ADDITION AND SUBTRACTION OF NEGATIVE NUMBERS - Expanding the metaphor “Arithmetic as Motion Along a Path”

ADDITION AND SUBTRACTION OF NEGATIVE NUMBERS USING EXTENSIONS OF THE METAPHOR “ARITHMETIC AS MOTION ALONG A PATH”

Negative numbers make up a domain of mathematics which is often considered difficult to teach and difficult to understand. According to some researchers abstract mathematical concepts are understood through conceptual metaphors. Using the framework of conceptual metaphors developed by Lakoff and Núñes (1997, 2000) a theoretical analysis is made of one of the characteristic metaphors for this domain in order to see what entailments an extension of the metaphor has. The analysis elicits three shortcomings of such an extension that might influence students’ comprehension: loss of internal consistency, loss of external coherence and enforcement of properties or structure that are not part of the original source domain.

“They [negative numbers] lie precisely between the obviously meaningful and the physically meaningless. Thus we talk about negative temperatures, but not about negative width” (Martínez, 2006, p. 6)

Recently some researchers have emphasized the important role of metaphors in learning and teaching mathematics (Frant et al., 2005; Lakoff & Núñes, 2000; Sfard, 1994; Williams & Wake, 2007). This was also the topic of working group at the Fourth Congress of the European Society for Research in Mathematics Education 2005. One of the questions for further research posed in this working group was: What are the characteristic metaphors, in use or possible, for different domains of mathematics? (Parzysz et al., 2005). The aim of this paper is to make a theoretical analysis of one of the grounding metaphors identified by Lakoff and Núñes (2000) and some extensions imposed on it by teachers when dealing with negative numbers. The chosen metaphor is “Arithmetic as Motion Along a Path” as it is used when the number line or thermometer is the source domain of the metaphor. 

“A metaphor is what ties a given idea to concepts with which a person is already familiar.” (Sfard, 1994)

In the theory of conceptual metaphors a metaphor is seen as a mapping from a source domain to a target domain (Kövecses, 2002; Lakoff & Johnson, 1980). If a representation or model is to function as a source domain of a conceptual metaphor, the source domain must be well known. Properties of the target domain

are then understood in terms of properties of the source domain. Most of the ‘metaphors we live by’ (cf. Lakoff & Johnson, 1980) are based on experiences that are products of human nature; physical experiences of our body, of the surrounding world or of interactions with other people. The target domain of a metaphor is never identical to the source domain. Metaphors make sense of our experiences by providing coherent structure between the two domains, highlighting some things and hiding others. The theory of conceptual metaphors implies that different metaphors are used to structure different aspects of a concept. A conceptual

metaphor is a mapping from entities in one conceptual domain to corresponding entities in another conceptual domain. Concerning mathematics Lakoff and Nunes (2000) claim that understanding basic arithmetic requires conceptual metaphors with nonnumeric source domains and that “abstraction of higher mathematics is a consequence of the systematic layering of metaphor upon metaphor” (p. 47). Basic arithmetic, they argue, is understood through four grounding metaphors:

– Collection of objects, combining, taking away (numbers as collections of objects)

– Construction of objects, combining, decomposing (numbers as constructed objects)

– Measuring lengths, comparing (numbers as length of segments)

– Motion along a path (numbers as points on a line)

These metaphors are sufficient for us to understand arithmetic with natural numbers, but in order to extend the field of numbers to include zero, fractions and negative numbers the metaphors also need to be extended, or “stretched “ (Lakoff & Núñes, 2000, p. 89). The authors go on describing a way of stretching the metaphors and blending several metaphors so that their entailments together create closure for arithmetic in the enlarged field of numbers. The analysis section of this paper will focus on the last of the grounding metaphors; “Arithmetic as Motion Along a Path”, and see what happens to it when it is extended to meet the requirements of negative numbers.

Using the theory of conceptual metaphors (Lakoff & Núñes, 1997, 2000) one could say that understanding the number line as a conceptual metaphor for arithmetic is by treating the number line as a source domain of the metaphor ‘Arithmetic as Motion Along a Path’. To make use of this metaphor it is necessary that the students are well acquainted with the source domain and the mappings between the source and the target domains. “A metaphor can serve as a vehicle for understanding a concept only by virtue of its experiential basis.” (Lakoff & Johnson, 1980, p. 18). A metaphor can be activated by a representation of some kind. Representing numbers by a drawn number line or an elevator going up and down are different ways of representing numbers using the “Motion Along a Path” metaphor. In some ways the number line and the elevator are different source domains, but in some ways they are similar since both refer to our experiences of motion along a path.

