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\chapter{Data Analysis and Interpretation}
\section{Analysis}
\section{Interpretation}
Here I want to put the ideas about definitions and abstraction. Without abstraction
definitions are more cumbersome to remember and operate with. This
discourages use of the axiomatic proof conceptions, because they are based on
definitions.
What about Valiant? His establishing of definitions in circuits in the mind by
conjunctions, and by disjunctions, of ideas. Without abstraction for definitions,
this is more cumbersome.
Is intuition helping or opposing our educational objectives? Can we get help from it?
\subsection{Productive and Counterproductive Beliefs}
What do they ``know'', and what do they ``know that isn't so''.
Some will be conscious, some will be unconscious.
\subsection{Productive and Counterproductive Momentum}
What are they trying to learn? Is it aligned with the departmental curriculum? the course goals?
\section{ Maybe duplicative?}
Three papers in this area were published:
\begin{itemize}
\item CCSCNE: Categorizing the School Experience of Entering Computing
Students \cite{smith2013categorizing}
\item FIE: Mathematization in Teaching Pumping Lemmas \cite{smith2013mathematization}
\item Koli Calling: Computer Science Students’ Concepts of Proof by Induction\cite{smith2014computer}
\end{itemize}
\subsection{Categories of Experience of Entering Students}
Undergraduate students beginning study of the computing disciplines present
various degrees of preparedness.\cite{smith2013categorizing} Some had no experience, some had had
informal experience, and some had had formal classes. The formal classes
extended fromusing applications to building applications. Informal experience
ranged from editing configuration files, such as background colors, to full time
jobs extended over multiple summers.
After publishing this paper, we encountered more related information. For
example, consistent with the work of Almstrum\cite{almstrum1996investigating}, we found that, about implications,
some students, who do know that any statement must and can, be
Table : Some example errors
Let x be empty
$|xy| \leq p, so xy = 0^p$\\
$|xy| \leq p; let x = 0^{p+r}, y = 0^{p+r}, 0 < r < p$\\
Let’s choose $|xy| = p$\\
$0^{p+1}0^b1^p \neq 0^{p+1}1^p \therefore xy^2z \not\in \mathcal{L}$
where $\mathcal{L} = \{0^i1^j, i \neq j\}$\\
we choose $s = 0^{p+1}1^p$ within $|xy|$\\
thus $\neq 0^p1^{p+1}$\\
Let $x = 0^a, y = 0^b1^a$\\
$x = 0^{p-h}, y = 0^h$\\
$x = 0^i, y = 0^i, z = 0^i1^j$
either true or false, thought implications must be true. Some interview participants
enjoyed a modified Moore method\cite{cohen1982modified} geometry class in middle school,
and relished opportunities to create proofs (not yet published). Other students
were not so well prepared.
\subsection{Representation/Symbolization in Pumping Lemmas}
We found that some students may lack facility in notation. For example, in the
application of the pumping lemma, students are expected to understand the
role of $i$, in the context that a string $s$, having component substrings $x$, $y$ and $z$,
can be used to generate other strings, of the form $xy^iz$, where $i$ gives the number
of copies of the substring $y$. Moreover, students are expected to understand that
the subdivision of a string of length $p$, $\sigma_1^a\sigma_2^{p-a}$, where $a \in \{0,1,\ldots,p\}$
uses $a$ as a parameter, a free variable, not one necessarily bound to a single instance
of a natural number, but a representation of a domain. An excerpt of the errors found on tests is shown in Table . Trigueros et
al. \cite[ p. 3]{jacobs2008developing} have observed that ``students are often unclear about the different
ways letters are used in mathematics''. We have seen this
lack of understanding in a situation in which it was proposed as evidence that
a single example, namely $\sigma_1^a\sigma_2^{p-a}$, formed a proof for a universally quantified
statement.
