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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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Representations 2

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Introduction

In a recent post I began a discussion of the mental, physical and mathematical representations of a mathematical object. The discussion continues here. Mathematicians, linguists, cognitive scientists and math educators have investigate some aspects of this topic, but there are many subtle connections between the different ideas which need to be studied.

I don’t have any overall theoretical grasp of these relationships. What I will do here is grope for an overall theory by mentioning a whole bunch of fine points. Some of these have been discussed in the literature and some (as far as I know) have not been discussed.  Many of them (I hope)  can be described as “an obvious fact about representations but no one has pointed it out before”.  Such fine points could be valuable; I think some scholars who have written about mathematical discourse and math in the classroom are not aware of many of these facts.

I am hoping that by thrashing around like this here (for graphs of functions) and for other concepts (set, function, triangle, number …) some theoretical understanding may emerge of what it means to understand math, do math, and talk about math.

The graph of a function

Let’s look at the graph of the function {y=x^3-x}.

What you are looking at is a physical representation of the graph of the function. The graph creates in your brain a mental representation of the graph of the function. These are subtly related to each other and to the mathematical definition of the graph.

Fine points

  1. The mathematical definition [2] of the graph of this function is: The set of ordered pairs of numbers {(x,x^3-x)} for all real numbers {x}.
  2. In the physical representation, each point {(x,x^3-x)} is shown in a location determined by the conventional {x-y} coordinate system, which uses a straight-line representation of the real numbers with labels and ticks.
    • The physical representation makes use of the fact that the function is continuous. It shows the graph as a curving line rather than a bunch of points.
    • The physical representation you are looking at is not the physical representation I am looking at. They are on different computer screens or pieces of paper. We both expect that the representations are very similar, in some sense physically isomorphic.
    • “Location” on the physical representation is a physical idea. The mathematical location on the mathematical graph is essentially the concept of the physical location refined as the accuracy goes to infinity. (This last statement is a metaphor attached to a genuine mathematical construction, for example Cauchy sequences.)
  3. The mathematical definition of “graph” and the physical representation are related by a metaphor. (See Note 1.)
    • The physical curve in blue in the picture corresponds via the metaphor to the graph in the mathematical sense: in this way, each location on the physical curve corresponds to an ordered pair of the form {(x,x^3-x)}.
    • The correspondence between the locations and the pairs is imperfect. You can’t measure with infinite accuracy.
    • The set of ordered pairs {(x,x^3-x)} form a parametrized curve in the mathematical sense. This curve has zero thickness. The curve in the physical representation has positive thickness.
    • Not all the points in the mathematical graph actually occur on the physical curve: The physical curve doesn’t show the left and right infinite tails.
    • The physical curve is drawn to show some salient characteristics of the curve, such as its extrema and inflection points. This is expected by convention in mathematical writing. If the graph had left out a maximum, for example, the author would be constrained (by convention!) to say so.
    • An experienced mathematician or advanced student understands the fine points just listed. A newbie may not, and may draw false conclusions about the function from the graph. (Note 2.)
  4. If you are a mathematician or at least a math student, seeing the physical graph shown above produces a mental image(see Note 3.) of the graph in your mind.
  5. The mathematical definition and the mental image are connected by a metaphor. This is not the same metaphor as the one that connects the physical representation and the mathematical definition.
    • The curve I visualize in my mental representation has an S shape and so does the physical representation. Or does it? Isn’t the S-ness of the shape a fact I construct mentally (without consciously intending to do so!)?
    • Does the curve in the mental rep have thickness? I am not sure this is a meaningful question. However, if you are a sufficiently sophisticated mathematician, your mental image is annotated with the fact that the curve has zero thickness. (See Note 4.)
    • The curve in your mental image of the curve may very well be blue (just because you just looked at my picture) but you must have an annotation to the effect that that is irrelevant! That is the essence of metaphor: Some things are identified with each other and others are emphatically not identified.
    • The coordinate axes do exist in the physical representation and they don’t exist in the mathematical definition of the graph. Of course they are implied by the definition by the properties of the projection functions from a product. But what about your mental image of the graph? My own image does not show the axes, but I do “know” what the coordinates of some of the points are (for example, {(-1,0)}) and I “see” some points (the local maximum and the local minimum) whose coordinates I can figure out.

