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1Volume 115| Number 1/2
January/February 2019
Research Article
https://doi.org/10.17159/sajs.2019/4652
© 2019. The Author(s). Published
under a Creative Commons
Attribution Licence.
Comparing mathematics knowledge of first-year
students from three different school curricula
AUTHORS:
Sonica Froneman1
Mariette Hitge1
AFFILIATION:
1School of Mathematical and Statistical
Sciences, North-West University,
Potchefstroom, South Africa
CORRESPONDENCE TO:
Sonica Froneman
EMAIL:
Sonica.Froneman@nwu.ac.za
DATES:
Received: 08 Mar. 2018
Revised: 21 May 2018
Accepted: 31 Oct. 2018
Published: 30 Jan. 2019
HOW TO CITE:
Froneman S, Hitge M. Comparing
mathematics knowledge of first-year
students from three different school
curricula. S Afr J Sci. 2019;115(1/2),
Art. #4652, 7 pages. https://doi.
org/10.17159/sajs.2019/4652
ARTICLE INCLUDES:
☒ Peer review
☐Supplementary material
DATA AVAILABILITY:
☐Open data set
☐All data included
☒On request from authors
☐Not available
☐Not applicable
EDITOR:
Linda Chisholm
KEYWORDS:
basic mathematics; school
mathematics curricula; tertiar y
mathematics preparedness
FUNDING:
None
Mathematics forms an integral part in the training of scientists and engineers. In recent history the South African
school system has experienced several changes in school curricula. In 1994 the traditional knowledge-based
curricula were replaced by an outcomes-based curriculum. Owing to implementation problems which resulted in
resistance from teachers and the general public, revisions followed of which the National Curriculum Statement
(NCS) and Curriculum Assessment Policy Statements had the most direct effect in terms of preparation for
tertiary mathematics. We report here on an investigation of the basic mathematical knowledge of three student
cohorts representing three curricula, namely the last cohort that received the traditional knowledge-based
curriculum, and the first cohorts that received the two outcomes-based curricula. The results indicate that
changes in the mathematical content of the curricula did not impact negatively on the basic mathematical
knowledge of students enrolled for tertiary mainstream mathematics. The only exception is Euclidean geometry,
for which certain topics were transferred to an optional paper in the NCS curriculum.
Significance:
•The introduction of outcomes-based curricula in South Africa initiated a discourse on the preparedness
of first-year students for programmes with mainstream mathematics.
•The availability of a homogeneous set of samples and a uniform test provided a unique opportunity
to compare the basic mathematical knowledge of first-year natural science and engineering students
entering university from three different exit-level school curricula.
Introduction
Role players involved in the training of scientists and engineers have a vested interest in the basic mathematics
knowledge of prospective science and engineering students. Changes in school curricula invariably influence the
preparedness of students for tertiary studies, especially where mathematics-intensive programmes are concerned.
Substantial changes were made to the South African school system since 1994 as a result of a change in the
government system.1 The racially and provincially segregated curricula of the apartheid era were replaced by a
unified national curriculum. Whereas the previous fragmented curricula were mainly specified in terms of content
knowledge to be learned using a transmission teaching model2, the new national curriculum was a skills-based
constructivist curriculum3, which was implemented with an outcomes-based management str ucture4. Another major
structural change to the curriculum was the merging of the higher grade (HG) and standard grade (SG) curricula
documents into a single curriculum document, presumably to diminish the emphasis on individual achievements
of a few in exchange for a more rounded approach to education for all learners. Owing to problems experienced
with the implementation of the new curriculum, revisions were introduced. Not all of these revisions were sustained
up to Grade 12, but, ultimately, three outbound Grade 12 cohorts can be distinguished in South Africa: (1) learners
who matriculated in the years up to 2007 who were exposed in Grades 10 to 12 to traditional knowledge-based
curricula (TKC); (2) matriculants of 2008 to 2013 who were exposed to a constructivist curriculum implemented
through an outcomes-based educational system (OBE) as summarised in the National Curriculum Statement
(NCS)5; and (3) those who matriculated between 2014 and the present who experienced a revised version of the
OBE curriculum, officially documented in the Curriculum Assessment Policy Statement (CAPS)6.