Some of our results were consistent with the framework described by Harel
and Sowder in 1998\cite{harel1998students}. We found students holding conceptualizations that
Harel and Sowder's 1998 model calls symbolization. Harel and Sowder have
identified another category of conceptualization, that correctly applied transformation
and axiomatic arguments. Some students expressed enthusiasm for
the power that inheres to building arguments with carefully specified component
ideas, in particular how the absence of ambiguity permitted arguments to
extend to great length while remaining valid. Not all of the students had developed
axiomatic conceptualizations of proof. About definitions, we collected
preliminary data on students' conceptualizations of definitions used in proofs.
Some students thought definitions were boring. Some students thought that
they could infer definitions from a few examples. Concerning executive function,
we found that some students do not state the premises clearly, and some
students did not keep track of their goal. About rules of inference, we found
Figure 5.3.1: Some categories / conceptualizations found among students of
introduction to the theory of computing, and published at FIE.
that some students apply invalid approaches to inference.
\subsection{Abstract Model for Proof by Mathematical Induction and Recursion}
Far from finding agreement that (a) theorems are true as a consequence of
the definitions and the premise, and that (b) proofs serve to show how the
consequence is demonstrated from the premise, axioms and application of
rules of inference, instead we found a variety of notions about proof, including
the well-known procedural interpretation \cite{tall2008transition,weber2004traditional,tall2001symbols}, and the well-known empirical
misconception \cite{harel1998students}. The conceptualization that definitions are not necessarily
of interest compared with the procedures seemed different in kind from the
concept image / concept definition discoveries of R\"osken et al. \cite{rosken2007integrating}.
Interviews with students revealed that some students saw generation of a proof
by mathematic induction as a procedure to be followed, in which they should
produce a base case, and prove it, and should produce an induction step, and
prove that. This was consistent with Weber [?, p. 4-426] who has stated ``in
the studies that I conducted, it was more often the case that undergraduates
applied procedures that were not meaningful to them.'' He went on to give a
quotation from a participant [?, p. 4-426] ``And I prove something and I look at
it, and I thought, well, you know, it's been proved, but I still don't know that I
even agree with it [laughs]. I'm not convinced by my own proof!'' Some of the
students interviewed did not know why this procedure generated a convincing
argument. Polya[?] has written a problem involving all girls being blue-eyed;
a similar problem appears in Sipse \cite{sipser2012introduction} about all horses being the same color.
The purpose of this exercise is to make students aware that the truth of the
inductive step must apply when the base case appears as the premise. In some
cases, this point was not clear to the students.
Students' conceptualizations of proof by mathematical induction can support
their choice to apply recursive algorithms. One student reported success at
both mathematical induction and recursive algorithm application without ever
noticing any connection. This student opined that having learned recursion
with figures, and proof by mathematical induction without figures, that no
occasion for the information to spontaneously connect occurred. Students reporting
ability to implement assigned problems recursively, but not the ability
to understand proof by mathematical induction also reported that ability to
write recursive programs did not result in recognition of when recursive solutions
might be applicable in general. Students reporting ability to implement
assigned problems recursively, and also the ability to prove using mathematical
induction also reported preferring to implement recursive solutions in
problems as they arose.
Our work on students' choices of algorithmic approaches was consistent with
work by other researchers in computer science education\cite{booth1997phenomenography} on conceptualizations
of algorithms. Our work served to unify that of mathematician educators
with computer science educators, by providing a plausible explanation why
the conceptualizations of recursive algorithms that were found, might exist.