Notes

1. This is metaphor in the sense lately used by cognitive scientists, for example in [6]. A metaphor can be described roughly as two mental images in which certain parts of one are identified with certain parts of another, in other words a pushout. The rhetorical use of the word “metaphor” requires it to be a figure of speech expressed in a certain way (the identification is direct rather than expressed by “is like” or some such thing.)  In my use in this article a metaphor is something that occurs in your brain.  The form it takes in speech or writing is not relevant.

2. I have noticed, for example, that some students don’t really understand that the left and right tails go off to infinity horizontally as well as vertically.   In fact, the picture above could mislead someone into thinking the curve has vertical asymptotes: The right tail looks like it goes straight up.  How could it get to x equals a billion if it goes straight up?

3. The “mental image” is of course a physical structure in your brain.  So mental representations are physical representations.

4. I presume this “annotation” is some kind of physical connection between neurons or something.  It is clear that a “mental image” is some sort of physical construction or event in the brain, but from what little I know about cognitive science, the scientists themselves are still arguing about the form of the construction.  I would appreciate more information on this. (If the physical representation of mental images is indeed still controversial, this says nothing bad about cognitive science, which is very new.)

References

[1] Mental Representations in Math (previous post).

[2] Definitions (in abstractmath).

[3] Lakoff, G. and R. E. Núñez (2000), Where Mathematics Comes From. Basic Books.

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The Mathematical Definition of Function

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Introduction

This post is a completely rewritten version of the abstractmath article on the definition of function. Like every part of abstractmath, the chapter on functions is designed to get you started thinking about functions. It is no way complete. Wikipedia has much more complete coverage of mathematical functions, but be aware that the coverage is scattered over many articles.

The concept of function in mathematics is as important as any mathematical idea. The mathematician’s concept of function includes the kinds of functions you studied in calculus but is much more abstract and general. If you are new to abstract math you need to know:

  • The precise meaning of the word “function” and other concepts associated with functions. That’s what this section is about.
  • Notation and terminology for functions. (That will be a separate section of abstractmath.org which I will post soon.)
  • The many different kinds of functions there are. (See Examples of Functions in abmath).
  • The many ways mathematicians think about functions. The abmath article Images and Metaphors for Functions is a stub for this.

I will use two running examples throughout this discussion:

  • {F} is the function defined on the set {\left\{1,\,2,3,6 \right\}} as follows: {F(1)=3,\,\,\,F(2)=3,\,\,\,F(3)=2,\,\,\,F(6)=1}. This is a function defined on a finite set by explicitly naming each value.
  • {G} is the real-valued function defined by the formula {G(x)={{x}^{2}}+2x+5}.

Specification of function

We start by giving a specification of “function”. (See the abstractmath article on specification.) After that, we get into the technicalities of the definitions of the general concept of function.

Specification: A function {f} is a mathematical object which determines and is completely determined bythe following data:

  • {f} has a domain, which is a set. The domain may be denoted by {\text{dom }f}.
  • {f} has a codomain, which is also a set and may be denoted by {\text{cod }f}.
  • For each element {a} of the domain of {f}, {f} has a value at {a}, denoted by {f(a)}.
  • The value of {f} at {a} is completely determined by {a} and {f} .
  • The value of {f} at {a} must be an element of the codomain of {f}.

The operation of finding {f(a)} given {f} and {a} is called evaluation.

Examples

  • The definition above of the finite function {F} specifies that the domain is the set {\left\{1,\,2,\,3,\,6 \right\}}. The value of {F} at each element of the domain is given explicitly. The value at 3, for example, is 2, because the definition says that {F(2) = 3}. The codomain of {F} is not specified, but must include the set {\{1,2,3\}}.
  • The definition of {G} above gives the value at each element of the domain by a formula. The value at 3, for example, is {G(3)=3^2+2\cdot3+5=20}. The definition does not specify the domain or the codomain. The convention in the case of functions defined on the real numbers by a formula is to take the domain to be all real numbers at which the formula is defined. In this case, that is every real number, so the domain is {{\mathbb R}}. The codomain must include all real numbers greater than or equal to 4. (Why?)

Comment: The formula above that defines the function G in fact defines a function of complex numbers (even quaternions).

Definition of function

In the nineteenth century, mathematicians realized that it was necessary for some purposes (particularly harmonic analysis) to give a mathematical definition of the concept of function. A stricter version of this definition turned out to be necessary in algebraic topology and other fields, and that is the one I give here.

To state this definition we need a preliminary idea.

The functional property

A set R of ordered pairs has the functional property if two pairs in R with the same first coordinate have to have the same second coordinate (which means they are the same pair).