Since the replacement of traditional, knowledge-based curricula by new, OBE-grounded curricula in countries such
as the USA, the United Kingdom, Canada, Australia, New Zealand and South Africa, discourses have developed
among academics and the general public on the expediency of these changes. Perceptions were that students
experienced difficulty with the transition to tertiary level, especially those enrolled in mathematics-intensive
programmes. For example, a survey done in the USA revealed that an increasing number of incoming students
needed remedial courses in mathematics.7 A report on the preparedness of Irish students for ter tiary mathematics
studies refers to ‘grade depreciation’, implying that grades achieved in state examinations were not comparable to
the same grades obtained 10 years earlier.8 However, it was not made clear to which extent the perceived lack of
mathematical knowledge could be attributed to the introduction of OBE.
In South Africa, lecturers have increasingly become aware of first-year mathematics students’ lack of understanding
of fundamental mathematical concepts.9-11 In the study of Engelbrecht and Harding9, the first-year mathematics
cohorts of 2005, 2006 and 2007 were compared using a mathematics achievement test which was designed to
determine the level of mathematical competency of students who did not necessarily excel in the final matric exam.
The questions were set on topics such as percentages, spatial geometry, parallel lines, word sums, ratio and
proportion, number concept and manipulation, functions, graphs and trigonometr y. The cohort of 2005 represented
students who did not experience OBE at school level, while the cohorts of 2006 and 2007 experienced OBE up to
Grade 9, but reverted back to traditional teaching in Grades 10 to 12. Engelbrecht and Harding9 concluded that,
except for geometry, word sums and ratio and proportion, the student performances were on par for these three
cohorts. In 2015 we published an ar ticle12 in which we compared the mathematical knowledge and skills of the 2008
cohort at our institution (the last cohor t exposed to traditional curriculum in Grades 10 to 12) with the 2009 cohort
(the first cohort exposed to the full OBE curricula to reach university) in terms of a framework consisting of three
mathematics knowledge types, namely procedural knowledge, proceptual knowledge and conceptual knowledge.
The framework was an attempt to investigate if the changes in the curricula impacted on the way mathematics was
taught and learned in Grades 10 to 12. The sample was all students enrolled for mathematics, including students
2Volume 115| Number 1/2
January/February 2019
Research Article
https://doi.org/10.17159/sajs.2019/4652
from science, engineering, commerce and education. The comparison
showed that the performance of the OBE cohor t was not as poor as has
been perceived initially; however, the expected outcome that students
from the OBE curriculum would have better conceptual understanding
because of the emphasis on exploration and searching for patterns only
materialised for some questions on the interpretation of graphs.
In the present article we follow up on these comparisons by including the
cohort of 2015, which was the first cohor t exposed to CAPS in Grades 10
to 12. The focus of this study is to investigate how changes in the school
curricula, especially differences in the mathematical content, influenced
the basic mathematical knowledge of natural science and engineering
students enrolled at our institution. We report on a quantitative empirical
study based on the results of multiple-choice tests written by all new
first-year mathematics students at our tertiary institution. The main
intention of the study was to investigate whether the mathematical
knowledge of the cohorts of 2008, 2009 and 2015, which represent
three different school curricula for Grades 10 to 12 – namely TKC, NCS
and CAPS – are significantly different.
Changes to mathematics curricula in South Africa
In order to gain insight into the influence of each curriculum on
the development of basic mathematical knowledge, we give a brief
overview of the foundations of these three curricula and their historical
implementation in South African schools. The management structure
of the new curriculum was grounded in the principles of OBE, where
assessment is based on ‘demonstration of outcomes’, ranging from
‘simple discrete content skills’ to the ‘highly complex open-ended life-
role performances required by adults in the real world’4(p.25). An OBE
system places greater emphasis on dispositions, resulting in formative,
criterion-based assessment, instead of summative assessment and
high-risk tests.13 Three varieties of OBE with rising hierarchical orders of
complexity14 and corresponding decreasing scales of modernism15 can
be distinguished. Traditional OBE is similar to a traditional knowledge-
based curriculum in terms of the organisation of learning content in
disciplines and an emphasis on academic development, but it differs
from the traditional knowledge-based curriculum in terms of its
assessment criteria, which are based on mastery of specified outcomes,
and on its learner-centred approach and emphasis on life-long learning.