Figure 5.3.2: Categories of Student Conceptualizations of Proof by Induction
that Recursion Works
Three papers in this area were published:
\begin{itemize}
\item CCSCNE: Categorizing the School Experience of Entering Computing
Students \cite{smith2013categorizing}
\item FIE: Mathematization in Teaching Pumping Lemmas \cite{smith2013mathematization}
\item Koli Calling: Computer Science Students' Concepts of Proof by Induction \cite{smith2014computer}
\end{itemize}
Table 5.3.3: The Outcome Space for Proofs by Induction
Category Description
1Following procedure The method is learned, without understanding
2Understands base case The idea that a base case is proved by an existence
proof, often with a specific example
3Understands implication The idea that an implication is proved by
assuming the premise is not used
4Does not understand connection
Sees the implication and proves it well, but
does not anchor the succession to a base
case
5Does understand the argument
Understands the argument
6Knows why recursion
works
Can tailor the argument to explain recursive
algorithms
7Appreciates data structures
supporting recursion
Can see the benefit to algorithm from recursive
data structure
\subsection{Categories of Experience of Entering Students}
Undergraduate students beginning study of the computing disciplines present
various degrees of preparedness\cite{smith2013categorizing}. Some had no experience, some had had
informal experience, and some had had formal classes. The formal classes
extended from using applications to building applications. Informal experience
ranged from editing configuration files, such as background colors, to full time
jobs extended over multiple summers.
After publishing this paper, we encountered more related information. For
example, consistent with the work of Almstrum\cite{almstrum1996investigating}, we found that, about implications,
some students, who do know that any statement must and can, be
either true or false, thought implications must be true. Some interview participants
enjoyed a modified Moore method\cite{cohen1982modified} geometry class in middle school,
and relished opportunities to create proofs (not yet published). Other students
were not so well prepared.
\subsection{Representation/Symbolization in Pumping Lemmas}
We have found students holding conceptualizations
that Harel and Sowder's 1998 model\cite{harel1998students} calls symbolization: We have found
that some students may lack facility in notation. For example, in the application
of the pumping lemma, students are expected to understand the role of $i$,
in the context that a string $s$, having component substrings $x$, $y$ and $z$, can be
used to generate other strings, of the form $xy^iz$, where $i$ gives the number of
copies of the substring $y$. Moreover, students are expected to understand that
the subdivision of a string of length $p$, expressed as $\sigma_1^a\sigma_2^{p-a}$, where $a \in \{0,1,\ldots,p\}$
uses $a$ as a parameter, a free variable, not one necessarily bound to a single instance
of a natural number, but a representation of a domain.
Trigueros et al.\cite[p. 3]{jacobs2008developing} have observed that ``students are often unclear about the different way letters are used in mathematics.''
We saw this
lack of understanding in a situation in which it was proposed as evidence that
a single example, namely $\sigma_1^a\sigma_2^{p-a}$, formed a proof for a universally quantified
statement. An excerpt of the errors found on tests is shown in Table .
Table : Some example errors\\
Let x be empty\\
$|xy| \leq p, so xy = 0^p$\\
$|xy| \leq p; let x = 0^{p+r}, y = 0^{p+r}, 0 < r < p$\\
Let’s choose $|xy| = p$\\
$0^{p+1}0^b1^p \neq 0^{p+1}1^p \therefore xy^2z \not\in \mathcal{L}$\\
where $\mathcal{L} = \{0^i1^j, i \neq j\}$\\
we choose $s = 0^{p+1}1^p$ within $|xy|$\\
thus $\neq 0^p1^{p+1}$\\
Let $x = 0^a, y = 0^b1^a$\\
$x = 0^{p-h}, y = 0^h$\\
$x = 0^i, y = 0^i, z = 0^i1^j$\\
Figure 5.3.3: Some categories / conceptualizations found among students of
introduction to the theory of computing, and published at FIE.
Some of our results were consistent with the framework described by Harel
and Sowder in 1998[?]. We found students holding conceptualizations that
Harel and Sowder's 1998 model calls symbolization. Harel and Sowder have
identified another category of conceptualization, that correctly applied transformation
and axiomatic arguments. Some students expressed enthusiasm for
the power that inheres to building arguments with carefully specified component
ideas, in particular how the absence of ambiguity permitted arguments to
extend to great length while remaining valid. Not all of the students had developed
axiomatic conceptualizations of proof. About definitions, we collected
preliminary data on students' conceptualizations of definitions used in proofs.
