ch5differential.tex

\chapter{Interpretation/Discussion}

In this chapter we discuss the results obtained in the analysis.

In a phenomenographic study, the results include categories of conceptualizations found in the collection of students. We infer categories of conceptualization from the units of meaning produced by the participants. A description of each category is part of the  result, and also pairwise relationships between some pairs of categories are part of the result. One expectation in a phenomenographic study is that arranging the categories as suggested by these relationships will make more easily seen a dimension of variation. The dimension of variation in turn may suggest distinguished points, called critical factors (also critical aspects\footnote{The phrases critical factors and critical aspects are both used. We prefer critical factors, as aspects might seem more specific to a component hierarchy of a learning objective, compared with critical factor which may admit more readily to a generalization/specialization hierarchy.}). Points along the dimension of variation are concepts, awareness of which is thought to be especially beneficial to students.

As we discuss the results, it can make sense to consider  alternate dimensions of variation, comparing them with the chosen. 
We consider alternate critical factors, both along alternate dimensions of variation, but also along the proposed dimensions of variation.

Critical factors are ideas that are necessary for a better conceptualization\cite{marton and pang?}. 
We conjecture that not all points along a dimension of variation are equally valuable.
Some ideas encounter more resistance from students than others. (Is a cite needed for this?)
Those ideas that are less readily adopted by students might be preferable as critical factors, in that these exercise the helping role of teachers.

 % % %recognition
 
\section{Interpretation of What Students Think Proofs Are}

Some students know that proofs are about truth, and are about demonstrating that something is true. The degree of refinement that distinguishes a derivation from a proof was not evident in the interviews. Some students know that such demonstrations involve logic, and make use of mathematical notation. Some students know that there are forms or patterns that can be used, such as proof by mathematical induction.

However, the importance (or lack thereof) that some students assign to definitions, and the lack of facility with representation in mathematical formulation, hinder some students, especially in the construction of proofs. Remembering that proofs show us the consequences of the definitions, and that definitions supply the justification for transformation steps, we can see that discounting the importance of definitions will have a strong effect on students' ability to create proofs.

Some students know that proofs contain sequences of logical statements.
There is a diversity of concepts for what a logical statement is.
Some students know that the use of a logical statement at a point within a sequence must be warranted by what has been established prior to that point.

Some students have expressed a preference, for occasions in which they are producing proof, for proofs whose construction is guided by what these students perceive as a procedure. It appears that these proofs involve less creativity or less searching for next statements. Some students characterize these proofs as having steps that can be followed. Some students refer to proofs that lack guiding steps as ``logic proofs''.

It may be that some students are not so comfortable with mathematic formulation that, as they search through the space of possible transformations to find helpful transformations, they are guided by a sense of distance between where they are and their goal.

We could propose an analogy between finding a desired destination along roads by ``dead reckoning'' vs. following the instructions from a source such as a list of directions or a GPS-enabled map software. In ``dead reckoning'', the driver has some notion of the relative placement of the present location and the goal, has a sense of direction, and can estimate the value of a road as it is discovered, according to its width, its proclaimed direction and sometimes the presence of a list of supported destinations. A person following directions can do so without this sense of relative placement.

Some students have observed that they began by trying every allowed transformation, to see what resulted, and that after spending hours doing this they developed some ability to predict what the result of a transformation would be.
Some students have remarked that they expect they would do better with more practice. 

In summary, we can consider plausible the implication ``If students did not appreciate the role of definitions, or if they did not have facility with abstracting from a specific situation to a mathematical formulation, they would have trouble developing a succession of warranted transformations.'' We have some student claims that they do not consider definitions to be valuable. We have also student claims that even when they can reason about concrete situations (instances with people and things), they cannot perform the same reasoning steps with symbols. These appear to be sufficient sources of difficulty.

