One of the strongest themes to emerge from all six interviews was an enthusiasm for teaching, and a willingness on the part of all colleagues to try a non-standard pedagogy that all but one had no experience with. None advocated using the Moore Method as the principle means of teaching undergraduate mathematicians—Lecturers E and F both explicitly warned against this—but even those who had been assigned the teaching (B, D, and E) were happy to ‘give it a go’. Lecturer D found themself working on proofs that had previously featured in their history of mathematics teaching, but that they had not considered as carefully before. Naturally our sample is self-selecting; someone showing little enthusiasm would be unlikely to be assigned the module and even less likely to agree to an interview; we must bear this in mind.
Another strong theme was the gulf between the expectation and reality of the mathematical knowledge that students had at their disposal. All members of staff had been warned that progress would be slow, especially at the beginning of a Moore Method course, yet all were surprised (and some somewhat frustrated) with exactly how slowly progress was made. The
‘shock’ expressed by Lecturer A was representative of all. Progress was made in all groups, though not uniformly so. Both Lecturers A and D each taught two groups dramatically different in the speed of their progress. Lecturer F eventually resorted to assigning questioners and answerers, and including an amount of exposition at the beginning of a class, but none of the others altered the intended approach to teaching to such an extent. It is telling that Lecturer F began with material that was considerably different to that seen in a traditional Moore Method course. Taking an existing paper and attempting to ‘plug the gaps’ is at odds with Coppin et al.
(2009, p. 57)’s advice to “progress from the simple to the complex”, and it is perhaps the case that the process suffered more as a result of this. Their method of teaching by the end of the module would be difficult to describe as a modified Moore Method, but something entirely different.
Besides progression through the problems themselves, student engagement in the whole process was difficult to begin with, especially in the first year that New Investigations was taught. The lack of examinations is unusual, and introducing the poster and mid-term notebook played an important part in encouraging students to engage with the module earlier. Moore was able to choose his students, and fostered a highly competitive environment that not all were comfortable with (see Section 5.2.4), but slow progress is inherent to the Method and covering a large amount of material was not an aim of the module.
There were two principle differences between ‘the’ Moore Method and the approach taken in New Investigations. Firstly, there was an emphasis on group work that is not present in the Moore Method until students have tried to solve problems for themselves. Students were not required to work in groups, however, though most chose to do so. Secondly, finding information outside course material was not proscribed but encouraged. Both of these changes reflected the aims of the module itself; students would otherwise not have the opportunity to work in groups during the undergraduate degree, and information gathering is an important skill for a graduating mathematician. In the second instance it was not the intention that students would be able to find solutions to their specific problems online, however; lecturers obfuscated problems and used non-standard definitions and terminology to prevent plagiarism.
In 2011/12 all the staff involved with the module had experience with teaching New
In-vestigations and Lecturer A was once again positive about the way it had run. The reduction in the numbers of staff teaching the module in 2012/13 aimed to make organising staff more straightforward, and allowed Lecturers A and C greater control. While changes to the module were made over the four years that it had been taught using the Moore Method, overall teaching remained unchanged. In that regard it appears as though the module has been a success for most, if not all, students.
Part III
Computer Aided Assessment
Chapter 8
The solution and comparison of equations
In Part I of this thesis we investigated the state of problem-solving teaching in universities in England and Wales. In Part II, we looked at the Moore Method, and its use in two universities in England. In this part we look at a separate area of mathematics education, namely computer-aided assessment (CAA), though with the same purpose—improving students’ problem-solving skills—in mind.
8.1 STACK and Computer-Aided Assessment
The work discussed in this part of this thesis focusses on STACK, a System for Teaching and Assessment using a Computer-algebra Kernel, is a computer-aided assessment (CAA) system for mathematics. A review of other computer-aided assessment software is beyond the scope of this thesis, we recommend Chapter 8 of Sangwin (2013, pp. 127–161). Figure 8.1 shows an example question with a student’s response and feedback. A student’s answer is entered as a mathematical expression, whose mathematical properties are then determined by STACK in order to mark it and give appropriate feedback.
Figure 8.1: An example STACK question
For many questions a student’s answer will have to be both algebraically equivalent to the correct answer, and in the appropriate form. A correct answer need not be unique, however, and STACK establishes the relevant properties of expressions to test objectively a student’s answer.
A distinguishing feature of STACK is that the feedback it gives may include calculations based directly upon the answer that the student has entered. Figure 8.1 gives an example of this type of feedback.
STACK establishes the properties of a student’s answer and generates necessary feedback using the computer-algebra system (CAS) Maxima. It may also use Maxima to randomly generate specific instances of a more general question, so in the example above it could produce several versions of the question with a stationary point at x ∈ {1, … ,10}.
The prototype test of correctness seeks to establish algebraic equivalence between a stu-dent’s and teacher’s answers, however this is limited to instances where there is only one correct answer (up to algebraic equivalence). The example in Figure 8.1 cannot be assessed using only algebraic equivalence. STACK provides a library of answer tests for assessing
different types of equivalence, and more than one test can be applied to a student’s answer (we cover this in Section 9.1).
The motivation for the work documented here was the desire to introduce a new answer test that would allow teachers to assess systems of equations, primarily because of their relevance to modelling and word questions (Sangwin, 2011). Assessing problem-solving is currently well beyond the capabilities of computer systems, but by requiring students to interpret information correctly and form it into coherent mathematics, modelling uses aspects of mathematical thinking also seen in problem-solving. In particular, we want to be able to assess systems of multivariate polynomial equations, i.e. polynomial equations in more than one variable, not something currently assessible in any other CAA systems. Before we introduce the mathematics involved in doing this, we begin by discussing the way in which STACK determines the correctness of an answer in general.