memos for logic, abstraction, structural relevance

Abstraction.tex

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+Some students report getting lost in the activity of abstraction. Two properties were associated with the idea of a solution to a problem: specificity of the problem and relation between the problem being solved, and any problem the students had encountered or anticipated encountering. Some students expressed a preference for solutions to specific problems. Some students express a preference for concrete problems. "I find it easier to work with things that are concrete rather than abstract. " Their immediate concern was to learn to solve a specific problem, before attending to a technique for solving more generally, i.e., solving problems of that type. Some students remark that they have noticed they are better prepared than other students to attend to a technique for a class of problems rather than a specific problem. These more able students explain that they know that a technique can be applied to another specific problem. Some students found pointing out analogies, where one problem instance had commonality with another instance, helpful. Some students observe that analogies are insufficient. In particular, students learning traveling salesman problems might not recognize these as more general problems on graphs. Some students observe that proof writing typically occurred at a level of abstraction higher than a single concrete instance, and that this made the material more difficult. Other students seemed to miss the import of operation at a higher level of abstraction, not seeing the multiplicity of instances of a general notion: "With recursion you can apply the principle to many different algorithms, trees, here, there, everywhere, but proofs, that's it. It's absolute and true you accept it and move on." and "face the fact that a lot of proof writing doesn't have anything to do with a specific problem, a specific circumstance". Combined with the difficulties we see students facing with mathematical formulation, one might consider the possibility that some students might be taking a superficial approach, manipulating formulation without attaching meaning at a lower level of abstraction. So, we have some students operating in concrete but not abstract, we might also have students operating at abstract but not concrete. If this were true it would not be surprising that some students feel anxiety or alarm at having to work with proof. "They gave us sequences and series and summations and they're like, write the equation for this and they were awful."

Memos.tex

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+Marton and Booth describe the idea of structural relevance. Briefly structural relevance is an organization of ideas. This organization of ideas is populated with notions of subject material related to the subject under discussion, whose relationship is important to the student. Thus, to the student of computer science studying proof, it is important that there is a relationship between proof by mathematical induction and recursive algorithms. For the student who knows that recursive algorithms are important, the organization of ideas about recursion can be used as scaffolding for the new concepts of proof by mathematical induction. Using the ideas of long term memory and working memory, we can say that long term memory about the utility of recursion can be used to scaffold ideas newly formed in short term memory. Some student remarks show that some students do not know of any relevance of the material on proving to computer science. They describe the course as not applicable, being instead mainly math oriented. Some opine that use of math is an artifact of the history of development of computer science, from the mathematical training of the originators. Some students believe that material on proof is related to computer science, but do not know of any specific examples. Some students found the interview made them aware of some connections. Instructors report using examples from computer science as proofs are illustrated. Some students reported a transition at which proofs became much more interesting because they were seen to be relevant. Some students are aware that proof can help them ascertain truths about algorithms, and some of these prefer to use spot checking of implementations instead. Others prefer explanatory diagrams as a means of gaining conviction. Some students appreciated that algorithms may be proved correct. One such remark included that a proof of a newly designed algorithm was helpful because a non-working implementation of a proved algorithm necessarily had bugs in the implementation. Some students appreciated that proof made clear what an algorithm could and could not do; it clarified the domain over which an algorithm worked. Some students knew that proofs can be used to show asymptotic resource utilization of algorithms, and lower bounds on asymptotic resource utilization for classes of algorithms. Some students regarded proofs related to algorithms as providing an enjoyable, shifted perspective on the algorithms, more theoretical compared with a practical implementation perspective. Some students felt that proofs came to life in the introduction to the theory of computing, having previously been strictly applied to mathematics rather than computer science. None of the students who were asked whether they initiated the synthesis of proof admitted to doing so. When asked whether starting with proof might be a good way to create an algorithm, students said they had not thought of that but believed it might be useful.
+So we see student opinion spanning from proofs being irrelevant to being co-relevant with implementation and not preferred, to being useful in isolating bugs to implementation vs. algorithm, to providing clarity on domain of relevance of an algorithm, and theoretical perspective on algorithms. Despite acknowledging utility, students claimed they did not avail themselves of proof unless assigned. Perhaps the investment of time and effort in creating a proof seems more than the perceived benefit.

ch6.tex

@@ -1008,6 +1008,8 @@ Some of these students have not yet acquired the perspective that proving theore
 Those students who felt they understood some proofs approached them by checking whether they felt each line of a proof was true.
 Some of these mentioned that a statement should be warranted by previous statements.
 
