Mathematical literacy is a real-world practical attribute yet students write a high-stakes examination in order to pass the subject Mathematical Literacy in the National Certificates (Vocational) (NC(V)). In these examinations, all sources of information are contextualised in language. It can be effortful for English second language students to decode text. The deliberate processing that is required saturates working memory and prevents these students from optimally engaging in problem solving. In this study, 15 items from an NC(V) Level 4 Mathematical Literacy examination are selected, as well as 15 student responses to each of these questions. From these responses, those which are incorrect are analysed to determine whether the error is due to insufficient mathematical literacy or a lack of English language proficiency. These results are used as an indication as to whether the examination is fair and valid for this group of students.

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Per Linguam 2012 28(2):65 -83

http://dx.doi.org/10.5785/28-2-531

P Vale, S Murray & B Brown

MATHEMATICAL LITERACY EXAMINATION ITEMS AND STUDENT

ERRORS: AN ANALYSIS OF ENGLISH SECOND LANGUAGE

STUDENTS' RESPONSES

Pamela Vale, Sarah Murray & Dr Bruce Brown

Rhodes University

___________________________________________________________________________

Mathematical literacy is a real-world practical attribute yet students write a high-stakes

examination in order to pass the subject Mathematical Literacy in the National Certificates

(Vocational) (NC(V)). In these examinations, all sources of information are contextualised in

language. It can be effortful for English second language students to decode text. The

deliberate processing that is required saturates working memory and prevents these students

from optimally engaging in problem solving. In this study, 15 items from an NC(V) Level 4

Mathematical Literacy examination are selected, as well as 15 student responses to each of

these questions. From these responses, those which are incorrect are analysed to determine

whether the error is due to insufficient mathematical literacy or a lack of English language

proficiency. These results are used as an indication as to whether the examination is fair and

valid for this group of students.

1. INTRODUCTION

In the Programme for International Student Assessment's (PISA) 2012 Mathematics

Framework, mathematical literacy is defined as 'an individual's capacity to recognise, do and

use mathematics, including to reason mathematically in a variety of contexts' (PISA Governing

Board, 2010: 5). For the purposes of the South African National Certificates (Vocational)

(NC(V)) qualification, Mathematical Literacy has been included as a subject which envisions

that students will become capable of 'managing situations and solving problems in everyday

life, work, societal and lifelong learning contexts' (Department of Education, 2007: 1) by

making use of mathematical concepts.

According to the Department of Higher Education and Training (DHET) (2012a), the

competence demonstrated by NC(V) students in Mathematical Literacy in 2011 was: NC(V)

Level 2, 29%; NC(V) Level 3, 27%; and NC(V) Level 4, 37%. The targets for 2012/2013 are

set at 10% higher, but this still leaves every level with a pass rate below 50%. Many factors

will have contributed to these poor results, but if the pass rate for NC(V) Level 4 is compared

to the results of the National Senior Certificate examination for which the pass rate was 85.9%

in 2011 (Department of Basic Education, 2011a), it is clear that there is cause for great concern

and a need for research into this subject in the NC(V) in particular.

This article will present the results of a small study conducted in an FET college. This study

formed the pilot which informed the research design of a larger scale descriptive case study.

Student errors in a mathematical literacy examination will be analysed according to what the

most likely cause of the error is. The aim is to reveal whether these students are losing marks

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because of their lack of English language proficiency or owing to their low level of

mathematical literacy. The results are not generalisable because of the small size of the sample

and can only be tentatively stated, but the description of these students' errors in relation to the

questions asked can reveal whether the examination questions seem fair and valid for these

particular students and thereby warrant a further similar investigation on a larger scale.

2. THE NATIONAL CERTIFICATES (VOCATIONAL)

The National Certificates (Vocational) were introduced as a 'sister qualification' (Umalusi,

2010a: 10) to the National Senior Certificate. Technical and vocational education has a history

of more than 100 years in South Africa (Wedekind, 2008), but there is no historical precedent

for the newly implemented NC(V) in the South African education system (Umalusi, 2010b).

The rationale for introducing this new curriculum was to provide an alternative qualification to

students that will equip them with both the theoretical background and practical experience

required to master a trade or technical skill (Umalusi, 2010a).

