Montessori was, first of all, a mathematician, who became a doctor, who became a teaching and learning designer and developer.

This shows in her elegant 'trinomial cube' (see our synaesthesia article), and many other learning objects (like my other favourite, the pink tower). What's important for intrinsic motivation is that the learning materials / objects should, as far as possible, be self-correcting, and in many ways, self-assessed.

If this can be achieved, the learner 'benchmarks' their own learning, and decides for themselves when the task is 'completed' (i.e. their learning task, not someone else's 'outcomes' or 'competencies'). And until they are happy, satisfied, and comfortable with where they have got to - en route to benchmarking their learning/task, they will continue - often for many repetitions.

So ... can we apply the same principles to learning at secondary or even tertiary education? Yes, university mathematics is a case in point.

At Portsmouth University, we implemented a computer-aided mathematics programme for first year mathematics students. They could choose how long they wanted to spend on their examinations: realistically, somewhere between 5 and 15 one-hour sessions, spread over a semester. On their first day, they were told that they should only answer those computer-generated questions that they were confident of getting right. If not, they should leave those questions, and go back to lectures, lecturers, tutorials, practice questions, text books, etc, and then return to attempt them again at a later stage. They had three attempts per question (with different values, of course), which they could use at anytime over the entire semester.

In effect, they managed their own assessment, at their own pace, question by question, over 15 weeks. The computer-based examination turned into a self-benchmarking exercise - it was no longer a test of what they could achieve on a particular day, but became a confirmation of their cumulative capability to do mathematics. The answers required ''writing' mathematics, not just choosing between possible answers.

Most of the students achieved the minimum pass rate quite early on in the process, but a large majority of them came back, of their own accord, to do 12 or even 15 hours of examination work, even though 5 hours sufficed for most of them to achieve the minimum to pass. And several of them came back right up to the 15th hour of examination because they were not satisfied with 95%, or 96%, and wanted to achieve 100%.

Then a new vice-chancellor arrived, and decided that the whole programme could not continue. Students now sit one, three-hour exam at the end of the semester, and that is it. His argument (if one can call it that) was that it would be 'unfair' for some students to do more hours on examination than others - despite the fact that the choice of how long to spend on examinations was open to all students equally. The programme died a quiet and unremarked death. External metrics and 'motivation' rules, OK (?)

It is all documented in an article or two, which demonstrates that the power of intrinsic motivation can lead to something as extraordinary as students voluntarily doing more than double the time in examinations that is required, and up to 15 hours of examinations in one semester.