\subsection{Alternate Dimensions of Variation}
The proposed dimension of variation is depth of understanding of the role of definitions, especially a generalization/specialization hierarchy of defined terms, vs. process steps, in proofs.
One alternative is that students are fully appreciative of the role of definitions, but lack some degree of facility using mathematical formulation. 
If this alternative were the case, students would be able to be articulate in their verbal descriptions of the relationships of ideas, while being unable to render these ideas in mathematical expressions.
Another alternative is that students are fully appreciative of the role of definitions, but lack some degree of ability to work with abstractions.
Another alternative is that students are fully appreciative of the role of definitions, but lack some degree of appreciation of the role of warrants that contribute to the formation of an argument from the succession of statements.

A lack of understanding of the warranted nature of a sequence of statements transforming the premise into the conclusion seems to offer an explanation of how it can be that otherwise capable students might think that experimentation is not redundant in the presence of proof. This might offer an explanation of how it can be that such students can believe that the argument proceeding from the assumption of the premise could establish that premise as true.

The proposed explanation that the student is not using words according to their definitions, and intended to say that the argument supports the claim, rather than saying an argument supports a premise.

\subsection{Alternate Critical Factors}

If we accept depth of appreciation of the role of definitions as the dimension of variation, we might ask where along this axis might the critical factors be.
We have seen student conceptualizations including that definitions are uninteresting, or can be inferred from a collection of examples. We have also seen, from students also majoring in math, a strong understanding of the role of definition in argumentation.

As an alternative critical factor we might choose the source of a warrant. We could consider that some conceptualizations include that warrants are based upon definitions and some conceptualizations do not know where warrants may arise.



% % % comprehension
\section{Interpretation of How Students Attempt to Understand Proofs}

Students do not always attempt to understand proofs they are shown.
"people have trouble with they see a proof, they see it, that's a theorem, that's a proof, that's true, i believe it, they don't look to see how is it a proof, everyone understands when you're staring at the screen, my recursion should work, my mathematical reduction works, but it's the steps in between that no one has an idea about, it's like a bridge, you start at a, you get to c, but b is the journey, and everyone skips that, they understand a, c but they don't"

 "while knowing what they can and can't do through proofs is of course important i just keep saying it gets a bit confusing in this class, nebulous sometimes"
 
  "the biggest thing that changed in my proof writing in the math side i didn't really have a good understanding of logical statements, like an if and only if"

When students attempt to understand proofs, they sometimes get stuck.  
They reported preferring a representation in code, which they could exercise in a development system. They did not know of a similar system that would help them tinker with or otherwise examine a proof.

It appears that, as students attempt to understand specific proofs, they try to find an example in which the symbols refer to concrete objects. In a proof about primes, some students will substitute specific primes to ``check the idea''.

Some students have had trouble transferring their understanding of proof patterns applied in one domain to proof patterns applied in a similar domain. In particular, some students who had studied proof by mathematic induction with natural numbers in one semester, when faced with understanding a proof by mathematic induction with natural numbers designating the level of a pushdown stack in a subsequent semester, claimed not to have understood proof by mathematic induction. These students did not detect the problem with the traditional ``all women are blue-eyed blondes'' argument.

%If construction by students is attempted without an appreciation of the role of definitions (students zone out during definitions, students prefer examples), is formulaic, i.e., by learned steps but not also problem solving (as can happen when proof by mathematic induction is selected as a technique because it is a known technique, rather than some sense of its appropriateness is signaled by the problem), that would be consistent with comprehension not involving warrants.

We consider the implication: ``If students could generalize from a specific problem involving unadorned natural numbers to a general argument involving natural numbers, and could specialize from natural numbers in general to natural numbers characterizing a (different) specific problem, then students would be able to transfer their understanding of one proof by mathematical induction using unadorned natural numbers to another proof by mathematical induction using natural numbers characterizing a different specific problem. We consider this implication to be believable. Then using student claims as evidence, we would have to conclude that either students have difficulty generalizing or students have difficulty specializing. (Given knowledge of the ease of learning specialization, and knowledge of the utility of good REM sleep in generalizing, we increase the confidence with which we regard generalization as the greater problem.)

When students are trying to understand a proposed proof, they sometimes wonder how the author knows that a statement is justified. Is it not the case that a statement is warranted by the definitions of the terms being represented, either by syntax alone or by the semantics of the expression?