The curriculum currently offers students a selection of 18 fields of study, all of which include

compulsory fundamental subjects comprising one language, Life Orientation and either

Mathematics or Mathematical Literacy.

The compilers of the NC(V) course need to develop sensitivity to the ever-changing industries

in which these students will be working and the course requires curricula which can rapidly be

adapted in response to this. Employability in a wider sense needs to be considered for this to be

achieved. A 'solid foundation of general education' (Gamble, 2003: 62) paves the way for

students to become employable in a variety of settings and enables them to adapt to changing

workplace demands independently. It is for this purpose that the fundamentals have been

included in the curricula of all 18 fields of study.

3. ENGLISH LANGUAGE PROFICIENCY AND MATHEMATICAL PROFICIENCY

English second language students face an additional challenge in their NC(V) studies, namely,

that of learning and being assessed in a language other than their home language. An FET

college audit (Cosser, Kraak & Winnaar, 2010: 13) revealed that the national percentage of

black students in FET colleges was 96% between 2007 and 2010. While race statistics do not

directly translate into language statistics, this does reveal that an overwhelming majority of

students at these colleges do not have English as their home language.

The stage of English language acquisition at which students are performing, as well as their

level of mathematical proficiency, will together determine how they cope with studying

Mathematical Literacy in English. When evaluating the assessment of English second language

students, it is essential for researchers to have an understanding both of how mathematical

thinking develops from infancy as well as the process of second language acquisition.

3.1. Second language acquisition and English as a medium of instruction

For many students whose second language is English, part of their learning will have been in

their home language. Transferring the mathematical skills they have developed in their home

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language into contexts presented in a second language therefore becomes more complicated.

For these students, the movement from informal spoken to formal written mathematical

language, for example, needs to occur at three levels: 'from spoken to written language, from

main language to English, and from informal to formal language' (Setati, 2002: 10). This results

in a complex and multi-layered challenge for vocational students.

The assessment of mathematical literacy, specifically in the final examinations, is text-based.

Questions that are cognitively undemanding to a native speaker will be more demanding for a

second language student (Cummins & Swain, 1986). This is particularly true for students who

are in the earlier stages of second language acquisition. The processing of speech becomes

highly automatic from the earlier stages of language acquisition; however, reading requires the

reshaping of these processes to 'interface with a new information source' (Crain & Shankweiler,

1988: 167). The processing of texts requires strategies of comprehension and production that

are distinct from those required for everyday oral interactions and may require specific and

sustained instruction (Cummins & Swain, 1986).

Despite the sophistication of the human brain, it is only able to process a certain limited

number of pieces of information when using working memory (Tuovinen & Sweller, 1999).

Cognitive load heory (Sweller, 1988) explains the interface between mathematical literacy and

English language proficiency in assessment. According to this theory, 'performance on

complex cognitive tasks depends on whether the amount of information presented to the

[student] equals or exceeds the availability of working memory' (Barbu, 2010: 4). What this

theory implies is that the devotion of cognitive resources to comprehending text would reduce

the cognitive resources available for mathematical problem solving (Barbu, 2010; Lucangeli,

Tressoldi & Cendron, 1998).

While working memory has a limited capacity, long-term memory is effectively limitless

(Tuovinen & Sweller, 1999). Only two to four elements can be manipulated in working

memory when the information that is presented is novel, as opposed to the seven elements

possible when information is more familiar (Van Merriënboer & Sweller, 2005). This expands,

however, when long-term memory is activated during the task. Long-term memory is a store of

knowledge as organised cognitive schemata. A schema 'categorises elements of information

according to [the way in which] they will be used' (Sweller, Van Merriënboer & Paas, 1998:

257). The entire schema then forms only one element to be held in working memory, allowing

for more elements to be accommodated and manipulated (Van Merriënboer & Sweller, 2005).

If practiced, these schemata can become automated, thus allowing for even more efficient use

of long-term memory during tasks (Van Merriënboer & Sweller, 2005). This is not possible for

working memory, where all processing is conscious (Sweller et al, 1998). Therefore the

activation of schemata, particularly those that are automatised, will increase the availability of

learned information during problem solving.

