4.4.1
Categories
The categories, developed in the traditional phenomenographic analysis, are listed in Table 4.4.1 and depicted in Figure 4.4.1.
4.4.2
Illustrative Quotations for Categories
Illustrative quotations for the categories are shown in Table 4.4.2.
This data comes from analysis of tests and interviews of students and teaching assistants.
Students claim to apply proof by mathematic induction without reference to the problem statement, citing reasons including “They taught us that the most.” and “I know I can carry out that process, it sort of checks itself.”
Teaching assistants state that proofs from class notes are provided as answers even when they do not match the problem that is posed.
Students sometimes attempt to adapt a pattern, but in an incorrect fashion. For example, a proof by contrapositive is blended with the notion of proof by contradic-
Figure 4.4.1: Outcome space from How Students Attempt to Apply Proofs When Assigned
Table 4.4.2: Illustrative Quotations for How Students Attempt to Apply Proofs (When Assigned)
Category Quote
Improve Effi-
ciency
My former proofs were wandering, I laugh at them now. Attempt Trans-
formations
I like proofs that are procedures, I don’t like “logic proofs”
Adapt Known
Proofs
a proof by contrapositive is blended with the notion of proof by contradiction, and some property is assumed for the in- verse of an implication, resulting in an attempt to derive a conclusion from a proof of an inverse.
Use Known Proofs
They taught us that (induction) the most.
I know I can carry out that process, it sort of checks itself.
tion, and some property is assumed for the inverse of an implication, resulting in an attempt to derive a conclusion from a proof of an inverse.
Students who succeed in proving report that over time they improve their proof attempts, and sometimes find their older proofs amusing for their wandering nature.
4.4.3
Relations
The relations are shown in Table 4.4.3.
The relation between use of known proofs, and the modification of a proof, is that modifications are conceivable.
The relation between working with modifications of patterns, and applying rules of inference outside the template of a pattern, is to step away from patterns.
The relation between creating sequences of application of rules of inference, and being concerned with short proofs is to develop a sense of efficiency.
Table 4.4.3: Relations for How Students Attempt to Apply Proofs (When Assigned)
Category Relation
Improve Efficiency View the transformations as a trajec-
tory that can be more direct. Attempt Transformations
Attempt Transformations Step away from patterns.
Adapt Known Proofs
Adapt Known Proofs Create a modification.
Use Known Proofs
4.4.4
Critical Factors
The critical factors are shown in Table 4.4.4.
The earliest critical factor is for the benefit of those students who have trouble even beginning a proof. They can start by examining the problem statement. It might resemble a problem they have seen solved. This can be illustrated by teaching solution of several problems that differ in concrete but not structural ways, urging the student to generalize.
The next critical factor is that proofs can be created, possibly by modification of a known pattern. Showing a set of proofs that differ only slightly, and involving class participation in creating modifications uses contrast to help students discern this point.
The next critical factor is that it is not necessary to use a pattern, that rules of inference can be used when they are justified. Contrasting those rules of inference that are justified from those that are not, and contrasting those rules of inference that are applicable in any given situation, from those that are not, might help students discern this constrained freedom.
Table 4.4.4: Critical Factors for How Students Attempt to Apply Proofs (When Assigned)
Critical Factor
View the transformations as a trajectory that can be more direct. Step away from patterns.
Create a modification.
There are students do not know how to begin; for them, a critical factor could be that one can examine the structure of the problem statement.
necessary. Contrasting long proofs with short ones, demonstrating the same claim, should help students discern this.
4.4.5
Dimensions of Variation
The dimension of variation begins with the student reiterating what has been shown in class. Deepening understanding is shown by evidence of modification of what has been shown, as far as a variation on a pattern that has been shown. Further understanding of the logic of transformation is demonstrated by correct (warranted) application of rules of inference. The most advanced conceptualization found includes that students can judge their proof attempts as to length, and seek improvements with more efficient transformations.
4.4.6
Validation
4.4.7
Outcome Space
The outcome space from the research question “How do students attempt to apply proof when assigned?” has as a foundation to use known proofs. This conceptualiza- tion was discovered from interviews of teaching assistants who reported that students submitted, as solutions to requests for proof, the same proof used in class, even though it was a proof of something else.
A critical factor for students to advance to the next level of understanding is that they can learn how to make valid modifications to existing proofs.
The conceptualization “Adapt Known Proofs” expands the understanding that proofs can be re-used. It allows students to see previous proofs as candidates for patterns, that can be modified. It allows that by making one or a succession of small valid changes, a student may begin to be creative in proof design.
A critical factor for students to advance to the next level of understanding is that by understanding what is a valid transformation, one does not need to depend upon patterns, however useful they may be.
The conceptualization “Attempt Transformation” incorporates the additional no- tion that one can create a proof without relying on a pattern. Thus, the critical factor that knowing valid rules of inference allows one to create valid proofs “from scratch” is a critical aspect, as well.
A critical factor for students to advance to the next level of understanding is that proofs carry out a succession of transformations that can make use of more or fewer steps, and that shorter proofs are often preferred.
The conceptualization “Improve Efficiency” implies an evaluative capacity on the part of the creator of a proof, that can compare two proof attempts on the basis of
Table 4.5.1: Illustrative Quotations for How Students Attempt to Apply Proofs (When Not Assigned)
Category Quote
Don’t seriously
apply proofs
outside of assign- ments
Q: do you ever decide on your own that you want to do a proof? A: no, I just tend I tend to just write code, it’s always been proof enough for me
Q: Do you ever find yourself doing proofs? associated with computer science? that haven’t been assigned? A: That have not been assigned? Q: Right, for fun, or because you want to know something? A: um, he-he, well, i did find myself doing proofs, they were silly proofs, just like things about like things stuff up, yeah since i didn’t have very solve it base, it was just like statements, not really just proof, just where you want to get to, so like the end result that you want to get to
4.5
Phenomenographic Analysis of How Students
Attempt to Apply Proofs (When Not Assigned)
4.5.1
Categories
There is only one category for student responses to this question. They do not attempt proofs when not assigned.
4.5.2
Illustrative Quotations for Categories
Some students claimed they never constructed proofs when not assigned. Illustrative quotations for the categories are shown in Table 4.5.1.
Table 4.6.1: Illustrative Quotations for Whether Students Exhibit Consequences of Inability With Proof (such as avoiding recursion)
Category Quote
Non-use of recur- sion
and then you split, what do i have to do to get to that point, so you have to actually find what are the required pieces for you to solve the problem so and every single piece, then you have to prove by itself, so that can i get to the second step, can i get to the third step, because if i lose the first proof, i will never get to the second, because i already established that my second piece depends upon my first piece, so i cannot move forward so i have to divide into small pieces and try to prove them.
Lack of confidence
I still feel very shaky with proofs, sometimes, still getting the hang of it, it hasn’t become second nature to me
it’s like I kind of understand like I can see why this would take how long it is but I don’t feel it very solid