Do They Know What We Are Talking About?Author(s): Lynne Hardcastle and Tony OrtonSource: Mathematics in School, Vol. 22, No. 3 (May, 1993), pp. 12-14Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215001 .

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DO

THEY

KNOW

WHAT

WE

ARE

TALKING

ABOUT

by Lynne Hardcastle, Cardinal Heenan High School and Leeds Advisory Service and Tony Orton, University of Leeds

The article by Utterburn and Nicholson (1970) revealed something of the extent of the problem of trying to use specialist vocabulary in mathematics lessons. Subsequently, a number of books on language issues in the teaching of mathematics have appeared, for example Shuard and Rothery (1984), Pimm (1987), Brissenden (1988) and Durkin and Shire (1991). The problem of whether our pupils know what we are talking about is still with us, however, and this article describes some attempts to investigate the extent of the problem within one school, with a view to increasing the awareness of the teachers as a first step towards improvement.

Stage 1 The preliminary experiment began by trying to eavesdrop on individual Level 8 pupils, in order to find out if they were using specialist terminology and if so whether it was being used correctly. In the first attempt, two pupils were placed in a problem-solving situation in which they were required to find out how much paper and string would be needed to wrap parcels of eight multilink cubes in the three different cuboidal configurations. Left alone with a tape recorder, so that no teacher could put words into their mouths, the pupils were encouraged to talk aloud as they exchanged ideas and thoughts. The pupils were successful, in that the problem was "solved", but the language experiment at first appeared to be something of a failure, because the pupils used very little specialist vocabulary. This was largely because they were quite clearly, and naturally, pointing at objects and their characteristics as they worked, so "this", "here" and "two of these", featured frequently in the recording! The second attempt was to interview individual pupils with the teacher wearing a blindfold. This simple but ingenious tactic proved to be much more effective and, in order to help the teacher to follow what was going on, pupils began to use words like "edge", "horizontal" and "face", in addition to more familiar words like "corner", "front" and "flat", none of which words had been used in the first attempt. When learning new vocabulary it is important that the learners are forced to speak them. As a result of the initial failure, a way of making pupils use the words we want them to use was devised, but it is not known yet to what extent this approach could be used between pupils and without a teacher taking part. Ways certainly do need to be found to compel pupils to use specialist vocabulary.

Stage 2 The second stage involved sixteen pupils from the whole range of ability. Two question sheets were given out and written responses to questions placed within familiar classroom contexts were collected.

275

Look at this number.

Is it odd or even?

Why?

How many digits does it have?

What is a digit?

Is it an integer?

What is an integer?

12 Mathematics in School, May 1993

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Look at this shape.

How many faces does it have?

What is a face?

How many edges does it have?

What is an edge?

How many vertices does it have?

What is a vertex (the singular of vertices)?

These tasks are shown in the boxes and responses were as follows:

Face The most common response (10 pupils) was "a face is a side (flat side, surface)". Other responses were, "it is a flat edge on a shape" (3), "it is where the edges meet" (2) and "the sides of it" (1).

Edge There were many responses here, with three pupils settling for each of "where two faces meet", "a corner" and "the rim or edge of the shape". Two pupils said "an edge is (might be) sharp". All other responses were individual and included "where the shape sides stop", "it is from corner to corner that goes straight down", "where the corners meet" and "like a corner inside out". The final response was a short essay commencing with "the edge is the straight part of the shape that is at each end and each side of the face".

Vertex It seems as if many pupils have a good idea what "face" and "edge" mean, though can't always express themselves very well, but the same could not be said of "vertex". Only four pupils admitted they had no idea, others clearly guessed, and that is perhaps potentially very dangerous. Some of the attempts were, "the hidden lines you can't see", "the edges going up or down", "a diagonal", "the corners going up in a vertical direction", "a line going up" and "opposite of horizontal".

Odd or even Most of the pupils got this answer right, but it is some of the explanations which were intriguing. One of the pupils who gave the incorrect answer said, "the first number is even so the rest is". Some who were correct referred to the numbers 7 and 5 as odd but the 2 as even, so was the number odd on a kind of majority vote, for these pupils?

Younger pupils have a great deal of difficulty in deciding which digits in a two or three digit number affect the decision as to evenness or oddness, and there still appear to be vestiges of this difficulty at age 12.

Digit Most pupils said that a digit is a number, but only five said it is "a single number", the remainder saying, in effect, "it has three digits".

Integer Only four pupils attempted to answer this, two saying "an integer is a number", and the other two saying "a number times the same number". The remaining twelve pupils made no attempt at this question.

Stage 3 The final stage of this experiment was modelled on the work of Otterburn and Nicholson, though here, being only twelve years old, the pupils were several years younger than in the earlier study. The words selected were those which had emerged from the successful rerun in Stage 1. Pupils were first invited to indicate recognition of a word, and were then asked to complete a symbol and/or diagram and/or explanation. Half of the complete year group took part, none of these 76 pupils having been involved in either of the earlier stages. In assessing performance, understanding was the only measure sought, so spelling, grammar and to some extent accuracy were ignored. Responses were classified as either "correct", or "blank" when there was either no response or nothing beyond the claim of recognition, or "confused" if the explanation was poor or muddled. The table lists the twelve words together with the percentage responses in each category.

Table of Results

Word % Correct % Blank % Confused

zero 79 8 13 square 76 5 18 rectangle 72 6 22 difference 57 17 26 area 49 18 33 digit 34 17 49 face 29 45 26 horizontal 18 29 53 volume 16 14 70 decimal point 16 29 55 vertical 13 18 68 edge 12 39 49

Discussion of Responses

Zero Some of the more unusual comments were, "zero is the first number there is", "north", "a number that has no value of its own" and "0 + 0 = 0".