Sweller and colleagues (Sweller et al, 1998) used reading as an example. After sufficient

practice, a reader is able to automatically process individual letters and words and, as a result,

is able to attend to the meaning of a text. Less experienced readers, such as an English

language learner who has not had sufficient practice, will need to devote working memory to

the processing of letters and words and will then be unable to hold additional information in her

or his working memory to comprehend meaning.

P Vale, S Murray & B Brown

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3.2 The nature of mathematical texts

A mathematical register refers to the specialised meanings of words from a reinterpretation of

vocabulary and phrases from the natural language (Cuevas, 1984). The mathematical register

constitutes the 'styles of meaning and modes of argument … rather than the words and natural

language structures' (K'Odhiambo & Gunga, 2010: 80). This mathematical speech and writing

requires that students become proficient in both ordinary and mathematical English (Setati,

2002), therefore requiring a certain level of linguistic competence in the language of instruction.

Patkin (2011:2) described mathematics as possessing 'unique linguistic forms' and making

frequent use of key terms, such as those signifying the four operations of addition, subtraction,

multiplication and division (Patkin, 2011). In fact, key terms are involved in signifying any

mathematical concept, as any important concept will result in the preponderance of terms.

The language of mathematics is 'informationally dense and structurally complex' (Hammill,

2010:1). Hammill (2010) list ed several characteristics of mathematical texts: complex ideas are

expressed in dense noun phrases; relationships are described by verbs; special terminology

often conflicts with common use of the words and logical connectives are used extensively.

These complex sentences have the effect of 'obscur[ing] the presence of people, distanc[ing]

the reader from the author, and portray[ing] the student as passive and mathematics as

impersonal' (Hammill, 2010:1). In this mathematical discourse, rhetorical information is not

explicitly stated, but is implied (Flick & Anderson, 1980). This leads to students with English

as a second language struggling to comprehend the whole meaning of a paragraph, despite

comprehending each of the individual sentences (Flick & Anderson, 1980).

It is difficult to separate the process of reading from the process of problem solving (Bergqvist

& Österholm, 2010). Lewis (1989) found that the majority of errors on solving mathematical

word problems occurred due to the misrepresentation of the problem structure as

communicated by the text and not from errors in computation.

Mathematical texts are not only characterised by particular technical terms and grammatical

structures but are frequently multimodal (Hammill, 2010). Texts almost always contain

symbolic notation and graphics in addition to the text (Hammill, 2010).

Symbols serve as a type of shorthand and are a means of condensing concepts into a

manageable form that can be manipulated (K'Odhiambo & Gunga, 2010). The reading of

symbols can be done as with English, from left to right. These symbols have their own syntax

but share the two-dimensional characteristics of diagrams (Hammill, 2010). They act as objects

and can be approached as such.

The reading of symbols requires some significant backtracking (Hammill, 2010). Hammill

(2010) commented that novice readers neglect to pay attention to parentheses and select rather

to focus on the individual operations. They 'respond strongly to the visual structure of symbolic

mathematics, independent of the semantic content' (Hammill, 2010: 4).

In mathematics, visual images are essential as a representation of abstract phenomena

(Hammill, 2010). What an image adds to the meaning of a text can frequently not be achieved

by written language alone. These modes, therefore, are distinct in terms of what they are able to

portray (Kress, 2000). Kress (2000:339) observed that 'Image is founded on the logic of

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display in space; writing (and speech even more so) is founded on the logic of succession in

time. Image is spatial and nonsequential; writing and speech are temporal and sequential'.

It is complex and difficult to acquire meaning from visual displays, particularly when the

student is offered no guidance in this regard (Hammill, 2010). The more challenging the

content of the text, the more likely the student is to focus on the accompanying visual displays,

and this can help to compensate for any difficulties a student may have in comprehending text

(Chen, 2011). Lowrie and Diezmann (2007) explained that in order to interpret graphics

successfully, students need to attend to and comprehend the mathematical content and context

as well as the graphics.

3.3 The relationship between English language proficiency and mathematical

proficiency

Barton and Neville-Barton (2003) estimated that academic performance varies according to

English language ability by up to 10%. Language as a 'vehicle for mathematics learning'

(Barton & Neville-Barton 2003: 4) is an important area for investigation and research as a

serious and complex disadvantage is faced by students who have a poor command of the

language of learning and instruction (Anthony & Setati, 2007; Barton & Neville-Barton, 2003).