Transitional OBE is more future orientated and accentuates the cultivation
of higher-order competencies, such as critical thinking, problem solving
and communication skills. Transformational OBE is the most complex
and extreme form of OBE which defies fixed curriculum outcomes based
on conventional subject areas, and strives to change the disposition of
learners. In the planning stages of the new South African curriculum,
a transformational OBE model was envisaged.16,17 However, the first
implemented version of OBE in South Africa – Curriculum 200518 –
was described by one researcher19(p.22) as a ‘potpourri of curriculum
proposals with largely unacknowledged origins’. However, with the
introduction of cross-curriculum critical outcomes, integrated learning
areas and integrated real-life problem settings, Curriculum 2005 could
be classified as a transitional model in the OBE hierarchy.
In practice, there was tension between the critical outcomes of
Curriculum 2005 and the formulation of its learning outcomes.15 At that
stage, most teachers were products of schooling in the old dispensation
and struggled to come to terms with the implementation of the new
curriculum.20,21 This difficulty resulted in revision of the original OBE
curriculum in an attempt to reduce the level of integration of subjects.
The Revised NCS for Grades R–922 was implemented in the foundation
phase from 2004, and the NCS for Grades 10–125 was introduced in
Grade 10 in 2006. These revisions did not address all the problems
teachers experienced, as complaints about implementation issues
and administration overload suffered by teachers persisted.23 Further
revisions were necessary to address these problems, and, in 2012,
new assessment criteria, referred to as the CAPS, were introduced
in Grade 10.6 In the revised document it was emphasised that the
basic philosophy of the curriculum remained unchanged and that the
adjustments only related to ‘what to teach and not how to teach’23. The
content organisation of the CAPS is similar to a traditional knowledge-
based curriculum, but the critical cross-curriculum outcomes were
retained. Some researchers labelled the CAPS curriculum as too
prescriptive and restrictive24,25, arguing that the most dramatic change
brought about by CAPS has been its shift in focus from assessment of
learning to learning for assessment. Signs are already there that South
Africa has moved away from OBE towards the US model of a standards
approach to education, with an overemphasis on external assessment
of learners in all the school phases.26 It is therefore not clear, and will
be difficult to determine, whether the changes envisaged by the OBE
curriclula had an effect on teachers’ practices in their classrooms.
In this article we rather focus on the effect of changes in the str ucture and
content of the curricula. Structurally, the differentiation between HG and
SG mathematics has been removed, with the unintended consequence
that fewer marks are available to differentiate among students who want
to enrol for mainstream mathematics. In order to better prepare learners
for their future role in society, themes of a statistical nature such as
data handling, descriptive statistics and financial mathematics – were
added to the core curriculum of mathematics. To make room for these
inclusions, absolute value theory and the remainder theorem were
omitted from the curriculum and Euclidean geometry was transferred to
a separate optional paper, to which a few new topics such as recursive
sequences, bivariate data and probability were added.5 One of the main
consequences of this move was that students in the first-year cohorts of
2009 to 2014 were only partly exposed to Euclidean geometry in Grades
10 to 12. Furthermore, the scope of some topics was reduced by limiting
assessment of formal proofs and definitions and the focus shifted to
the application of rules and theorems in problem-solving situations. For
example, the factor theorem was used to find the roots of higher-order
polynomials, and the logarithmic rules were applied to solve for the time
period of an investment.
The main objective of the OBE mathematics curriculum was to deliver
learners who are able to ‘transfer skills from familiar to unfamiliar
situations’5(p.5). The emphasis shifted from knowing mathematics
as facts, rules and principles, to the interpretation and application of
conceptual representations such as graphs and algebraic patterns.