Examples

  • The set {\{(1,2), (2,4), (3,2), (5,8)\}} has the functional property, since no two different pairs have the same first coordinate. It is true that two of them have the same second coordinate, but that is irrelevant.
  • The set {\{(1,2), (2,4), (3,2), (2,8)\}} does not have the functional property. There are two different pairs with first coordinate 2.
  • The graphs of functions in beginning calculus have the functional property.
  • The empty set {\emptyset} has the functional property .

Example: Graph of a function defined by a formula

The graph of the function {G} given above has the functional property. The graph is the set

\displaystyle \left\{ (x,{{x}^{2}}+2x+5)\,\mathsf{|}\,x\in {\mathbb R} \right\}.

If you repeatedly plug in one real number over and over, you get out the same real number every time. Example:

  • if {x = 0}, then {{{x}^{2}}+2x+5=5}.  You get 5 every time you plug in 0.
  • if {x = 1}, then {{{x}^{2}}+2x+5=8}.
  • if {x =-2}, then {{{x}^{2}}+2x+5=5}.

This set has the functional property because if {x} is any real number, the formula {{{x}^{2}}+2x+5} defines a specific real number. (This description of the graph implicitly assumes that {\text{dom } G={\mathbb R}}.)  No other pair whose first coordinate is {-2} is in the graph of {G}, only {(-2, 5)}. That is because when you plug {-2} into the formula {{{x}^{2}}+2x+5}, you get {5} every time. Of course, {(0, 5)} is in the graph, but that does not contradict the functional property. {(0, 5)} and {(-2, 5)} have the same second coordinate, but that is OK.

How to think about the functional property

The point of the functional property is that for any pair in the set of ordered pairs, the first coordinate determines what the second one is. That’s why you can write “{G(x)}” for any {x } in the domain of {G} and not be ambiguous.

Mathematical definition of function

A function{f} is a mathematical structure consisting of the following objects:

  • A set called the domain of {f}, denoted by {\text{dom } f}.
  • A set called the codomain of {f}, denoted by {\text{cod } f}.
  • A set of ordered pairs called the graph of { f}, with the following properties:
    • {\text{dom } f} is the set of all first coordinates of pairs in the graph of {f}.
    • Every second coordinate of a pair in the graph of {f} is in {\text{cod } f} (but {\text{cod } f} may contain other elements).
    • The graph of {f} has the functional property. Using arrow notation, this implies that {f:A\rightarrow B}.

Examples

  • Let {F} have graph {\{(1,2), (2,4), (3,2), (5,8)\}} and define {A = \{1, 2, 3, 5\}} and {B = \{2, 4, 8\}}. Then {F:A\rightarrow B} is a function.
  • Let {G} have graph {\{(1,2), (2,4), (3,2), (5,8)\}} (same as above), and define {A = \{1, 2, 3, 5\}} and {C = \{2, 4, 8, 9, 11, \pi, 3/2\}}. Then {G:A\rightarrow C} is a (admittedly ridiculous) function. Note that all the second coordinates of the graph are in {C}, along with a bunch of miscellaneous suspicious characters that are not second coordinates of pairs in the graph.
  • Let {H} have graph {\{(1,2), (2,4), (3,2), (5,8)\}}. Then {H:A\rightarrow {\mathbb R}} is a function.

According to the definition of function, {F}, {G} and {H} are three different functions.

Identity and inclusion

Suppose we have two sets A and B with {A\subseteq B}.

  • The identity function on A is the function {{{\text{id}}_{A}}:A\rightarrow A} defined by {{{\text{id}}_{A}}(x)=x} for all{x\in A}. (Many authors call it {{{1}_{A}}}).
  • The inclusion function from A to B is the function {i:A\rightarrow B} defined by {i(x)=x} for all {x\in A}. Note that there is a different function for each pair of sets A and B for which {A\subseteq B}. Some authors call it {{{i}_{A,\,B}}} or {\text{in}{{\text{c}}_{A,\,B}}}.

Remark The identity function and an inclusion function for the same set A have exactly the same graph, namely {\left\{ (a,a)|a\in A \right\}}.

Graphs and functions

  • If {f} is a function, the domain of {f} is the set of first coordinates of all the pairs in {f}.
  • If {x\in \text{dom } f}, then {f(x)} is the second coordinate of the only ordered pair in {f} whose first coordinate is {x}.