Some students do not understand that the semantics of recursive programming is the same as the semantics of proof by mathematic induction. Some students come to appreciate the correspondence as they discuss it in an interview. Some students saw these subject explained with mathematical notation but not figures in the case of mathematic induction and with figures but not mathematical notation in the case of recursion. The difference between these two representations was given by the students as the reason they did not recognize the semantics of these as being the same. Discussing these together allowed some of these students to generalize from the disparate representations to the common idea. This suggests that the thinking about the semantics of the proof by mathematic induction can be superficial. Student interviews suggest that the depth of understanding is sometimes insufficient to convince the student of the argument.

Could it be that the students' declared preference for proofs they can supply by following a procedure, such as proof by mathematic induction, derives from their ability to provide a proof without involving the depth of understanding necessary when they must thoughtfully navigate the space of allowable transformations.

\subsection{Alternate Dimensions of Variation}
The proposed dimension of variation is degree of pattern matching and reduction of generalization.
Student conceptualizations include that comprehension of a proof can be obtained by finding the explanation for the corresponding proof text among class notes, by recognizing that a proof follows a pattern that has been explained in class. 
Students also try to understand proofs by considering specific elements of the domain to which the proof applies.
Students reach for understanding of an unfamiliar line of reasoning by trying to find an analogy in a more familiar specific domain. A proof may invoke that a relationship is transitive, and a student may try to appreciate the reasoning by considering a family tree.

An alternate dimension of variation is facility with mathematical formulation.
Generalization occurs when mathematical formulation is used, so these separate ideas may behave in similar ways.
We would like to distinguish between a dimension of variation for mathematical formulation alone, that does not include generalization, and the dimension of generalization, that includes mathematical formulation.
One tool is the effectiveness of reasoning by analogy.
What is intended here by reasoning by analogy is that we can help a student see a relationship, $R$, among entities if a relationship with the same structure is already known for some other entities. 
This process offers the opportunity to use generalization.
When the known entities and their relationship is discussed, it can happen that the relationship can be seen separately from the whole.
The relationship, seen separately, is now a more general idea, from which, with changed entities, another example can be specialized.
Discussion with some students has shown that some students can separate out relationships from a specific case into a more general idea.
On the other hand, some students, even when presented with two concrete sets of entities enjoying a similar relationship, do not readily separate the relationship from its realization with entities.
In the use of mathematical formulation, we have seen conceptualizations that attach to the notion of variable, that is can be used to refer to a subset of a domain, as a free variable does. We have also encountered conceptualizations that a variable may not refer to a locus or other set of points, rather it must necessarily represent a single point, that may, by manipulation, be discovered, thus there exists a dimension of variation of conceptualization of generalization, even within the domain of mathematical formulation.


\subsection{Alternate Critical Factors}
Attempt at visualization is a very interesting critical factor because it fits in well with the work of Harel and Sowder\cite{harel1998students} who distinguished the categories perceptual and transformational. 
Our students are thinking of a proof that carries out a process, to which idea we can readily, but not necessarily correctly, associate the idea of transformation of a representation.
An alternate critical factor could be analysis into parts.  These parts could be stages in time, but they might also be static components, lemmas.

 % % % applications
\section{Interpretation of Reasons for Teaching Proof}

Some students do not know why proofs appear in the curriculum.
Though we might expect this to be remediated upon the student seeing proofs used in subsequent classes, sometimes students don't recognize proofs when they see them in subsequent classes.

Some students are thought not to know why proofs are taught, because they propose such reasons as ``The professors were math majors, and feel we should know some math.''

%Some students are thought not to recognize that proof is about coming to be aware of the truth value of an expression, given the awareness of a related expression. 

Some students know that proofs are used for justifying claims about the correctness and resource consumption of algorithms. No student was found who considered provability to be a guide for algorithm construction.

%Some students' conceptions of what proofs are for are at a lower level of detail than others. For example, a student reported that proof by counterexample is for proving something is not true.

