Reading skills and mathematics

Abstract

This article considers the relationship between poorly-developed reading skills and academic performance in mathematics. It discusses some aspects that underpin all successful reading and considers these in relation to the reading difficulties experienced by a group of foundation phase mathematics students. The project investigating these difficulties was divided into a testing phase and an intervention phase. This article reports on the testing phase. South African Journal of Higher Education Vol.16(3) 2002: 196-206

Full text

196
Reading skills and mathematics
C A Bohlmann & E J Pretorius
University of South Africa
ABSTRACT
This article considers the relationship between
poorly-developed reading skills and academic
performance in mathematics. It discusses some
aspects that underpin all successful reading and
considers these in relation to the reading diffi-
culties experienced by a group of foundation
phase mathematics students. The project inves-
tigating these difficulties was divided into a
testing phase and an intervention phase. This
article reports on the testing phase.
INTRODUCTION
The persistent problem of poor academic perfor-
mance of many students at primary, secondary
and tertiary level is disturbing, particularly in science
and mathematics. The conceptual complexity and
problem solving nature of these disciplines make
extensive demands on the reasoning, interpretive and
strategic skills of students, especially when carried
out in a language that is not the student's primary
language. It is well known that South African
students have achieved the questionable distinction
of being at the bottom of the list in the TIMSS report
(Third International Mathematics and Science Study
1999). Although there has been considerable con-
troversy surrounding these tests, such as bias towards
Western (particularly North American) learners, the
fact remains that, even in comparison with the rest of
Africa, South African learners performed extremely
poorly.
There are obviously many factors, both extrinsic and
intrinsic to a learner, that contribute to poor academic
performance. Many studies, for instance, have been
undertaken on the role of language in mathematics,
and we know a lot about the ways in which poorly
developed language skills undermine students' math-
ematical performance. But to what extent does read-
ing ability influence a student's ability to comprehend
and do mathematics? This question is particularly
important in the context of print-based distance
education where students need to be good readers
in order to ``read to learn''. As long ago as 1987 it was
pointed out (Dale & Cuevas 1987) that proficiency in
the language in which mathematics is taught,
especially reading proficiency, was a prerequisite for
mathematics achievement. When students have diffi-
culty reading to learn, it is often assumed that their
comprehension problems stem from limited language
proficiency. This reflects an underlying assumption
that language proficiency and reading ability are
basically ``the same thing''. If this were so, then all
mother-tongue speakers should automatically be
good readers in their mother tongue. This is patently
not so. Furthermore, if language proficiency and
reading ability were basically ``the same thing'', then
improving the language proficiency of students
should improve their reading comprehension. Re-
search shows that this does not readily happen (eg
Hacquebord 1994). Reading is more than fluency in
articulating what is written; it is also more than
understanding the sum of the meanings of individual
words. Language proficiency and reading are clearly
related. However, they are, conceptually and cogni-
tively, uniquely specific skills that develop in distinct
ways and that rely on specific cognitive operations. It
is important to recognise this distinction because it
has pedagogical implications.
In this article we report on a study undertaken jointly
by the Departments of Mathematics and Linguistics at
the University of South Africa (Unisa), in which the
reading skills of mathematics students were tested
and the relationship between reading ability and
performance in mathematics was examined. Reading
ability, in the context of this study, refers to ``reading
comprehension'' ability. In reading research, a dis-
tinction is made between ``decoding'' and ``compre-
hension''. ``Decoding'' involves those aspects of
reading activity whereby written signs and symbols
are translated into language. ``Comprehension'' refers
to the overall understanding process whereby mean-
ing is assigned to the whole text. The interaction
between decoding and comprehending in skilled
readers happens rapidly and simultaneously. Most
researchers and practitioners of reading agree that
comprehension cannot effectively occur until decod-
ing skills have been mastered (eg Perfetti 1988).
However, skill in decoding does not necessarily imply
skill in comprehension. Many readers may readily
decode text but still have difficulty understanding
what has just been decoded (Daneman 1991; Yuill &
Oakhill 1991). In this study, students were tested on
comprehension skills and not on decoding skills,
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since it was assumed that, by tertiary level, decoding
skills are well established and automatised.
MATHEMATICS DISCOURSE AND READING
Mathematics is taught and understood via the
sublanguage of mathematical discourse, or the
mathematics register (Frawley 1992; Dale & Cuevas
1987). In general the mathematics register is abstract,
non-redundant (Prins 1997) and conceptually dense.