Examples

The set {\{(1,2), (2,4), (3,2), (5,8)\}} has the functional property, so it is the graph of a function. Call the function {H}. Then its domain is {\{1,2,3,5\}} and {H(1) = 2} and {H(2) = 4}. {H(4)} is not defined because there is no ordered pair in H beginning with {4} (hence {4} is not in {\text{dom } H}.)

I showed above that the graph of the function {G}, ordinarily described as “the function {G(x)={{x}^{2}}+2x+5}”, has the functional property. The specification of function requires that we say what the domain is and what the value is at each point. These two facts are determined by the graph.

Other definitions of function

Because of the examples above, many authors define a function as a graph with the functional property. Now, the graph of a function {G} may be denoted by {\Gamma(G)}.  This is an older, less strict definition of function that doesn’t work correctly with the concepts of algebraic topology, category theory, and some other branches of mathematics.

For this less strict definition of function, {G=\Gamma(G)}, which causes a clash of our mental images of “graph” and “function”. In every important way except the less-strict definition, they ARE different!

A definition is a device for making the meaning of math technical terms precise. When a mathematician think of “function” they think of many aspects of functions, such as a map of one shape into another, a graph in the real plane, a computational process, a renaming, and so on. One of the ways of thinking of a function is to think about its graph. That happens to be the best way to define the concept of function.  (It is the less strict definition and it is a necessary concept in the modern definition given here.)

The occurrence of the graph in either definition doesn’t make thinking of a function in terms of its graph the most important way of visualizing  it. I don’t think it is even in the top three.

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Mental, Physical and Mathematical Representations

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For a given mathematical object, a mathematician may have:

  • A mental representation of the object. This can be a metaphor, a mental image, or a kinesthetic understanding of the object.
  • A physical representation of the object. This may be a (physical) picture or drawing or three-dimensional model of the object.
  • A mathematical definition and one or more mathematical representations for the object. Such a representation is itself a mathematical object.

The boldface things in this list are related to each other in lots of ways, and they are fuzzy and overlap and don’t include every phenomenon connected with a math object.

I have written about these things ([1], [2], [3], [4]). So have lots of other people. In this post I summarize these ideas. I expect to write about particular examples later on and will use this as a reference.

Two Examples

The following examples point out a few of the relationships between the ideas in boldface above. There is much more to understand.

Function as black box

The idea that a function is a black box or machine with input and output is a metaphor for a function.

A is a metaphor for B means that A and B are cognitively pasted together in such a way that the behavior of A is in many ways like the behavior of B. Such a thing is both useful and dangerous, dangerous because there will be ways in which A behaves that suggest inappropriate ideas about B.

The function as machine is a good metaphor: for example functional composition involves connecting the output of one machine to the input of another, and the inverse function is like running the machine backward.

The function as machine is a bad metaphor: For example, it is wrong to think you could build a machine to calculate any given function exactly. But you can still imagine such a machine, given by a specification (it outputs the value of the function at a given input) and then, in your imagination, connecting the input of one to the output of another must perforce calculate the composite of the corresponding functions.

Like any metaphor, this is a mental representation. That means the metaphor has a physical instantiation in your brain. So a metaphor has a physical representation.

Different people won’t have quite the same concept of a particular metaphor. So a metaphor will have lots of slightly different physical representations, but mathematicians form a community, and communication between mathematicians fine-tunes the different physical instantiations so that they correspond more closely to each other. This is the sense in which mathematical objects have a shared existence in a community as Reuben Hersh has suggested.

A function is a mathematical object, which can be rigorously specified as a set of ordered pairs together with a domain and a codomain. There is a cognitive relationship between the concepts of function as math object and function as black box with input and output.

Triangle

A triangle can be drawn, or created on a computer and a physical image printed out. You may also have a mental image of the triangle.

The physical and the mental images are not the same thing, but they are definitely related. The relationship is mediated by the neuronal circuitry behind your retinas, which performs a highly sophisticated transformation of the pixels on your retina into an organized physical structure in your brain, connected to various other neurons.

This circuitry exists because it helps us get a useful understanding of the world through our eyes. So a picture of a triangle takes advantage of pre-existing neuron structure to generate a useful mental representation that helps us understand and prove things about triangles.

This mental representation also lives in a community of mathematician. Like any community, it has subgroups with “dialects” — varying understanding of representation.

For example, a mathematician who looks at the triangle below sees a triangle that looks like a right triangle. A student sees a triangle that is a right triangle.

