Interpretation of Analysis from Section 4.10

Here we interpret the results from section 4.10, see Figure 4.10.1. For the conve- nience of the reader, the categories were, from earliest to most advanced, did not yet recognize that argumentation by logical deduction is the subject when learning how to prove, followed by the more comprehensive conceptualization that includes the idea of reasoning, but (significantly) encapsulated within processes that can be reused without necessarily understanding; followed by a more comprehensive concep- tualization in which proofs can be constructed outside of patterns, but the process of warranting is not soundly understood; followed at last by understanding justification of steps and valuing efficiency.

It seems worthy of note that some students do not see, at least not right away, that argumentation is being taught.

Some students do not pay attention evenly to the material presented in lecture. Some students choose to pay more attention to examples, and even ignore definitions. It can easily be problematic for students if they choose naively the material to which they attend most.

Argumentation

One interpretation, consistent with the data, is that students don’t pay as much attention to the nature of a proof as an argument as they pay attention to the apparent

similarity to programs.

When a proof has a form that students can memorize, such as proof by mathemat- ical induction, they can memorize the form and write proofs as sequences of steps, as they describe. When a proof does not have this form, they know they need to use logic, but have been inarticulate about how they use logic. They do not speak of warrants, and they do not mention finding structure, either in proofs they are trying to understand or in statements they are trying to prove, and no student related the structure of the statement they were trying to prove to the structure of the proof they were constructing.

By feeling uncomfortable with logic, not paying attention to definitions, and not noticing structure, and not having the purpose of constructing an argument, they are at a disadvantage in creating proofs.

It could be that students are scaffolding their learning of proof with their knowl- edge of programming. Though the Curry-Howard isomorphism provides strong sup- port for scaffolding learning of proof by studying how compilers and/or interpreters transform programs from source code into more platform-specific expressions, this is not generally the approach to this scaffolding that students are taking. Instead, we find some students observing that some proofs, for example, proofs by mathematic induction, seem to proceed according to a process. We have seen the student concep- tion “logic proofs”, placed in contrast with proofs, production of which can seem like a process. One significant difference is the creativity called for by the so-called logic proofs, compared with the process-like proofs. Another difference is that a proof con- structed by following a process might appear successful in an assessment even though unconvincing to its writer.

provided by every available rule of inference. This impractical approach, when tried, suggests that some students might not, early in their practice, understand the syntax of the logical formulations with which they are working. This would be consistent with the work of Almstrum[8], and observations by Sheehy (reported herein).

When we ponder a preference for examples over definitions, in the context of scaffolding learning of proof with learning of programming, the utility of examples in learning programming is of interest. Examples, such as “Hello, world” are commonly used in learning programming.

It may be useful to consider an approach to definitions from examples described by Carnap[48, page 137]. He describes starting with the idea of parallel lines in a fixed plane. From the idea of parallel lines, equivalence classes of parallel lines can be obtained. From the idea of differing equivalence classes, the idea of direction can be abstracted.

This furnishes an example of a dimension of variation. By noticing that direction can vary, the concept of the direction of a parallel line is made distinct from the concept of parallel lines. We may conjecture, in the manner of Marton and Pang[177] that, if every parallel line had the same direction, no distinction between lines being parallel and lines having the direction would occur.

If we were to build upon the preference of some students to work with examples, or even to appreciate the task they take upon themselves when they attempt to proceed in this way, we might wish to explore the nature of learning about proofs through examples.

To introduce proof techniques, we use domains such as natural numbers, integers, and graphs. This is deliberate, so that time need not be spent acquainting students with the domain from which examples are chosen. We use these domains in instruction

and assessment.

In later courses we employ these proof techniques with other domains, such as grammars and state machines.

Argumentation Without Much Reasoning

Whether or not students have accepted that reasoning is what they are doing, they can learn to assemble arguments from parts of arguments. Even when they do accept that reasoning is what they want to do, they do not necessarily understand the reasoning that is inherent in the proof pattern they are constructing. Especially when they can get a full score for creating a proof, they might not prioritize finding out what the reasoning is, that makes their successful construction a proof. Even when they want to know how the construction proves the proposition, they may not find time to address their uncertainty.

Uncertain About Justification

For proofs that are not created using a pattern, what some of the students call “logic proofs”, reasoning must be used. In this conceptualization, students understand that individual rules of inference are allowed steps; students may not understand when a rule may be used, or when application of a rule is apt to help them in their goal. Semantics are sometimes helpful in warranting proof steps. When students are not appreciative of the role of definitions, they may be hindered in creating steps in proofs.

Have the Idea, Need Practice

With this conceptualization, students are conscious of deliberate reasoning expressed in mathematical formulation, using inference rules warranted by the circumstances. Some students expressed that they had seen their ability to write proof improve over their academic careers, and expected continued improvement.