Other than developing an understanding of the mathematical register, in order to acquire

mathematical knowledge, students need to participate in the negotiation of meaning in the

classroom, which also requires competent use of English (Anthony & Setati, 2007). Cuevas

(1984: 138) noted that the 'mastery of mathematical concepts presupposes some facility with

the language used to express, characterise, and apply those concepts'.

In South Africa, this is particularly important as '[t]he majority of [students] … learn

mathematics in a language that they are not fluent in' (Setati & Barwell, 2008: 2). Research has

revealed that the majority of students who are unsuccessful in Grade 12 are not studying in

their home language (Setati & Barwell, 2008). There are additional factors which contribute to

this picture, but it is undeniable that English proficiency is related to proficiency in

mathematical literacy in the South African context.

Sarah Howie's (2005) secondary analysis of the South African results of the Third International

Mathematics and Science Study supports the above contention. Her key findings include that

'pupils could not communicate their answers in the language of the test' (Howie, 2005: 178)

and that 'pupils who [spoke] the language of the test more frequently at home … attain[ed]

higher scores on the mathematics test' (Howie, 2005: 178). Howie's (2005) analysis indicated a

variety of statistically significant factors affecting the mathematics results of the student

participants: achievement on an English language proficiency test; socio-economic status;

perception of mathematics; exposure to English; the teachers' view of their professional status;

the mathematics teachers' beliefs about mathematics; school location; extent of English use in

the classroom; teachers' time spent working and teachers' preparedness for lessons. The data,

however, specifically supports her conclusion that the 'language component represented in a

number of the variables' (Howie, 2005: 184) has a strong effect on achievement in mathematics.

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4. WHAT IS MATHEMATICAL LITERACY?

Mathematical Literacy was introduced as a subject in South Africa in the Further Education and

Training (FET) phase in order to build on the mathematical content of the preceding General

Education and Training Phase. The emphasis is on the use of these skills in applied contexts,

which increases in difficulty and complexity as the student progresses through the FET phase,

demanding higher levels of understanding and analysis (Venkatakrishnan & Graven, 2006).

Therefore, a shift is envisioned from abstract mathematics to the concrete application of

mathematical skills (Venkatakrishnan & Graven, 2006).

The Subject Guidelines: Mathematical Literacy Level 2 (DHET, 2012b: 1), defined

mathematical literacy as 'an attribute of individuals … that involves managing situations and

solving problems in everyday life, work, societal and lifelong learning contexts by engaging

with mathematical concepts … presented in a wide range of different ways'. The Department

of Basic Education (2011b: 10) listed the competencies that comprise mathematical literacy as

'the ability to reason, make decisions, solve problems, manage resources, interpret information,

schedule events and use and apply technology'.

The range of different forms of presentation, and information to be interpreted in the NC(V)

subject, are described as 'collected and organised data obtained from numbers, tables and

graphs' (DHET, 2012b: 5), all of which are framed by a given context. The way in which these

contexts and concepts are presented in examinations is by 'written inscription and language

[which is] used to create, record and justify [mathematical] knowledge' (Anthony & Setati,

2007: 218). Since language proficiency in English is not specified in the scope of the

Mathematical Literacy curriculum, it should not interfere with students' success in the

assessment of their mathematical literacy proficiency.

Polya is considered by many to be the 'father of the modern focus on problem solving in

mathematics education' (Passmore, 2007: 1). In his pioneering work, How to Solve It (Polya,

1957), he outlined a process that has informed conceptions of problem solving (Organisation

for Economic Co-operation and Development (OECD), 2003) and mathematical literacy (PISA

Governing Board, 2010) still used today. Polya's (1957) four-stage process of problem solving

consists of

1. Understanding the problem

In this first step, an understanding of the unknowns, data and conditions of the problem

needs to be established.

2. Devising a plan

A connection needs to be found between the unknowns and the given data to arrive at a

plan for finding the solution.

3. Carrying out the plan

4. Looking back

The solution obtained by carrying out the plan must be examined for its correctness.