This is “sees” in the sense of what their brain reports after all that processing. The mathematician’s brain connects the “triangle I am seeing” module (in their brain) to the “looks like a right triangle” module, but does not connect it to the “is a right triangle” module because they don’t see any statement in the surrounding text that it is a right triangle. The student, on the other hand, fallaciously makes the connection to “is a right triangle” directly.

In some sense, a student who does not make that connection directly is already a mathematician.

A triangle also exists as a mathematical object in your and my brain. It is described by a formal mathematical definition. The pictures of triangles you see above do not fit this definition. For one thing, the line segments in the pictures have thickness. But the pictures trigger a reaction in your neurons that causes your brain to cognitively paste together the line segments in the drawing to the segments required by the formal definition. This is a kind of metaphor of concrete-to-abstract that connects drawings to math objects that mathematicians use all the time.

Note that this “concrete-to-abstract metaphor” itself has a physical existence in your brain.  It drops, for example, the property of thickness that the line segments in the drawing have when matching them (in the metaphor) with the line segments in the corresponding abstract triangle.  On the other hand, it preserves the sense the all three angles in the triangle are acute.  The abstract mathematical concept of triangle (the generic triangle) has no requirement on the angles except that they add up to pi.

Summary

The discussions above describe a few of the complex and subtle relationships that exist between

  • Mental representations of math objects
  • Physical representations of math objects
  • Formally defined math objects and their formally defined representations.

I have purported to discuss how mathematics is understood (especially in connection with language) in several articles and a book but only a few of the relationships I just described are mentioned in any of those articles. Perhaps one or two things I said caused you to react: “Actually, that’s obviously true but I never thought of it before”. (Much the way I had mathematicians in the ’60’s tell me, “I see what you mean that addition is a function of two variables, but I never thought of it that way before”.) (I was a brash category theorist wannabe then.)

A lot of research has been done on understanding math, and some research has been done on mathematical discourse. But what has been done has merely exposed the fin of the shark.

References

[1] Images and metaphors (in abstractmath).

[2] Representations and Models (in abstractmath).

[3] Mathematical Concepts (previous blog).

[4] Mental Representations in Math (previous blog).

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Operation as metaphor in math

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Operation: Is it just a name or is there a metaphor behind it?

A function of the form {f:S\times S\rightarrow S} may be called a binary operation on {S}. The main point to notice is that it takes pairs of elements of {S} to the same set {S}.

A binary operation is a special case of n-ary operation for any natural number {n}, which is a function of the form {f:S^n\rightarrow S}. A {1}-ary (unary) operation on {S} is a function from a set to itself (such as the map that takes an element of a group to its inverse), and a {0}-ary (nullary) operation on {S} is a constant.

It is useful at times to consider multisorted algebra, where a binary operation can be a function {f:S_1\times S_2\rightarrow S_3} where the {S_i} are possibly different sets. Then a unary operation is simply a function.

Calling a function a multisorted unary operation suggest a different way of thinking about it, but as far as I can tell the difference is only that the author is thinking of algebraic operations as examples. This does not seem to be a different metaphor the way “function as map” and “function as transformation” are different metaphors. Am I missing something?

In the 1960’s some mathematicians (not algebraists) were taken aback by the idea that addition of real numbers (for example) is a function. I observed this personally. I don’t think any mathematician would react this way today.

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The emergence of space as a character

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This is an update of a post from a couple of years ago.

Until computers came along, there was no such thing as a space character. The space between printed words was simply a space. In computing, each letter is represented by a certain number, and starting in the early days the space was represented by the number 32 (in decimal notation). In that sense, the way we thought of the space between printed words shifted from empty space to an object represented by empty space.

In the late eighties, I was at a church service on the Sunday when they talk about the budget. After the talk, ten members of the congregation marched up front each carrying a sign with one letter on it. They arranged themselves to spell

GIVE MONEY

This was concrete evidence that we had changed the way we think about spaces between words. The congregation of this upscale church included many engineers and other professional people.

The space character is used in Mathematica to denote multiplication: One writes “x y’’ to mean x times y. This allows multiletter variable names without ambiguity. “distance time’’ would be the product of distance and time. When you have some experience with Mathematica, you think of space between variables as a genuine symbol meaning multiplication.

Space is used in other places in math with a kind of positive meaning; for example, “sin x’’ means the result of evaluating the sine function at x. But I don’t believe most mathematicians think of that space as a symbol. I didn’t until I thought of writing this comment. I am not at all sure it is useful to think of it that way.

