TOPOLOGY AND LANGUAGE

1. USING SIMPLE WORDS FOR SOME TOPOLOGICAL NOTIONS

1. Introduction

This article is about topology and the use of words which are found in both the English language and the Krio language of Sierra Leone. The enquiry was conducted at Njala University College during the 1991/92 and 1992/93 sessions. Those who participated were children in the final year of the primary school and first year of secondary school, undergraduate students and adults. Reference to those whose influence has contributed to this work was briefly made in proposing a method of enquiry in mathematics education (1).

Perhaps the most decisive influence is due to Bishop's article titled Visualising and Mathematics in a Pretechnological Culture and Wheeler's contribution, Drawings and Representations, in (3). These two highlight the need for a cultural factor such as language to be given the attention it deserved. Wheeler, in particular, contends that `neither the Gestaltists, nor the Geneva school give much attention to the power of cultural factors, and particularly language, in describing what is seen and how it is represented'.

What is reported in this article is confined to the use of ordinary everyday words in Krio for real-life situations bearing objects which can be described as topological. Following the authors of the A.T.M., the word `folk' will be used to describe such words (4). In the Sierra-Leone context, the notion of an `ordinary everyday language or folk word' needs to be explained. Like most African countries, the country is endowed with languages, at least ten. Among these is the Krio language, the national lingua franca. A striking feature of Krio is that at least 80 percent of its vocabulary is derived from a single language, English (5). Children, especially those in urban areas, are caught up in a situation in which outside the classroom they speak Krio, while inside the classroom they speak English because it is the medium of instruction at all levels of the educational system.

When linguistic difficulties arise in the classroom teachers sometimes resort to the use of the Krio language to bring notions `down-to-earth'. Concerned teachers of English, on the other hand, claim that Krio interferes with the learning of English because of the similarities and differences which exist between the two languages in phonology, syntax, and meanings of words. The first two are not matters of concern in this article.

The primary concern is about the use of words and factors underlying the choice of words. The notions for which folk words serve as names or labels are often considered to be elementary. The word `elementary' is underlined to emphasize the point and commonly-held belief that they are obvious and intuitive. Consequent upon this belief, they have been introduced into elementary mathematics. More often than not, these notions turn out to be deeper than one envisages and deeper ideas, according to Hardy, are usually hard to grasp (6). We shall discuss this later in the paper when we consider the notion of a curve. We now turn to a discussion of how information was obtained.

2. Method of Observation

We proceeded by obtaining a large number of examples. The information obtained ranged from the analysis of children's written work (and drawings) to interviews with adults. However, the main source of information was task-based interviews as described by Davis and his colleagues (7). The protocol involved an interviewer who would present a task or pose a question to which the child would respond under the agreement that she/he would talk aloud as much as possible, explaining what she/he was doing and thinking about. At the end of the interview, the child might be asked to explain fully a few points that had been obscure.

Sometimes an episode would be reviewed, adding whatever components seemed to deserve mention. Some of the episodes were tape-recorded and notes were taken by the interviewer and an additional observer when one was present.

The children were encouraged to think in their natural fashion. This, of course, meant that we had to do without a tape-recorder in some cases, or we had to conduct interviews in Krio so that the children could express themselves freely and fluently.

The excerpts given in this article and others in this series are English versions of interviews conducted in Krio. We shall consider first the use of simple words to name notions related to objects such as points, line segments, neighbourhood and curves.

3. Naming Objects

Several episodes were recorded for the task requiring children to name objects. In each case the children were asked to produce first a drawing of an inelastic string stretched between its ends on a table and then to name the drawing. The string had ink marks along its length and on the two ends, and these marks were also to be shown/represented in the drawing. The names frequently used for the ink marks were `scratches', `points', `full-stop' and `dots'. All of these labels reflected to a great extent the variety and differences in representation of ink marks on the string. The string itself was represented by a line segment with distinct end points but they simply called it a `line'.