Learners were encouraged to use graphical representations to solve
problems and to search for connections between different topics. For
example, in the topic of functions, more emphasis was placed on the
connection between algebraic equations and the subsequent shifting
of graphs. Another example is the topic of series and sequences in
which the general recursive formulation in terms of geometric and
arithmetic sequences were reformulated in terms of linear, quadratic and
exponential sequences, which could be linked to functional graphs.
With the introduction of the CAPS, the content of the optional paper,
which included topics of Euclidean geometry, was reintegrated into
the core curriculum, while only the single topic of linear programming
was omitted. These additions resulted in an overall increase in the
mathematical content of the curriculum. In the CAPS document, the
specific mathematical content to be learned was more clearly defined
and examples were given for each specific curriculum statement.
Methodology
Research question
The empirical study was undertaken to address the research question:
Are there practical significant differences in the basic mathematical
knowledge of first-year natural science and engineering students who
matriculated from traditional knowledge-based curricula (TKC), the new
outcomes-based curriculum (NCS), and the recently revised outcomes-
based curriculum (CAPS)?
Research design
The research can be categorised as a quantitative investigation using a
comparative analysis of variance (ANOVA)-test analysis of the results of
a diagnostic test written by first-year mathematics students in order to
quantify the differences in basic knowledge between students from three
different school cohorts.
Mathematics knowledge of first-year students
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Sample
The sample was conveniently selected and consisted of all bona fide
first-year students enrolled for mainstream mathematics in engineering
and natural sciences programmes at our tertiar y institution in 2008, 2009
and 2015. Only students who wrote the relevant matriculation exam in
the previous year were considered in the analyses. For Grades 10 to 12,
the 2008 cohort was exposed to traditional knowledge-based curricula
(TKC), the 2009 cohort to the new outcomes-based curriculum (NCS),
and the 2015 cohort to the recently revised outcomes-based curriculum
(CAPS). The admission requirement for enrolment into mainstream
mathematics modules for the 2008 cohort was 60% for SG or 50% for
HG mathematics in the final matriculation examination. For both the 2009
and 2015 cohorts, the requirement was Level 4 (50%) for mathematics.
The sample sizes for the respective cohorts were n=287 for the 2008
cohort, n=357 for the 2009 cohor t and n=455 for the 2015 cohort.
The demography of students enrolled for 2008, 2009 and 2015 did
not differ considerably. This similarity gave us an opportunity to
investigate the effect of changes in the school curriculum on a relatively
homogeneous sample of students regarding academic achievement at
school level, but with exposure to different school curricula.
Ethical considerations and data collection
Permission was obtained from the director of the School for Computer,
Mathematical and Statistical Sciences to conduct the research. Ethical
clearance was obtained from the Research Ethics Committee of the
Faculty of Natural Sciences (reference number NWU-0007-14-S3).
The first-year cohorts of 2008 and 2009 wrote the diagnostic test in
the first week of their respective first semesters. The participation was
voluntary and individual results were not made public. Since 2011, all
first-year students enrolling for mainstream mathematics on campus
have been required to attend a mathematics refresher course before
the commencement of classes. The first-year cohort of 2015 wrote the
diagnostic test on the first day of the mathematics refresher course. Only
data from bona fide first-year students who matriculated in South Africa
from the relevant curricula were considered for the data analysis.
Selection of test items
The mathematical knowledge test used in the empirical study for the
2008 and 2009 cohorts consisted of 40 multiple-choice questions:
24 items were selected from a test developed to predict students’
success in tertiary studies in mathematics27, and 16 items extracted
from previous question papers were added to the core test to broaden its
scope in terms of specific topics in mathematics. The 2015 cohort wrote
the pre-test of the mathematics refresher course; this test consisted of
35 questions. A total of 25 items appeared in both tests and could be
used in the comparative analysis.
Classification of test items in terms of topic areas
The 25 items were grouped into different topics of mathematics, namely
algebraic knowledge, functions and graphs, trigonometry, geometry
and differentiation. Because of the multidimensionality of mathematical
knowledge it is not always possible to classify certain items. For
example, the simplification of can be classified as algebraic
knowledge, but it also requires an understanding of the function concept.