When lead type was used in hand typesetting, there were different sizes of blank lead slugs  to put in between letters. With linotypes, a different technique was used: wedges were shoved down between the letters to force the line of type to be right justified.

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Multivalued Functions

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Multivalued functions

I am reconstructing the abstractmath website and am currently working on the part on functions. This has generated some bloggable blustering.

The phrase multivalued function refers to an object that is like a function {f:S\rightarrow T} except that for {s\in S}, {f(s)} may denote more than one value. Multivalued functions arose in considering complex functions such as {\sqrt{z}}. Another example: the indefinite integral is a multivalued operator.

It is useful to think of a multivalued function as a function although it violates one of the requirements of being a function (being single-valued).

A multivalued function {f:S\rightarrow T} can be modeled as a function with domain {S} and codomain the set of all subsets of {T}. The two meanings are equivalent in a strong sense (naturally equivalent). Even so, it seems to me that they represent two different ways of thinking about multivalued functions.: “The value may be any of these things…” as opposed to “The value is this whole set of things.”) The “value may be any of these…” idea has a perfectly good mathematical model: a relation (set of ordered pairs) from {S} to {T} which is the inverse of a surjective function.

Phrases such as “multivalued function” and “partial function” upset some uptight types who say things like, “But a multivalued function is not a function!”. A stepmother is not a mother, either.

I fulminated at length about this in the Handbook article on radial category. (This is conceptual category in the sense of Lakoff, Women, fire and dangerous things, University of Chicago, 1986.). The Handbook is on line, but it downloads very slowly, so I have extracted the particular page on radial categories here.

Functions generate a radial category of concepts in mathematics. There are lots of other concepts in math that have generated radial categories. Think of “incomplete proof” or “left identity”. Radial categories are a basic mechanism of the way we think and function in the world. They should not be banished from mathematics.

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Function as map

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This is a first draft of an article to eventually appear in abstractmath.

Images and metaphors

To explain a math concept, you need to explain how mathematicians think about the concept. This is what in abstractmath I call the images and metaphors carried by the concept. Of course you have to give the precise definition of the concept and basic theorems about it. But without the images and metaphors most students, not to mention mathematicians from a different field, will find it hard to prove much more than some immediate consequences of the definition. Nor will they have much sense of the place of the concept in math and applications.

Teachers will often explain the images and metaphors with handwaving and pictures in a fairly vague way. That is good to start with, but it’s important to get more precise about the images and metaphors. That’s because images and metaphors are often not quite a good fit for the concept — they may suggest things that are false and not suggest things that are true. For example, if a set is a container, why isn’t the element-of relation transitive? (A coin in a coinpurse in your pocket is a coin in your pocket.)

“A metaphor is a useful way to think about something, but it is not the same thing as the same thing.” (I think I stole that from the Economist.) Here, I am going to get precise with the notion that a function is a map. I am acting like a mathematician in “getting precise”, but I am getting precise about a metaphor, not about a mathematical object.

A function is a map

A map (ordinary paper map) of Minnesota has the property that each point on the paper represents a point in the state of Minnesota. This map can be represented as a mathematical function from a subset of a 2-sphere to {{\mathbb R}^2}. The function is a mathematical idealization of the relation between the state and the piece of paper, analogous to the mathematical description of the flight of a rocket ship as a function from {{\mathbb R}} to {{\mathbb R}^3}.

The Minnesota map-as-function is probably continuous and differentiable, and as is well known it can be angle preserving or area preserving but not both.

So you can say there is a point on the paper that represents the location of the statue of Paul Bunyan in Bemidji. There is a set of points that represents the part of the Mississippi River that lies in Minnesota. And so on.

A function has an image. If you think about it you will realize that the image is just a certain portion of the piece of paper. Knowing that a particular point on the paper is in the image of the function is not the information contained in what we call “this map of Minnesota”.

This yields what I consider a basic insight about function-as-map:  The map contains the information about the preimage of each point on the paper map. So:

The map in the sense of a “map of Minnesota” is represented by the whole function, not merely by the image.

I think that is the essence of the metaphor that a function is a map. And I don’t think newbies in abstractmath always understand that relationship.

A morphism is a map

The preceding discussion doesn’t really represent how we think of a paper map of Minnesota. We don’t think in terms of points at all. What we see are marks on the map showing where some particular things are. If it is a road map it has marks showing a lot of roads, a lot of towns, and maybe county boundaries. If it is a topographical map it will show level curves showing elevation. So a paper map of a state should be represented by a structure preserving map, a morphism. Road maps preserve some structure, topographical maps preserve other structure.