Cutting the string with a pair of scissors gave rise to bits of string with spaces between them, sketches of which were still labelled `lines'. In one case the space between the two line segments was referred to as a `hole'.

The question as to what else in real-life looked like a line was taken up with the children. The examples we obtained from them included `lines for drying clothes', `electric cables from one pole to another' and `lines between goal posts'. The occurrence of the word `line' in each of the examples was suggestive of its use as a folk word. It is of interest too to note that the word evoked in the minds of children not only physical and perceptible objects bearing features of straightness, width and length but also actions and other objects related to these perceptible objects.

Another usage of the word `line' to suggest actions such as support or agreement in an argument, especially among teenagers, is illustrated by the following Krio sentences.

`You de in layn with me' meaning, `you support me or agree with me'.

`Us layn you de?' meaning, `which line do you belong to?'

Line segments drawn on a paper tend to set a different context for children's discourse about lines. The task that highlighted this was a diagram of two intersecting line segments shown below with dots for easy identification of the common point and the end points of the segments.

It was presented to the children and they were asked to talk about the line segments and the points on them. Talking about a line segment in relation to another or about points on them simply reduced to finding names for points. For example, the point common to both segments was named `middle point' and the end points were referred to as `other points' in order to distinguish them from the `middle points'. What is of interest here is the use of the word `middle' for the common point. On a line segment the word `middle' can be used by children to describe more points than one, and even notions such as `betweenness', `neighbourhood', `interior' and `boundary'.

Middle points are not simple between other points, they separate or divide the line segments into two equal parts - one part consisting of points on the left side and the other part of points to the right. Other points are referred to only in as much as they are separated from the others by the `middle points'. No particular expressions such as `end points' or `boundary points' are used for them as in the case of the middle points. Points to the left can be joined to each other, to middle points, and to points on the right; and then all the points can be joined in a single line.

Producing sketches of real-life objects associated with the word `line' and talking about these sketches seem to suggest that notions of intervals and line segments are within the experiences of children even though they cannot associate them with words such as `intervals' or `neighbourhood', and express them explicitly and precisely using the language of a topologist. Children use the word `middle' instead of the word `neighbourhood' for points in some particular position in relation to others in a line segment. These points and others on the line segment are joined or connected by the same line segment. But what will children have in mind for a notion like a ray? Is there a simple word they can possibly use or choose to use for this notion? Perhaps a Krio word derived from English? Here is an excerpt from one interview to support the observation that they do not use a single word for a ray and what the word `ray' triggers are objects of finite length.

Four children - Mustapha, Zainab, Hannah and Josephine were involved in a discussion which started with the following question.

Question: Can you think of anything that looks like a line that starts at a point and goes on without end?

Zainab: A road from a town to a far away place.

Hannah: It is like a light from a lamp.

Josephine: It is like a long road.

Question: Can you think of anything that will illustrate the idea of a line which extends in opposite directions without end?

Hannah: The highway of Taiama junction.

Josephine: I cannot imagine anything that looks like that.

Hannah: The Taia river.

Mustapha: The electric cable in the streets.

Question: Suppose the line continues in both directions without end, what name will you give to it?

Mustapha: It is the longest line.

Zainab, Hannah and Josephine: It is a line that has no end, no beginning.

The first question was exploratory and it required the children to think of real-life situations with features of a line having an initial point and extending indefinitely. Hannah's response was impressive; it was precisely what was expected. Nothing else can best illustrate the notion of a ray than the light from a lamp. That Hannah was capable of thinking of such an example was remarkable. The second question was intended to find out whether they knew of notations, representations or conventions in use for drawing or representing a line extending in both directions without end (or indefinitely). As the excerpt shows, all the children refer to objects with finite length and width. Representations of which we have seen they are capable of making. Even Hannah who offered `light from a lamp' in response to the first question suggests in the case of the second question `the Taia river' an object which starts at a point and ends somewhere.