For this reason, the items were compared on an individual basis.
Comparison criteria
The differences between the means of the individual questions of the
three cohorts were compared using the ANOVA test for Cohen’s effect
sizes, as given by Ellis and Steyn28(p.51). Effect sizes (d-values) are
independent of the sample size and provide a measure of the practical
significance of the differences between the means. According to Cohen,
d-values are interpreted as follows:
• Small effect:
• Medium effect:
• Large effect:
Although the sample used is a convenience sample, p-values giving the
statistical significance of the differences between the cohorts are repor ted
for the sake of completeness. A p-value smaller than 0.05 indicates that
the difference between the cohorts is statistically significant.
Results
The results are presented in Tables 1 to 6 for the different topic areas.
The abbreviations ‘T’, ‘N’ and ‘C’ indicate the cohorts from the TKC
(2008 cohort), NCS (2009 cohort) and CAPS (2015 cohor t) curricula,
respectively. The abbreviation ‘T/N’, for example, indicates the practical
difference between the scores for the TKC and NCS curricula. An
asterisk in the p-column indicates an acceptable statistically significant
difference (p<0.05), and asterisks in the d-column indicate a small
(*), medium (**) or large (***) effect in practical difference. A positive
d-value in a column indicates that the group mentioned first performed
better, while a negative d-value indicates a better performance by the
group mentioned second.
Algebraic knowledge
The results for algebraic knowledge are listed in Table 1. The cohorts all
performed well (>55%) on most of the questions, with the exception
of Question 4 (<51%) on simplifying a surd and Question 25 (<36%)
on applying rules of logarithms. When comparing d-values for the rest
of the questions, five other questions (Q1, Q3, Q13, Q16 and Q17)
yielded small (*) to medium (**) practical differences between cohorts.
The TKC cohort performed slightly better than the other two cohorts in
solving general linear equations (Q1), and when applying long division
to determine the remainder of a polynomial (Q16); better than the NCS
cohort when simplifying fractions (Q3), and better than the CAPS cohort
when solving fractional equations (Q13). Questions 2 and 17 are both
set on the application of exponential rules. Question 2 was a direct
application of the rules where the expression (2x3)-2 had to be simplified,
whereas in Question 17 students first had to use higher-order thinking
skills to analyse the question before applying the appropriate rule. The
question stated that they had to determine a third of 315. This higher-
order thinking could explain the lower performance of all cohor ts on this
question in relation to Question 2. The CAPS cohort per formed best in
both these questions.
Functions and graphs
The results for functions and graphs are given in Table 2. The two
questions on the quadratic function (Q6 and Q21) were answered well
by all the cohorts (>68%), with the NCS group receiving the lowest
scores. The CAPS cohort per formed the best in determining the inverse
of a function using algebraic manipulation (Q19), while the NCS cohort
performed best in the question on linking a graph of an absolute value
function to the given equations (Q9). All cohorts per formed poorly in
linking an inequality to its graphical representation (Q32), with the CAPS
cohort per forming the worst.
Trigonometry
The results for trigonometry are given in Table 3. The questions on
trigonometry tested the application of right-angled trigonometry (Q12)
and finding the period from trigonometric equations (Q27). There was
no practically significant difference in performance on the question on
right-angled trigonometry (Q12). The CAPS cohort performed best in the
question on the period of the tangent function (Q27).
Geometry
The results for geometry are given in Table 4. For the items listed in
Table 4, the students had to apply the named theorem to perform
numerical tasks. In the geometry section, the performance of the NCS
cohort was the poorest. The practical difference between the scores for
the NCS and the other two cohorts, for the questions on angles in the
same segment (Q29), angles in the centre and circumference (Q30),
and a line perpendicular to a chord (Q31), ranged from a small (0.40)
to medium (0.60) effect. In the question on the theorem of similarity of
triangles (Q35), the CAPS cohort per formed best with a small effect
(0.27) in practice.
