Cuarto escenario: 32 pinos, 32 hayas

Based on the analysis and interpretation of my data I identified the goal of intuitive understanding. With reference to my theoretical model (Figure 5.2) I associated this goal with the actions of presenting a mathematical problem (having the ‘outer’ form of an example), and formulating all problems, definitions and concepts in the context of the vector space Rn. I also associated the action of verbalising intentions with this goal.

69

In my interpretation and as I discussed in Section 5.4.3, the goal of engagement with mathematics was a necessary step towards achieving the goal of an intuitive under- standing.

In research meetings the lecturer said that he wanted students to ‘get a feel’ (M2, 44:06) or ‘develop a feel’ (M8, 06:56) for the concepts in linear algebra, and ‘to make summaries of that sort’ (M2, 18:29), meaning summaries of an informal or intuitive nature. In a lecture in Week 2, the lecturer presented the method of Gaussian elimination for solving linear equation systems. In a research meeting, when talking about teaching the topic, he said,

But the reason why I’m asking them actually to solve things by hand is so that they get a feel of what can happen, and so that they get a feel for how the algorithm works, and what the different cases are that can hap- pen. Ultimately we’re going to diagnose the solvability, inconsistency, unique solvability or number of free parameters of a linear equation system from the echelon form. And we will see that those are all possibilities. (M2, 44:06; italics my emphasis)

I interpreted the lecturer’s comment “so that they get a feel ” as expressing the goal of intuitive understanding. I associated the action of presenting mathematical problems with the goal of an intuitive understanding. With this action the lecturer used a mathematical problem as a psychological tool. I referred to a mathematical prob- lem as the formulation of a question or exercise to students in mathematical terms (i.e. mathematical notation or language).

The second action that I associated with the goal of an intuitive understanding was the action of formulating in the context of vector space Rn. The lecturer decided to formulate all linear algebra concepts, all definitions and theorems, etc. in Semester 1 in the context of the vector space Rn. The vector space Rn denotes the space of all n-tuples, or the space of all n-component column vectors. Formulating linear algebra concepts in the context of the vector spaceRnwas an action designed to direct students towards an intuitive understanding (“develop a feel”) which the lecturer expressed in a research meeting as follows:

So that’s . . . what I’m aiming for is to talk about these linear combinations, and linear independence, these crucial concepts, in the context of column

vectors where most people feel comfortable they can calculate with them. . . . That’s why I asked them today, and I ask them again on the problem sheet, “Write this vector as a linear combination of the other vectors. Can it be done?”, because I’m hoping that students . . . will develop a feel for what it means that one vector is a linear combination of others, and that vector is not. And also, that’s also why I’m putting the emphasis on, where I can, putting the emphasis on really what we’re talking about . . . (M8, 07:34)

By focussing on the vector space Rn the lecturer wanted to direct students’ attention away from the computational and towards the more conceptual aspects of linear algebra. (See Section 6.2.3 for a detailed discussion of this action in relation to the concept of a subspace.) I interpreted the vector spaceRnas a material tool, rather than a psycholog- ical tool. The course notes were written often using, for example, only column vectors in R2 and R3. This also resulted in mathematical language that was ‘simpler’ than it would have been if the course notes had been formulated in the context of abstract vector space theory. The vector spaceRn as a tool affected the formation of the course notes and the language used (in the course notes and in lectures) and hence was a material tool.

The third action that I associated with the goal of intuitive understanding was the action of verbalising intentions. That is, the lecturer made the goal of an intuitive understanding explicit in lecture by telling his students of his goal. For example, in a lecture in Week 6 the lecturer said,

And also you should have had a look at your problem sheet where I put down a couple of problems asking you to do something with these terms. And the purpose of these problems and the purpose of the new problems for this week is that you get some experience in working with these objects and these concepts, and you get a feel for how they fit together and how they work. (L15, 05:23; italics my emphasis)

Thus I associated three actions with the second goal in my model, the goal of an in- tuitive understanding. As discussed in Section 5.4.3 and as set out in the diagram on page 62, I considered the goal of engagement with mathematics as contributing and supporting the goal of an intuitive understanding. That is, attaining the goal of engage- ment with mathematics was a pre-requisite for attaining the goal of an intuitive understanding.

71