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math, understanding math

Curiosity

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Science Daily recently reported on a new study [1] that shows that intellectual curiosity is a good predictor of academic performance.  A few days ago I published the post Liberal-Artsy people.  Now I know that what I was talking about are people with intellectual curiosity!  In the earlier post, I contrasted them with what I called “B.Sc.” types, who are narrowly focused and are not interested in asides in math class about the connections with some concept and other concepts, stories about the discoverer of the concept, the meaning of the name of the concept, and so on.

So better names would be “IC people” instead of Liberal Artsy people and “NF people” (Narrow Focus people) instead of B.Sc.  This is better terminology because it isn’t the type of undergraduate degree they have that matters but their attitude toward knowledge of the world.

There are things to say about these concepts with respect to research mathematicians.  I have known a good many over the years.  (My advice to young people who want to do math research is: Hang around people who know more than you do.)  My impression is that most of the very best mathematicians are IC people who are interested in all sorts of things, not just their branch of math.

Even so, some of the best mathematicians are narrowly focused.  This has always been the case.  Isaac Newton was evidently IC but Kurt Gödel was apparently NF.  (He had no interest in things outside math.  On the other hand, he did find a new model of general relativity, so he was willing to look at others parts of math besides logic.)

I have known some NF mathematicians.  When I wanted to tell them about something they might say, “I have enough trouble keeping up with my field”.  The ones that I knew were mediocre and rarely published much beyond writing up their dissertation.  I suspect that the famous NF mathematicians were simply brilliant enough to get away with being NF.

Perhaps the sort of NF student whose eyes glaze over when

  • you mention Evariste Galois’s tough and short life, or
  • talk about how group theory can be used to classify crystals, or
  • mention that “tangent” comes from the Latin word for “touching”

are doomed to the same mediocrity.  But undoubtedly some of those NF students will turn out to do great things, and some of the IC students will wind up dilettanting through life and never coming close to achieving their potential.

Don’t prejudge students.

[1] S. von Stumm, B. Hell, T. Chamorro-Premuzic. The Hungry Mind: Intellectual Curiosity Is the Third Pillar of Academic Performance. Perspectives on Psychological Science, 2011; 6 (6): 574 DOI: 10.1177/1745691611421204

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exposition, language of math, math, Mathematica, representations, understanding math

Thinking about abstract math

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The abstraction cliff

In universities in the USA, a math major typically starts with calculus, followed by courses such as linear algebra, discrete math, or a special intro course for math majors (which may be taken simultaneously with calculus), then go on to abstract algebra, analysis, and other courses involving abstraction and proofs.

At this point, too many of them hit a wall; their grades drop and they change majors.  They had been getting good grades in high school and in calculus because they were strong in algebra and geometry, but the sudden increase in abstraction in the newer courses completely baffles them. I believe that one big difficulty is that they can't grasp how to think about abstract mathematical objects.  (See Reference [9] and note [a].)   They have fallen off the abstraction cliff.  We lose too many math majors this way. (Abstractmath.org is my major effort to address the problems math majors have during or after calculus.)

This post is a summary of the way I see how mathematicians and students think about math.  I will use it as a reference in later posts where I will write about how we can communicate these ways of thinking.

Concept Image

In 1981, Tall and Vinner  [5] introduced the notion of the concept image that a person has about a mathematical concept or object.   Their paper's abstract says

The concept image consists of all the cognitive structure in the individual's mind that is associated with a given concept. This may not be globally coherent and may have aspects which are quite different from the formal concept definition.

The concept image you may have of an abstract object generally contains many kinds of constituents:

  • visual images of the object
  • metaphors connecting the object to other concepts
  • descriptions of the object in mathematical English
  • descriptions and symbols of the object in the symbolic language of math
  • kinetic feelings concerning certain aspects of the object
  • how you calculate parameters of the object
  • how you prove particular statements about the object

This list is incomplete and the items overlap.  I will write in detail about these ideas later.

The name "concept image" is misleading [b]), so when I have written about them, I have called them metaphors or mental representations as well as concept images, for example in [3] and [4].

Abstract mathematical concepts

This is my take on the notion of concept image, which may be different from that of most researchers in math ed. It owes a lot to the ideas of Reuben Hersh [7], [8].

  • An abstract mathematical concept is represented physically in your brain by what I have called "modules" [1] (physical constituents or activities of the brain [c]).
  • The representation generally consists of many modules.  They correspond to the list of constituents of a concept image given above.  There is no assumption that all the modules are "correct".
  • This representation exists in a semi-public network of mathematicians' and students' brains. This network exercises (incomplete) control over your personal representation of the abstract structure by means of conversation with other mathematicians and reading books and papers.  In this sense, an abstract concept is a social object.  (This is the only point of view in the philosophy of math that I know of that contains any scientific content.)