All these, as examples of rays, are indications that the words `ray' and `line' are in currency but they evoke in the minds of children entities possessing finite length. Objects they were capable of experiencing and were within their imagination. Asking for something other than these was to ask for something outside/beyond their experience. Josephine clearly sums up the situation when she says that she cannot imagine anything that looks like a line extending in both directions without end.

A line without end is inconceivable and therefore children resort to the use of words and examples suggestive of part rather than the whole object called a ray. The meaning attached to the word `line' has to do with both the words they used and the examples they gave -roads, rivers, highways, electric cables. Those who kept to the use of the word `line' itself will only stress its length or direction in their description of it. But this is far from using a single word to name it. It is clear, from these examples, that they are not exposed to notations and conventions illustrating these notions. Children's representations of a ray or line are simply sketches of line segments, roads and electric cables.

The diagram of the two intersecting lines can highlight new dimensions depending on the kind of discourse one encourages children to be involved in. When, for instance, it is used as a starting point for a discussion of two roads crossing in a village with houses at the point of crossing and at the ends of the roads, the points are no longer thought of in terms of `middle points' and `other points' but as representing objects like houses and people living in these houses. These people are referred to as neighbours, suggesting closeness, friendliness, etc.

Relating the points and lines to marks on a string, the points are likely to be thought of in terms of dots and the closeness or nearness between points disappears; those in the middle can be clearly differentiated now from those not in the middle. This observation brings one to the fundamental issue of thinking about, feeling and sensing the existence of objects and the way we talk about or represent them.

Questions which force us to think and talk about ourselves and objects seem to have a hidden corollary. And this has to do with the question: `What is in some ways different from the other?'

This process of differentiation which this corollary is suggesting, it seems to me, occurs in every language. It applies even more obviously in children's discourse between themselves or between an individual and a newly encountered group. The issue, if at all it exists, has to do not only with the question or its corollary but also with the answers offered, which invariably employ a figure of speech termed synecdoche, a metaphor which substitutes a part for a whole. Deployment of the synecdoche was found in the expressions used by children to describe/represent objects associated with the notion of a line, ray or line segment.

(The reader may want to compare the observations which follow with those of Mason and Pimm on generic names, generic examples (8).) The word used for these notions was `line'. You ask them to tell which objects in real-life have properties or features which characterise mathematical objects like rays and lines, they would give none that would be satisfactory to a topologist.

For line segments we have given the examples - roads, cables and rivers. These objects, although they belong to the class of objects with width or thickness, these attributes are ignored, length is the only attribute of significance to them, others such as width, the meandering of a river or winding of a road at points about its length were ignored or not stressed. These objects were all classified as examples of `lines'. The process of differentiation does not seem to be quite clear from either the examples and drawings taken so far for rays and lines. The word `line' is used to name rays and line segments. In the next section we give examples to illustrate the kind of differentiation children are capable of making in the case of curves in a plane.

4. Describing Curves in a Plane

Sketches of two strings - one with loose ends and the other with ends tied together to form a loop - were the subject of discussion for groups of children. The sketch of the string with loose ends was named a `curve', `crooked line' or `bent line', etc. and that of the string with ends tied together was called a `circle'. The terms `crooked line' or `bent line' suggest some form of differentiation between a line (i.e. straight line) and a curve whereas the use of the word `circle', when the drawings were all far from being circular, suggests the use of a word for a ? to name different objects belonging to ? class.

The task of putting points and stating the number of points that could be put on a curve or a `crooked line' and the circle occupied much of the discussion with children. The children could not settle down for a definite number of points. Various suggestions were made ranging from "many points" to `as many points as one would like to put on either the circle or the curve' which they referred to as "bent or crooked line". The children, in the case of the "crooked line", observed that it had a starting point. The "circle", on the other hand, had neither an end point nor a starting point. In the following excerpt from one group discussion, Mohammed makes some observations in an attempt to differentiate a line or curve from a circle.

MS: That is a good observation, what else can you say about them?

Mohammed: There is also an end point.

