5.3 Activity and motive
5.3.1 The EAG approach
Based on my analysis, I have used the EAG approach as an overall design for developing and explaining the teaching process. The approach was closely linked to the motive of the activity.1 I could discuss the EAG approach at the action-goal level of analysis if it were a strategy in pursuit of a (achievable) goal. However, this was not the case. Activity in my study related to the teaching of linear algebra ‘inductively’. Thus the EAG approach was inherent in Activity, and not separate from it. It represented a framework for the didactical decisions that the lecturer made in relation to his planning and designing of the linear algebra module. I discuss the individual strategies that the lecturer designed in support of an ‘inductive’ approach at the action-goal level analysis in Section 5.4. Based on this analysis I describe the process of teaching linear algebra concepts based on the presentation of examples, and develop a theoretical model of the lecturer’s design for teaching linear algebra.
The lecturer based his approach on ‘presenting an example first, followed by a definition or theorem’. In Section 5.3 I had termed this approach ‘EAG’ where the initials stand for ‘Example - Argument - Generalisation’, and describe the process of:
we introduce an Example,
we make an Argument on the example, and then
we Generalise to an observation (a definition or theorem).
The lecturer’s aim was to present an example, and then to make an argument on the example in order to derive a rule or an observation. The observation could highlight a generality in what was observed in the example. This is my interpretation of comments that the lecturer made in research meetings. For example, the lecturer made his approach explicit (in a research meeting), when he said,
Generally speaking, I decided that I would focus on doing the development of the argument on examples, and then trying to abstract a general fact from the example, as I have done in most cases so far. And so then, what I am doing is go[ing] through the example, and then highlight[ing] the important facts on the example, and then condens[ing] them into a general observation.2
1The motive drives Activity. The motive also gives rise to goals (forming a dialectical, i.e. mutually
constitutive relationship) which are realised in and through actions.
2This terminology agreed with the use of the term ‘observation’ in the course notes. The lecturer
used the term ‘observation’ instead of ‘theorem’ in the course notes. The observation that is referred to here is a theorem, but unlike the theorem, the observation is not proved (see M12, 20:48).
And I have several times mentioned to students that this is what we’re doing, and that it’s a good idea to see an example not as an isolated example but rather as a representative of a big class. (M9, 11:58)
and
. . . And the way that the observations work is that, most of the time, we go over an example first and then we extract a general statement from the example we have gone through. And the entire thing is based on the obser- vation that the same argument we have just used in the example will also apply in a very large class of other examples, which makes the step from the specific to the general. And I have discussed that step with students in class many times. (Slight pause.) And again, I don’t know how much of that actually sticks. I repeat it because I think it’s important, and then I’m hoping that students go away and start thinking about things the way I demonstrate in class. (M15, 56:10)
The lecturer made his approach explicit also to his students in lecture. For example, in Week 9 he said to his students,
We haven’t seen very many proofs in this lecture. Most of the time we have derived our observations from examples. And that’s a good thing to do. It’s a good thing always to have a look at the examples, and see if there are any general statements that we can derive from them, anything that we can learn from the examples. (L25, 43:05)
All the lecturer’s intentions and strategies that I list at the action-goal level in Section 5.4, I interpreted as describing the EAG approach as indicated above.
I demonstrate the EAG approach with an example that the lecturer presented to students in Week 5 (Example 3.14, in lecture L14). The outline of the example (e.g. in this case the vectors and the questions posed) were written out in full in the student version of the course notes. After each question (a) to (d) there was a blank space where students could enter the solution to the example. The lecturer presented the solution in the lecture. With this example the lecturer sought to demonstrate, or derive, the theorem “The range of a matrix is a subspace”. I have reproduced Example 3.14 below without the solution. (A copy of the full solution of Example 3.14 is reproduced on page 58.)
57
Example 3.14. Consider an unknown 2 × 3 matrix A. We know that A satisfies Ax1 = b1 and Ax2= b2, where b1= 2 3 ! , b2 = −1 5 ! , x1 = 1 3 −7 , x2 = 3 −3 2 .
(a) Is b1 in the range of A? Is b2 in the range of A?
(b) Is b1+ b2 =
1 8
!
in the range of A?
(c) Take the number λ = 3. Is λb1 =
6 9
!
in the range of A?