The things we call “maps” in math are usually morphisms. For example, you could say that every simple closed curve in the plane is an equivalence class of maps from the unit circle to the plane. Here equivalence class meaning forget the parametrization.

The very fact that I have to mention forgetting the parametrization is that the commonest mathematical way to talk about morphisms is as point-to-point maps with certain properties. But we think about a simple closed curve in the plane as just a distorted circle. The point-to-point correspondence doesn’t matter. So this example is really talking about a morphism as a shape-preserving map. Mathematicians introduced points into talking about preserving shapes in the nineteenth century and we are so used to doing that that we think we have to have points for all maps.

Not that points aren’t useful. But I am analyzing the metaphor here, not the technical side of the math.

Groups are functors

People who don’t do category theory think the idea of a mathematical structure as a functor is weird. From the point of view of the preceding discussion, a particular group is a functor from the generic group to some category. (The target category is Set if the group is discrete, Top if it is a topological group, and so on.)

The generic group is a group in a category called its theory or sketch that is just big enough to let it be a group. If the theory is the category with finite products that is just big enough then it is the Lawvere theory of the group. If it is a topos that is just big enough then it is the classifying topos of groups. The theory in this sense is equivalent to some theory in the sense of string-based logic, for example the signature-with-axioms (equational theory) or the first order theory of groups. Johnstone’s Elephant book is the best place to find the translation between these ideas.

A particular group is represented by a finite-limit-preserving functor on the algebraic theory, or by a logical functor on the classifying topos, and so on; constructions which bring with them the right concept of group homomorphisms as well (they will be any natural transformations).

The way we talk about groups mimics the way we talk about maps. We look at the symmetric group on five letters and say its multiplication is noncommutative. “Its multiplication” tells us that when we talk about this group we are talking about the functor, not just the values of the functor on objects. We use the same symbols of juxtaposition for multiplication in any group, “{1}” or “{e}” for the identity, “{a^{-1}}” for the inverse of {a}, and so on. That is because we are really talking about the multiplication, identity and inverse function in the generic group — they really are the same for all groups. That is because a group is not its underlying set, it is a functor. Just like the map of Minnesota “is” the whole function from the state to the paper, not just the image of the function.

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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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Templates in mathematical practice

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This post is a first pass at what will eventually be a section of abstractmath.org. It’s time to get back to abstractmath; I have been neglecting it for a couple of years.

What I say here is based mainly on my many years of teaching discrete mathematics at Case Western Reserve University in Cleveland and more recently at Metro State University in Saint Paul.

Beginning abstract math

College students typically get into abstract math at the beginning in such courses as linear algebra, discrete math and abstract algebra. Certain problems that come up in those early courses can be grouped together under the notion of (what I call) applying templates [note 0]. These are not the problems people usually think about concerning beginners in abstract math, of which the following is an incomplete list:

The students’ problems discussed here concern understanding what a template is and how to apply it.

Templates can be formulas, rules of inference, or mini-programs. I’ll talk about three examples here.

The template for quadratic equations

The solution of a real 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}

This is a template for finding the roots of the equations. It has subtleties.

For example, the numerator is symmetric in {a} and {c} but the denominator isn’t. So sometimes I try to trick my students (warning them ahead of time that that’s what I’m trying to do) by asking for a formula for the solution of the equation {a+bx+cx^2=0}. The answer is

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

I start writing it on the board, asking them to tell me what comes next. When we get to the denominator, often someone says “{2a}”.

The template is telling you that the denominator is 2 times the coefficient of the square term. It is not telling you it is “{a}”. Using a template (in the sense I mean here) requires pattern matching, but in this particular example, the quadratic template has a shallow incorrect matching and a deeper correct matching. In detail, the shallow matching says “match the letters” and the deep matching says “match the position of the letters”.

Most of the time the quadratic being matched has particular numbers instead of the same letters that the template has, so the trap I just described seldom occurs. But this makes me want to try a variation of the trick: Find the solution of {3+5x+2x^2=0}. Would some students match the textual position (getting {a=3}) instead of the functional position (getting {a=5})? [Note [0]). If they did they would get the solutions {(-1,-\frac{2}{3})} instead of {(-1,-\frac{3}{2})}.

Substituting in algebraic expressions have other traps, too. What sorts of mistakes would students have solving {3x^2+b^2x-5=0}?