It features more complex and more compact relation-
ships than normal discourse (Crandall, Dale, Rhodes
& Spanos 1980). Furthermore, mathematics is a
discipline characterised by precision, conciseness
and lack of ambiguity. The reading of mathematics
texts thus requires close attention to detail. Parts of
mathematics texts tend to be procedural in that they
provide instructions and explanations on how to carry
out a task or algorithm. Other activities aim to develop
conceptual knowledge. Mathematics texts are also
hierarchical and cumulative, in the sense that under-
standing each statement or proposition is necessary
for understanding subsequent statements. If a parti-
cular step in a method, procedure or argument is
misunderstood or overlooked, this has severe con-
sequences for overall comprehension. Reading
mathematics texts also requires integration of all the
information in the text. A mathematics reader thus
needs to interact with the text, be alert and attentive,
be sensitive to comprehension failure as soon as it
happens, and be capable of applying repair strategies
when comprehension failure occurs. Reading rate
adjustment and multiple readings are also necessary
because of the conceptual density of, and the
interpretative demands made by, mathematical sym-
bols and graphic aspects, such as charts, tables and
graphs.
Rationale for study
In the Department of Mathematics at Unisa there has
been significant growth in the number of students
enrolling for the Mathematics Access Module, which
is provided through the medium of English. These
students generally have a school-leaving certificate,
which does not permit them to study at a university;
or they may have a university exemption matric
certificate (which does give them access to uni-
versity) but without the necessary mathematics
symbol to study in the Science Faculty. The student
population is extremely heterogeneous, in that some
students may not have studied mathematics beyond
Grade 9, while some may have studied it up to Grade
12 but obtained less than 40% on the Higher Grade or
less than 50% on the Standard Grade.
Many of the students who enrol for this module are
characterised by weak academic, linguistic and
mathematical ability. In spite of recognition of their
difficulties and some student support activities (lim-
ited, due to the constraints of distance education and
the financial resources of both the university and the
students) over the two years of the existence of the
current module, student performance has been un-
acceptably low.
It is well recognised that mathematical thought is best
developed by interaction between learners and
teachers, and between learners and learners, mediated
in a variety of ways. In our particular context, where
student access to technological resources cannot be
guaranteed, and where distance limits the extent to
which group learning opportunities can be utilised,
text becomes the primary resource for learning. In the
light of these factors it was decided to examine more
closely the reading abilities of the students registered
for the Mathematics Access Module. The two main
research questions that inform the study are:
1. Is there a relationship between reading ability
and academic performance in mathematics?
2. What specific kinds of comprehension problems
do students experience during the reading of
mathematics texts?
READING COMPREHENSION: ANALYTIC
FRAMEWORK
This article focuses on reading as a way of construct-
ing meaning. In the study the reading process is
assumed to involve the simultaneous interaction of
several component skills. Readers are viewed as
active participants who construct meaning while they
read by building up a coherent mental representation
of the text. They add to and modify this representation
as they encounter new information in the text. This
representation is not identical to the text; rather, it is a
mental map of what the text is about. It is constructed
from explicit information in the text and from the
inferencing and integrative processes that readers
engage in to link up items of information in the text to
one another, and to link items of information in the
text to items of information stored in long-term
memory, the storehouse of one's background knowl-
edge. If readers cannot create a coherent representa-
tion of what the text is about, then meaningful
comprehension does not occur.
In order to assess the extent to which readers are
constructing meaning as they read, it is important to
examine those aspects of a text that typically engage
readers in perceiving links between items of informa-
tion. We now take a closer look at some of the
processes that contribute to meaning construction
during reading.
Anaphoric resolution
Anaphora basically involves repeated reference in
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discourse. In other words, a referent relating to a
person, entity, event, state or idea that is introduced
into a discourse is referred to again, either by means
of repetition of the same linguistic item (eg noun or
proper noun) or by means of another linguistic item
(eg pronoun or synonym). Consider, for example, the
repeated reference in the following mathematics text:
Multiplication is denoted by the symbols x, . or
( ). The result of multiplying two numbers a
and bis called the product of aand b, and we
usually write axbor a.b. Because aand b
represent two different numbers (and are not
two digits that make up a single number) there
is no confusion if we write ab instead of axb.It
is easy to see that we cannot do this when
numerals are used, because 23 does not mean
263 (Singleton & Bohlmann 2000:54).