MS: Yes, since there is an initial point, I suspect that there is a terminal point. But what else?

Mohammed: There are many points on the line, some are in the middle and two outside the middle.

MS: Is the curve similar to a straight line?

Mohammed: They are not similar because one is crooked. The crooked line is short and the straight line is long.

MS: What can you say about these Mohammed? Are they similar?

Mohammed: Yes, they are similar because you can put points on both the curve and the straight line. In both cases the end points are joined by the same line.

MS: That is very good. Now is there any similarity between this (pointing to the simple closed curve) and the line?

Mohammed: Yes, I can put points on both the circle and the line. But the circle has no end points, it is just like a hole. It has points that are joined by one just like the line.

The expression `simple closed curve' was not used as it was not familiar to the children. We wanted to use a vocabulary that was no stranger to them. As the word `equivalent' is not commonly used in Krio we decided to use `similar'. The words forming the expression `simple closed curve' used by topologists may be known separately to the children because they occur in the Krio language but not collectively. The Krio words for notions such as those of a circle and curve take on the same meanings in the Krio language just as in English. Outside school, children use the word `curve' and it may refer to objects considered to be round and circular or to even a relationship which is best described by the following Krio sentence.

`He is not in my circle', meaning in English `He is not one of my friends' (5). Despite the fact that the word is in currency in Krio, the sketches of the children for the string with ends tied together fell far short of what a westerner would call a circle or associate with the word `circle'. Yet children called their drawings circles. Asking them to tell the number of regions/parts a circle would divide the surface of a paper on which it was drawn surprisingly led to different answers.

Most of the responses were that the circle would divide the surface of the paper into two parts, the region inside the circle and that outside the circle. Those who claimed there was a third part included the circumference of the circle in their account. One response which generated both surprise and curiosity was one offered by one child who said the circle would divide the surface of the paper into the following parts:

above the circle

below the circle

inside the circle

outside the circle

left of the circle

right of the circle

The difficulty of the child could not have been a linguistic one as the interview was conducted in Krio. It seems to me that it has to do with part-whole relationship.

This response led to a further probe into what the children understood by the terms `interior', `exterior', and `boundary' of a circle. Although these exist in Krio, children hardly use them. They would use the words `inside' and `outside'. For the word `boundary' they would either use `middle' or `dividing line'. In the next section, examples in which these terms are associated with a circle drawn on a rubber sheet and on paper will be discussed.

5. Rubber Sheet Geometry

A piece of rubber sheet is a very simple device for illustrating intuitive notions of topology. Topological transformations are thought of in terms of `premathematical' metaphors such as stretching, twisting and bending without tearing the rubber sheet. Talking about topological terms i.e. `interior', and `exterior', and `boundary' using these metaphors take on a much more meaningful dimension than simply talking about these terms without reference to objects and their deformations. Children would have the experience of seeing the objects being transformed and they would observe for themselves that positions of points in the interior and exterior of the circle and even those on the boundary (i.e. the circumference) remained unchanged. The transformed circle would, however, be called something else - `egg shape', `oblong shape', `stretched circle', etc., but not simply a circle. Mohammed, for example, remarked that points on the circle moved to the left and right but points inside it remained inside just as those outside it, but they were moving around the `stretched circle'. Similar remarks were made when the rubber sheet was bent or twisted. The difference was that not every part of the transformed circle was visible to the children. Consequently, what was seen was no longer called a circle. This observation is likewise true for other transformations of the circle. Here is what Kalilu has to offer as a name for the transformed circle when asked the following question:

MS: What name will you give to it?

Kalilu: It is an egg circle.

MS: Compare it with the curve. What do they have in common and what are the differences?

Kalilu: The ends of this (pointing to the curve) are not joined, that one (egg circle) is joined; it has no ends.

MS: That is good Kalilu. Now how far can points inside the circle move about?

Kalilu: They can only move in this space. (He points to the interior of the circle.)

MS: Why do you think that the points cannot move to the exterior?

