Category Archives: Proposals for research

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Offloading chunking

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In my previous post I wrote about the idea of offloading abstraction, the sort of things we do with geometric figures, diagrams (that post emphasized manipulable diagrams), drawing the tree of an algebraic expression, and so on.  This post describes a way to offload chunking.  

Chunking

I am talking about chunking in the sense of encapsulation, as some math ed. people use it.  I wrote about it briefly in [1], and [2] describes the general idea.  I don't have a good math ed reference for it, but I will include references if readers supply them.  

Chunking for some educators means breaking a complicated problem down into pieces and concentrating on them one by one.  That is not really the same thing as what I am writing about.  Chunking as I mean it enables you to think more coherently and efficiently about a complicated mathematical structure by objectifying some of the data in the structure.  

Project 

This project an example of how chunking could be made visible in interactive diagrams, so that the reader grasps the idea of chunking.  I guess I am chunking chunking.  

Here is a short version of an example of chunking worked out in ridiculous detail in reference [1]. 

Let \[f(x)=.0002{{\left( \frac{{{x}^{3}}-10}{3{{e}^{-x}}+1} \right)}^{6}}\]  How do I know it is never negative?  Well, because it has the form (a positive number)(times)(something)$^6$.    Now (something)$^6$ is ((something)$^3)^2$ and a square is always nonnegative, so the function is (positive)(times)(nonnegative), so it has to be nonnegative.  

I recognized a salient fact about .0002, namely that it was positive: I grayed out (in my mind) its exact value, which is irrelevant.  I also noticed a salient fact about \[{{\left( \frac{{{x}^{3}}-10}{3{{e}^{-x}}+1} \right)}^{6}}\] namely that it was (a big mess that I grayed out)(to the 6th power).  And proceeded from there.  (And my chunking was inefficient; for example, it is more to the point that .0002 is nonnegative).

I believe you could make a movie of chunking like this using Mathematica CDF.  You would start with the formula, and then as the voiceover said "what's really important is that .0002 is nonnegative" the number would turn into a gray cloud with a thought balloon aimed at it saying "nonnegative".  The other part would turn into a gray cloud to the sixth, then the six would break into 3 times 2 as the voice comments on what is happening.  

It would take a considerable amount of work to carry this out.  Lots of decisions would need to be made.  

One problem is that Mathematica doesn't provide a way to do voiceovers directly (as far as I know).  Perhaps you could make a screen movie using screenshot software in real time while you talked and (offscreen) pushed buttons that made the various changes happen.

You could also do it with print instead of voiceover, as I did in the example in this post. In this case you need to arrange to have the printed part and the diagram simultaneously visible.  

I may someday try my hand at this.  But I would encourage others to attack this project if it interests them.  This whole blog is covered by the Creative Commons Attribution – ShareAlike 3.0 License", which means you may use, adapt and distribute the work freely provided you follow the requirements of the license.

I have other projects in mind that I will post separately.

References

  1. Abstractmath article on chunking.
  2. Wikipedia on chunking
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Skills needed for learning languages and math

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Learning a language involves a variety of skills, and so does learning math. Some skills are apparently needed for both, but others are distinct.

Learning languages and learning math

Some years ago I sat in on a second year college Spanish class. Most of the other students were ages 18-24. The students showed a wide spectrum of ability.

  • Some were quite fluent and conversed easily. Others struggled to put a sentence together.
  • Some had trouble with basic grammar, for example adjective-noun agreement (number and gender). I would not have thought second year students would do that. Some also had trouble with verbs. Spanish verbs are generally difficult, but second year students shouldn’t have trouble with using “canta” with singular subjects and “cantan” with plural ones.
  • Some had trouble reading aloud, stumbling over pronunciation, such putting the accent in the right place in real time and pronouncing some letters correctly (“ll”, “e”, intervocalic “s”). The rules for accent and pronouncing letters are very easy in Spanish, and I was surprised that second year students would have difficulty with them. But the speech of most of them sounded good to me.

I can read Spanish pretty well, but have had very little practice speaking or writing it. I comprehend some of what they say on Univision (soap operas are particularly easy, but I still miss more than half of it), but then I am hard of hearing. I used to have a reasonable ability to speak and understand street German; judging from experience I think it would come back rapidly if we went to live in a German-speaking city again. I can easily read math papers written in Spanish or in German, but I couldn’t come close to giving a math lecture in either language.

Some find learning rules of pronunciation that are different from English very hard, like the Spanish students I mentioned above. I know that some people can’t keep “ei” and “ie” straight in German, and some Russian students find it hard to get used to the Cyrillic alphabet. I find that part of language learning easy. I also find learning grammar and using it in real time fairly easy. I have more difficulty remembering vocabulary.

Learning the new sounds of a language is an entirely different problem from learning the rules of pronunciation.

Mathematical ability

Some difficulties that students have with the symbolic language of math [1] are probably the same kind of difficulties that language students have with learning another language.