The highlighted noun phrase mentions a specific
convention in the text, and it is referred to again by
means of the determiner this. This determiner takes its
interpretation from the conditional clause referring to
the notion of writing ab instead of axb. The
underlined item in the text is called an anaphor, and
the shaded entity it refers back to is called the referent.
Successful anaphoric resolution contributes to the
construction of a coherent text representation be-
cause it enables the reader to identify and track topic
continuity in the text, shifts in the topic, and to make
connections between elements in the discourse.
Skilled readers should be able to resolve anaphors
with almost total accuracy (Webber 1980).
Text-semantic relations
Coherence in a text derives to a large extent from the
semantic and rhetorical relations underlying units of
discourse. These relations can be represented in
``general conceptual terms, abstracting away from
the context-specific content of the segments'' (San-
ders, Spooren & Noordman 1992:2). These concep-
tual relations include four basic types, viz additive (X
and Y), temporal (X then Y), causal (X, as a result Y)
and contrastive relations (X. However, Y). Although
these relations have been variously referred to as basic
thought patterns or logical relations (Brostoff 1981),
semantic relations (Fahnestock 1983) and relational
propositions (Mann & Thompson 1986), they all
basically refer to the same concept and will hence-
forth collectively be called text-semantic relations.
These relations underlie the way we perceive the
world and the way we think.
Perception of these relations is fundamental to the
construction of coherent mental representations of
text. Research findings indicate that knowledge of
such relations distinguishes skilled readers from their
less skilled counterparts (eg Meyer, Brand & Bluth
1980; Geva & Ryan 1985). Skilled readers perceive
these relations, albeit unconsciously, and this enables
them to see the connectedness between items of
information in the texts as they read.
In this study, attention was specifically focused on
causal and contrastive relations since they are a
common feature of mathematic discourse. In order
for a semantic relation to exist between text units
there must, minimally, be a binary relationship
between two (or more) text units: ``binary'' in that
one unit is semantically linked to the other and in
some way completes the meaning of the other. For
instance, in causal relations, the one member func-
tions as an antecedent [X] and the other as a
consequent [Y].
Causal and conditional relations
There are several notions of causality, ranging from
strict formal definitions of a deductive nature to
intuitive but fuzzy lay notions. The concept of
causality refers basically to the relationship between
two events or states of affairs, such that the first one,
the antecedent X, brings about or enables the second,
the consequent Y. Although there are many different
types of causal relations (Pretorius 1994), in this
study causal relations were divided quite broadly into
two categories, viz. general causal relations and
conditional causal relations.
The first category included relations where the
antecedent causes the consequent or the antecedent
comprises a premise or reason for a consequent result.
In other words, the contents expressed in a prior unit
of text provide a basis from which a conclusion is
drawn. In expository writing, the supporting state-
ment(s) X can comprise an observation, causal
statement, argument, or evidence, from which a
deduction or conclusion Y is drawn. As Fahnestock
points out, the sentences comprising a premise-
conclusion link in everyday logic are not always
expanded into valid syllogisms, ``but they are in-
tended by the writer and meant to be taken in by the
reader as pairs of supporting and supported state-
ments'' (Fahnestock 1983:404). Consider the follow-
ing extended causal argument with the causal
conjunctives in italics.
Consider the logarithmic function y=f(x)=
log
a
x.
We know that D
f
={x:x40}.
Thus for the function gdefined by g(x)=
log
a
(2x+ 1), we have
D
g
={x:x47
1
/
2
}. Hence x =71 is not
an element of D
g
.
The second category of causal relations included a
specific type of causality involving conditional causal
clauses. This distinction was made because of the
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high frequency of if ± then clauses in mathematical
texts. Traditionally, in syntax and logic, conditional
clauses describe a situation in which it is claimed that
``something is, or will be, the case, provided that, or
on condition that, some other situation obtains''
(Flew 1984:70). This kind of logical relation is
regarded as a subcategory of the causal relation since
there is a dependency relation between the X and Y
members, such that Y does not occur unless X occurs.
