Resultados y análisis del escenario final

I identified the acquisition of mathematical language as one of the lecturer’s goals for student learning. With reference to my theoretical model (Figure 5.2, on page 61) I associated this goal with the actions of verbalising intentions and designing con- tent for coursework/problem sheets.

The lecturer expressed acquiring mathematical language as a goal for students’ learning. In research meetings he talked about students needing to ‘learn the language of linear algebra’ (M12, 02:03; M6, 12:22; M7, 01:51), to ‘get fluent in that notation’ (M5, 04:08), and to be ’able to read a definition’ (M5, 27:15). For example, in a research meeting he said,

Yes, of course it’s [learning the language is] important because they’re [the students are] supposed to start reading mathematics on their own. And our students are very slow at that, and many probably even when they graduate, couldn’t take up a mathematics book and read it. But that’s why I’m putting a lot of emphasis on that language. (M12, 02:03)

and,

And what we are going to see over the next couple of weeks is a reformulation of linear equation systems in ever-new guises. That means there is a lot of new language coming up, which students will probably struggle to absorb. (M6, 12:22)

In lectures he emphasised the importance of mathematical language when addressing his students. For example, in a lecture in Week 5 he said,

That’s [Subspace is] an important new idea and also an important new word that you need to learn to use. And there’re a couple of more new words that I need to give you. . . . So these are important observations and important new words. And you will, as we go on through this chapter, you will find a lot of new words that I need to introduce to you, a lot of new words in which we talk about solutions to linear equation systems, and it’s very important that you learn how to use these words properly and to speak that language because this is the language in which we will be able to formulate observations, theorems that are much more general than we observe here.

And if you look at the problem sheet for this week there are a lot of things that are phrased in the new terminology that I’m introducing here and that I will introduce tomorrow. The calculations that I’m asking you on this problem sheet are all very easy. All questions on that problem sheet either ask you to do a certain matrix operation or they ask you to solve a linear equation system. So the calculations are things that you really know how to do. But the important thing is that you spend the time to find out what that language means. If you understand the language that we’re using for linear algebra you’ll almost certainly be able to do everything very well. But it’s important that you be able to understand the language we’re using and to use it properly. So please, pay attention to the new terms and the new ideas that we’re going to introduce over this chapter. (L12, 43:25)

The lecturer was referring to the terminology of linear algebra, the formulation of con- cepts in terms of definitions and theorems and general mathematical notation. For example, students needed to understand the notions of subspace, spanning set and lin- ear independence. The lecturer stressed that the formal definitions were important and necessary as they provided a precise formulation of the concept or idea that was being discussed. In a research meeting the lecturer referred to the formalism of mathematics as ‘where the power of mathematics came from’ (M15, 44:28).

The lecturer made the goal of acquisition of mathematical language explicit in lectures when addressing students. For example, in lecture L12 (43:25) which I have already quoted above or in lecture L15 when he said,

I am hoping that you reviewed your lecture notes and you are comfortable now speaking in these terms and using this language. (L15, 05:12)

Making his goals explicit to his students related to the first action that I listed at the beginning of this section, the action of verbalising intentions.

The second action that I associated with the goal of students acquiring mathematical lan- guage was the action of designing content for coursework/problem sheets. With the term ‘coursework’ I referred to all assessed examinations and tests, while ‘problem sheets’ referred to formative and non-assessed exercises. In designing mathematical con- tent for coursework and problem sheets, the lecturer’s aim was for students to practice and acquire correct and precise mathematical language. In a research meeting, he said,

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So, this is on the one hand, this is, because I think, as I said, that reading text is an important skill and engaging with the language. On the other hand also this coursework is intended to send the message to the students, I keep saying ‘Using, being able to use the language is important’, and I mean that. And for that reason I’m also willing to assess you on that, if only to 1% of the module mark, . . . (M15, 1:06:16)

With reference to the theoretical model (Figure 5.2) that I created based on my analysis of the lecturer’s comments, I considered the goal of an intuitive understanding as a pre- requisite for students acquiring mathematical language. Similarly, and in turn, acquiring correct mathematical language (and in particular the language of linear algebra) was a necessary step towards students’ conceptual understanding of the topic of linear algebra.