Kalilu: Because they will not pass this line, it is the dividing line.

MS: Where can a point outside move?

Kalilu: It can move above, below, right and left of the circle.

MS: Really, but what about a point on the line you called the dividing line, where can it move?

Kalilu: A point there can only move on the line itself.

Kalilu's response to the question about points exterior to the circle should strike a chord about an earlier response of a child for whom the circle divided the sheet of paper into five parts. The only constraint to the movement of the exterior points is the circle. It seems from this discussion that Kalilu prefers to use the expression `dividing line' for the circumference (i.e. the boundary between points inside and outside the circle) to emphasize the fact that it separates the two regions.

Intuitive though these transformations may appear when conceived as premathematical metaphors, the actions which children take to make sense of them and talk about their observations can be very complex.

These actions involve stressing certain features to the exclusion of others and the expressions used to describe their observations reflect this complexity.

A figure of speech already mentioned (synecdoche) is also employed by the children; for example, using `circle' for the `circumference' of a circle, `middle' or `inside' for its `centre'. Situations like these were similarly encountered in the case of line segments and curves. The word `line' was used for `line segment' or `intervals' or `rays'. The expression `straight line' or `crooked line' were only used to stress perceived differences between objects.

The word `curve' was also used for line segments that were not straight or considered to be bent.

More comprehensive terms were used for a less comprehensive, or vice versa, as a whole for a part or a part of a whole, class or subclass. Another figure of speech - metonymy -which the children employed was one in which the name of an attribute or adjunct was substituted for that of the thing meant, i.e. `middle' for `betweenness', or `indeterminacy' for `boundary'. Later on we will encounter examples which will show that `middle' is used for even the interior and centre of a circle.

6. The `Middle' of a Circle

An episode that illustrates the preceding comments in many ways involved four Krio speaking school children and five Krio speaking (native) adults with very good proficiency in both Krio and English. The diagram below was presented to them with points labelled with letters. The following questions were posed first in Krio. The English translation given below each

question was worked on in collaboration with the five Krio speaking adults after they had all answered the questions.

(1) KRIO: Us point den de middul the sakul?

ENGLISH: Which of the points are in the middle of the circle?

(2) KRIO: Us point den de unda the sakul?

ENGLISH: Which of the points are under/below the circle?

(3) KRIO: Us point den de pan tap the sakul?

ENGLISH: Which of the points are top/below the circle?

(4) KRIO: Us points den de innsay the sakul?

ENGLISH: Which of the points are inside the circle?

(5) KRIO: Us points den de outsay the circle?

ENGLISH: Which of the points are outside the circle?

For the Krio speaking children, aged 11 to 17 years, the points in the middle of the circle were identified as A, B and C. While the points above (on top or on) the circle were G and F. The points under (or below) the circle were J and E. The points inside were A, B and C; and then points outside were H, D, J, E and I.

The five adults gave answers different from those of the children for some of the questions. For the first question they were very much cautious unlike the children. We observed that they were searching diligently for a particular point, i.e. the centre of the circle. As there was none, the points A, B and C were ruled out. So, there was no point in the middle of the circle. One of them still believing that there must be such a point continued the search and identified D as a point in the middle of the circle, meaning that `D was half-way between the highest and lowest point of the circle'.

The adults also identified the points G and F as being above (on or on top) of the circle; J and E as being below (or under), and A, B and C as being inside the circle. For the points outside all of them agree that G, I, J and H were outside but one or two of the adults also thought that F, D and E were outside. What was clear from this episode was that while the adults were mindful of the overlaps or gaps in Krio the children were not. A single Krio word `middle' as it relates to points can mean any of the following: circumference, exterior and centre. Also of significance is the issue of overlap as pointed by Bishop in (2). For the Krio-speaking adults, as well as children, the middle of a circle is the same thing as its centre. There is confusion for some as to which points are outside the circle, including those on the circumference of the circle.