One of the main differences between conditional
causal clauses and ordinary causal clauses lies in the
truth-conditional status of the former, where the truth
of the consequent is guaranteed by the truth of the
antecedent. Consider the two examples of conditional
relations below:
Consider 715x42. Both the statements 71
52 and 7142 are true. The first inequality,
7152, is true because 71is less than 2. The
second inequality will be true if either 7152
or 71 = 2 is true. Obviously 71 = 2 is not
true, but since 7152 is true, we may write
7142. (Singleton & Bohlmann 2000:23).
Research has established that the recognition of
elements in a text that are causally linked to each
other is an important determinant of text comprehen-
sion (cf Trabasso, Van den Broek & Suh 1989;
Pretorius 1996).
Contrastive relations
Another category of text-semantic relations to be
examined was that of contrastive relations. In a
contrastive semantic relation, the second statement
in the ``pair'' carries information that counters the
information in the first part. It presents an opposing or
unexpected point to what has been stated, a conces-
sion to or qualification of a previous statement, or a
denied implication. It provides a less expected
alternative to what has already been stated. Fahne-
stock (1983:415) points out that such text-semantic
relations reflect ``the processes of distinguishing,
making exceptions, conceding or contrasting by
which thinking, and the prose which represents
thinking, is carried on''. Skilled reading naturally
includes the ability to perceive and follow disconti-
nuative turns in the text. In order to understand
contrastives, one needs to understand the line of
argument presented in the first pair part in order to
recognise that the second pair part reflects an
opposing, contrasting or qualifying point. For exam-
ple:
When we work with natural numbers we find
that adding two natural numbers results in a
natural number but subtracting one natural
number from another does not necessarily result
in a natural number. For example, 3 + 5 = 8,
and 3, 5 and 8 are all natural numbers.
However,875 = 3 and 3 EN, but 578
is a calculation that cannot be performed in N
(Singleton & Bohlmann 2000:70).
In the above example, the argument concerns the
additive operation involving natural numbers, which
is then qualified. The view that follows (signalled in
italics by but and later by however) is a contrastive
one in the sense that it qualifies the previous phrase
by pointing out that the subtractive operation does
not necessarily result in a natural number.
Research has shown that negative statements tend to
be processed more slowly than positive statements,
and the same seems to be true for contrastive text-
semantic relations. Because they present a contrast-
ing point of view to that just given, they are not
understood as readily as their continuative counter-
parts (Fahnestock 1983:406). Research also indicates
that in both L1 and L2 learners, contrastive relations
are usually acquired later than additive, temporal and
causal relations.
Sequencing
Sequencing refers to the way in which propositions in
a text follow one another. The most obvious type of
sequencing is temporal sequencing, where sentences
in a discourse are linked by means of a temporal
sequence. Temporal sequences are also often tied up
with causal relations, since many states/events/
arguments are the result of prior causes or reasons
and follow them temporally (although their actual
order of presentation in a text may be reversed). The
sequencing of events, states or arguments also
includes causal and contrastive sequences, as well
as the ordering of examples after explanations (ie
exemplification sequence).
The concepts and arguments dealt with in mathema-
tical texts are often presented within a developmental
or logical perspective. It is important for a reader to
keep track of sequence in order to understand the
relationships between successive propositions. The
ability to sequence ideas is an important reasoning
skill and, as Lesiak & Bradley-Johnson (1983:212)
point out, this skill ``can affect performance in several
academic areas''.
Graphics
Graphics refers to tables, graphs, schemas and other
visual aids that occur in texts to represent and
complement verbal information. Graphic information
forms an integral part of mathematics texts. The ability
to understand the information represented in these
visual forms and to relate it to the information in the
text form a crucial part of mathematics reading
comprehension.
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Low-frequency vocabulary
There are statistical differences in the distribution of
words that occur in oral and written contexts. A
distinction is commonly made in vocabulary studies
between three main categories of words, viz common
or high-frequency words, academic words and
technical words. High-frequency words are those
that occur in our everyday conversations and com-
prise a core of about 5 000 to 6 000 basic words. A
much larger group of words makes up the rest of the
English vocabulary, but comprises low-frequency
words. Academic words refer to a group of about
800±1 000 words that occur commonly in academic
discourse. These are words that occur across dis-
cipline boundaries, eg words such as hypothesis,
proponent, assumption, paradigm, posit, etc. These
words seldom occur in everyday conversations, and
are hence categorised as low-frequency words. The
final group of low-frequency words, technical words,
are discipline-related words that occur with high
frequency in a specific discipline and reflect the ``tools
of the trade'' within that discipline, but outside of the
discipline are seldom used. Technical terms in
mathematics include integers, logarithm, operation,
square root, exponents, etc.