When I have taught elementary logic, I usually have a scattering of students who can’t keep the symbols {\land} and {\lor} separate. (See Note [a].) Some even have the same trouble with intersection and union of sets. This is sort of like differentiating “ie” and “ei” in German, except that the latter distinction runs into cognitive dissonance [2] caused by the usual English pronunciation.

Of course, both language students and math students have immense problems with cognitive dissonance in areas other than symbol-learning. For example, many technical words in math have meanings different from ordinary English usage, such as “if”, “group”, and “category”. Language students have difficulties with “false friends” such as “Gift”, which is the German word for “poison”, and very common words such as prepositions, which can have several different translations into English depending on context — and many prepositions in other European languages look like English prepositions. (Note [b]).

On the other hand, some types of mathematical learning seem to involve problems language students don’t run into.

Substitution, for example, appears to me to cause conceptual difficulties that are not like anything in learning language. But I would like to hear examples to the contrary.

If {f(x) = x^2+3x+1}, then {f(x+1)= (x+1)^2+3(x+1)+1}. Is there anything like this in natural languages? And simplifying this to {x^4+5 x^2+5} is not like anything in natural language either — is it?

Is there anything in learning natural languages that is like thinking of an element of a set? Or like the two-level quantification involved in understanding the definition of continuity?

Is there anything in learning math that involves the same kind of difficulty as learning to pronounce a new sound in another language? (Well, making a speech sound involves moving parts of your mouth in three dimensions, and some people find visualizing 3D shapes difficult. But that seems like a stretch to me).

A proposal for investigation

Students show a wide variety of conceptual skills. Some skills seem to be required both in learning mathematics and in learning a foreign language. Others are different. Also, there is a difference between learning school math and learning abstract math at the college level (Note [c]).

TOPIC FOR RESEARCH

  • Identify the types of concept formation that learning a foreign language and learning math have in common.
  • Determine if “being good at languages” and “being good at mathematics” are correlated at the high school level.
  • Ditto for college-level abstract math.

Undoubtedly math teachers and language teachers have written about certain specific issues of the sort I have discussed, but I think we need a systematic comparative investigation of skills involved in the tasks of learning languages and learning math.

I have made proposals for research concerning various other questions with math ed, particularly in connection with linguistics. I will install a new topic “Proposals for research” in my “List of categories” (on the left side of the screen under “Recent posts”) and mark this and other articles that contain such proposals.

Notes

[a]. That is why, in the mathematical reasoning sections of abmath, for example [3], I use the usual English wordings of mathematical assertions instead of systematically using logical symbolism. For many students, introducing symbols and then immediately using them to talk about the subtleties of meaning and usage puts a difficult burden on some of the students. (I do define the symbols in asides).

This may not be the right thing to do. If a student finds it hard to learn to use symbols easily and fluently, should they be studying math?

[b]. I once knew a teenage German who spoke pretty good English, but he could not bear to use the English possessive case. That’s because German young people (assuming I understand this correctly) hate to say things like “Das Auto meines Vaters” and instead say “Das Auto von meinem Vater”. Unfortunately this resulted in his saying in English “The car from my father”, “The girlfriend from my brother” and so on…

[c]. I have been concerned primarily with understanding the difficulties students have when starting to study abstract math after they have had calculus. I have seen many students ace calculus and flunk abstract algebra or logic. There is a wall to fall off of there. The only organization I know of concerned with this is RUME, although it is involved with college calculus as well as what comes after.

References

[1] The symbolic language of math.

[2] Cognitive dissonance.

[3] Conditional assertions.

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Syntax Trees in Mathematicians’ Brains

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Understanding the quadratic formula

In my last post I wrote about how a student’s pattern recognition mechanism can go awry in applying the quadratic formula.

The template for the quadratic formula says that the solution of a quadratic equation of the form ${ax^2+bx+c=0}$ is given by the formula

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

When you ask students to solve ${a+bx+cx^2=0}$ some may write

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

instead of

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2c}$

That’s because they have memorized the template in terms of the letters ${a}$, ${b}$ and ${c}$ instead of in terms of their structural meaning — $ {a}$ is the coefficient of the quadratic term, ${c}$ is the constant term, etc.

The problem occurs because there is a clash between the occurrences of the letters “a”, “b”, and “c” in the template and in the equation to solve. But maybe the confusion would occur anyway, just because of the ordering of the coefficients. As I asked in the previous post, what happens if students are asked to solve $ {3+5x+2x^2=0}$ after having learned the quadratic formula in terms of ${ax^2+bx+c=0}$? Some may make the same kind of mistake, getting ${x=-1}$ and ${x=-\frac{2}{3}}$ instead of $ {x=-1}$ and $ {x=-\frac{3}{2}}$. Has anyone ever investigated this sort of thing?

People do pattern recognition remarkably well, but how they do it is mysterious. Just as mistakes in speech may give the linguist a clue as to how the brain processes language, students’ mistakes may tell us something about how pattern recognition works in parsing symbolic statements as well as perhaps suggesting ways to teach them the correct understanding of the quadratic formula.