+{Some students report that a dynamic quality, such as the transformations of data they see in algorithms and in coding, helps them understand, compared to a static quality, that they associate with a proof (as well as with a theorem).}
+
 \begin{figure}[h]
 \centering
 \includegraphics[width=0.7\linewidth]{/home/theresesmith/Documents/2015Fall/Research/Thesis/howThemes}
@@ -1025,6 +1027,14 @@ Some of these mentioned that a statement should be warranted by previous stateme
 
 \newpage
 \section{What do students think  a proof is for?}
+
+{
+It goes together that students who don't generalize/abstract well might not notice any application of proof to what they care about, (presuming they have an idea already, what they care about).  
+I.e., when there is a problem with abstraction, we can expect that some students will not know what is the point, in terms of interest to them.
+They will not find structural relevance on their own.
+Moreover, when we operate at an abstract level, we should remember that some students will not see any connection between that abstraction and a specific example (say, on a test) to which they are supposed to apply it. More generally, they may not be able to see to what context the abstraction applies.
+If they don't have an idea of a suitable context for use, they might not use.
+}
 Some students think that proofs are not applicable to what they do. They think they do not need to know it. Because they do not need to know it, they logically conclude that learning to produce a "proof" procedurally is enough, because, it earns full credit.
 
 Some students combine the learning about proof with the subject matter that is used to exercise proof techniques; they think proof is for demonstrating facts about numbers.
@@ -1076,11 +1086,30 @@ Some students have asked what lemma means.
 Some students knew that lemmas were built for use in larger proofs.
 Some students were interested to hear about Dr. Lamport's structure in proofs.
 
+\subsection{Abstraction/Generalization Relating to Proofs}
+Students recognize a difference between programs and algorithms.
+This recognition is more subtle than a distinction between expression in pseudocode and expression in a specific language.
+Students recognize that a program might be solving a single version of a problem: values bound to parameters can be expected to change (Single assignments to parameters constitute an instance of a problem.), and conditional execution can differ based on the parameters, but there is a sense in which one problem is being solved.
+Algorithms, by contrast, are thought to address a wider domain of problems. Moreover, they are considered material to be reused, either in whole or in part, for more problems as people have insight.
+
+We can compare this study of and reuse of at least parts of with learning to play chess. Students of chess attend to previous games as instructive, as having relevance to future games, as sources of inspiration. 
+
+Taking this attitude toward algorithms and chess games as one, which we might call a pro-synthesis attitude, we can contrast it with an attitude that some students have expressed about proof. Proof is said to be once-only, over as soon as the theorem has been demonstrated.
+
+One thing we might infer from this attitude is that there are some students who are not internally motivated to learn to fashion mathematical arguments. Is it a foregone conclusion that if they wanted to learn to develop such arguments, they would not regard proofs they have been shown to be over?
+
+Some students observe that mathematics includes a language that aids in thought. They also say that this language is infrequently used (by them); some say they find the language of algorithms is sufficient for their purposes.
+
+Perhaps the wealth of defined terms in mathematics is so much larger than the number of keywords in any particular programming language, or in pseudocode, that students perceive a barrier to fluency in even a subset of mathematical discourse.
+
+It may be significant that in programs or algorithms we don't need warrants to justify writing a next statement. It is only the case that the variables on the right hand side should be declared and preferably initialized, and the operations invoked should have those data types in its domain.
+
 \newpage
 \section{What do students think it takes to make an argument valid?}
 Some students, when prompted about rules of inference, felt that when all statement transformations were warranted, an argument was valid.
 Some students stated that, when the target of the proof was true, the proof was valid, converse error.
 
