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Every unsuccessful problem solver is unsuccessful in his or her own way: affective and cognitive factors in proving

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Abstract

It is widely recognized that purely cognitive behavior is extremely rare in performing mathematical activity: other factors, such as the affective ones, play a crucial role. In light of this observation, we present a reflection on the presence of affective and cognitive factors in the process of proving. Proof is considered as a special case of problem solving and the proving process is studied adopting a perspective according to which both affective and cognitive factors influence it. To carry out our study, we set up a framework where theoretical tools coming from research on problem solving, proof and affect are present. The study is performed within a university course in mathematics education, where students were given a statement in elementary number theory to be proved and were asked to write down their proving process and the thoughts that accompanied this process. We scrutinize the written protocols of two unsuccessful students, with the aim of disentangling the intertwining between affect and cognition. In particular, we seize the moments in which beliefs about self and beliefs about mathematical activity shape the performance of our students.

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Notes

  1. The full text of the questionnaire is reported as an appendix in (Hadamard 1954).

  2. There is evidence that already in 1914 Hadamard and Polya worked together.

  3. It is well known that there is a problem of terminology, see the discussion in the book (Leder et al. 2002). We adopt the view of McLeod (1992), who claims that “the affective domain refers to a wide range of beliefs, feelings, and moods that are generally regarded as going beyond the domain of cognition. H. A. Simon (1982), in discussing the terminology used to describe the affective domain, suggests that we use affect as a more general term; other terms (for example, beliefs, attitudes, and emotions) [are] used as more specific descriptors of subsets of the affective domain.” (p. 576)

  4. We underline the difference between the representational systems cited before (affective system, cognitive system, ...) and the representations as semiotic resources (symbols, graphs, ...).

  5. In (Harel & Sowder 1998) a person’s proof scheme consists of what constitutes ascertaining and persuading for that person. These schemes are grouped in three main classes: external (ritual, authoritarian, symbolic), empirical (inductive, perceptual), analytical (transformational, axiomatic). In this study we will be concerned with ritual proof scheme (mathematical arguments are judged only on the basis of their surface appearance) and symbolic proof scheme (proof is carried out using symbols without reference to their meaning).

  6. Cognitive units are defined by Barnard and Tall (1997, p. 41) as “a piece of cognitive structure that can be held in the focus of attention all at one time”.

  7. These students follow a curriculum which is oriented to the profession of mathematics teacher in secondary school. To become a mathematics teacher it is necessary to have a degree in scientific or technological disciplines and afterwards to follow a two years training program.

  8. Two proofs are reported in the Appendix.

  9. The case study of Frank is performed through the analysis of different sets of data: questionnaire on beliefs, on-line motivation questionnaire on the problem, video of the think aloud solving process, video based stimulated recall interview.

  10. We point out that when we mention panic, we do not refer to an attitude towards mathematics in general, rather we refer to a state of feeling confined to the moment of resolution of the given problem.

  11. We note that Booh uses the term least rather than greatest, but he refers to divisors and not to multiples.

  12. Indeed, the definition of coprime numbers requires also the fact that 1 is the only common factor (this means that it is the greatest common factor, and not the least!).

  13. Leo Tolstoy, Anna Karenina (1877; trans. Constance Garnett).

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Acknowledgements

Research program supported by MIUR (PRIN 2005019721_002 ‘Meanings, conjectures, proofs: from basic research in mathematics education to curricular implications’).

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  1. Dipartimento di Matematica dell’Università di Genova, via Dodecaneso 35, 16146, Genoa, Italy

    Fulvia Furinghetti & Francesca Morselli

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  1. Fulvia Furinghetti
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Correspondence to Fulvia Furinghetti.

Appendix: Two proofs of the statement

Appendix: Two proofs of the statement

Proof by contradiction:

Let m and n be two coprime numbers.

suppose that m and m  +  n are not coprime. Then there exist a  ∈  N, a  ≠  1, p, q  ∈  N such that m  =  ap, \(m + n = aq\). Since m  +  n  >  m it is q  >  p.

Then \(m + n = ap + n = aq\)

$$\begin{array}{*{20}l} {aq - ap = n} \hfill \\ {a\left( {q - p} \right) = n.\,{\text{Since}}\,q > p,\,q - p\,{\text{is}}\,{\text{a}}\,{\text{natural}}\,{\text{number}}{\text{.}}} \hfill \\ \end{array} $$

Hence m, n are not coprime, which contradicts the hypothesis.

Direct proof:

Let m and n be two coprime numbers.

Let a∈ N be a common factor of m and m  +  n

Then, there exist p, q ∈ N such that m  =  ap, \(m + n = aq\). Since m  +  n  >  m it is q  >  p.

Hence \(m + n = ap + n = aq\)

$$\begin{array}{*{20}c} {aq - ap = n} \\ {a\left( {q - p} \right) = n.} \\ \end{array} $$

Since q  >  p, q−p is a natural number. Thus, a is a common factor of m and n.

Since m, n are coprime, a must be 1.

Then, if a is a common factor of m and m  +  n, a is 1, that is to say m and m  +  n are coprime.

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Furinghetti, F., Morselli, F. Every unsuccessful problem solver is unsuccessful in his or her own way: affective and cognitive factors in proving. Educ Stud Math 70, 71–90 (2009). https://doi.org/10.1007/s10649-008-9134-4

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