The authors of the framework for the PISA 2012 (PISA Problem Solving Expert Group,

2010) assessment of problem solving have derived their four problem-solving processes

from Polya's (1957) work. These processes do not necessarily form a linear information

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processing system but represent parallel processes (PISA Problem Solving Expert Group,

2010). They are

1. Exploring and understanding

The problem situation is observed and interacted with in the search for information,

limitations and obstacles in order to build a mental representation.

2. Representing and formulating

A coherent mental representation is formed by mentally organising information and

integrating it with prior knowledge. This includes constructing tabular, graphical,

symbolic or verbal representations; formulating hypotheses and critically evaluating

information.

3. Planning and executing

Planning involves goal setting and devising a strategy to achieve the goal state and then

carrying out this plan.

4. Monitoring and reflecting

As progress is made towards the goal state, intermediate and final results are checked

with remedial action taken where necessary. On reflection, assumptions are critically

evaluated and alternative solutions considered.

(PISA Problem Solving Expert Group, 2010: 20)

The PISA Governing Board (2010) described the process of solving a mathematical problem

embedded in a context as mathematical literacy in practice. This is illustrated in the figure

below.

Figure 1: Mathematical literacy in practice

(PISA Governing Board, 2010: 6)

The process of solving such a problem begins when a problem is posed in a given real-world

context, for example, the area of a floor to be tiled needs to be calculated by a contractor. The

problem needs to be translated into a mathematical one by formulating a model. In this case,

the model would require a procedure to calculate the area as well as knowledge of the relevant

measurements. Once the model has been derived and the required information gathered, intra-

mathematical work is done to arrive at a numerical solution. This numerical result must be

interpreted in order to make the decisions for which the calculation was performed. A

contractor may need the information from this example for a variety of decisions, such as to

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calculate the cost to tile this floor, in which case this information will become necessary for the

next problem-solving cycle.

5. THE ASSESSMENT OF MATHEMATICAL LITERACY

The assessment of Mathematical Literacy in the NC(V) is divided into two components (DHET,

2012c). The Internal Continuous Assessment component is weighted as 25% of the final result,

and consists of internally controlled formative tasks undertaken throughout the year. Two

external examinations are written at the end of the academic year, which are together weighted

as 75% of the final result.

5.1. Fair assessment

The latest assessment guidelines for Mathematical Literacy (DHET, 2012c) emphasise the

principles of validity, as well as fairness and transparency in assessments. Methods referred to

as unfair are, among others 'bias based on ethnicity, race, gender, age, disability or social

class … and comparison of students' work with other students, based on learning styles and

language' (DHET, 2012c: 3).

The concept of bias and unfairness in Mathematical Literacy assessment can be extended to

include instances where assessments require a high level of language proficiency in order for

students to comprehend questions and formulate responses. It is possible that students whose

home language is the language of learning and teaching and those who have already

experienced this language as the medium of instruction may be favoured. This has an impact on

the validity of assessments, as the language proficiency of the student could become what is

assessed as opposed to the outcomes of the subject being examined.

Umalusi (2010c) monitors the quality of examinations and assessments for the NC(V). Their

evaluation of the November 2009 NC(V) examinations revealed that poor editing forced

external moderators to 'grapple with poor language usage, incorrect spelling and typing errors'

(Umalusi, 2010c: 17). In addition, Umalusi (2010c) found that some editors were unaware that

the changes they made with regard to language and sentence structure could change the

meaning of the questions. As only a sample of examination papers is externally moderated in

this way, the implication is that many more poorly constructed examination papers find their

way into the examination room. A student with limited language proficiency in the language of

the examination will almost certainly be unable to accurately respond to such poorly

constructed questions and their results, and consequently, may not accurately reflect their

knowledge of that particular subject.

5.2. Assessing English second language students

When assessing English second language students, Cummins and Swain (1986:141) pointed

out that 'educators' implicit assumptions with regard to the nature of language proficiency are

by no means innocuous'. Academic difficulties can be created by educators who are ignorant of

the fact that English second language students require much longer than English home

language students to attain grade and/or age appropriate levels of academic language

proficiency (Cummins & Swain, 1986). Halliday (2010) emphasised that assessment is a social

practice, involving power relations, and therefore the principles of justice need to be

deliberately applied by considering students' language proficiency when interpreting results.