Syntactic Structure

“Structural meaning” refers to the syntactic structure of a mathematical expression such as ${3+5x+2x^2}$. It can be represented as a tree:

(1)

This is more or less the way a program compiler or interpreter for some language would represent the polynomial. I believe it corresponds pretty well to the organization of the quadratic-polynomial parser in a mathematician’s brain. This is not surprising: The compiler writer would have to have in mind the correct understanding of how polynomials are evaluated in order to write a correct compiler.

Linguists represent English sentences with syntax trees, too. This is a deep and complicated subject, but the kind of tree they would use to represent a sentence such as “My cousin saw a large ship” would look like this:

Parsing by mathematicians

Presumably a mathematician has constructed a parser that builds a structure in their brain corresponding to a quadratic polynomial using the same mechanisms that as a child they learned to parse sentences in their native language. The mathematician learned this mostly unconsciously, just as a child learns a language. In any case it shouldn’t be surprising that the mathematicians’s syntax tree for the polynomial is similar to the compiler’s.

Students who are not yet skilled in algebra have presumably constructed incorrect syntax trees, just as young children do for their native language.

Lots of theoretical work has been done on human parsing of natural language. Parsing mathematical symbolism to be compiled into a computer program is well understood. You can get a start on both of these by reading the Wikipedia articles on parsing and on syntax trees.

There are papers on students’ misunderstandings of mathematical notation. Two articles I recently turned up in a Google search are:

Both of these papers talk specifically about the syntax of mathematical expressions. I know I have read other such papers in the past, as well.

What I have not found is any study of how the trained mathematician parses mathematical expression.

For one thing, for my parsing of the expression $ {3+5x+2x^2}$, the branching is wrong in (1). I think of ${3+5x+2x^2}$ as “Take 3 and add $ {5x}$ to it and then add ${2x^2}$ to that”, which would require the shape of the tree to be like this:

I am saying this from introspection, which is dangerous!

Of course, a compiler may group it that way, too, although my dim recollection of the little bit I understand about compilers is that they tend to group it as in (1) because they read the expression from left to right.

This difference in compiling is well-understood.  Another difference is that the expression could be compiled using addition as an operator on a list, in this case a list of length 3.  I don’t visualize quadratics that way but I certainly understand that it is equivalent to the tree in Diagram (1).  Maybe some mathematicians do think that way.

But these observations indicate what might be learned about mathematicians’ understanding of mathematical expressions if linguists and mathematicians got together to study human parsing of expressions by trained mathematicians.

Some educational constructivists argue against the idea that there is only one correct way to understand a mathematical expression.  To have many metaphors for thinking about math is great, but I believe we want uniformity of understanding of the symbolism, at least in the narrow sense of parsing, so that we can communicate dependably.  It would be really neat if we discovered deep differences in parsing among mathematicians.  It would also be neat if we discovered that mathematicians parsed in generally the same way!

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Where does the generic triangle live?

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I recently posted the following question on MathOverflow.  It is a revision of an earlier question.

Motivation 1

There is such a thing as a generic group. In category theory this is done by constructing “theory” of the group, which is a category in a certain doctrine. Functors (in that doctrine) to Set, or more generally to any topos, are groups. The barest such theory (as usually seen) is the Lawverean algebraic theory of groups. This theory is a category containing an object and operations making it a group object in that category, and the theory is the smallest such category that contains all finite limits. There are fancier ones; the fanciest is the classifying topos for groups, which is in some sense the initial topos-with-group object. Since in a topos, you have full-scale first order intuitionistic logic, the classifying topos for groups allows you to reason about the generic group inside the classifying topos and the theorems you prove will be true for all groups. (This is only an approximation of the actual situation.) In particular you can’t prove it is abelian and you can’t prove it isn’t; the logic clearly does not have excluded middle.

Motivation 2

You can prove that a triangle that has two angles that are equal must be isosceles (has two sides that are equal). You can do this with Pappus’ proof: Look at the triangle, flip it over the perpendicular from the odd angle to the other side, look at it again, and the side-angle-side theorem shows you that the “two” triangles are congruent, so two sides much be equal. This appears to me to be true without requiring the parallel postulate. So the theorem and the proof must be true not only in Euclidean 2-space but in any surface of constant curvature. (Here I am getting into territory I know very little about, so this particular motivation may be totally misguided.)

The Question

So what I want is a classifying space of some sort that contains the generic triangle in such a way that maps of the correct sort to any surface of constant curvature are triangles, and so that Pappus’ proof can be carried out in the classifying space. The space doesn’t have to be a topos or a category at all. I have no clue as to what sort of structure it would be.

Note 1: Even the Lawvere theory of groups has its own internal logic — in this case equational logic. You certainly cannot prove the generic group is or is not abelian with equational logic.

Note 2: It does not seem reasonable to me that Pappus’ proof would work in a surface with variable curvature. But maybe there is some trick to define “angle mod curvature” that would make it true.

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