Mathematics teachers often use visual representations, such as manipulatives or drawings, here referred to as representations or models as source domains of metaphors. Thus, the number line is not simply a representation of numbers, but rather a representation of numbers through the metaphor “Motion Along a Path”. The model will highlight properties of numbers that are similar to properties of moving along a path, whereas other properties will be out of focus. When a representation is visualised by a person without being visual per se, it can be referred to as a mental model.

There are many pedagogical models and representations in use to illustrate negative numbers, the most common ones in Sweden being the number line (often as a thermometer) or the use of money (gaining and borrowing). These models function as source domains of different grounding metaphors. Researchers argue

about whether the use of such models help student understanding or not. According to Linchevski & Williams (1999) some researchers argue against using models for negative numbers, whereas they themselves claim that subtraction with negative numbers can be understood through the use of models. Whether the use of several different models simultaneously will confuse or help the student is also a matter of consideration (Ball, 1993; Kilborn, 1979). Gallardo (1995) suggests teaching negative numbers using discrete models, where whole numbers represent objects of an opposing nature, rather than using the number line. Frant et al. (2005) show that teachers sometimes use metaphors without being aware of it, and sometimes are aware of using them, but not of the possible difficulties involved. The importance of being aware of the limitations of a model was shown by Kilhamn (in press). 

The following section will show how the number line and the thermometer can serve as a source domain for addition and subtraction involving negative numbers through the “Arithmetic as Motion Along a Path” metaphor, and what happens to the metaphor when it is extended. Analysing the metaphor in the manner attempted in this paper was first done by Lakoff and Núñes (2000, p. 72–73) and further elaborated by Chiu (2001, p. 118–119) The line, drawn horizontally as a number line or time line, or vertically as a thermometer, is seen as a path and addition and subtraction are seen as motions along this path.  So far the mapping is coherent with our embodied experiences. There are, however, aspects of the target domain with no corresponding aspects in the source domain. What teachers and textbooks tend to do in these cases is to extend the metaphor by reversing the direction of the metaphor and turn the target domain (M) into the source domain (S).” The teachers’ source domain is mathematics and the target is daily life because they try to think of a common space to communicate with the students” (Frant et al., 2005, p. 90). Situations are made up in domain S to fit the situations in domain M and thus create external coherence between domain S

and M. Unfortunately many of these made up situations lack experiential basis and seem quite ‘unnatural’, and it is difficult to keep the domain internally consistent.

                                      Mapping of the number line/thermometer metaphor

Source domain (S)                                                                                   Target domain (M)

Center point on the line (here called the origin)                                          zero

A point on the line relative to the origin                                                      a number

A point to the right or above the origin                                                       a positive number

A point on the line to the left or below the origin                                         a negative number

A distance on the line                                                                                 absolute value of a number

Motion to the right or upwards                                                                   addition

Motion to the left or downwards                                                                subtraction 

Let us look at the expression a + b = c. In domain M; a, b and c are all numbers. In domain S (the metaphor of Arithmetic as Motion Along a Path) only a and c are understood as points on the line, whereas b is not a point on the line but a number of steps to move; it is seen in connection with the operation sign. We now have two different types of numbers. In this source domain there is a great difference between a point on a line and a motion along the line. The motion is always positive or zero. You either move or you don’t move. There is no such thing as anti-motion. Motion can be in different directions, but the direction is settled by the

operation sign. Now assume that b is a negative number. What would that be in domain S when the idea of moving a negative number of steps is not part of our experiential basis? There are two common ways of solving this by extending the metaphor. The first way is to impose direction and start speaking of motion forward and backward relative to the person moving. The default direction is to face right. In order to achieve consistency we need to make some changes to the mapping. The metaphor now has to involve somebody (or something) with front and back. Since the temperature scale or the passing of time lack front and back they are of no use in this mapping. 