Some students thought definitions were boring. Some students thought that
they could infer definitions from a few examples. Concerning executive function, we found that some students do not state the premises clearly, and some
students did not keep track of their goal. About rules of inference, we found
that some students apply invalid approaches to inference.
\subsection{Abstract Model for Proof by Mathematical Induction and Recursion}
Far from finding agreement that (a) theorems are true as a consequence of
the definitions and the premise, and that (b) proofs serve to show how the
consequence is demonstrated from the premise, axioms and application of
rules of inference, instead we found a variety of notions about proof, including
the well-known procedural interpretation [?], and the well-known empirical
misconception [?]. The conceptualization that definitions are not necessarily
of interest compared with the procedures seemed different in kind from the
concept image / concept definition discoveries of R\"osken et al. [?].
Interviews with students revealed that some students saw generation of a proof
by mathematic induction as a procedure to be followed, in which they should
produce a base case, and prove it, and should produce an induction step, and
prove that. This was consistent with Weber [?, p. 4-426] who has stated ``in
the studies that I conducted, it was more often the case that undergraduates
applied procedures that were not meaningful to them.'' He went on to give a
quotation from a participant [?, p. 4-426] ``And I prove something and I look at
it, and I thought, well, you know, it's been proved, but I still don't know that I
even agree with it [laughs]. '’m not convinced by my own proof!'' Some of the
students interviewed did not know why this procedure generated a convincing
argument. Polya[?] has written a problem involving all girls being blue-eyed;
a similar problem appears in Sipser[?] about all horses being the same color.
The purpose of this exercise is to make students aware that the truth of the
inductive step must apply when the base case appears as the premise. In some
cases, this point was not clear to the students.
Students' conceptualizations of proof by mathematical induction can support
their choice to apply recursive algorithms. One student reported success at
both mathematical induction and recursive algorithm application without ever
noticing any connection. This student opined that having learned recursion
with figures, and proof by mathematical induction without figures, that no
occasion for the information to spontaneously connect occurred. Students reporting
ability to implement assigned problems recursively, but not the ability
to understand proof by mathematical induction also reported that ability to
write recursive programs did not result in recognition of when recursive solutions
might be applicable in general.(Say something about how this is consistent with the procedural conceptualization, bifurcation in Tall's writing.) Students reporting ability to implement
assigned problems recursively, and also the ability to prove using mathematical
induction also reported preferring to implement recursive solutions in
problems as they arose.
Our work on students' choices of algorithmic approaches was consistent with
work by other researchers in computer science education[?] on conceptualizations of algorithms. Our work served to unify that of mathematician educators
with computer science educators, by providing a plausible explanation why
the conceptualizations of recursive algorithms that were found, might exist.
Figure 5.3.4: Categories of Student Conceptualizations of Proof by Induction
that Recursion Works
\section{Helping Students Discern Abstraction}
Recall that variation theory holds that students cannot discern a thing unless
contrast is provided. Pang has pointed out that [], for persons aware of only
one language, ``speaking'' and ``speaking their language'' are conflated. Only
when the existence of a second language is known, does the idea of speaking
become separated from the idea of speaking a specific language.
(Here is a specialization (Hofstadter), formation of a new conjunct (Valiant), see Besold 2015)
Abstraction is important in computer science, and is worthy of investigation.
