Section (iii) Mathematical Induction and the Triangle Inequality: Cultivating More Fruitful Uses of Intuition and Hindsight as Features of the Shift to More Expert Mathematical Practices
Context: See
Extract 6.3Structure: In the following the tutor presents a 'back-to-basics' proof of the Base Statement of the Mathematical Induction for the triangle inequality as opposed to the geometric proof given in the lectures. The students then discuss the role of the £ sign and the necessity to start this inductive proof from n=2 instead of n=1.
The Episode: A Factual Account. See
Extract 6.3An Interpretive Account: The Analysis
A Cognitive Shift Towards More Rigorous Mathematical Practices. This Episode, mostly through the tutor's exposition on
• the need to reduce the interference of intuition with the construction of mathematical arguments, and,
• the foundational requirement for economy of principles and conciseness in mathematical reasoning,
exemplifies the change of mathematical culture that the novices are required to go through. The moral for the novices here is captured in the tutor's words on intuition: 'it's the right way to understand it and remember it, but it's not the right way to prove it'. The substantially different role of intuition in the expert and the novice mathematician's mind is illustrated in this Episode if one juxtaposes the left and right columns of the following table:
|
Jack's and Andrew's reference to the lecturer's use of a geometrical picture in his proof for the triangle inequality for two numbers |
(a) the tutor's use of the same picture — as a tool for comprehension — in his response to Andrew's question on the meaning of £ in the triangle inequality and (b) the tutor's pragmatic response to where the proof by mathematical induction of the triangle inequality should start from (the triangle inequality 'is a basic assertion for two. It then has some information in it. For one it doesn't') |
For Jack and Andrew the lecturer's picture is a convincing argument for the case n=2 whereas for the tutor it is a vivid illustration of how the triangle inequality works and only a starting point for the mathematisation of the argument (the tutor looks at the proof for n=2 in view of the prospective proof for n=k).
A Mathematical Mind in Genesis. Like in Extract 6.1, Andrew's evolving mathematical persona emerges as that of
• a competent executioner of mathematical operations. The competence demonstrated in Andrew's elaboration on the eight possibilities in the proof of the triangle inequality for two numbers is remarkable: not only does he point out to the tutor the possibility of reducing the number of cases to be written down but also helps the tutor realise what this number is.
• a practitioner. When the tutor criticises the students' use of squaring and of a geometric picture for the proof of the triangle inequality, Andrew's first reaction is to wonder whether this 'is not the right thing to do'. The strong practical action-centred semantics of words such as 'thing' and 'to do' and in particular the use of the word 'right' is an indication that Andrew is in search of rigorous recipes.
• an observant formalist. These are the two observations that Andrew makes in this Episode that initiate dynamic interaction among the tutor and the two students:
Observation 1. In A1-A3 Andrew seems to think that the £ symbol in the triangle inequality for n=2 is used redundantly because what actually is intended is to say that, depending on the sign of x and y, |x+y| is equal to one of ±(x±y), whereas £ gives the impression that the whole range of values less than |x|+|y| is covered.
Observation 2. In A6 he wonders whether starting the proof by Mathematical Induction from n=1 invalidates the proof which the tutor suggests they start from n=2.
As far as O1 is concerned, as Jack and the tutor observe, £ might be 'discarding information' but 'it is true' as well. It is also a representation that can be uniformly transferred to the context of complex numbers and vectors. Therefore by hindsight it is a functional and economical symbol to use. Similar lack of mathematical experience seems to have led Andrew to O2: the general form of Mathematical Induction allows the proof of statements for all nÎ À where n>k and k is a natural number. Both students in this Episode seem to think that Mathematical Induction compulsorily covers all natural numbers: hence Jack started his proof from n=1. The idea seems to linger in Andrew's mind in A7: he still seems to believe that to use the statement for n=2, one has to prove the statement for n=1.
In any case Andrew's observations are acute and insightful and, even though they are not flawless, they convey an impression that Andrew is being efficiently enculturated into the new status-quo of rigour and precision of university mathematics. In other Episodes, nevertheless, evidence is provided of how a similar attitude, when taken to pedantic extents, occasionally has hampered his efficiency and his mathematical growth.
Conclusion: In the above, the novices are in need of, and engaged in, a cognitive shift from unrigorous to rigorous forms of mathematical thinking. In particular a discussion of the meaning of £ in the triangle inequality and the Base Clause of Mathematical Induction gave rise to the students' metamathematical queries and highlighted how mathematical experience empowers hindsight as well reinforces a more fruitful use of intuition.