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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
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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.
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September/October 2020
Research Article
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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
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September/October 2020
Research Article
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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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