Almost all IB mathematics teachers are curious about how well students will take to the new guidelines for IB coursework. There are many aspects of this change which I could talk about at length, but I’ve decided to discuss one part which I feel needs to be considered to help to ensure that the exploration is, as the IB put it, “stimulating and rewarding” for all students involved.
The IB have changed from providing two pieces of coursework, one investigation task and one modelling task, to one piece of exploratory coursework. The biggest part of the change is that the student will now be charged with the task of posing a fruitful problem to explore which “should be commensurate with the level of the course, that is, it should be similar to that suggested by the syllabus.” (Internal Assessment Criteria and Additional Notes [OCC document], 2012)
The part which interests me is that the mathematics should be similar to that in the course. Whilst I understand why the IB have stipulated this as a requirement, I also feel that the IB may consider taking a leaf out of Hirst and Weir’s (1981) paper on “Undergraduate Investigations in Mathematics.” Hirst and Weir write about their experience facilitating third year mathematics graduates during a module in which the students spent the entire time working on an individual exploration. These explorations were either devised by the student or, more frequently, posed by the lecturers themselves. Below is Hirst and Weir’s account of the explorations provided to the undergraduate mathematicians:
University teachers will be perhaps surprised to hear that I have been using sources devised for schools, or B.Ed. courses, and conclude that the students are not therefore tackling sufficiently demanding problems, or “proper final year mathematics”. I have to emphasise that I deliberately choose situations where the technical content is at a reasonably low level, since I want the students to explore, to ask questions, to open up their own problems to look for patterns, to formulate conclusions, to provide proofs, to be critical and to evaluate their work. This is where the demands of the work lie, so that the context must not inhibit the students by conceptual difficulties or the manipulative intricacies of the situations. As Maclone (1973) points out in any case, it is not a major cause of concern that mathematics graduates are lacking in the purely factual knowledge required in their later professional lives. The value of investigational work comes from the handling of all these other dimensions of mathematical activity.
Hirst and Weir (1981, pp. 375) (See Book and Article Reviews for Reference)
Based on this reasoning, should the IB prescribe that the mathematics used in the individual exploration be similar to the mathematics learnt in the course of study?
Let’s say that a student wanted to investigate a task in more detail which they had already met during their GCSE years. As a standard example, take the diagonals of a rectangle task, in which the aim is to find a formula for the number of squares the diagonal of the rectangle passes through for a rectangle with arbitrary length and width (Dan Meyer’s visual treatment):
Let’s say a student investigated this in their last year of GCSE and wanted to try to understand why the formula is the way it is. Maybe they would like to extend the work they did to look at trapezia – possibly on isometric triangle paper. This is mathematics which the student has previously engaged with and could take further.
It seems to me that we have timed-tests (worth 80% of the final IB mark) which assess students’ knowledge and understanding of the mathematics in the course. It is my contention that a coursework element is better served to assess the aims of the course of study, not necessarily the content. If a student wants to explore a topic which is similar to, or above the level of the course, then that’s absolutely great, but why restrict a student who may be “stimulated and rewarded” by an exploration which is at a lower technical content to the current mathematical content they are studying? If it is fruitfully explored, why does it matter? I realise that exploring syllabus content in more detail will benefit many students for the timed-tests. However, some students may benefit more from the experience if they explore topics in more depth which they are already familiar with, and can concentrate on the exploration and creativity involved in mathematics rather than technical know-how.