+Some students express an interest in the nature of the feedback on their proof attempts.
 
 
 
@@ -1122,6 +1151,33 @@ Some students, and some instructors, do not emphasize that a single presentation
 
 \section{Combined Description}
 
+The interviews show a dimensions of variation over several capabilities we might wish students to develop, so as to have sufficient achievement levels at proof.
+
+We wish students to be able to recognize when proofs are being given, to comprehend the proofs well enough both to achieve intrinsic conviction from reading them,  and to find them explanatory when the curriculum is using proofs to explain. We wish them to be able to avail themselves of proof synthesis   sufficiently to ascertain whether algorithms they might compose are suitable for a problem they are addressing, and to persuade others, if they wish to publish algorithms or applications thereof.
+
+\subsection{Abstraction}
+
+To recognize a mathematically formulated proof, the student needs to understand the properties of a proof.  Some students claim they have not seen proofs, after taking discrete math and while taking introduction to the theory of computation. We have encountered conceptualizations similar to the "ritual proof" category described by Harel and Sowder\cite{harel1998students}. In our students, lack of practice with abstraction offers a partial explanation.
+
+\input{Abstraction.tex}
+
+\subsection{Structural Relevance}
+
+Among those students who know that proofs are being taught, there are some students who are not sure why the curriculum includes a course on proofs.
+The opportunity to motivate the study of proof techniques by reference to their expected use has been lost for these students.
+Besides motivation, there is the scaffolding offered by concepts about other parts of the curriculum, that could be employed, if students discerned the relevance of proof.
+
+\input{Memos.tex}
+
+\subsection{Logical Deduction}
+
+In order for students to comprehend the argument exhibited in those mathematical proofs they will encounter in the curriculum, it is important that they recognize logical deduction. We know from Almstrum\cite{almstrum1996investigating} that high school students taking the educational testing service's advanced placement exam in computer science have more difficulty with problems involving logic than with those problems that do not. 
+
+The combination of finding something, such as logical deduction, difficult and not being aware it is relevant can be expected, for students whose time is thoroughly committed, to result in avoidance.
+
+\input{logic.tex}
+
+
 \subsection{Generalization meets  preference for examples over definitions}
 Harel and Sowder\cite{harel}, and also Polya\cite{polya} have described a category of conceptualization in which students work with examples,   concrete examples, in preference for definitions. An axiomatic, Hilbert-style approach is not appreciated in this conceptualization.
 When dealing with this conceptualization, 

logic.tex

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+We want the students to apply logic in the process of reasoning. We know from Almstrum that logic is difficult for students preparing for CS major. It is not logical to assume a general rule is true on the basis of a few examples, yet some students have difficulty with this. There is a more general difficulty with quantifiers. Some students seem to have attempted to memorize when counterexamples suffice, rather than to understand. Sometimes students attempt to substitute solution finding over solution creation. Some students feel that use of logic distinguishes proofs they can attempt procedurally (using steps) from other proofs, which they call "laws of logic proofs". Some students know they should not jump to conclusions, but are not always sure how to avoid doing so. Patterns are recognized in use of rules of inference. Proof by contradiction is one such pattern. Students sometimes begin a pattern, or declare a pattern, but do not follow it to its conclusion. For example, a student might start proof by contradiction but omit to conclude that a variable assumed false must therefore be true. Some students wonder whether the law of the excluded middle is always valid (with boolean logic). Some students say they find it difficult to follow the exposition of a proof. It may be that difficulties with abstraction are connected with difficulties logic, because use of logic, application of rules of inference, makes use of abstraction. The rules are presented at a level of abstraction. Some students do operate at the level of abstraction of rules of inference. They call logic cool and beautiful. Interestingly, some differentiate between proofs and "normal math".