Inquiry into students' conceptualizations of formalization using symbols, symbolization,
has shown similar results among students of mathematics and of
computer science [?, ?]. Student populations contain the conceptualization that
proofs ought to be expressed using symbols, and some proof attempts show
that not all students are able to formalize meaningfully. Mathematics and computer
science pedagogies differ on the recommended style of variable names in
symbolization. In mathematics, there is a preference for single letter variable
names, and in computer science it is recognized that longer variable names assist
readers in understanding. In mathematics the use of single variable names
is preferred because it is thought to contribute to cultivating students' ability
to learn abstraction. If, in computer science education, we apply variation
Table 5.3.5: The Outcome Space for Proofs by Induction\\
Category Description\\
1 Following procedure The method is learned, without understanding\\
2 Understands base case The idea that a base case is proved by an existence
proof, often with a specific example\\
3 Understands implication The idea that an implication is proved by
assuming the premise is not used\\
4 Does not understand connection
Sees the implication and proves it well, but
does not anchor the succession to a base
case\\
5 Does understand the argument
Understands the argument\\
6 Knows why recursion
works\\
Can tailor the argument to explain recursive
algorithms\\
7 Appreciates data structures
supporting recursion\\
Can see the benefit to algorithm from recursive
data structure\\
theory, we gain confidence in the idea that students may discern the process
of abstraction as we vary the names of the variables. We could imagine deriving
code from a requirement about a specific class, and using corresponding
variable names, and we could show the process of promoting the code into a
more general class in the inheritance hierarchy, changing the variable names to
correspond to the more general domain of objects. Thus we can borrow from
the approach used by mathematics education, but make it more explicit, taking
advantage of computer science's explicit treatment of inheritance hierarchies in
object oriented code. Seeking evidence of students' conception of abstraction,
we could examine overridden methods to see whether variable names in more
and less general implementations bear that relation to one another.
\section{Helping Students Discern Abstraction}
\section {Algebra}
In middle or high school algebra students became familiar with the use of letters in equations,
and solving equations which resulted in individual values, or no value, being attached to the letters.
Ideally the ability to understand expressions, to formulate pre- and post conditions would be acquired.
As we have seen that this occurs sometimes, but does not always occur,
there may be benefit to some students to review this idea.
We might choose to emphasize abstraction in this process.
\section{Geometry}
In high school geometry, formal proofs of geometric properties are covered.
Students are exposed to a form for argument, and are given examples of use of rules of inference to perform logical deduction.
We have seen that sometimes this process is appreciated in enough generality to be recognized
as an example of argumentation.
We have seen as well, that some students found this process entirely specific to geometry,
doubting that it had broader application.
\section{Seeing a Broader Context}
It may be that some students do not see a separation between the activity of formalization on the one hand,
and the application area of finding solutions to equations on the other.
It may be that some students do not see a separation between the activity of deducing using logic on the one hand, and the application area of learning geometric facts on the other.
It seems consistent with neglecting opportunities for abstraction, that these separations are not always seen,
Speciation, an idea which uses abstraction, providing a hierarchy of properties animals and plants might have, was not recognized early or universally. So, it is
not surprising that abstraction, which involves choice about which details to defer, and which to regard as significant, is not always obvious.
In the machine learning perspective, features can be learned.
What do I want to say, it takes some effort to recognize features?
There might be a way to formulate choice of features such that some better efficiency is gained by thinkingof the features in that order vs. another order.
(Such as, we never have to think about some features for some parts of the tree.)
If we think about knowledge being organized in neural networks, such that abstraction has a physical manifestation, we can see that ideas between which there is little distance (by some measure, neurons, glia?) in the tree will more frequently elicit one another by linkages at the metabolic level.
At the neural level, modifications for efficiency are constantly taking place (Do we have this from Kandel and Squire?).
We might wish to exploit this in the way we teach, to exhibit the abstraction deliberately, to minimize the amount of neural connection remodeling that would occur in the process of providing an efficient neural connection remodeling that would occur in the process of providing an efficient neural representation.
Being able to learn by analogy testifies to the utility of having a neural representation that corresponds to abstraction.
Students who are working without hierarchical organization of concepts are at a disadvantage.
\section{Mathematics tests in high school that involve proving}
What can we learn from students of computer science who excelled in reasoning to this level?
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