Mathematics knowledge of first-year students
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Mathematics knowledge of first-year students
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Table 1: Question by question comparison for algebraic skills
Question Mean (%) d-value
p-value
Number Description T N C T/N N/C T/C
1 Solving a linear equation 80 62 69 0.38*-0.15 0.24*0.000*
2 Rules of exponents 72 74 80 -0.02 -0.15 -0.17 0.013*
3 Fractional expression 87 79 83 0.20*-0.10 0.11 0.016*
4Simplifying 48 37 50 0.24*-0.27*-0.03 0.000*
13 Fractional equations 80 77 70 0.09 0.14 0.22*0.002*
16 Long division 83 57 62 0.54** -0.10 0.44*0.000*
17 Rules of exponents 63 71 79 -0.17 -0.17 -0.33*0.000*
22 Roots of parabola 89 87 86 0.07 0.04 0.11 0.268
25 Rules of logarithms 36 27 29 0.18 -0.04 0.15 0.029*
Average for algebraic skills 71 63 68
T, TKC (2008 cohort); N, NCS (2009 cohort); C, CAPS (2015 cohort)
Note: An asterisk in the p-column indicates an acceptable statistically significant difference (p<0.05), and asterisks in the d-column indicate a small (*) or medium (**) effect in
practical difference. A positive d-value in a column indicates that the group mentioned first performed better, while a negative d-value indicates a better performance by the group
mentioned second.
Table 2: Question by question comparison for functions and graphs
Question Mean (%) d-value
p-value
Number Description T N C T/N N/C T/C
6 Intersection of line and parabola 85 71 89 0.30*-0.41*-0.13 0.000*
9 Shifted graph of absolute value function 66 77 61 -0.22*0.31*0.10 0.000*
19 Finding inverse of function algebraically 52 53 75 -0.03 -0.43*-0.46*0.000*
21 Symmetry axis of parabola 75 68 75 0.15 -0.15 0.00 0.031*
32 Finding inequality from graph 32 28 19 0.09 0.18 0.26*0.000*
Average for functions and graphs 62 59 64
T, TKC (2008 cohort); N, NCS (2009 cohort); C, CAPS (2015 cohort)
Note: An asterisk in the p-column indicates an acceptable statistically significant difference (p<0.05), and an asterisk (*) in the d-column indicates a small effect in practical difference.
A positive d-value in a column indicates that the group mentioned first performed better, while a negative d-value indicates a better performance by the group mentioned second.
Table 3: Question by question comparison for trigonometry
Question Mean (%) d-value
p-value
Number Description T N C T/N N/C T/C
12 Right-angled trigonometry 82 78 76 0.09 0.04 0.13 0.153
27 Period of tangent function 49 53 62 -0.09 -0.18 -0.27*0.000*
Average for trigonometry 66 66 69
T, TKC (2008 cohort); N, NCS (2009 cohort); C, CAPS (2015 cohort)
Note: An asterisk in the p-column indicates an acceptable statistically significant difference (p<0.05), and an asterisk (*) in the d-column indicates a small effect in practical difference.
A positive d-value in a column indicates that the group mentioned first performed better, while a negative d-value indicates a better performance by the group mentioned second.
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Differentiation
The results for differentiation are given in Table 5. The questions on
differentiation included two questions on limits (Q26 and Q33) and two
questions on application of differentiation (Q24 and Q34) (Table 5).
Although determination of maxima and minima of parabola is included
in all curricula, all cohorts performed poorly on the question based on
the maximum of a horizontally shifted parabola (Q24). The TKC cohort
performed best in determining the limit of a hyperbole (Q26), and the
CAPS cohort per formed best in the question based on the interpretation
of the derivative as the slope of a tangent line (Q34).
Overall performance
The results for the average performance on all the questions are given
in Table 6. There are two sets of comparisons: one including geometry
results and one without the geometry results. The former shows a
decline in the overall performance of the NCS curriculum with a medium
effect of 0.56 with respect to the TKC curriculum, and of 0.55 with
respect to the CAPS curriculum. When disregarding the effect of the
shift of Euclidean geometry to an optional paper in NCS curriculum, the
effect becomes smaller, namely 0.38 and 0.29, respectively.