Research has shown that in the learning context,
knowledge of low-frequency words is associated
with academic success. Research indicates that
students with smaller vocabularies typically know a
higher percentage of high-frequency words (Corson
1983; Cooper 1996) which occur predominantly in
oral contexts. Students who do little reading in
English have poor exposure to low-frequency words.
From the few vocabulary studies that have been
undertaken in South Africa, we see disconcerting
findings on the low vocabulary levels, particularly of
low-frequency words, of L2 students who study
through the medium of English. For example, Cooper
(1996) found a relationship between vocabulary
levels and academic performance, with weak students
having significantly fewer low-frequency words.
Vocabulary knowledge is claimed by many to be one
of the best predictors of reading comprehension (eg
Davis 1968). Intuitively this makes sense, for words
form the building blocks of meaning. However, this
does not explain why differences in vocabulary
knowledge arise in the first place. A typical feature
of skilled readers is their large vocabulary, while weak
readers typically have low vocabulary levels. How do
skilled readers come to acquire so many more words
than their unskilled counterparts? In fact Daneman
argues that differences in vocabulary size are ``the
result of differences in reading skill rather than the
primary cause of such differences'' (Daneman
1991:525). In other words, students' scores on a
vocabulary test containing both high- and low-
frequency words can indirectly indicate the extent to
which they are ``readers''.
METHODOLOGY
In this section we describe the subjects, materials and
procedures undertaken during the pilot phase and we
focus on the nature of the reading tests and the results
that they yielded.
Framework of the research project
The study spanned a period of two years. During the
first phase (2000) appropriate reading tests were
designed specifically to test mathematical compre-
hension. These were first piloted on a small group of
students (25) before the larger group (n = 960) was
tested. The pilot study was undertaken to get a ``feel''
for the reading and language proficiency levels of the
Mathematics Access students and to trial the reading
tests that were designed. During the second phase
(2001) an experimental and control group were set
up and an intervention programme was undertaken
for the experimental group, based on the results of the
reading test of 2000. This article reports on the first
phase of the project.
Subjects
The pilot study consisted of a group of 25 students
who were registered for the Mathematics Access
Module (MAT011±K) and who came to campus
everyday. They completed a battery of tests compris-
ing four main components (to be explained below).
They also wrote a standardised language proficiency
test and a general reading test based on non-
expository texts (2 newspaper reports) but which
included similar items to those of the mathematics
reading test. The purpose of the pilot tests was to
identify potential problems with the selected texts and
the test items, and to test the reliability of the test
items. Once these tests had been administered and
refined, they were then sent to all Mathematics
Access Module students in 2000.
Material
In order to build up an in-depth profile of the
students' reading ability, a series of tests was
designed that focused on specific reading skills. The
texts that were used were authentic mathematics texts
taken from the six study guides that form the basis of
the Access Module.
The final test comprised the following four sections:
Section A A questionnaire that obtained biographic
details about the student such as matric
performance, attitudes towards reading,
their perceptions of their own reading
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skills, and information concerning their
reading practices.
Section B This section focused specifically on ana-
phoric references. This comprised 21
separate paragraphs of mathematics texts
in which a total of 26 anaphoric items
were tested.
Section C This section focused specifically on vo-
cabulary assessment via a discrete, con-
text-embedded test. It comprised 15
multiple-choice items, testing high-fre-
quency, academic and technical terms
respectively.
Section D This component of the test, comprising
56 items, focused mainly on testing the
comprehension of text-semantic rela-
tions, sequencing and graphics. Items
that tested the readers' ability to infer
sequencing consisted of the re-ordering
of scrambled paragraphs. This method of
testing comprehension has been shown
to have high psychometric value (Page
Â
1990:118).
The test was posted to all the students registered for
the Access Module as part of an extra, voluntary but
credit-bearing assignment. To encourage participa-
tion, five prizes of R200 each were offered, with the
winners' student numbers being randomly drawn. In
all, 402 responses were received. Owing to the
distance education context, there were no controls
regarding the time it took the students to complete the
tests. Although the students were asked to be as
honest as possible in answering the tests and to do it
on their own, there were also no controls over the
sources they might have used to help them.
RESULTS
In this section we report on the reliability of the
reading test and various results.
