Report

When students first take up proof at university, there are some who are not so sure what the main message is. They might react to arguments about concrete entities that they would have known, with confidence, that conclusion without argument. To some, it is not clear that it is the argument, to which they are trying to direct their attention. Some students use argument forms “It must be x, because if it were not x some contradiction would occur.” We can infer from their reactions to the teaching of proof, that it is not the proof technique that claims their attention. (“Do programmers have to know these prime number facts?”) Students have a difficulty moving from the stage of an argument reasoning about concrete entities, to the stage of an argument reasoning about abstract entities. It could be generalization from the concrete to the more abstract, but it could also be, not paying attention or resisting paying attention, or not making the effort to pay attention, to the argument.

Even for students who are conscious that the subject is an argument, they are not so clear how to make one, especially if it is not one of those for which there are process steps (“Logic proofs”). Some students are also not clear on why the product of the process steps forms a convincing argument.

Moreover, not all students are clear that a proof shows unequivocally, suggesting that the proof could be “backed up by experimental data”.

Also, some students say they do not know what they would use a proof for. If it were not unequivocal, then what benefit would it have over code? Another insight contributing to this view is that students offer alternative activities to proof, that produce examples. When a statement to be shown is an existential statement, an example is fine. For disproving a universal statement, a counterexample is fine. For proving a universal statement, unless it is a generic particular, the example is not adequate. This casts doubt over whether students are really understanding universal quantifiers. There is other evidence that students have trouble with quantifiers, as some students do not succeed in negating statements with quantifiers.

Student have been disappointed that all the applications of proofs that they saw were about things (theorems) that were already known (by someone) to be true. Some students, at least early in their careers as students, think proofs are supposed to show some new fact, and are disappointed to see that the product is (“only”) a different representation of an implied fact. Students seem not to appreciate what benefit a new representation can have.

Also, students do not necessarily see the connections between multiple represen- tations. When data structures are taught with tree diagrams for binary search trees, some students do not know how to represent the same idea in code.

Some students also do not see any connection between the tree diagram and recursive definitions and proof by mathematical induction.

Some students do not see any benefit in spending time on UML R _{diagrams, either} class hierarchy, components or sequence diagrams.

Some student learn how to manipulate mathematical formulation without knowing how to apply mathematical formulation to problems they may encounter (such as discovering whether a context supports the use of an algorithm).

Maybe students are not noticing that proofs for universal statements are possible, as well as proofs for existential statements (“Think code is perfectly satisfactory for the purposes for which we have seen proofs applied.”)

Some students are not aware that proof can be used to determine whether a situation is suitable for the application of an algorithm they have learned. One consequence of this is that they make less use of these algorithms they have learned than they otherwise might.

The students who were dual majors in math knew that proofs were arguments by logical deduction, that careful definitions enabled the construction of arguments, and that the results of proof were unequivocal.

One graduate student, whose undergraduate education was elsewhere, who was not a mathematics major, used proofs in papers, because it was required in that publishing venue. She said she did not use proofs at any other time. She developed algorithms without using proofs, and in the process of publishing the algorithms, furnished proofs, adjusting the algorithm if the proof seemed as if it would be too long.

Chapter 6

Validity and Reliability

In the first section of this chapter we address the components, in the goals of va- lidity and reliability, that are appropriate, according the Denzin and Lincoln[65], for research in the social constructivist paradigm. In the second section we highlight those portions of our methodology which were directed towards those goals. In the third section we describe these efforts as they took place in support of some published papers and a manuscript in preparation.

6.1

Introduction

Several researchers have written advice about how to pursue validity and reliability. Correspondingly, several requirements exist, at differing degrees of formality, for a qualitative research study to be valid.

Denzin and Lincoln[64, 163, 164, 65] are widely cited; we have adopted their advice. We also follow the recommendations of Merriam[190].

This is intended to differ from quantitative research, in which the perspectives of the majority are thought to predominate, discounting viewpoints contributed espe- cially by minorities.

Following the advice of Merriam[190], in this section we report on how (using the methods described in Chapter 3) we developed a degree of confidence in our findings, through the multiplicity of practices and perspectives we used, including the software programs we used to manage and organize data, the parallel pipelined nature of collecting and analyzing data as the research proceeded, and our style of inductive and comparative analysis.

6.2

Validity and Reliability Goals in the Social