7. Re-interpreting the Notion of a Curve

We never really asked the question: `What is a curve?' We worked with objects, their names and representation. A circle is an example of what the topologist will refer to as a jordan (or simple closed) curve. It does not only add to examples we have encountered but it also confirms a statement made by Klien years ago, namely: `that everybody believes that he knows what a curve is until he has had enough experience with mathematics to see the countless possible exceptions that can lead to confusion' (9). The language to express the classical view of a curve was that of a `path of a continuously moving point' (10). This view was quite appropriate until the two related notions `thinness' and `thickness' - presented a problem credit for the resolution of which has been give to Peano and Moore. Their argument is that, if an interval and its image are considered, the image of an interval may well be a square and its interior, a cube or sphere with its interior. With any of these they have demonstrated that the path of a continuously moving point need not be a thin curve-like set. It may well be a square or sphere.

Established independently by Mazurkiewicz and Hahn is the fact that any set which can be characterised as a locally connected continuum in a Euclidean space of any dimension can be represented as the continuous image of the unit closed interval. This characterisation or representation of a curve gives yet another interpretation of the notion of a curve which essentially rests on other topological notions.

Some Remarks

What is the point about these examples? They buttress the remarks which now follow.

First, in mathematics sometimes the attempt to describe what seems to be a simple notion leads to a complicated and artificially looking definition i.e. the notion being defined/ described is far more complex and profound than one first envisages (11). The name for the object remains simple and the notion assumes complexity with time.

Second, in the logical development of any branch of mathematics, each definition of a notion or relation involves another notion or relation. Therefore the only way to avoid a viscous circle is to allow certain primitive notions and relations to remain undefined (12).

In my work with students, a primitive notion-like point which is taken as obvious and undefined has been the source of difficulties for many of the students. On meeting, for the first time, the definition of an accumulation point they encounter difficulties similar to those observed by Davis and his colleagues (7), and this difficulty arises primarily, when they try to think about points using pre-mathematical metaphors i.e. dots and stones, the metaphor cannot be made to match the abstract situation. So, incorrect cognitive apparatus is used again and again. Students try to think about `smaller' or `larger' points like the children in this enquiry or try to compress the points in an interval more tightly. More examples will be given later on words and attributes which children associate with the notion of a point.

Acknowledgements

I am deeply indebted to my ex-students for their suggestions and assistance in collecting the examples used in this and other articles. I am grateful in particular to Mackotie Sisay for helping with the interviews and the preparation of material used in this enquiry.

References

(1) Sesay, M (1992) Unfolding as a method on enquiry in mathematics education, Chreods 4.

(2) Bishop, A J (1979) Visualising Mathematics in a Pretechnological Culture, Educational Studies in Mathematics, 10, pp 135-146.

(3) Wheeler, D (1970) Drawing and representation. In A.T.M. Mathematical Reflections, Cambridge University Press.

(4) ATM (1967) Notes on Mathematics for Elementary Schools, Cambridge University Press, p 317.

(5) File, C N & (1980) A Krio-English Dictionary, Oxford University Press.

Jones, E D

(6) Hardy, H G (1967) A Mathematician's Apology, Cambridge University Press.

(7) Davis, R B, (1984) The Roles of `Understanding' in the Learning of

Young S & Mathematics. Part II of a Final Report of

McLoughlin P National Science Foundation Grant. University of Illinois.

(8) Mason, J & (1984) Generic examples: seeing the general in the particular.

Pimm, D Educational Studies in Maths, 15, No 3.

(9) Klein, F (1939) Elementary Mathematics from an Advance Standpoint Geometry, Dover Publications, pp 173-4.

(10) Whyburn, G T (1968) What is a curve? P J Hilton (ed). Studies in Modern Topology, Vol 5. The Mathematical Association of America, pp 24-25.

(11) Cronin-Scanlon, J (1967) Advance Calculus, D C Health and Company, Boston.

(12) Coxeter, H S M (1969) Introduction to Geometry, John Wiley and Sons Inc.