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Furthermore, mathematical literacy examinations do not merely require the processing of text,

but also the interpretation of multi-modal information presented in symbolic notation, diagrams,

graphs and tables (Hammill, 2010).

According to the Department of Basic Education (2011b:10), it is 'essential that the contexts

students are exposed to [must be] authentic and relevant, and relate to daily life, the workplace

and the wider social, political and global environments'. It is further mentioned that 'students

must be able to work with actual real-life problems and resources, rather than with problems

developed around constructed, semi-real and/or fictitious scenarios' (Department of Basic

Education, 2011b: 10). Problem solving should not have, as a pre-requisite, an excellent

command of language, but in final, high-stakes examinations, it is precisely this limited

command of language in second language students that may prevent them from solving the

given problems accurately.

Figure 2 gives a graphic depiction of how language proficiency can affect problem solving in

an examination setting. In examinations, problems are linguistically created, and sources of

information are contextualised through the use of written scenarios. The question requires

decoding by reading the text as well as viewing any multimodal information. The level of

students' language proficiency will either allow them to understand the context and resume the

mathematical literacy problem solving process or will limit their understanding, causing them

to enter the problem-solving process with inaccurate information. In the same way, once the

solution has been calculated, the raw number obtained must be encoded back into the context

through writing or presenting information in tables, graphs or symbolic notation.

Figure 2: Problem solving in examinations

(Adaptation of PISA Governing Board, 2010: 6)

5.3. Error analysis

The analysis of errors has a long history in mathematics education. Students may respond

incorrectly in a great variety of ways, but several models have been developed to assist in

broadly identifying where the student has gone wrong. Clements (1980) pointed out that it is

difficult to distinctly categorise the source of any one particular error, as they closely interact

and overlap. It is, however, possible to apply a theoretical hierarchy of errors to broadly

classify what might have led to the student's incorrect response.

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Data from researchers who have constructed various analysis hierarchies confirm that a

frequent source of errors in written mathematical tasks is difficulty in 'reading and reading

comprehension' (Clements, 1980: 13). For ESL students, the linguistic item variables, together

with their own personal competence in the language, will impact on their ability to respond

correctly to any item. This leads to several recognisable errors.

Newman (1977) has proposed the following error categories into which student errors on

mathematical tasks can be grouped: reading; comprehension; transformation; process; encoding.

Reading errors arise when a student is not able to understand the actual words or phrases used

in the problem. Comprehension errors are closely related, being those that are due to the

student not understanding what it is that the item required. This could be related to the language

proficiency of the student, but could also be an indication that the student has not yet fully

comprehended the mathematics concept involved. A transformation error occurs when a student

is unable to decide what needs to be done in order to solve the problem. Processing errors

involve the inability to carry out the method that has been identified as appropriate during

transformation, and an encoding error results from the inability to communicate the solution,

whether in written or spoken form.

Each error occurs at a specific stage in the problem-solving process. If Newman's (1977) error

categories are mapped over Figure 3, the model describing problem solving in examinations, it

becomes clearer where these errors can be placed in this process.

Figure 3: Sources of error in mathematical literacy examinations

(Adaptation of PISA Governing Board, 2010: 6)

Through careful inspection of student responses to examination items, it is possible to detect

which factor/s contributed to the error as well as to determine at what stage of the problem-

solving process this error occurred. The particular focus of this research is to distinguish

between decoding and encoding errors, as well as mathematical calculation errors, in order, to

compare number of errors related to poor mathematical literacy as opposed to those related to

weak English language proficiency.

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6. RESEARCH METHODOLOGY

6.1. Preliminary investigation

The stimulus for the research question was a significant drop in the class average of an NC(V)

Level 3 Mathematical Literacy class when a test containing a particularly large amount of

reading was administered. A focus group was held with 7 students of varying abilities from this

class to investigate the possible reasons for this decrease. One question in particular had been

problematic, with no student achieving more than 25% for it. When questioned, all students

reported that they had struggled to understand the text. Other contributions to the discussion

included the following:

Some vocabulary was not understood (this vocabulary was not part of the mathematical

terminology that students are required to know)

A diagram would have made the text more comprehensible

The introductory text to each question was considered confusing.

