The first project task was to determine the extent to which explicit problem-solving teaching is found in university mathematics departments in England and Wales. Given the tight timescale, an online questionnaire distributed to heads of departments via email was decided on as the fastest and most effective approach.
A list of mathematics departments was collated from the information available on the HoDoMS website (http://www.coventry.ac.uk/ec/HODOMS/); email addresses for the heads of each department were then found via their websites. A questionnaire was authored, kept short to encourage a large number of responses, and distributed with Google Docs. Because interpretations vary, it began with the following description of problem-solving as:
“
…any substantial task or activity that calls for original, lateral or creative thinking by students; brings several ideas or techniques together in a surprising way; intro-duces something new; illuminates some topic, e.g. with a helpful counterexample.This was followed by five questions on problem-solving in the respondent’s department:
1. Does your institution offer a module in any of its Mathematics Degree Programmes which requires students to engage in problem-solving?
• No
• Yes, an optional module
• Yes, a compulsory module
2. What is the module code and title (choose the best example in your programme)?
3. Is problem-solving the central aim of the module?
4. What year is the module taken in?
5. Please give a brief description of the module you offer.
Respondents’ department and contact details were asked for to determine who had answered the quesitonnaire, this was also used for follow-up contact with those who agreed to it.
Individual emails were sent to the heads of department of the 59 English and Welsh HEIs offering a Mathematics BSc. These included a brief description of the project and a link to the questionnaire. After a month, attempts were made to contact by telephone those who had not responded to either the email or questionnaire.
4.2.1 Summary of Questionnaire Responses
The questionnaire was completed by 35 heads of department or their representatives. Responses indicated an absence of explicit problem-solving teaching in the mathematics degrees of English and Welsh HEIs. Twelve of the respondents acknowledged that they had no explicit problem-solving in either compulsory or optional modules. In half of these cases, problem-problem-solving was mentioned in the published aims of the programme.
Of the remaining 22 departments, 5 offered an optional module to students while 17 stated that problem-solving was included in one or more compulsory modules. From the module descriptions in the responses, and further information from departments’ module handbooks, these were divided as follows:
• Seven modules on modelling, which were seen as distinct from problem-solving.
• Five project modules (four of them final-year), involving students researching and reporting on various topics, but not explicitly solving problems.
• Three modules on operational research, numerical methods, and numerical analysis that again did not include explicit problem-solving.
• One response naming no particular module but stating that problem-solving was inherent to a mathematics degree.
• Six modules (four optional, two compulsory) where explicit problem-solving was taught as the central aim of the module.
The responses to the questionnaire demostrate the difficulty in discussing ‘problem-solving’
in the context of a mathematics degree. The description we used was necessarily narrow, given the project remit, and while we used this in the questionnaire ‘problem-solving’ was interpreted more broadly by many respondents. The six modules identified as including a significant amount of explicit problem-solving in the sense that we intended subsequently became the subjects of our case-studies. They were as follows:
University of Birmingham Problem-solving is taught to year students in their first-term using the Moore Method. There are no lectures, and the only materials students are given are the problems they are required to solve and definitions they might need to solve them. No examples or model answers are available to students. There are currently two groups using questions on different topics – geometry and set theory. Students do not get to choose which group they are in. They meet for two hours per week to present and discuss their solutions. The module is optional for BSc students but now expected for students enrolled on the 4-year Mathematics MSci programme. This is discussed in Chapter 6.
Durham University Delivered in the first-term of the first-year, Durham’s compulsory mod-ule, entitled Problem-Solving, is based on Thinking Mathematically (Mason et al., 2010).
It is taught using both lectures and problems classes, in which students work through problems in groups, recording the progress they are making by using rubrics2.
University of Macondo (Pseud.) Macondo’s compulsory second-year module, Investigations in Mathematics, sees students working in groups of around a dozen on different mathe-matical topics at an appropriate level. It is taught using a modified Moore Method by several members of staff working in parallel. This is discussed in Chapter 7.
University of Manchester The Mathematics Workshop is a first-year, first-term module again using Thinking Mathematically, though to a lesser degree than at Durham. Students begin the year working in computer labs, before concentrating on modelling and problem-solving after the mid-term break. A compulsory module, the Workshop has both lectures and classes.
Queen Mary The only third-year module in the study, Queen Mary’s Mathematical Problem-Solving is an optional module taken by a dozen students each year. Each student has a different set of problems from a range of topics in pure mathematics. Although there are no lectures and only a single class each week, students may seek help from members of staff at other times.
Warwick Warwick’s first-year Analysis 1 module has been taught using problem-solving for the past 15 years. This unique example among our case-studies is a core module taken by
2Discussed in Section 4.6
all its 320 first-year mathematics students, taught by problem-based learning. Students have one lecture and four hours of group work in problems classes each week.
Five of the universities deemed to have an eligible problem-solving module were members of the Russell Group, or (in Queen Mary’s case), about to become members. Macondo is a member of the 1994 Group of universities. Were the questionnaire to have had more responses, we may have found other suitable modules and been able to choose a better cross-section of UK mathematics departments. As it is, we must note that this is the case and use those data available.