Skip to content

Commit

Permalink
Browse files Browse the repository at this point in the history
analysis on how students attempt to comprehend proof
  • Loading branch information
theresesmith committed Oct 19, 2015
1 parent 7b87ea7 commit 28cef1b
Show file tree
Hide file tree
Showing 2 changed files with 54 additions and 7 deletions.
60 changes: 53 additions & 7 deletions ch3.tex
Expand Up @@ -365,7 +365,7 @@ The audio portion of all interviews was collected by electronic recorder and sub
Mathematics, as it is currently practiced, is a delicate interplay between monastic contemplation and blowing stuff up with dynamite. (Ellenberg)\cite[page 223]{ellenberg2014not}
\end{quote}Probably I don't want to keep this, but it's fun at the moment.

Students exhibit an understanding of proof at the "black box" level, i.e., there is understanding of the role of proof, without considering any internal structure . When a proof exists, we can know that the thing the proof proves is true, in the context that applies. We can "use that theorem".
Some students exhibit an understanding of proof at the "black box" level, i.e., there is understanding of the role of proof, without considering any internal structure . When a proof exists, we can know that the thing the proof proves is true, in the context that applies. We can "use that theorem". Other students, though, do not have this idea consolidated yet. For example, if we consider proof by exhaustion applied to a finite set of cardinality one, we can associate to it, the idea of a test. Students, assigned to test an algorithm for approximating the sine function, knew to invoke their implementation with the value to be tested, but did not check their result, either against the range of the sine function, or by comparison with the provided sine implementation, presenting values over 480 million.

Moving the the "white box" level, we find a spectrum of variation in student understanding.
The most opaque end of this spectrum has been called "magic incantation". In this conceptualization we find those ideas of statements that are not clear, and use of mathematical symbols that is not understood.
Expand All @@ -377,18 +377,64 @@ The audio portion of all interviews was collected by electronic recorder and sub
Another waystation on this dimension of variation is "sequence of statements". A more elaborate idea is "sequence of statements where each next statement is justified by what when before". A yet more complete concept is "finite sequence of statements, starting with the premise and ending with what we want to prove, and justified in each step." A more profound conceptualization was found "finite sequence of statements, starting with axioms and premises, proceeding by logical deduction using (valid) rules of inference to what we wanted to prove, that shows us a consequence of the definitions with which we began, an exploration in which we discover the truth value of what we wanted to show".

A few categories, such as those above, serve to identify a dimension of variation. When our purposes include discovering which points we may want to emphasize, we can examine the categories seeking to identify how they are related and how they differ.
It can certainly be that having more categories provides more critical aspects. For example, Harel and Sowder\cite{harel} offered extrinsic and intrinsic conviction, and their most advanced definitional/transformational class as broader categories, and (how many?) useful subcategories of these, yielding (how many?) critical aspects that suggest what teachers could usefully vary, to help learners discern items that would advance their knowledge.

It can certainly be that having more categories provides more critical aspects. For example, Harel and Sowder\cite{harel} offered extrinsic and intrinsic conviction, and their most advanced definitional/transformational class as broader categories, and (how many?) useful subcategories of these, yielding (how many?) critical aspects that suggest what teachers could usefully vary, to help learners discern items that would advance their knowledge.


\subsubsection{Phenomenographic Analysis of How Students Attempt to Understand Proofs}

\subsubsection{Phenomenographic Analysis of How Students Attempt to Understand Proofs}
Some students are attempting to understand proofs while not recognizing that they are studying a proof.
"Were we studying proofs today?" "No" "Were we discussing certain contexts, and why certain ideas will always be true in those contexts?" "Yes" "Doesn't that seem like proof, then?" "Yes"\\
"So, you're taking introduction to the theory of computing this semester. Do you seen any use of proofs in that course?" "No"\\


Some students read proofs.

Some students look up the definitions of terms used in the proofs and some do not.
Some students think (or hope) they can solve problems involving producing proofs, without knowing the definitions of the terms used to pose the problem.
Some students are aware that definitions are given, but "zone out" until examples are given. When examples are given, the students attempt to infer definitions themselves. Some students will compare the definitions they infer with the mathematical community's definitions. Some students do not.

Some students think that the reading of proofs is normally conducted at the same speed as other reading, such as informal sources of information.

Of students who read proofs attentively, some try to determine what rule of inference was used in moving from one statement to the next, and some do not.

Some students notice that lemmas can be proved and then used as building blocks in a larger argument, and some do not.

Some students can identify the forms of proof learned in discrete systems, when they see them employed in proofs, such as the combination of arguing by contrapositive and modus ponens. Some cannot.

Some students can identify these forms in an argument if the argument is made about concrete objects, such as cars or specific people. Some of these students have difficulty transferring this ability with concrete objects to application to abstract entities such as sets, algorithms or symbols.

Some students who achieve with difficulty the ability to recognize the application of rules of inference in one argument about abstract entities, become quicker at recognizing arguments of similar form about other abstract entities, and some continue to achieve with difficulty, as if learning the first argument did not facilitate learning the second argument.

Students would attend to diagrammatic representation of proofs, such as a block digram depicting machine descriptions packaged as input for yet other machines to process, but were not observed to employ such diagrams.

Students have been seen to employ decision tree diagrams.

Students would attend to algorithm representation of proofs, such as a recursive process that determines a prime factorization, but were not observed to employ such algorithmic descriptions.

Except when assigned to do so, students were not observed to attempt to solve simpler problems, such as by imposing partitioning into cases. Except when assigned to do so, students were not observed to attempt to solve more general problems, as is sometimes helpful.\cite[that pin dropping probability problem]{ellenberg2014not}

\subsubsection{Phenomenographic Analysis of Reasons for Teaching Proof}

Excerpts of student transcripts were selected on the basis of being related to this question. A dimension of variation emerged from the data, such that the excerpts seemed readily organized along this dimension.

\begin{table}
\begin{tabular}{|p{7cm}|p{8.5cm}|}\hline
\begin{longtable}{|p{7cm}|p{8.5cm}|}\hline

\endfirsthead

\multicolumn{2}{c}%
{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
\hline \multicolumn{1}{|c|}{\textbf{Category}} &
\multicolumn{1}{c|}{\textbf{Representative}} \\ \hline
\endhead

\hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline
\endfoot

\hline \hline
\endlastfoot

% \begin{tabular}{|p{7cm}|p{8.5cm}|}\hline
Category & Representative\\\hline\hline


Expand Down Expand Up @@ -419,8 +465,8 @@ The audio portion of all interviews was collected by electronic recorder and sub
& I know that recursion has the same structure as proof by mathematical induction. \ldots If I had an algorithm with a recursive data structure like a tree, and I had to prove something like termination about it, I'm not sure what approach I would use, it would depend.\\\hline

Some students see that they could employ proof to explore whether an algorithm can be expected to solve a problem in a given context that includes bounds upon resources that are available for consumption. & mostly design the algorithm first, we had some expectation of what that complexity results would be and then we try to find an approach to prove.
\end{tabular}
\end{table}
%\end{tabular}
\end{longtable}



Expand Down
1 change: 1 addition & 0 deletions preamble.tex
Expand Up @@ -53,6 +53,7 @@
\usepackage{algorithmic}
\usepackage{float}
\usepackage{multicol}
\usepackage{longtable}

\theoremstyle{plain}
\newtheorem{thm}{Theorem}[section]
Expand Down

0 comments on commit 28cef1b

Please sign in to comment.