Notably, a student who had achieved above 80% for the test commented that he had struggled

significantly. He mentioned that it had taken him an unusually long time and many attempts at

some questions because he had not easily understood what was required of him. This was

evident in his answer book, where the answer had been scratched out and attempted more than

once.

If their errors were broadly grouped into those attributable to reading comprehension and

limited language proficiency versus those attributable to deficient mathematical proficiency, the

following results were obtained:

Figure 4: Error analysis of focus group students' test responses

For most students, more percentage points were lost due to lack of comprehension than to

mathematical errors.

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6.2. Research design

Following from the preliminary investigation, the aim of the research was to discover what

types of errors English second language NC(V) students are making in Mathematical Literacy

examinations, and at what stage in the problem-solving process these errors are occurring.

6.2.1 Research goal and question

The research aimed to address the following two research questions:

Question 1: What type of errors are students making?

Question 2: How do the number of mathematical literacy-related errors compare to the

number of language proficiency-related errors?

6.2.2 Participant and sample selection

The selected examination paper had been moderated and was approved as complying with the

guidelines for assessment published by the Department of Education (2007). It could therefore

be assumed a suitable paper for analysis.

From this examination paper, 15 questions were selected. These questions were selected on the

basis of their reading requirements, varying from a single sentence to a paragraph. They also

represented a range of difficulty levels with regard to their mathematical content.

In all, 15 English second language NC(V) Level 4 students' scripts were selected for analysis. 1

These students all achieved between 30 and39% (elementary achievement) in the examination,

but represented a wider range of language proficiency, with English First Additional Language

results of 31% (elementary) to 51% (adequate). The scripts selected were examined to as certain

whether the student had attempted the questions or not. Those who had left out more than 30%

of the examination were eliminated to allow for the largest possible number of analysable

errors.

6.2.3 Analysis of data

After correct responses had been eliminated, 131 errors were available for analysis. Each error

was attributed to one of the following categories:

Decoding: Text (reading)

Decoding: Graphics, tables and symbolic notation (viewing)

Encoding: Text (writing)

Encoding: Graphs, tables and symbolic notation (presenting)

Mathematical proficiency

Carelessness

Errors attributed to reading were those in which a lack of comprehension of the text contained

in the problem was evident in the way in which data was interpreted and used in the calculation.

Viewing errors were those in which a lack of comprehension of symbolic notation, tables or

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graphs had led to the error. It is possible to distinguish between reading and viewing errors

because of the nature of mathematical texts, although both are decoding errors.

Where students struggled to translate their solutions into written form, the error was attributed

to limited ability to encode information in writing. These students understood the question and

were able to execute calculations correctly but failed to accurately explain the solution in

writing, where this was required. Similarly, where students showed comprehension of the

question and an ability to perform calculations but were unable to summarise this information

accurately in symbolic form or in a table or graph, the error was attributed to a lack of ability to

present data accurately, also an encoding error.

In addition to decoding and encoding errors are those in which mathematical calculation errors

are made. Errors can be attributed to this category if the student demonstrates comprehension

of the problem by constructing the correct mathematical model, but is unable to execute the

calculation accurately. Errors may also occur due to carelessness. This was listed as the cause

in cases where students demonstrated complete understanding of the problem as well as the

ability to construct a mathematical model of the problem in context and calculate the result, but

still made a small error due to a lack of careful calculation.

7. RESULTS

The number of items available for analysis was 225, of which 131 were errors analysed

according to what the underlying cause was most likely to be. The results of the error analysis

revealed inaccurate mathematical calculations to be the dominant source of error, accounting

for 52% of the total errors. Decoding errors were also prominent: 38% of the errors were

attributed to decoding, of which 26% were reading comprehension errors and 12% were

viewing errors owing to a lack of comprehension of symbolic notations, tables or graphs.

Encoding accounted for 7% of the errors.

