Principle 4: Communicate explicit evaluative criteria for improving the quality of word use and mathematical justifications in classroom talk

The discussion of evaluative criteria brings the specificity of the mathematics content and the context in which teachers work to the fore. The mediation of the language practices of word use and justifying, and evaluative criteria for these has without doubt been the most difficult aspect of the MTF to mediate.

The underlying mathematics is what a variable is. We know from extensive research focused on the notion of a variable in the teaching and learning of algebra (e.g. Rhine, Harrington, & Starr, 2019) that learning the meaning and use of‘variable’ is not trivial. The recognition of p being a variable, and thus, ANY number, is necessary for a full explanation. Our structuring principles have enabled us to work with and from teachers’ productions to the specificity' of the underlying mathematics, and towards making available and explicit, the range of possible evaluative criteria. To do this, we needed to mediate different word uses and justifications and their value for learning these specific mathematics. We asked teachers to examine the words written on their newsprint and across explanations, and to identify and then distinguish formal words, words that name mathematical objects (e.g. like and unlike terms, variable, unknown, constant, number) from informal and colloquial words, and how these played a part in producing a fuller mathematical explanation. We also worked with the criteria for producing different justifications, distinguishing non-mathematical from those that are partial or fully mathematical.Table 5.1 provides the range of explanations offered by teachers over the years, the nature of the criteria transmitted in their justifications, and thus implications and exemplars for what and how these could be improved.

TABLE 5.1 Criteria for justifying why 5y> + 4 is not 9p

Justification and explanation

Evaluative criteria

p is a number so if we try 2, then we have 5p + 4 = 14 ;9p = 18 so 5p + 4 ф 9p

The variable p represents any number, so in the equation we should be able to substitute any value for p to make the equation true ALWAYS:

If p = 2, then we have 5p + 4 = 14; 9p = 18 so 5p + 4 ф 9p

If p = 0, then 5p + 4 = 4; 9p = 0; 5p + 4 ф 9p But if p = 1, then 5p + 4 = 9; and 9p = 9 then 5p + 4 = 9p

so 5p + 4 is not ALWAYS equal to 9p, only when p = 1

5p+4=p + p + p + p + p+ + + + 5p+4=p+p+p+p+p+p+p+p+p These are not the same

We can think of 5p as 5 pencils, but 4 is just a number. When we add, we won’t get 9 pencils

We can’t add pencils to numbers They are unlike terms

We can think of p as a box with a number of sweets, but we don’t know how many sweets are in the box.

Is 50 + 4 the same as 9Щ? i.e. Is 5 boxes of sweets plus 4 more sweets the same as 9 boxes of sweets?

p is a number, rather than a variable This justification is partial. Only a single case is tested.

It can be countered with what if p = 1? p is a variable and so represents any number This means that if the equation 5p + 4 = 9p is a true equation, it should hold for any value substituted for p Here we show that it is not true for example if p = 2 or 0 or any other number we may test except 1 As we don’t know the value of p, this is not a true equation.

This is a full explanation. It rests on the properties of a variable, p here is treated as an object This justification is partial, relying on a ‘rule’ or the visual representations that highlights p and 1 as different objects The justification is mathematically incorrect. We need to distinguish between p representing pencil and p representing the number of pencils Justification is through an everyday metaphor

This justification is ‘non’ mathematical This justification is mathematical, and one often used in our schools as it draws on familiar representations of unknown numbers. It draws in everyday notions of eating sweets and enables discussion of whether and how you know who has more. We describe this justification as partial, as it still invokes the notion of ‘a number’.

Depending on how the session unfolded, and the range of validating criteria the groups of teachers produced, the course presenter would conclude this activity with a discussion of different justifications and the criteria they reflect. Figure 5.5 is an example of such a summary discussion.

Activity 1 was followed by further activities with different tasks, each of which leads to building explanatory communication through word use and justifications. Each is designed and enacted with the same structuring principles as described above. For example, one of the tasks is centered on solving for .v in the equation 6 - x = 10. This task has also been developed following a lesson interestingly with Grade 10 learners, where solutions for x included 16, -16, 4 and -4. Here teachers are not asked to produce explanatory communication, but they were given four different imagined teachers’ explanations to compare and contrast, looking at the words used and justifications given as to which of the solutions was correct and why. We have designed a range of explanations to provoke discussion on words used and evaluative criteria transmitted. The learner task is “easy” for teachers and teaching the solving of equations like this part of their practice. This particular equation and its solution enables critical discussion of deliberate attention to words used when dealing with integers, and specifically how the common use of’minus’ for either or both of “subtract” and “negative” can be confusing rather than helpful for learners. Here too we are able to focus on deriving as opposed to stating steps if the procedure is followed as this is often in colloquial language (e.g. we take x or 4 to the other side); and on how difficult it is to construct an everyday context for which this equation is a model. Critically, we are able to communicate how important for both the teacher and learners to be able provide discursive elaborations such as “we

Distinguishing validating criteria for differing justifications are looking for a number for x that makes this equation true, that when I subtract it from 6 the answer is 10”

FIGURE 5.5 Distinguishing validating criteria for differing justifications are looking for a number for x that makes this equation true, that when I subtract it from 6 the answer is 10”.

In each iteration of the TM1 course, each of the activities in this session, and especially Activity 1, has provoked heated discussion, particularly in relation to whether letter as object (through fruit or any other real object) “works”. Many teachers who produced such justifications appreciate their limitations and expressed this in the discussion. At the same time, others who similarly show appreciation remained adamant that this was a useful strategy', notwithstanding the future obstacles it might create. More important for them was that it worked and connected with their students, reflecting the strong view of making mathematics accessible, and this meant what was ‘easier’ for learners.

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