 content was disappointing

for both quintile groupings. However, both groupings displayed similar

performance patterns. Notwithstanding the caveats mentioned above, a

mean score of 54.8% for the quintile 5 group indicates that even towards

the end of Grade 9 there are learners in top-performing schools who still

have difficulty manipulating algebraic symbols.

Fundamental to the notion of epistemological obstacle in the context

of solving equations is that learners accept the need to replace their

inappropriate arithmetic approaches with an algebraic approach.

Although most learners attempted algebraic approaches for all three

items, approximately 24% of the sub-sample used only arithmetic

approaches. This may also suggest they do not have a relational view of

the equal sign. Furthermore, the error analysis shows that, on average,

each learner in the sub-sample made more than one equation error and

one letter error across Q9b and Q9c.

The error analysis reveals that errors made by quintile 5 learners in solving

linear equations with letters on both sides stem more from difficulties in

manipulating algebraic expressions and dealing with negativity than in

executing the standard procedure for solving equations. Of the errors

reported here, 92 (43.8%) errors relate to negatives/subtraction in some

way. Furthermore, nearly half (45.6%) of these negativity errors involved

the incorrect simplification of two like terms to a single term. While these

findings confirm some of what we found in a previous study on learners’

algebra performance in lower quintile schools4, the insight, at least for

quintile 5 schools – that difficulties with negatives and subtraction are

more common with like terms – is a new empirical finding, although not

necessarily surprising.

Implications for curriculum and teaching

From the above findings there are two clear implications for the

curriculum and four implications for teaching.

An analysis of the Senior Phase Mathematics curriculum21 suggests

that two problems highlighted in this study may have their roots in the

curriculum itself. For example, the content of integers is split over Grades

7 and 8, a split exacerbated by the move from primary to secondary

school. The Grade 8 curriculum assumes learners come with knowledge

of adding and subtracting integers, that this merely requires revision and

that teachers should focus on multiplication and division of integers in

Grade 8. While teachers may ignore this ‘advice’ in their own classrooms,

the official teaching support materials such as annual teaching plans,

scripted lesson plans and learner workbooks will follow the curriculum

closely and thus fall prey to the poorly conceived plans for teaching and

learning integers. The evidence from this study and prior research shows

clearly that all aspects of integers need detailed attention in Grade 8.

A similar problem arises with equations. There is considerable focus

on solving equations by inspection and insufficient attention to formal

equation operations in Grade 8. Also, there is no explicit recognition of

the importance of attending to equations with letters on both sides. By

Grade 9, it is assumed that learners have mastered this work and can

move on to more complex linear examples as well as quadratic and

exponential equations. Given this breadth of equation types, teachers

may overlook the need to deal with simple linear equations with letters

on both sides, in the rush to cover the other types. The curriculum needs

to foreground the cognitive shifts in moving from equations with a letter

on one side to equations with letters on both sides, with additional time

allocated to consolidate these procedures, thus supporting learners to

navigate and overcome the epistemological obstacle they encounter

when they have to operate on the letter in solving equations. The

overburdened curriculum could be eased by removing quadratic and

exponential equations from Grade 9 as they are dealt with in detail in

later years.

Implications for teaching follow closely from the curriculum implications.

Firstly, teachers should pay explicit attention to helping learners develop

an equivalence view of the equal sign, even in Grades 8 and 9. Without

an equivalence view, learners will continue to have difficulty in solving

equations of all kinds. Secondly, Grade 8 teachers should pay attention

to all four arithmetic operations on integers, with particular attention to

subtraction. Attention to fluency with negative numbers should continue

into Grade 9. Thirdly, continual attention must be given to fluency in

algebraic manipulation, particularly with examples involving subtraction

and negatives. This study suggests that such a focus will improve

learners’ performance on equation solving. Fourthly, teachers need to

appreciate the cognitive shift necessary to solve equations with letters

on both sides and take time to deal with the case of a

χ

+b=c

χ

+d. They

should also include equations with more than two terms on each side,

e.g. 4–2

χ

+3=3

χ

+1–

χ

. This provides practice in algebraic simplification

as well as in performing inverse operations.

Conclusion

This study shows that difficulties with introductory algebra are not

restricted to learners in lower quintile schools. Furthermore, it makes

three important contributions. Firstly, there are similarities in the

performance patterns of Grade 9 learners across quintiles on a test

of number, algebra and function spanning content of Grades 7 to 9.

Secondly, it reveals and confirms learners’ specific difficulties in working

with negatives and subtraction in relation to algebra. Thirdly, it highlights

the specific insight that while few learners were making errors with

addition of like and unlike terms, there was a proliferation of errors in

working with like terms and negatives.

While many learners in quintile 5 schools overcome these difficulties

and perform well in mathematics by Grade 12, the same cannot be said

for the majority of learners in lower quintile schools. The curriculum

recommendations proposed above suggest that specific curriculum

changes are necessary in the topics of integers and equations. These

may help to address the ways in which the curriculum contributes to

learner difficulties with negative numbers and aspects of algebra.

The recommendations for teaching address similar issues. However,

opportunity for teachers to implement the recommendations requires

some flexibility in curriculum pacing to address learners’ errors and

backlogs.

Acknowledgements

This work is based on research supported by the National Research

Foundation (NRF) of South Africa (grant numbers 105901, 115261,

71218) and the First Rand Foundation (FRF). All opinions, findings and

conclusions or recommendations are that of the author and the NRF

and FRF accept no liability whatsoever in this regard. I am grateful

to my colleagues in the WMCS project for many hours of productive

discussions on our research and development work, particularly in

relation to the teaching and learning of algebra.

Grade 9 learners’ algebra performance

Page 6 of 7

7Volume 116| Number 9/10

September/October 2020

Research Article

https://doi.org/10.17159/sajs.2020/8125

Competing interests

I declare that there are no competing interests.

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