Most students on the verge of abstract math don’t make mistakes with the quadratic formula that I have described. The thing about abstract math is that it uses more sophisticated templates

  • subject to conditions
  • with variations
  • with extra levels of abstraction

The template for proof by induction

This template gives a method of proof of a statement of the form {\forall{n}\mathcal{P}(n)}, where {\mathcal{P}} is a predicate (presumably containing {n} as a variable) and {n} varies over positive integers. The template says:

Goal: Prove {\forall{n}\mathcal{P}(n)}.

Method:

  • Prove {\mathcal{P}(1)}
  • For an arbitrary integer {n>1}, assume {\mathcal{P}(n)} and deduce {\mathcal{P}(n+1)}.

For example, to prove {\forall n (2^n+1\geq n^2)} using the template, you have to prove that {2^2+1\geq 1^1}, and that for any {n>1}, if {2^n+1\geq n^2}, then {2^{n+1}+1\geq (n+1)^2}. You come up with the need to prove these statements by substituting into the template. This template has several problems that the quadratic formula does not have.

Variables of different types

The variable {n} is of type integer and the variable {\mathcal{P}} is of type predicate [note 0]. Having to deal with several types of variables comes up already in multivariable calculus (vectors vs. numbers, cross product vs. numerical product, etc) and they multiply like rabbits in beginning abstract math classes. Students sometimes write things like “Let {\mathcal{P}=n+1}”. Multiple types is a big problem that math ed people don’t seem to discuss much (correct me if I am wrong).

Free and bound

The variable {n} occurs as a bound variable in the Goal and a free variable in the Method. This happens in this case because the induction step in the Method originates as the requirement to prove {\forall n(\mathcal{P}(n)\rightarrow\mathcal{P}(n+1))}, but as I have presented it (which seems to be customary) I have translated this into a requirement based on modus ponens. This causes students problems, if they notice it. (“You are assuming what you want to prove!”) Many of them apparently go ahead and produce competent proofs without noticing the dual role of {n}. I say more power to them. I think.

The template has variations

  • You can start the induction at other places.
  • You may have to have two starting points and a double induction hypothesis (for {n-1} and {n}). In fact, you will have to have two starting points, because it seems to be a Fundamental Law of Discrete Math Teaching that you have to talk about the Fibonacci function ad nauseam.
  • Then there is strong induction.

It’s like you can go to the store and buy one template for quadratic equations, but you have to by a package of templates for induction, like highway engineers used to buy packages of plastic French curves to draw highway curves without discontinuous curvature.

The template for row reduction

I am running out of time and won’t go into as much detail on this one. Row reduction is an algorithm. If you write it up as a proper computer program there have to be all sorts of if-thens depending on what you are doing it for. For example if want solutions to the simultaneous equations

2x+4y+z = 1
x+2y = 0
x+2y+4z = 5

you must row reduce the matrix

2 4 1 1
1 2 0 0
1 2 4 5

(I haven’t yet figured out how to wrap this in parentheses) which gives you

1 2 0 0
0 0 1 0
0 0 0 1

This introduces another problem with templates: They come with conditions. In this case the condition is “a row of three 0s followed by a nonzero number means the equations have no solutions”. (There is another condition when there is a row of all 0’s.)

It is very easy for the new student to get the calculation right but to never sit back and see what they have — which conditions apply or whatever.

When you do math you have to repeatedly lean in and focus on the details and then lean back and see the Big Picture. This is something that has to be learned.

What to do, what to do

I have recently experimented with being explicit about templates, in particular going through examples of the use of a template after explicitly stating the template. It is too early to say how successful this is. But I want to point out that even though it might not help to be explicit with students about templates, the analysis in this post of a phenomenon that occurs in beginning abstract math courses

  • may still be accurate (or not), and
  • may help teachers teach such things if they are aware of the phenomenon, even if the students are not.

Notes

  1. Many years ago, I heard someone use the word “template” in the way I am using it now, but I don’t recollect who it was. Applied mathematicians sometimes use it with a meaning similar to mine to refer to soft algorithms–recipes for computation that are not formal algorithms but close enough to be easily translated into a sufficiently high level computer language.
  2. In the formula {ax^2+bx+c}, the “{a}” has the first textual position but the functional position as the coefficient of the quadratic term. This name “functional position” has nothing to do with functions. Can someone suggest a different name that won’t confuse people?
  3. I am using “variable” the way logicians do. Mathematicians would not normally refer to “{\mathcal{P}}” as a variable.
  4. I didn’t say anything about how templates can involve extra layers of abstract.  That will have to wait.
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