Other students connect learning proof by induction with explaining why algorithms work: "yes, of course, the first thing i thought of when i saw induction, was recursion"

Some students were asked whether they ever employed proof for any reason other than having been assigned. No student said they employed proof for any purpose other than responding to an assignment to provide a proof. When students said they never used recursion because they never knew whether a situation warranted the use of recursion, the interviewer suggested that proof might be a useful tool for understanding the situation. Students acquiesced. Given the difficulty with transfer, it appears that students do not choose to exercise this transfer.
\subsection{Alternate Dimensions of Variation}
The proposed dimension of variation is degree of specificity of purpose, with initial ideas being vague but more advanced ideas appreciating some particular applications of proof.

An alternative dimension of variation could be to provide deeper understanding of the nature of computation. Engaging in proof gives students the opportunity to contemplate the operations they carry out in algorithm construction with respect to how the complexity of an algorithm may change as aspects of the computation change, as from 2-SAT to 3-SAT.
\subsection{Alternate Critical Factors}
There is a big gap between obtaining bounds on resource consumption and deriving mathematical formulation of intuitive ideas.
The effects of aspects of algorithms on the resulting complexity class might be a valuable stepping stone in between.

% % % more application
\section{Interpretation of How Students Attempt to Apply Proofs (When not assigned)}
We can also look at how students attempt to apply proofs when assigned.
What are conceptualizations of application of proofs?
They need to use proofs, like pumping lemma, to solve problems posed to them, such as "Can we say this language is not regular?".
Rather than understanding what the proof says about features a regular language must possess, (Scouts must be thrifty, brave, clean and reverent.) and inverting the statement to find that a language not having these features cannot  be regular (If a person is not thrifty or not brave or not clean or not reverent, that person cannot be a scout.), students seem to prefer to learn a procedure. Such a procedure would include, test for all possible segmentations, show for any such segmentation, it is not the case that segment $y$, when replicated any natural number of times, contributes to a string that is present in the language. Some students stumble in the application of the procedure, especially if the procedure is described using a mathematical formulation. Some students try to memorize and reproduce  the mathematical formulation without understanding it.

Quantifiers introduce additional complexity and challenge for students.

There is insufficient evidence of students applying proofs when not assigned.
\subsection{Alternate Dimensions of Variation}
We don't have enough information about this.

\subsection{Alternate Critical Factors}
We don't have enough information about this.
% % % yet more application
\section{Interpretation of Whether students exhibit consequences of inability (such as avoiding recursion)}
Students claim to be unsure of when certain algorithmic approaches, such as recursion, are appropriate, and do not think of proof as helpful in a determination.
\subsection{Alternate Dimensions of Variation}
We do know that students say they eschew making use of algorithmic techniques they know, for want of recognizing when these are applicable.
Some students say their approach to choosing among algorithms they know includes consulting their relatives and friends.
We could postulate a dimension of variation about the relative value of the methods students use when they reason about the applicability of algorithms to situations.

\subsection{Alternate Critical Factors}
We could arrange algorithms according to some guidance about how generally useful they are, and discuss variations in situations that make algorithms more or less useful. 

\section{Interpretation of How familiar and/or comfortable are students with different (specific) proof techniques: induction, construction, contradiction?}
Students are familiar, in the sense of name recognition, with proof by mathematical induction, contrapositive, contradiction and cases.
Students see these as patterns for proof construction, i.e., steps that can be carried out. 
Students are aware that, but are not always possessed of intrinsic conviction that the patterns achieve the desired result.
Thus the students do not always achieve confidence in what Carnap\cite[p. 21-22]{carnap1958introduction} calls "the psychological content (the totality of associations)" of the consequence of the logical deductions.
When some proof is subsequently viewed in a context where it is intended to be explanatory, its power over the students' conceptions is correspondingly diminished.

\subsection{Alternate Dimensions of Variation}
The proposed dimension of variation is that comfort derives from degree of similarity to a process, as in, students are more comfortable with proof by induction because there is a process that they have learned for producing its parts. Students have somewhat less comfort with proof by contradiction, which also has a process, but also has a scope within which a premise obtains, and an outer scope in which it does not. Students have even less comfort with proofs for which ``application of logic'' is the only guidance they perceive.