Reliability test
The alpha (Cronbach) model reliability test, available
on the SPSS programme, was applied to each
component of the test. This is an analysis of internal
consistency, based on the average inter-item correla-
tions. It provides an overall index of the internal
consistency of a test and identifies problem items that
could be eliminated from the test. Reliability scores of
between 0.60 and 0.70 are regarded as satisfactory,
while scores above 0.80 are regarded as desirable.
Sections B, C and D of the final mathematics reading
test had alpha reliability coefficients of 0.85, 0.70 and
0.80 respectively.
The relationship between reading skill and
mathematical performance
In order to get some idea of the relationship between
reading ability and performance in mathematics, the
students were categorised into different groups,
based on the following two criteria:
1. Their overall reading scores, ie the sum of their
scores for Sections B, C and D of the reading test,
converted into percentages; these groups are
referred to as ``reading groups''.
2. Their results in the final mathematics examina-
tion, expressed as a percentage; these groups are
referred to as ``academic groups''.
On the basis of the results in the overall reading
scores, the following four reading categories were
identified:
1. Students with a comprehension level below 45%.
These students have major reading comprehen-
sion problems.
2. Students with comprehension levels of between
46%±59%. These students are reading at frustra-
tion level and have a fragmented understanding
of the texts they read.
3. Students with comprehension levels of between
60%±74%. These are students with reading
comprehension problems who cope to some
extent and who would probably benefit from
reading instruction.
4. Students who scored 75% or higher. These are
students who are fairly skilled readers and usually
understand most of what they read.
The descriptive statistics in Table 1 provide an
overview of the overall reading scores as well as the
scores for each component of the reading test across
the different reading ability groups.
It is not surprising that the mean scores in each
subcomponent of the test improve as reading ability
improves. What is interesting, however, is that the
mean mathematics scores also improve as reading
ability improves. In other words, these figures show
that the worse the reading ability of the student, the
worse their performance in mathematics tends to be.
Further analyses showed a wide range in mathematics
performance of the skilled reading group (Group 4),
and conversely, limited mathematics performance of
the very poor reading group (Group 1), with scores
that tend to cluster in the low 20%. What these results
suggest is that while high reading scores do not
guarantee mathematical success, a low reading score
does limit mathematical achievement. In other words,
poor reading ability in this group seems to function as
a barrier to effective mathematical performance.
So far we have examined mathematical performance
as a function of reading ability. However, it is also
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interesting to consider reading scores as a function of
mathematical performance. The mean percentage for
the mathematics examination was 34.5%, with a
median of 31%. What was striking about the results
was that very few students scored between 60% and
70%. Percentages tended to reflect bad failures (20%±
29%), failures ranging between 30%±50%, and passes
above 70%. On the basis of these results, the
following three academic groups were categorised,
viz. Fail (0%±29%), At Risk (30%±59%) and Pass
(60%±100%). The bar graph in Figure 1 shows the
mean reading score across the three mathematics
groups.
A one-way ANOVA was used to further explore the
relationship between the three different academic
groups with regard to overall reading scores. The
ANOVA determines whether the differences between
the specified groups are larger than the differences
within the specified groups. The analysis yielded F(2,
305) = 20,002, p0.000. A Scheffe test showed
significant differences between all three groups in the
overall reading scores. Differences between the three
different groups according to the different compo-
nents of the reading test are reflected in Table 2 on the
next page.
As these results show, in all three components of the
reading test, there were consistently significant
differences between the students who failed badly
(below 30%) and those who were at risk (30%±59%).
If one excludes the vocabulary component from the
reading test and considers only Sections B and D,
then we see significant differences between all three
groups. Likewise, the overall scores on the reading
test (ie Sections B, C and D) also show significant
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Table 1
Mean reading comprehension scores for the four reading ability groups
Reading
Group 1
n=45
Reading
Group 2
n=78
Reading
Group 3
n=95
Reading
Group 4
n=90
B: Anaphoric references 28.6 52.5 68.2 80.8
C: Vocabulary 45.4 58.4 73.7 80.8
D: Logical relations, graphs 37.2 49.4 61.4 78.8
Mathematics exam 24.8 32.5 33.4 41.7
Figure 1
Mean reading comprehension scores for the three academic groups

203
differences between all three groups. These results
provide robust evidence for differences in reading
ability in relation to academic performance ± students
who pass their mathematics examination are students
with higher reading scores than students who are at
risk or who fail. Students who get less than 30% for
mathematics are typically students with poor reading
skills. What these results show is that the stronger a
student's reading ability, the better his/her chances of
performing well in the mathematics examination may
be. Students whose overall scores in the reading test
were 45% or lower typically failed their mathematics
examination hopelessly. Students whose overall
scores in the reading test were 75% or above typically
passed their mathematics examination with 60% or
more.