Notes

[a]  Before you object that abstraction isn't the only thing they have trouble with, note that a proof is an abstract mathematical object. The written proof is a representation of the abstract structure of the proof.  Of course, proofs are a special kind of abstract structure that causes special problems for students.

[b] Cognitive science people use "image" to include nonvisual representations, but not everyone does.  Indeed, cognitive scientists use "metaphor" as well with a broader meaning than your high school English teacher.  A metaphor involves the cognitive merging of parts of two concepts (specifically with other parts not merged). See [6].

[c] Note that I am carefully not saying what the modules actually are — neurons, networks of neurons, events in the brain, etc.   From the point of view of teaching and understanding math, it doesn't matter what they are, only that they exist and live in a society where they get modified by memes  (ideas, attitudes, styles physically transmitted from brain to brain by speech, writing, nonverbal communication, appearance, and in other ways).

References

  1. Math and modules of the mind (previous post)
  2. Mathematical Concepts (previous post)
  3. Mental, physical and mathematical representations (previous post)
  4. Images and Metaphors (abstractmath.org)
  5. David Tall and Schlomo Vinner, Concept Image and Concept Definition in Mathematics with particular reference to limits and continuity, Journal Educational Studies in Mathematics, 12 (May, 1981), no. 2, 151–169.
  6. Conceptual metaphor (Wikipedia article).
  7. What is mathematics, really? by Reuben Hersh, Oxford University Press, 1999.  Read online at Questia.
  8. 18 Unconventional Essays on the Nature of Mathematics, by Reuben Hersh. Springer, 2005.
  9. Mathematical objects (abstractmath.org).

 

 

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exposition, math, understanding math

Liberal-artsy people

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I graduated from Oberlin College with a B.A. as a math major and minors in philosophy and English literature, with only three semesters of science courses.  I was and am "liberal-artsy".   As professor of math at Case Western Reserve University,  I had lots of colleagues in both pure and applied math who started out with B.Sc. degrees. We did not always understand each other very well!

Caveat: "Liberal-artsy" and "Narrowly Focused B.Sc. type" (I need a better name) are characteristics that people may have in varying amounts, and many professors in science and math have both characteristics.   I do, myself, although I am more L.A. that B.Sc.  Furthermore, I know nothing about any sociological or cognitive-science research on these characteristics.  I am making it all up as I write.  (This is a blog post, not a tome.)

I recently posted on secants and  tangents.  These articles were deliberately aimed to tickle the interests of L.A.  students.

Liberal-artsy types want to know about connections between concepts.  In each post, I wrote on both common meanings of the words (secant line and function, tangent line and function) and the close connections between them.  Some trig teachers / trig texts tell students about these connections but too many don't.   On the other hand, many B.Sc. types are left cold by such discussions.  B.Sc. types are goal-oriented and want to know a) how do I use it? b) how do I calculate it?  They get impatient when you talk about anything else.  I say point out these connections anyway.

L.A. types want to know about the reason for the name of a concept.  The post on secants refers to the metaphor that "secant" means "cutting". This is based on the etymology of "secant", which is hidden to many students  because it is based on Latin.  The post makes the connection that the "original" definition of "secant" was the length of a certain line segment generated by an angle in the unit circle. The post on tangents makes an analogous connection, and also points out that most tangent lines that students see touch the curve at only a single point, which is not a connotation of the English word "touch".

Many people think they have learned something when they know the etymology of a word.  In fact, the etymology of a word may have little or nothing to do with its current meaning, which may have developed over many centuries of metaphors that become dead, generate new metaphors that become dead, umpteen times, so that the original meaning is lost.  (The word "testimony" cam from a Latin phrase meaning hold your testicles, which is really not related to its meaning in present-day English.)

So I am not convinced that etymologies of names can help much in most cases.  In particular, different mathematical definitions of the same concept can be practically disjoint in terms of the data they use, and there is no one "correct" definition, although there may be only one that motivates the name.  (There often isn't a definition that motivates the name.  Think "group".)  But I do know that when I mention the history of a name of a concept in class, some students are fascinated and ask me questions about it.

L.A. types are often fascinated by ETBell-like stories about the mathematician who came up with a concept, and sometimes the stories illuminate the mathematical idea.  But L. A. types often are interested anyway.  It's funny when you talk about such a thing in class, because some students visibly tune out while others noticeably perk up and start paying attention.

So who should you cater to?  Answer:  Both kinds of students.  (Tell interesting stories, but quickly and in an offhand way.)

The posts on secants and tangents also experimented with using manipulable diagrams to illustrate the ideas.  I expect to write about that more in another post.

For more about the role of definitions, check out the abmath article and also Timothy Gowers' post on definitions (one of a series of excellent posts on working with abstract math).


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