An alternate dimension of variation is that comfort derives from the frequency with which they have practiced a certain kind of proof. Students have remarked that they begin a proof by selecting the technique they have practiced most often, without regard for the problem statement.

\subsection{Alternate Critical Factors}
We could emphasize the kinds of transformations that are made.
An applicable rule of inference allows us to make a transformation.
We might raise the students' comfort level by increasing their ``sense of direction'', their sense of how distant the presently known representation is from the desired representation, and developing a sense of whether application of any given rule can move the representation in the desired direction.
% % % synthesis
\section{Interpretation of Which  structural elements students notice in proofs}

From (Random1 from CPS) we see ``How did we come up with the proof above? Parts of it are \emph{determined} by the structure of the formulas we have, while other parts require us to be \emph{creative}.''

It would help students to be able to recognize and exploit structure of the statement to be proved. They could be guided by the structure of what they are to prove, in choosing some parts of the proof they are to construct. Therefore it is of interest to know whether they notice structure within example proofs they are shown.

Because some students do not recognize structures, such as implication introduction, in proofs, and do not correlate these with the structure of statements to be proved (``When I need to do a proof I try proof by mathematical induction because we practiced that most often.'') it lends more confidence to the conjecture that certain approaches to proofs are memorized as monolithic processes, as might occur when learning is scaffolded by learning a proof as a single algorithm, rather than understood as arguments, with parts that can be assembled, where the parts, being smaller arguments, might be thought of as smaller algorithms.
There is some significance to the assembly of the smaller parts.
Some part of it may be pattern matching,\cite{CPS random1 } but there may be a requirement for reasoning, for creating an argument.

Students were familiar with the idea of proof by contradiction, though when it was described as preparing a context in which an additional premise was available, and therefore similar to introduction of special cases, which also introduce contexts with additional premises, students did not find this to be a familiar idea.

One interpretation of this is that some students are comfortable with the idea of proofs having structure, beyond the practiced structure of proof by mathematic induction and proof by contradiction. 

When students consider whether a rule of inference is applicable to, and helpful for the transformation of logical representation they are attempting to accomplish, we might ask whether there are two levels of abstraction. One for the rule of inference in isolation, and another for that same rule of inference when in use in the specific proof.


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{Alternate Dimensions of Variation}


\subsection{Alternate Critical Factors}

% % % analysis
\section{Interpretation of What do students think it takes to make an argument valid?}
There is the conceptualization that many theorems exist and they can be invoked, in combination.
There is the conceptualization that established patterns of proof, such as proof by mathematical induction, and proof by contradiction, and proof by modus ponens exist.
There is the conceptualization that later statements must be justified by earlier statements, without clarity as to how this justification is achieved.
There is the conceptualization of subsequent statements being warranted on the basis of the definitions of the terms being transformed.


\subsection{Alternate Dimensions of Variation}

The proposed dimension of variation is extent of what students understand to confer validity. 
An alternate dimension of variation is the degree of judgment about how large a transformation step may be attempted with complete confidence.
\subsection{Alternate Critical Factors}
The granularity of a transformation could be chosen according to what theorem provers use, or invariants applied to algorithms.

 
 
\section{Interpretation of Combined Data}


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.

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 mathematical 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 none 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 knowledge of programming.
Though the Curry-Howard isomorphism provides strong support 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 conception "logic proofs", placed in contrast with proofs that seem like processes.
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 constructed by following a process might appear successful in an assessment even though unconvincing to its writer.

Some students report beginning with a premise and trying every transformation 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\cite{}, 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\cite[page 137?]{carnap1958introduction}. 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. 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.

We know some students have trouble with generalization. 
Some students can reason about relationships among concrete entities, but have trouble
expressing the same relationship in mathematical formulation.
For students who have trouble generalizing, so that they do not necessarily abstract away the particular domain, the idea of moving from one example, to the idea of member of a domain, to the idea of different member of the same domain can be an initial challenge. Some students have articulated that this is the maximum level of difficulty of homework that they expect. Changing domains, as from natural numbers devoid of context, to, for example, depth of a stack (which is of course a natural number), as occurs in Sipser\cite{sipser2012introduction} in a proof about the language of pushdown automata and context free grammars, has caused some students difficulty.