Cluster analysis
To further explore the relationship between different
aspects of reading skill and academic performance, a
cluster analysis was applied to the data. Cluster
analysis is a multivariate statistical technique used to
identify and classify groups (clusters) in terms of the
characteristics they possess. Hair, Anderson, Tatham
and Black (1995:423) describe each object in a
cluster as being ``... very similar to others in the cluster
with respect to some predetermined selection criter-
ion. The resulting clusters of objects should then
exhibit high internal (within-cluster) homogeneity
and high external (between-cluster) heterogeneity.''
The selection criteria for the cluster analysis were the
different aspects of reading skill that the test assessed,
viz
1. Pronominal anaphors (eg it and they)
2. Determiner anaphors (eg this and these)
3. Paraphrase anaphors (eg this process)
4. Low-frequency words (typical of mathematical
discourse)
5. Causal relationships (eg Since X, Y; If X ... then
Y)
6. Contrastive relationships (eg Although X, Y; X,
however, Y)
7. Understanding information in graphs
8. Performance in the mathematics examination
K-means cluster analysis was performed for four
different cluster values. Three and five cluster solu-
tions were also performed for these eight criteria but
convergence was not consistently attained among
these cluster assignments, with no clear patterns
emerging from the clusters. The four-cluster option
thus appeared optimal.
Problem areas in the reading of mathematics
texts
The second aim of the study was to identify some
typical reading problems that students experience
when reading mathematics texts. We shall look at all
nine major subcomponents of the reading test to
determine which aspects proved to be the most
challenging for the students. It is interesting that a
similar pattern is reflected in each subcomponent of
the reading test as the students move up the
academic/mathematics groups. As can be seen, the
sequencing task was the most difficult across the
groups, followed by contrastive and then conditional
text-semantic relations. This information is presented
visually in the stacked bar graph in Figure 2.
SAJHE/SATHO VOL 16 NO 3 2002
Table 2
ANOVA results: Differences in reading skill between the three academic groups
Group 1
Fail
(n =154)
Group 2
At Risk
(n=131)
Group 3
Pass
(n=24)
Sig.
level*
Section B: Anaphoric references 1 * 2 *0.00
Section C: Vocabulary 1 * 2 *0.01
Section D: Semantic relations, graphs, 1 *
sequencing 1 * 2 *0.00
Section B + D 2 *0.03
2 * 3 0.03
Overall score: Section B + C + D 1 * 2 *0.00
2 * 3 0.02
* = significant difference between groups

204
DISCUSSION
While the importance of reading in the language,
social and human sciences seems undisputed, it has
often been assumed that success in mathematics and
science requires primarily logico-deductive and nu-
merical skills, and consequently the role that reading
plays in constructing and understanding complex
concepts and in problem solving in these disciplines
is often underestimated or overlooked. It is of course
important not to over-simplify the problem. Many
people with poor reading skills are good mathemati-
cians; many people with excellent reading ability do
not cope with mathematics. There are obviously many
different variables involved, not least of which are the
issues of motivation, patience, persistence and other
cognitive aspects uniquely (perhaps) associated with
mathematical argument. Yet the analysis of the read-
ing test administered to the Mathematics Access
Module students in 2000 shows a robust relationship
between reading ability and academic performance in
mathematics. Students who failed their mathematics
examination had considerably poorer reading skills
than those who passed. Reading ability does not of
course guarantee performance in mathematics but the
results do suggest that poor reading ability functions
as a barrier to effective mathematics performance.
Weak readers are only achieving reading comprehen-
sion levels of 50% or less, which effectively means
that half of what they read they don't properly
understand, with dire consequences for their aca-
demic performance.