Notice how complicated the structure of domain S becomes. When teachers use this mapping to explain addition and subtraction in their classroom they often get confused and loose track of which way they are facing and moving (Kilborn, 1979). The metaphor also comes into conflict with other metaphors that are deeply rooted in our culture such as ‘positive is up and forwards’, ‘negative is down and backwards’. ‘Adding on is up’ (we grow up, we build up etc). ‘Right is forward’ (writing flows from left to right). The overall external coherent system of metaphors is here being violated and therefore this superficially created metaphor might cause confusion.

                                         Extended mapping of the number line metaphor

Source domain (S’)                                                                                  Target domain (M)

Standing somewhere on the line facing right

or up                                                                                                        default setting

Standing on the centre point on the line                                                      zero  

Standing on a point on the line relative to the

origin                                                                                                        a number 

Standing on a point on the line to the right or

above the origin                                                                                        a positive number 

Standing on a point on the line to the left or

below the origin                                                                                        a negative number 

A distance on the line                                                                                absolute value

Facing right or upwards and moving forward                                             adding a positive number

Facing right or upwards and moving backward                                          adding a negative number  

Turning around and moving forward                                                          subtracting a positive number

Turning around and moving backward                                                       subtracting a negative number

Holding on to the thermometer as a metaphor  entails a different extension. Adding a negative number is interpreted to be the same as subtracting a positive number; 3 + (−2) = 3 – 2, an interpretation which lacks meaning in the source domain. The mapping is no longer internally consistent since addition sometimes is seen as a motion along the line and sometimes need to be transformed into a subtraction. The subtraction (a – b) can be interpreted as the difference between a and b, or the motion needed to get to a from b. This is coherent with the mathematical definition of subtraction saying that (a – b) is the number x that solves the equation b + x = a. The great difficulty here is that in our experience of the difference between two temperatures or two points on a line, we always conceive of that difference as an absolute value. The difference between the two numbers -4 and +6 is always spoken of as 10. If we are to understand that 6 – (−4) and (−4) – 6 yield different answers we need to incorporate into the mapping the aspect of direction. 6 – (−4) =10 because the temperature rises from (−4) to 6, whereas (−4) – 6 = (−10) because the temperature falls from 6 to (−4). The direction here is from b to a, which is from right to left in the expression. This is an ‘unnatural’ direction since we often intend the reverse interpretation: 2 – 8 usually reads ‘from two to eight’. Direction has been identified as one of the critical features for learning subtraction of negative numbers (Kullberg, 2006). It is, however, a feature closely connected to the use of certain metaphors and the interpretation of subtraction as difference. See table 3. This extension of the metaphor is inconsistent since for the subtraction (a – b = c), a, b and c have different sources depending on whether they are positive or negative. If b is positive then a and c are referred to as temperatures and b is the change, but if b is negative then a and b are referred to as temperatures and c is the change.

                                   Extended mapping of the thermometer metaphor

Source domain (S)                                                                             Target domain (M)

Centre point on the line (here called the origin)                                     zero

A point on the line relative to the origin                                                 a number

A point on the line above the origin                                                      a positive number  

A point on the line below the origin                                                      a negative number

A distance on the line                                                                          absolute value

Motion up (rising)                                                                               adding a positive number  

Motion down (falling)                                                                         subtracting a positive number  

A distance with direction between two points

on the line                                                                                          subtraction

To add a negative number a different metaphor is needed that will make it plausible that a + (–b) = a – b since there is no way a temperature can go up a negative number of degrees.

As we have seen, extending the metaphor may make the metaphor less functional for three different reasons:

– Loss of consistency within the metaphor itself

– Loss of coherence with connected metaphors

– Properties or structure being forced on the source domain that is not part of the experiential basis.

Expanding the metaphor “Arithmetic as Motion Along a Path” in either of the two ways described here entails difficulties. It remains to be shown whether what is gained by the extension compensates for what is lost in comprehension and applicability. Nevertheless it is essential that teachers are aware of the consequences of extensions like the ones described in this paper. There are of course alternative ways of extending the metaphor, and other models, referring to different grounding metaphors, that also need to be extended in order to cover up subtraction of negative numbers. A further analysis of such metaphors could be a helpful tool for teachers in their struggle of teaching this topic. Analyzing what happens to a grounding metaphor that is extended to include the operations multiplication and division with negative numbers is a different task again, but none the less important.