RSTAQ: Tips

Developing mathematical language

The acquisition of formal mathematics language has special significance because of the differences between the classroom and home situations.

Opportunities to explore, experiment and confirm the use of formal mathematical language within classroom settings are fewer than those associated with the development of 'natural' language in home and community settings.

Much of the formal mathematics language is introduced and used only within the confines of the classroom or specific situations. This fact needs to be remembered and taken into account when mathematics programs are being developed. The development of mathematics language should be included as part of program development.

The following framework is suggested to assist in the planning of appropriate activities so that students may develop an understanding of the formal language and use it with meaning. Experience has shown that formal mathematical language is often used in a rote fashion without meaning. There are two considerations to bear in mind when planning language activities. They are:

Formal mathematics language should be linked and attached to concepts that have already been developed by the students.

Only one new element should be introduced at a time.
An 'element' in this context refers to vocabulary, sentence type, symbols, pictorial representations, diagrams or tables.

1. Student language

Mathematical activities in a variety of contexts relevant to the student's situations are planned and implemented. During this early phase of concept development the students' language is accepted and used within reason.

2. Modelling

When the concept has been developed to an appropriate level, for example, subtraction, a language element may be introduced and linked to the student's existing language through modelling:
Student: Ten, take away eight is two.
Teacher: That's correct. Ten, subtract eight is two.
Sufficient time should be allowed for the student to hear (or in the case of the symbol '-', see) and become familiar with the new element before others, for example, 'minus', are introduced.

3. (a) Self-generation

As students are exposed to the language they become familiar with it. They begin to experiment and have their use of it confirmed. They assimilate the language and begin to use it with understanding in appropriate settings.

3. (b) Focussed activity

If, after a reasonable time span, some students are not self-generating the new element, then more focussed activities through DIRECTED MODELLING may be undertaken. Directed modelling has two phases:

Phase one:
The student's response is accepted. The new element is modelled by the teacher but with a preface. For example: 'That's correct. In mathematics we say "subtract" or write it this way using the minus sign - '.
Time is allowed for assimilation. If, after a period, the new element is not being self-generated, then the next phase may be adopted.

Phase two:
The student's response is accepted and extension is attempted. For example:
Teacher: That's correct. Can you say (write) it another way? Or
That's correct. Can you remember how we say (write) it in mathematics?
If no response, the directed modelling is continued and additional time is allowed before phase two of step three is reintroduced.

4. Extension

Once the language element has been assimilated, new ones can be introduced, for example, 'minus', but this element should also be linked to other known terms, 'subtract' and 'take away'.
Informal language, acceptable in informal situations, will fade out over time and formal language will become the norm in formal situations.

A systematic approach to the introduction of formal mathematics language is necessary for some students to develop understanding and to use it appropriately in relevant contexts.

The framework here is only one way of helping those students. If adopted it should be used flexibly and modified as necessary to meet the needs of individual students.

Experience and research (Cohen and Stoven, 1981; Toohey, 1984) have indicated that many students are unable to solve mathematical word problems because of the use of inappropriate reading strategies rather than difficulties with the mathematics involved in the programs.

Some students need focussed learning activities in order to develop appropriate reading strategies for the solving of mathematics word problems. The word problems need to be read a number of times. Each reading has a different purpose and therefore requires different strategies.

First reading:
Read the total passage to identify the MAIN IDEA, for example, identify the PROBLEM.

Second reading:
Read the passage to examine the information and to organise it in some form, such as a listing, what is KNOWN and what is to FIND or a diagrammatic representation. Parts of the passage may need to be reread and reflected upon. A system of marking or highlighting may be used.

Third reading:
Reread the passage to check that all relevant information has been collected and sorted and that irrelevant information has been discarded. Between the third and fourth reading the problem is solved. Some rereading of sections may be necessary as part of this process.

Fourth reading:
Reread the problem to consider the outcomes, for example, to match the solution to the problems, confirm decisions and check reasonableness of the outcome.

Fifth reading:
Reread the passage to write the solution in sentence (paragraph) form in terms of the original problem. Multi readings may be necessary at each of the five stages.

M. A. Toohey