Directions: Journal of Educational Studies Vol. 21 No. 2 Dec. 1999
The Language Factor in Mathematics Education1
Sala Bakalevu
Language is an important factor in the learning and teaching of
mathematics. While for most students a mathematics lesson is
generally a language lesson within the mathematics part, the
sequence seems more complicated for second-language learners.
For many Pacific Islanders, learning in English and Mathematical
English creates serous cognitive difficulties. Using appropriate
language(s) in context must be a consideration.
Introduction
Towards the end of last year (1998), a few weeks after the Fiji
national examinations, some letters appeared in the Letters to the
Editor column of the Fiji Times concerning the mathematics
examinations. Two in particular are relevant – firstly, one which
came from a Form 6 Fijian student, and secondly, the reply to it
from someone who had taught mathematics.
In her letter “Maths Exam” (FT 3/12/98), the student complained
that the sixth form mathematics examination was “full of words”.
She deemed the word problems unfair, and seemed to think that
“wordless problems” would have been better and would serve the
same mathematical purpose. A week later, a reply to this complaint
appeared in the same column. The second writer expressed his
disappointment at the student’s comments and argued that, as a
tool, the value of mathematics is in its use to solve the “word
problems” that come with real life situations. While he acknowledged
the language difficulty of second language learners, the writer
insisted that making meaning of mathematical based problems rather
than mechanistic ‘monkey maths’ must be the aim of mathematics
1. Paper presented at the Pacific Teacher Educators’ Conference held at
the University of the South Pacific, 18-22 January, 1999.
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Directions: Journal of Educational Studies Vol. 21 No. 2 Dec. 1999
learning. He attacked teaching which did not generate
understanding.
While I concur with the latter’s suggestions about meaningful
mathematics teaching, I do not think that he fully appreciates the
frustration and stress of learning in a second language generally,
and mathematics in particular. As a Fijian who learnt and taught
mathematics in English as a second language, I believe there is a
case to answer. In this paper I explore the value and critical role of
language in making sense of mathematics. I begin with an analysis
of the nature of the language of mathematics, and then highlight
some linguistic aspects in mathematics learning that will help us to
appreciate the extra burden borne by second language learners.
The language of mathematics
Mathematicians have for long made us believe that mathematics is
a neutral entity that is culture-free, that its concepts are the same
everywhere. This culture-free perspective has been challenged
strongly over the years, particularly by indigenous groups who have
claimed marginalisation in mathematics learning. Bishop (1988)
clarified the relationship between culture and mathematics and
expressed the view that mathematicians confuse the ‘universality
of truth’ of mathematical ideas with the cultural bias of mathematical
knowledge and expressions. He defines mathematics as a cultural
product that has a built-in context. This has important implications
for language, to be discussed later.
Mathematics is also a powerful language of communication. It is a
kind of hybrid language that is made up of ordinary English (OE)
and mathematical English (ME), and matched with the western
view of the world. Mathematical English comprises the
mathematical register and the language of symbol notation.
According to Pimm (1991), the mathematical register that comprises
the technical terms of mathematics developed largely from ordinary
English. The practice in its development has been to redefine simple,
ordinary words rather than coin new ones. This has resulted in:
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Directions: Journal of Educational Studies Vol. 21 No. 2 Dec. 1999
i.
“multi-meaning” words which have a simple meaning in
natural language, but also a precise, more complex
mathematical definition, symbol and use. E.g. set, root,
function.
ii.
“word combinations” which carry different
meanings from the sum of their everyday use.
E.g. pie chart, simple interest, square foot.
(Halliday 1978)
The symbol system is mainly separate from that of natural language
because it uses a different, more complex rule-governed writing
system. Schweiger (1994) locates it also as mostly based on a
western heritage, even though the symbols are mainly non-
alphabetic, having being formed through a complex form of encoding.
He traced some symbolic symbols to the Roman, Greek, Hebrew
and other western traditions. Others are believed to be of Asian
origin, particularly Indian. There are more types whose origins are
not as obvious.
Symbols are characteristic of mathematical texts and are also a
powerful aspect of mathematics. For many students, however,
symbolic representation contributes to a great deal of confusion in
mathematics and, as Pimm (1991:20) suggests, it is “the very
`concreteness’ of the symbols and the absence of obvious
mathematical referents [that] can lead many pupils to believe that
the symbols are the mathematical objects”.