Square The majority of those who were correct apparently needed to draw a diagram. Only 9 per cent of the sample accurately described the essential properties of a square, and if assessment of responses had been based on the description alone this word would have been bottom of the list. Only

Mathematics in School, May 1993 13

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one pupil described a square as "the multiplication of a number by itself"

Rectangle As with "square", most pupils relied on a diagram, although 21 per cent of the sample did describe the angle properties. Among the more unusual comments were, "like a square but longer crossways", "a three-sided shape", "when two sides are the same as well", and "when two sides facing each other are the same all the way round".

Difference Much has been written about the potential for confusion associated with this word. Nevertheless, more than half of the pupils were able to explain the mathematical meaning associated with subtraction, which was a surprise since the word was not explicitly used in lessons. Unacceptable responses included, predictably, "when one thing is different from another", "two numbers that are different like 4 and 5", "something is not the same as the rest" and "it means two different things".

Area The main reason for the low correct percentage is that many pupils did not explain the word at all, they simply performed an area calculation. The topic, an important one at this level, had clearly been interpreted only as a number crunching exercise. Confusion with "perimeter" did occur, but only with weaker pupils. A smiall number of pupils confused "area" with "volume" and used three- dimensional shapes to illustrate.

Digit The very large number of confused responses included those in which "digit" was defined as a number and/or examples were given as 1, 2, 3, 4,... that is apparently without end and excluding zero. Incorrect definitions included "digit is a shape", "where it is not a whole number", "numbers that are used on clocks", and "digit is numbers that have dots in".

Face There were many nil responses to this word. Most of the correct ones were illustrated by means of a. diagram, usually a cube, with an arrow pointing to the correct feature. Written explanations often referred to "side". No- one explained that the shape they had drawn had faces all round, including some which were hidden. Confused answers included "front of the head", "to face someone", "clockface tells us the time", "a square has four faces", and "one plus zero equals faces of a circle".

Horizontal/Vertical Many of the responses suggested that pupils had learned that loose definitions of "horizontal" and "vertical" were acceptable, as in responses involving "from left to right" and "from top to bottom of a page". It is common in teaching graphical work to refer to horizontal and vertical axes, but this can easily promote a misconception (see Orton, 1992) and, for some of the pupils involved, this appears to be what had happened. In other words, the poor success rate is at least in part the consequence of expecting more accurate definitions in the research than those which are often implied by teaching. The best answers were "like the horizon", "lying down" and "flat", for "horizontal", and "upright" and "on its base standing up", for "vertical".

Decimal point The greatest difficulty with this was the inability to explain the part that numbers "after" the decimal point played in the value of the number. Many children, especially weaker

ones, recognised the words but could only explain them in the context of money, for example "it is to do with pounds and money", and "it separates numbers like a2.50". Even more able pupils were unsure, and this is illustrated by these responses, "it separates numbers: 1.5 isn't the same as 15", "a dot which changes a number's value", "the point is to mark where it splits", "after it are the things that don't make up a number", "it comes after a whole number and before the amounts of little numbers", and "it separates two numbers where the number on the right is lower". Unusual responses included, "you put it in the date", "sometimes used to read time", "it separates two digits from each other" and "the decimal point is between whole numbers and is smaller than them".

Volume The apocryphal "it's a knob on the television set" was not given by any pupil! However, responses were not good on the whole, largely because many pupils gave examples of volume calculations (c.f. area) or drew a diagram like a jug or measuring cylinder without any explanation. Some pupils said, "it's like a jug of water", or "the amount of something", or "it is in a cup", and many others used the word again in the definition, for example "it is the volume of something".

Edge Rather more than a third of the pupils gave no response to this, which is surprising given that pupils could have fallen back on everyday examples even if they didn't know mathematical ones. Responses from other pupils basically fell into two categories, they were either correct or they described what in effect is the perimeter of a plane shape, this presumably coming from everyday usage.

Summary Overall, only thirty-nine per cent of responses were accepted as correct, and a further forty per cent were confused. These figures are somewhat alarming considering so few words were surveyed. If these results are projected across the whole spectrum of essential specialist mathemat- ical vocabulary it suggests that, when such vocabulary is included in what we say, Year 8 pupils only know what we are talking about for around forty per cent of the time. A further forty per cent of the time they might well think they know what we are talking about but may only have a hazy or incomplete notion. Of course, it must be admitted that the methods used here to measure and quantify difficulties may not be perfect, but they are certainly no worse than methods available to busy teachers, on the basis of which quick diagnoses have to be made. Whatever the difficulties of measuring the extent of problems caused by particular words, the above results suggest that particular language is likely to form an additional barrier to understanding over and above any difficulties caused by ideas or concepts. Clearly, the evidence suggests that helping pupils with the language of mathematics should be an important part of mathematics teaching.

References Brissenden, T. (1988), Talking About Mathematics, Blackwell. Durkin, K. and Shire, B. (Eds) (1991), Language in Mathematical

Education, Open University Press. Hardcastle, L. (1992), Mathematical Language Used by Children in

Problem-solving Situations, Unpublished MEd dissertation, University of Leeds.

Orton, A. (1992), Learning Mathematics, Cassell. Otterburn, M. K. and Nicholson, A. R. (1976), The Language of (CSE)

Mathematics, Mathematics in School, 5, 5. Pimm, D. (1987), Speaking Mathematically, Routledge. Shuard, H. and Rothery, A. (Eds) (1984), Children Reading Mathematics,

John Murray.

14 Mathematics in School, May 1993

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