(d) Is the zero vector 0 in the range of A?
Earlier in the course the lecturer had introduced the null space of a matrix A, that is the solution set of the homogeneous equation system Ax = 0. He had shown that the null space has similar properties to the set of all n-component vectors: It is closed under addition and scalar multiplication and contains the zero vector. With this observation the lecturer introduced the definition of the subspace. (See Sections 6.2.2 and 6.2.4 for a detailed discussion of this example.)
In relation to Example 3.14, I have interpreted the EAG approach as follows:
- We introduce an Example:
(i) With Example 3.14 the lecturer posed four questions for students to answer. The formulation of the example included an unknown matrix A and four vectors for which the lecturer gave concrete, numerical values.
- We make an Argument on the example:
(ii) The four questions (a) to (d) were designed to lead students to recognise the corre- spondence between the answers to the questions and the definition of a subspace.
- We Generalise to an observation (a definition or a theorem):
(iii) As a result of (ii), students were to arrive at, and recognise that the range of a matrix is a subspace. This was then summarised in what the lecturer called ‘Observation 3.15’. (See page 58 for the full solution of this example.)
Example 3.14. Consider an unknown 2× 3 matrix A. We know that A satisfies Ax1= b1 and Ax2= b2, where b1= 2 3 , b2= −1 5 , x1= 13 −7 , x2= −33 2 . (a) Is b1in the range of A? Is b2in the range of A?
Solution:
b1∈ range A because the equation system Ax = b1is solvable (x1is a solution).
b2∈ range A because the equation system Ax = b2is solvable (x2is a solution).
(b) Is b1+ b2= 1 8 in the range of A?
Solution:Yes. The equation system Ax = b1+b2is solvable, and x1+x2=
40 −5 is a solution because A(x1+ x2) = Ax1+ Ax2= b1+ b2.
(c) Take the number λ = 3. Is λb1=
6 9
in the range of A?
Solution: Yes. The equation system Ax = λb1is solvable, and λx1=
34 −21 is a solution because A(λx1) = λ Ax1= λb1.
(d) Is the zero vector 0 in the range of A?
Solution: Yes. The equation system Ax = 0 solvable, and x = 0 is a solution because A0 = 0.
In this example, we have verified that the range of a matrix has the three properties of Observation 3.5. We can therefore conclude:
Observation 3.15. The range of a matrix is a subspace.
59
‘Observation 3.15’ is the theorem “The range of a matrix is a subspace”. As stated in Section 5.1 the lecturer chose the terminology of ‘Observation’ (rather than a ‘Theorem’) because he did not give a formal proof as this point in the course. In general, proofs were provided in the second semester, when results were re-visited in the context of abstract vector space theory.
By proceeding in this manner the lecturer sought to provide students with a gentle introduction to mathematical reasoning. In a research meeting he said,
. . . And because I know that many of our students are unfamiliar with the way how mathematical arguments are usually phrased, and how the structure of a mathematical exposition in definitions, theorems and examples, for most of our students is something entirely new and something they find difficult. So I decided to go for an almost exclusively example-based development and go for these observations as an indication of that, which in my first semester linear algebra, I find very easy to do because I know there’s going to be a second semester which is going to make a second tour through more or less the same material in the context of abstract vector spaces, and on a much more formal level. (M9, 15:36)
In the quotation above, the lecturer refers to his knowledge of students and, based on his teaching experience in a previous year, the difficulties that students had with the module. I interpreted going “for an almost exclusively example-based development” and going “for these observations” as indicating a ‘bottom-up’ or ‘inductive’ style of teaching. This was an overall approach and contributed to the rationale of his teaching. A ‘bottom-up’ or ‘inductive’ style contrasts with a more traditional ‘DTP’ (definition- theorem-proof) style.
In this section I have discussed the lecturer’s ‘inductive’ teaching style which I termed the ‘EAG’ approach. It was an examples-based approach where theorems and defini- tions were stated after students had worked on an example first. In contrast, in the more traditional ‘DTP’ style, theorems and definitions are usually stated first, followed by examples that explain or exemplify the theorem (or definition), or test students’ un- derstanding of the theorem (or definition).
I now present the second level of analysis. This is my interpretation of the meetings data in respect of the actions and the goals of Activity. I develop a theoretical model of the teaching process that links goals with associated actions.