The problem with mathematical word problems
Mathematics education begins and proceeds in language,
it advances and stumbles because of language, and its
outcomes are often assessed in language. (Durkin and Shire
1991:3)
Many mathematical tasks are presented to students as word
problems. Like the ones that the student complained about in her
letter, word problems comprise a couple of sentences giving pieces
of related information, and it is the student’s task to perform
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Directions: Journal of Educational Studies Vol. 21 No. 2 Dec. 1999
appropriate mathematical operations and supply the answer. Where
word problems are drawn from real life situations, the task in learning
is to decode the written language into mathematical language, solve
it mathematically, and then reverse the processes. Begg (1998)
considers a typical mathematics problem in four stages:
1.
Real Problem Mathematical Problem
4. 2.
Real Solution Mathematical Solution
3.
In the diagram, step (1) or ‘problem comprehension’ is about
translating each sentence of the structure (concept formation), and,
like step (4), is overtly language based. Step (2), which involves
using a learnt algorithmic procedure, has now been taken over by
computers and calculators. This leaves steps (1), (3) and (4) that
involve ‘translating a real problem into a mathematical model,
reinterpreting the mathematical solution back into the real context
and checking the reasonableness of the solution in the real world’
(Begg 1998:5) as students’ tasks. The critical point here is that
what students comprehend from the words of the question help
them decide on the mathematical strategy to adopt. Research
suggests that, very often, incorrect answers for word problems are
due to students performing “correct arithmetic computations on
incorrect representations of the problems” (Lewis & Mayer 1987:
364).
Miscomprehension of the information mainly in the form of ‘relational
sentences’ and ‘compare problems’, and the use of logical
connectives are common language problems in mathematics
problem-solving. Relational sentences include three term series
problems or transitive inferencing problems (Galligan, 1997) such
as: “Tevita is heavier than Jone, and Manasa is lighter than Jone.
Who is heaviest?” The two statements suggest a particular ordering
of the three terms, Tevita, Jone and Manasa, which is arrived at by
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Directions: Journal of Educational Studies Vol. 21 No. 2 Dec. 1999
the integration of the two individual two-term arrangements arising
from each of the two statements, into a single three term
arrangement.
‘Compare statements’ are those that define one variable in terms
of another. E.g. “There are six times as many men as women.”
“He is half as heavy as his son.”
Another type of ‘compare statement’ is represented by these two
different representations of the same problem.
i. Jone has 5 marbles. Tomu has 8 more than Jone. How many
marbles does Tomu have?
ii. Jone has 5 marbles. He has 8 marbles less than Tomu. How
many marbles does Tomu have?
Research has revealed students’ difficulties with integrating
statements into single terms, as well as sorting relational statements
of word problems. It seems that problems that have these unique
combinations are more difficult to solve than those that do not.
The difficulty for second language learners
Mathematics education is undergoing change both in form and focus.
Change in our understanding of how people learn has shifted focus
to students’ thinking processes. The importance of students making
sense and conceptualising has led to the concern for increased
student participation in learning, including their ability to discuss,
explain and clarify their ideas.
In mathematics, I place language as the most obvious difficulty
which our students encounter. The unique register and special
written form of the language of mathematics could cause many
students of mathematics, monolingual English language background
students and second-language learners alike, to experience difficulty
with the language of mathematics texts and discourses. It has been
suggested that for most students a mathematics lesson is generally
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Directions: Journal of Educational Studies Vol. 21 No. 2 Dec. 1999
a language lesson within the mathematics part. However, the
sequence seems more complicated for second-language learners.
For us, mathematics has always been taught in English. For the
majority of our students, English is a second language, and for
many others it is a third and not encountered until the child begins
school. For all of us, having to learn in a language which is alien
and often not reinforced outside the school is a serious handicap.
In class, students encounter problems of understanding the teacher
and the material, and also experience the stress of struggling to
express themselves in English. Whereas it has often been suggested
that our children rarely talk or communicate in class out of respect
for the teacher, I argue that a lack of facility with the English
language and the shame of poor articulation are greater barriers
that hold them back. Looking back on my experience as a
mathematics teacher, I remember well the difficulty of helping
students make meaning of mathematical problems, particularly at
the upper secondary level, where mathematics is more abstract.
Some concepts just never registered. My colleagues and I followed
the text closely for fear of teaching the wrong thing. Fear of
misrepresentation often restrained us from straying away from the
given statements and exercises (foreign as they were). When
students found it difficult to grasp a point, we faced the problem of
finding alternative ways of expressing it while keeping the meaning
as close as possible to the original. In the end, against our best
intentions, many resorted to drill and practice methods.
Howard (1995) has proposed that the demand on indigenous
students coming from a cultural background very different from
that of the school is heavier than that borne by students whose
culture and language is closer to that which dominates in the school.