The amount of generalization needed to appreciate that the recursive programming technique, which solves problems of a larger size in terms of problems of the same form of smaller size, is similar to proof by mathematical induction, in which the truth value of a statement indexed by a larger natural number is implied by the truth value of a statement of the same form indexed by a smaller natural number does not occur spontaneously in some students. This amount of generalization has occurred during interviews with some students.

Czeisler\cite{czeisler} reports how students improve on performance over several days, when a good night's sleep occurs in between each day.
Somebody\cite{} reports that REM sleep, which occurs in the latter part of a good night's sleep, is used for the generalization process.
When one considers that some students start their homework just before it is due, and frequently do not provide themselves with a good night's sleep, it is not very surprising that generalization would be difficult.
It might also be worthy of note that the type of long-term memory that is formed in a condition of fear is of a different nature than one formed without fear (Is this in the new Ballard brain hierarchy book from MIT press? Also, Damascio, which is not new.), and that long term memories formed in fear contain much irrelevant detail. It might be that fear impacts the ability to generalize.

Students who choose to discount the importance of definitions and instead engage to learn from examples might depend upon their ability to generalize on their own, which in turn, might not help them sufficiently.

Attempts to explicitly support generalization, by rendering concepts found in a chapter as UML class symbols, and arranging these symbols in an inheritance hierarchy were not appreciated by some students. Some (better) students have asked why we spend time teaching them with UML diagrams. It would appear that conscious, as well as unconscious, generalization, is not easily used by some students.

Without generalization, without the efficiency in representing concepts obtained by use of mathematical formulation, students can be hindered in their efforts. These efforts include creating transformations of formulation as is needed in proofs that involve un-choreographed logical steps.

One example of difficulty students might have in generalizing could be, that the specific occupants of places in a logical sentence cannot be generalized.
For example ``My father was born in Jamaica, of Jamaican parents, so I know that he cannot be elected president of the United States.''
One step in generalization would be: ``Person a was born in Jamaica, of Jamaican parents, so I know that he or she cannot be elected president of the United States.''
A student has reported that this step is believable, but requires concentration, because the idea 'a' is not fixed. (Perhaps it is that bound variables are ok, but free variables are troublesome.)
Likewise, ``Person a was born outside of the US in country b, of parents neither of whom were US citizens, so I know that he or she cannot be elected president of the United States.'', requires further concentration. (Two free variables are more disturbing than one.) However, this topic is currently in the news, so it is understandable.
Sentences about things not in the news, having two free variables, are confusing.

Even when the construction is first offered with bindings to concrete entities, and a second construction is offered with free variables, the confusion persists:
First construction:\\
Time travel is not possible.\\
A deLorean (commercially available) car plus a Mr. Fusion can perform time travel.\\
Because we cannot have time travel, something in the construction deLorean and Mr. Fusion is bogus.\\
The deLorean is commercially available.\\
The Mr. Fusion must be unattainable.\\
The first construction is thought to be understood.

$A_{TM}$ is not possible.\\
A simple program construction and $E_{TM}$ gives $A_{TM}$.\\
The simple program construction is clearly ok.\\
$E_{TM}$ must be unattainable.\\
The second construction is found to be elusive.

One interpretation might be that generalization is difficult. Another interpretation is that the sense making of the first instance is that, yes, the reader knows that Mr. Fusion is unattainable, but it is not the argument that has convinced the reader  of this.
It may be that the argument, noticed or not, is not the agency by which the reader is convinced. It might be some other feature of the specific entities populating the concretely populated argument, so ``Well, everybody already knows that one can't have a Mr. Fusion, because if we could, we wouldn't be driving with gas.'' 





\subsection{Alternate Dimensions of Variation}

\subsection{Alternate Critical Factors}