Mathematics knowledge of first-year students
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Table 4: Question by question comparison for geometry
Question Mean (%) d-value
p-value
Number Description T N C T/N N/C T/C
29 Angles in the same segment 92 64 91 0.58** -0.57** 0.02 0.000*
30 Angles at the centre and circumference 25 8 30 0.40*-0.48*-0.09 0.000*
31 Line ^ to chord 84 60 89 0.49*-0.60** -0.16 0.000*
35 Theorem on similarity 69 61 81 0.17 -0.42*-0.27*0.000*
Average for geometry 68 48 73
T, TKC (2008 cohort); N, NCS (2009 cohort); C, CAPS (2015 cohort)
Note: An asterisk in the p-column indicates an acceptable statistically significant difference (p<0.05), and asterisks in the d-column indicate a small (*) or medium (**) effect in
practical difference. A positive d-value in a column indicates that the group mentioned first performed better, while a negative d-value indicates a better performance by the group
mentioned second.
Table 5: Question by question comparison for differentiation
Question Mean (%) d-value
p-value
Number Description T N C T/N N/C T/C
24 Maximum value of y=–(x–2)234 34 32 0.00 0.05 0.05 0.680
26 Limit of hyperbole 50 30 27 0.40*0.06 0.46*0.000*
33 Finding limit for case 47 42 55 0.11 -0.27*-0.16 0.000*
34 Differentiation as slope of tangent line 52 54 63 -0.04 -0.19 -0.23*0.001*
Average for differentiation and limits 46 40 44
T, TKC (2008 cohort); N, NCS (2009 cohort); C, CAPS (2015 cohort)
Note: An asterisk in the p-column indicates an acceptable statistically significant difference (p<0.05), and an asterisk (*) in the d-column indicates a small effect in practical difference.
A positive d-value in a column indicates that the group mentioned first performed better, while a negative d-value indicates a better performance by the group mentioned second.
Table 6: Comparison on overall performance
Mean (%) d-value p-value
Geometry T N C T/N N/C T/C
Included 66 57 66 0.56** -0.55** 0.01 0.000*
Excluded 65 59 63 0.38*-0.29*0.10 0.000*
T, TKC (2008 cohort); N, NCS (2009 cohort); C, CAPS (2015 cohort)
Note: An asterisk in the p-column indicates an acceptable statistically significant difference (p<0.05), and asterisks in the d-column indicate a small (*) or medium (**) effect in
practical difference. A positive d-value in a column indicates that the group mentioned first performed better, while a negative d-value indicates a better performance by the group
mentioned second.
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Mathematics knowledge of first-year students
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Discussion of results
The statistical comparison of responses to the individual questions in
general did not yield considerable differences for the three cohorts.
Responses for some questions were statistically significantly different
according to the p-values, but most of these differences were not
significant in practice according to the d-values. There was a slight drop
in performance for the NCS cohort in relation to the TKC and CAPS
cohorts, but the practical difference was small. The comparison of the
questions testing algebraic knowledge showed small differences in
practice for questions set on application of exponential rules, solving
linear equations, working with different forms of quadratic equations,
and solving and simplifying fractional equations and expressions,
respectively. This was also the case for functions and graphs,
trigonometry and differentiation. Noticeable exceptions were the
questions set on the topic of geometry (with the NCS cohor t performing
poorest), the question on finding the remainder after division (with the
TKC cohort performing best), and the question on finding the inverse of a
function (with the CAPS cohort per forming best). The poor performance
of the NCS cohort in the geometry questions can be attributed to the
transfer of some topics of Euclidean geometry to an optional paper. In
our sample, only 35% of the students of the 2009 NCS cohort wrote the
optional paper in the NCS examination of 2008. In the NCS curriculum,
the remainder theorem was omitted and students from this cohort had
to rely on the algorithm for long division to find the remainder. Although
the remainder and factor theorems are briefly mentioned in the CAPS
curriculum, this cohort also struggled more than the students from TKC
with the application of the remainder theorem. The high score of the
CAPS cohort in the question on finding an inverse function with algebraic
manipulation is difficult to explain, as finding an inverse function is
included in all three curricula.
