By Henry Joseph Monck Mason

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I clarify that, in the context of the proposed categorization, the number of cases involved in a task refers to the cases covered in the relevant statement that is mentioned in the task—as opposed, for example, to the number of cases that one might have to consider in constructing a proof for that statement. 1 A classification of proving tasks with illustrative examples. Number of cases involved in a proving task Purpose of a proving task Justification of a statement Refutation of a statement A single case (a) Prove that 33 plus 36 is a multiple of 3 (b) Prove that 73 is a prime number (c) Disprove that 33 plus 36 is a multiple of 6 (d) Disprove that 51 is a prime number Multiple but finitely many cases (e) Prove that the sum of any two multiples of 3 between 30 and 50 is a multiple of 3 (f) Prove that there are four prime numbers between 10 and 20 (g) Disprove that the sum of any two multiples of 3 between 30 and 50 is a multiple of 6 (h) Disprove that there are as many prime numbers between 1 and 10 as there are between 21 and 30 Infinitely many cases (i) Prove that the sum of any two multiples of 3 is a multiple of 3 (j) Prove that six times any prime number is a multiple of 3 (k) Disprove that the sum of any two multiples of 3 is a multiple of 6 (l) Disprove that the sum of any two prime numbers is a prime number a prime number.

The Set-up of the Investigation In this chapter I set up the investigation that I undertake in subsequent chapters. 1 I also describe the data sources and analytic method used in the book. Some of the issues discussed in this chapter can be relevant throughout school mathematics, though most of the examples I offer relate to the elementary school level. A Categorization of Proving Tasks A major finding in the literature on whole-number arithmetic has, I believe, broader implications for the design of mathematics tasks.

The episode presents a situation where students’ exploration of a single case led naturally not only to a proving task concerning that particular case, but also to a proving task concerning the broader class of cases to which the particular case belonged. ” I invited everyone in the class to consider the validity of Sam’s decisive assertion and to see if they could explain why he seemed to be so sure. The question I was asking was, How does he know that is true? ” And Carl responded, moving to the level of a mathematical generalization, “You don’t have to do that.

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