ISSN 1011±3487
Table 3
Summary of cluster descriptors
Cluster 1
n=73
Cluster 2
n = 111
Cluster 3
n=70
Cluster 4
n=17
Pronominal anaphor 55.3 64.6 65.1 85.3
Determiner anaphor 57.8 65.6 68.2 77.6
Paraphrase anaphor 53.2 60.4 65.0 81.5
Low-frequency words 59.5 68.9 72.1 85.9
Causal relations 51.1 61.0 64.8 76.8
Contrastive relations 42.7 50.0 53.7 72.1
Graphs 57.4 71.7 77.4 89.7
Mathematics exam percentage 16.6 30.8 50.7 71.4
Figure 2
Performance in subcomponents of reading test according to academic groups

205
Skilled readers resolve anaphors with at least 95%±
100% accuracy. The results of the reading test
indicate that many of these mathematics students
have problems with anaphoric resolution, with aca-
demically weaker students being consistently less
successful than academically stronger students. Diffi-
culty in this area of reading implies a lack of
specification of content matter; in other words,
readers who do not resolve anaphors successfully
keep ``missing the point'' about the focus of a
paragraph. This affects their comprehension, and
hence their ability to read to learn.
Linguistic research has shown that text-semantic
relations ``knit'' ideas and arguments in a text together
and give it coherence. The ability to perceive such
relations while reading enables a reader to construct a
rich and coherent representation of what the text is
about. The Pass students were much better at
perceiving such relations than the At Risk students,
who in turn were better than the Fail students. It is
interesting to note that the conditional and contras-
tive relations in particular seemed to pose problems.
Given that conditional and contrastive propositions
are a common characteristic of mathematics dis-
course, the poor level of comprehension of such
relations among these mathematics students is cause
for concern.
The results indicated that the sequencing items posed
the greatest challenge to the students. Although such
tasks are not typically part of what mathematics
students do, their performance on such tasks assesses
the extent to which they pay attention to semantic
and logical clues in a text to help them construct
meaning while they read. These results indicate that
the weaker students regularly miss vital clues that aid
in constructing and keeping track of meaning in a
text. This kind of response indicates an immature level
of reading behaviour. Given that the reading of
mathematics texts requires precision, and demands
comprehension of each successive unit of text, failure
to attend to explicit semantic and logical clues can
cause a reader to miss the point of an argument and
hence construct an erroneous and fragmented repre-
sentation of the text.
These findings suggest that special attention should
be given to helping students develop a deeper
understanding of causal, conditional and contrastive
relations in mathematics discourse and the different
linguistic markers that are used to explicitly mark such
relations, so that they come to understand the
relationships in a meaningful and productive manner.
In this way they can build up a schema of each
relationship in their minds. Such schemata become
bridges between recognising the nature of the
argument in the text and constructing a meaningful
representation of it during the reading process.
Drawing students' attention to the way in which
anaphors function in discourse could also be helpful,
especially for the weaker readers.
Reading is important in the learning context because
it affords readers independent access to information
in an increasingly information-driven society. It is also
a powerful learning tool ± a means of constructing
meaning and acquiring new knowledge, and con-
solidating, modifying and expanding knowledge
bases. Students need to be good readers in order to
be able to ``read to learn''. If students have not
properly mastered this learning tool their potential for
success in the learning context is handicapped. We
make sense of our world by constructing meaning
largely through language. In the distance-learning
context, where much of the access to mathematical
information occurs via the written word, it is
important to take into account the reading skills of
students. It is essential that they be given all possible
assistance in improving their proficiency in reading,
understanding and using English in the context of
mathematics. It is perhaps necessary at the foundation
level to consider administering tests designed for the
reading of mathematical texts, not to function as a
gate-keeping device, but to advise potential students
that they should first upgrade their reading skills
before studying mathematics. For this purpose it is
necessary to design a reading module/course that will
specifically help to develop those kinds of reading
skills that are needed to make sense of the complex
language of mathematics discourse. As Freitag
(1997:18) points out, ``underdeveloped reading skills
can keep our students from realizing their full
potential and developing into the mathematical
learners they are capable of becoming''.
The reading skills needed to comprehend mathe-
matics texts and word problems are the tools with
which students access, learn and apply mathematical
concepts and skills. If education is ``designed to
empower individuals to become active participants in
a technologically based world economy, true em-
powerment meaning becoming academically compe-
tent participants'' (Garaway 1994), then we need to
focus on reading as a fundamental skill underlying
academic competence. If students can be given
opportunities to improve their reading in the context
of mathematics, they should have a better chance of
success.
SAJHE/SATHO VOL 16 NO 3 2002

206
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