The differences between investigations and inquiries
When running workshops for experienced maths teachers, I hear the claim that Inquiry Maths is just another name for investigations. On one occasion, a teacher appeared exasperated as she accused me of "re-inventing the wheel" and declared that "we've come full circle in maths teaching." Evidently, I had failed to distinguish between investigations and inquiries, but, more importantly, I had also failed to understand that the colleague remembered a time when the investigation classroom was very different to what we know of it today.
The reason the investigation survived at all during this period, it could be argued, was its inclusion in the GCSE specification as coursework. However, coursework investigations became so structured by the requirements of the exam board's mark scheme that they fell into disrepute. No longer was a piece of coursework a reflection of a student’s independent thinking, but rather a reflection of how well the teacher knew the mark scheme. Coursework was scrapped in 2007.
Investigation classrooms were not always like this. During my PGCE year in the early 1990s, I visited a maths department that taught the whole curriculum through investigations. It was one of the last, if not the very last, school in the country to do so. It was like no other department I have visited or worked in since. Students were allowed to investigate or not, depending on whether the teacher’s questions about a starting point had aroused their curiosity. They investigated individually, occasionally having discussions with the teacher, until they discovered the mathematical concept for which the starting point had been designed. There was no whole-class instruction, which seems unbelievable today.
This is how I imagine the classroom Marion Bird describes in her 1983 booklet on generating mathematical activity. Marion refers to the activities she uses as 'inquiries' and I would class some of them as inquiries in the sense I use the word today. Splitting decominoes, for example, starts with a diagram and, even though Marion sets an initial question, she allows the activity to develop into multiple pathways that encompass different forms of mathematical reasoning. However, another one of her activities - The greatest number of intersections - has become a classic investigation in which students have to draw more diagrams, tabulate results, identify a pattern and discover the generalisation. This is the inductive method of a science experiment in which more results confirm the hypothesis or lead to its revision.
The exclusively inductive approach is not consistent with the combination of induction and deduction that characterises mathematics, and certainly not with the deductive nature of mathematical proof. Polya explains in How to Solve It and Mathematics and Plausible Reasoning how "deduction completes induction." While the mathematician finds an interesting result through plausible, experimental, and provisional reasoning, the result of this creative work is established definitively with a rigorous proof. It is the certitude given by a proof that makes further results unnecessary. (I have discussed the limitations of the inductive approach in an article from 2008.) The table below summarises the differences between investigations and inquiries as I see them.
August 2013 (updated June 2016)
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