The following table summarises the results of the error analysis:

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Table 1: Error attributions

Mathematical

Calculations

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A graphic display of these results, with the categories of presenting and person variables

excluded, is included below:

Figure 5: Bar graph showing percentage of errors per error attribution category

Definitions of mathematical literacy all contain an element of decoding information presented

in tables, graphs, numbers and symbolic notation. According to the model used for this analysis,

where mathematical literacy is lacking, the error attribution of 'viewing' is made. Therefore, if

the aim of the research is to separate errors based on language proficiency versus errors made

due to poor mathematical literacy, viewing needs to be incorporated into the mathematics

category to create the category mathematical literacy. If this change is made, the rearranged

result becomes that depicted below:

Figure 6: Percentages of mathematical literacy, reading and writing errors

We can further simplify the picture if reading and writing is collapsed into one category:

language proficiency. This allows a comparison between the percentage of the mathematical

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literacy errors made and the percentage of errors due to limited language proficiency. This is

shown in the graph below.

Figure 7: Mathematical literacy versus language proficiency in the attribution of errors

8. DISCUSSION AND CONCLUSION

The results of the error analysis of these students' scripts clearly reveal that a large number of

errors can be attributed to lack of language proficiency. Mathematical literacy, as expected, is

the largest category of error, but having 32% of the total errors due to limited language

proficiency is cause for concern. This implies that the examination paper is not valid for this

group of students as a measure of their mathematical literacy as a sizable part of the

examination has succeeded only in revealing the students' level of English language

proficiency. This calls into question the fairness of the assessment for the student population, a

principle outlined in the assessment guidelines for the subject (DHET, 2012c) as being of

utmost importance.

The English First Additional Language results for the students in the sample ranged from

31to 51%. A score of 51% categorises the student as achieving an ' adequate result' (DHET,

2012b), but this is still low, considering the academic language demands placed on second

language students by learning and being assessed in English. It is therefore not surprising that

language proficiency has affected their results, but considering that all of the definitions of

mathematical literacy classify it as a practical, real-world ability, the processing of English text

should not determine whether a student passes or fails the subject. With the result of the final

examination comprising 75% of the final mark for NC(V) Mathematical Literacy, and in light

of the results of this study, it is entirely possible that this could be the case for many students.

8.1. Concluding thoughts

Where a lack of validity and fairness occurs in the final examinations, the consequences for

students are great because of the high-stakes nature of this component. All assessment items in

Mathematical Literacy, therefore, need to be carefully constructed with regard to the language

demands if students' abilities and subject knowledge are to be fairly assessed. Considering the

practical focus of mathematical literacy definitions, as well as the fact that the majority of the

South African population are not home language English speakers, the assessment of

mathematical literacy needs to be reconsidered. If written examinations are confounded by

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language problems for English second language students, either their high-stakes nature needs

to be changed or a practical examination must to be introduced. Creative solutions need to be

found to address this dilemma.

END NOTES

1 Permission to access the examination scripts of the 2011 Level 4 class was obtained from the Head of Department. The

identity of the students whose scripts were examined was unknown to the researcher. The only stage at which the student's

name was used was to eliminate those students whose mother tongue was not isiXhosa as this needed to be done according

to a class list and the demographic forms completed by each student at the beginning of the year. Once this was completed,

each student's script which had been selected was allocated a code for identification, and no further reference to their

names was required.

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BIOGRAPHICAL NOTE

Pamela Vale is a PhD student of the Faculty of Education at Rhodes University. Her research interest is

primarily assessment of literacy in adult education, particularly that of English second language students.

She is supervised by Ms Sarah Murray and Dr Bruce Brown. Email address: pamvale83@gmail.com

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This article presents the results of a study which investigates differences in reading performance on passages of scientific discourse containing implicit and explicit definitions. Both EFL and American students were tested, and differences in performance were examined with respect to 1) implicit-explicit information, 2) level of EFL proficiency, and 3) the EFL vs. American (native) distinction. It was generally found that both American and EFL students found implicit definitions more difficult to comprehend. However, the differential in the difficulty levels remained relatively constant for both groups. Thus, while level of English proficiency increased, differences in scores on the implicit and explicit definitions remained the same. These results suggest that difficulty in comprehending implicitly stated information may not be a problem unique to students of EFL, but rather a more general reading problem which affects native speakers as well.