There were other notable results obtained for some questions, albeit not
in terms of a practical difference in results. Although the absolute value
function was omitted in the NCS and CAPS curricula, students from these
cohorts could apply their knowledge of functions to identify the correct
shifted graph of the given absolute value function and surprisingly the
NCS cohort obtained the highest score for this question. Despite a
scaling down in the rules of logarithms in the NCS and CAPS curricula,
all three cohorts per formed poorly in the question on logarithms. A low
score was also obtained for the question on the simplification of a surd.
A common denominator of these two questions is knowledge about
algebra of functions, namely that in general f(x+a) f(x)+f(a) or that
f(xa) a f(x). All three cohorts failed to intuitively identify the maximum of
a quadratic function of the form y=–(x–2)2 as zero. All these questions
required a conceptual understanding of the mathematics involved and in
spite of the intention of the OBE curricula to foster higher-order cognitive
skills, all the cohorts per formed poorly on these type of questions.
The comparison of the total scores of the three cohorts indicates a lower
performance of the NCS cohort relative to the TKC and CAPS cohor ts. This
can partly be attributed to the omission of Euclidian geometry from the
NCS curriculum, and the reintegration of geometry in the core curriculum
of the CAPS curriculum. After the exclusion of the geometry results from
the overall scores, the NCS cohort still performed slightly poorer.
Conclusions and recommendations
In this study, the mathematical knowledge which is mostly learned in
Grades 10 to 12, and which we presume will have the greatest impact
on success in tertiary mainstream mathematics, was compared for
three cohorts representing three different exit-level school curricula. The
comparisons of the overall results and the results of individual questions,
which reflect topics of basic mathematical knowledge of Grades 10 to
12, show that in general there was little or no difference in practice for
these cohorts. The only mentionable difference was in the domain of
Euclidean geometry, in which the NCS cohort performed poorer, which
can be directly attributed to the transfer of some topics of Euclidean
geometry to an optional paper. The results signal that the omission of
certain basic topics can be detrimental to the preparation of learners
for tertiary studies where knowledge of these mathematical topics is
important. If learners were not exposed to the gradual build-up of basic
knowledge of a domain, it would be difficult to remedy the situation within
a short duration. Developers of school mathematics curricula should
be sensitive to the requirements of tertiary educational institutions
regarding the basic mathematical knowledge needed by natural science
and engineering students.
The introduction of new outcomes-based curricula led to the perception
that students from these curricula enrol at tertiary institutions with poorer
basic mathematical knowledge than those from traditional knowledge-
based curricula. The results of this study indicate that this perception is
not necessarily true. The samples in our study were fairly homogeneous
in terms of demography, schooling and selection criteria, and the main
difference was in their exposure to different school curricula. We suggest
that other factors, such as school management or general societal
changes or technological innovations, should be considered as an
explanation for so-called grade depreciation. Finally, we want to point
out that the results of the study do not reflect on changes envisaged
by OBE curricula regarding teaching practices or the development of
higher-order thinking skills. More qualitative studies will be needed to
investigate these factors.
Acknowledgements
We thank Prof. Gilbert Groenewald, Director of the School for Computer
Science, Statistics and Mathematical Sciences at the Potchefstroom
Campus of the North-West University at the time of the study, for granting
permission to conduct the empirical study and for financial support;
Cum laude Language Practitioners for language and reference editing;
and the Statistical Consultation Services at the Potchefstroom Campus
of the North-West University for reviewing the statistical calculations.
Authors’ contributions
S.F. initiated the study, formulated the introduction and theoretical
overview, and was responsible for the literature section. M.H. performed
the statistical calculations. Both authors contributed equally to the
data collection, data analysis, interpretation of the statistical analyses,
construction of the tables, discussion of the results and the conclusion,
and write-up of the empirical study.
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