This is because indigenous learners have to:
• adjust to the culture of the schools
• meet the challenges of a wider community culture
• develop their knowledge of the school language
• become conversant in English, and
• learn the language of mathematics including words, signs and
symbols. (Howard 1995:9)
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Verbalising algorithmic steps in solving problems is important for
clarity of thought and understanding. In the mathematics classroom,
our students have not only to verbalise in a second language, English,
but also they face the additional difficulty of mastering the special
register of the language of mathematics.
Possible way forward
I began this paper with a discussion of a Fijian student’s anguish
over word problems and the difficulty of learning in a second
language. I propose that improvement in linguistic skills as well as
the development of local contexts of learning could pave the way
to greater understanding.
Teacher education must help mathematics teachers not only to
know the content areas of mathematics but also the contexts,
including the linguistic resources involved. To be able to understand
the mathematical text, the reader has to have a context for it and
thus be able to consider the technical terms correctly and see the
relationships. This includes some understanding of English and
accepted patterns of reasoning in that culture. In schools,
mathematics teachers have to be language teachers and teach the
language of mathematics as an integral part of mathematics. They
could ask their language counterparts to spend time on the linguistic
skills involved constructing ‘relational sentences’, ‘compare
statements’ and logical connectives that are important in
mathematics. It is also important to understand that our students’
miscomprehension has much to do with the way they verbalise in
the first language whose structures are different.
Local contexts – ethnomathematics
All use of language has a context (Halliday 1989). Halliday gives a
useful discussion of the importance of context for understanding.
He perceives the context as ‘a kind of bridge between the text and
the situation in which the text occurs that enables the reader to
understand the text and interpret its meaning . . . If the reader or
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listener is not able to bring to the passage appropriate assumptions
derived from the context of situation, then there is no information
and no communication’. For our students, this leads to frustration
and possible miscomprehension.
I believe that greater emphasis needs to be put on the context and
application of problems. We need to ask ourselves the questions:
How much of the mathematics we teach has application to students’
reality? Whose reality is contained in mathematics texts? For the
Fijian student in question as with others like her, learning in context
is important for concept formation and for the context be relevant
and meaningful. The identification of mathematical ideas in different
cultures which come under the study called ethnomathematics
suggests that there are ‘other’ mathematical systems determined
by different languages and culture modes (Barton 1997; Gerdes
1995; D’Ambrosio 1995). In my research project (Bakalevu, 1998),
an analysis of Fijian society from a mathematical perspective
highlights Fjiian mathematical ideas and ways of mathematising
that are different from western ideas and ways.
Fijian ways of mathematising
Fijian students come to school from a society that has its own
ways and means of mathematising. While there is no category
‘mathematics’ in Fijian culture, there are mathematical ideas
embedded in aspects of the socialisation processes and activities
of the people. In my project, I focussed on the systems of counting,
measuring, economics, navigation, kinship and design, all of which
depict people’s ways of solving problems. They are, in the main,
different from the systems in the school and in mathematics.
Some important findings of the investigation suggest that:
• the Fijian is not prone to quantifying to the same extent as
required in mathematics,
• Fijian methods of numeration suggest that Fijian students may
have a limited mental map of the world of numbers and
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operations, as well as a less well-developed number sense than
most others,
• Fijian students may have a limited conceptualisation of formal
measurement units,
• the Fijian system of counting certain items in tens could be useful
context for developing place value, decimals and percentages
etc.,
• there are numerous Fijian descriptors that are more commonly
used than numerals,
• Fijian contexts of dinau and kerekere (redistribution and
reciprocity) make sense to the Fijian (even if the rest of the world
thinks they are not economically sound) and could be used to
establish the ideas of negative numbers (normally difficult to
visualise) , profit and loss, percentages, multiplicity, factors, etc.,
• there is a lack of a full mathematical lexicon in the Fijian register,
• the concepts of estimation and approximation are part of everyday
transactions and pronounced extensively in a descriptive language,
• The process of mat weaving has ‘hidden’ mathematical skills
while the more obvious concepts of periodicity, symmetry and
tessellations are easily demonstrated.
• Tapa printing presents an excellent context for most geometric
transformations.
These findings are important in that they are the ideas that Fijian
children live with, understand and value. The contexts also have the
relevant linguistic resources. According to constructivist ideals, they
are the ideas that students will hold onto and retain longest. The
challenge for mathematics teachers and educators is to plan activities
in contexts which enable students to make connections between
the mathematical ideas in their culture, and those in the mathematics
classroom and the broader culture. Systems need to be in place to
assist mathematics teachers to do that effectively. In support of this
programme, Barton (1997) makes the point that indigenous learners
who know two mathematical systems, speak two languages and
have been introduced to two ways of knowing and seeing are better
off than those with a monofocus. It makes perfect sense to encourage
and support them to propogate both systems.
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References
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