Category Archives: Mathematica

computer science, exposition, language of math, math, Mathematica, representations, Uncategorized, understanding math

A visualization of a computation in tree form

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Live evaluation of expressions in TreeForm 3, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The code is ad-hoc.  It might be worthwhile for someone to design a package that produces this sort of tree for any expression. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

This demonstration shows the step by step computation of the value of the expression $3x^2+2(1+y)$ shown as a tree.  By moving the first slider from right to left, you go through the six steps of the computation. You may select the values of $x$ and $y$ with the second and third sliders.  If you click on the plus sign next to a slide, a menu opens up that allows you to make the slider move automatically, shows the values, and other things.

Note that subtrees on the same level are evaluate left to right.  Parallel processing would save two steps.

A previous post related to this post is Making visible the abstraction in algebraic notation.

  

 
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Making visible the abstraction in algebraic notation

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Handmade Exp Tree.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

Algebraic notation

Algebraic notation contains a hidden abstract structure coded by apparently arbitrary conventions that many college calculus students don't understand completely. This very simple example shows one of the ways in which calc students may be confused:

  1. $x+2y$
  2. $(x+2)y$

Students often mean to express formula 2 when they write something like \[x\!\!+\!\!2\,\,\,\,y\] (with a space).  This is a perfectly natural way to write it. But it is against the rules, I presume because in handwriting it is not clear when you mean a space and when you don't. 

Formula 1 can also be written as $x+(2y)$, and if it were usually written that way students (I predict) would be less confused.   Always writing it this way would exacerbate the clutter of parentheses but would allow a simple rule:

Evaluate every expression inside parentheses first, starting with the innermost.

Using trees for algebra

Writing algebraic expressions as a tree (as in computing science)

  • makes it obvious what gets evaluated first
  • uses no parentheses at all.

An example of using the tree of an expression to do calculations is available in Expressions.nb (requires Mathematica) and Expressions.cdf (requires CDF player only) on my Mathematica website.  I could imagine using tree expressions instead of standard notation as the normal way of doing things. That would require working on Ipads or some such and would take a big amount of investment in software making it intuitive and easy to use.  No, I am not going to embark on such an adventure, but I think it ought to be attempted.  (Brett Victor has many ideas like this.)

Transforming algebraic notation into trees

The two manipulable diagrams below show the algebraic notation being transformed into tree form.  I expect that this will make the abstract structure more concrete for many students and I encourage others to show it to their students.  Note that the tree form makes everything explicit.>

After I return from a ten-day trip I will explore the possibility of making the expression-to-tree transformer turn the expression into an evaluable tree as in Expressions.nb and Expressions.cdf.  In the I hope not to distant future students should have access to many transformers that morph expressions from one form into another.  Such transformers are much more politically correct than Optimus Prime.

Offloading chunking and Computable algebraic expressions in tree form are earlier posts related to this post.

 

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Metaphors in computing science I

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(This article is continued in Metaphors in computing science II)

Michael Barr recently told me of a transcription of a talk by Edsger Dijkstra dissing the use of metaphors in teaching programming and advocating that every program be written together with a proof that it works.  This led me to think about the metaphors used in computing science, and that is what this post is about.  It is not a direct answer to what Dijkstra said. 

We understand almost anything by using metaphors.  This is a broader sense of metaphor than that thing in English class where you had to say "my love is a red red rose" instead of "my love is like a red red rose".  Here I am talking about conceptual metaphors (see references at the end of the post).  

Metaphor: A program is a set of instructions

You can think of a program as a list of instructions that you can read and, if it is not very complicated, understand how to carry them out.  This metaphor comes from your experience with directions on how to do something (like directions from Google Maps or for assembling a toy).   In the case of a program, you can visualize doing what the program says to do and coming out with the expected output. This is one of the fundamental metaphors for programs. 

Such a program may be informal text or it may be written in a computer language.

Example

A description of how to calculate $n!$ in English could be:  "Multiply the integers $1$ through $n$".  In Mathematica, you could define the factorial function this way:

fac[n_] := Apply[Times, Table[i, {i, 1, n}]]

This more or less directly copies the English definition, which could have been reworded as "Apply the Times function to the integers from $1$ to $n$ inclusive."  Mathematica programmers customarily use the abbreviation "@@" for Apply because it is more convenient:

Fac[n_]:=Times @@ Table[i, {i, 1, 6}]

As far as I know, C does not have list operations built in.  This simple program gives you the factorial function evaluated at $n$:

 j=1;  for (i=2; i<=n; i++)   j=j*i; return j;  

This does the calculation in a different way: it goes through the numbers $1, 2,\ldots,n$ and multiplies the result-so-far by the new number.  If you are old enough to remember Pascal or Basic, you will see that there you could use a DO loop to accomplish the same thing.

What this metaphor makes you think of

Every metaphor suggests both correct and incorrect ideas about the concept.  

  • If you think of a list of instructions, you typically think that you should carry out the instructions in order.  (If they are Ikea instructions, your experience may have taught you that you must carry out the instructions in order.)  
  • In fact, you don't have to "multiply the numbers from $1$ to $n$" in order at all: You could break the list of numbers into several lists and give each one to a different person to do, and they would give their answers to you and you would multiply them together.
  • The instructions for calculating the factorial can be translated directly into Mathematica instructions, which does not specify an order.   When $n$ is large enough, Mathematica would in fact do something like the process of giving it to several different people (well, processors) to speed things up.
  • I had hoped that Wolfram alpha would answer "720" if I wrote "multiply the numbers from $1$ to $6$" in its box, but it didn't work.  If it had worked, the instruction in English would not be translated at all. (Note added 7 July 2012:  Wolfram has repaired this.)
  • The example program for C that I gave above explicitly multiplies the numbers together in order from little to big.  That is the way it is usually taught in class.  In fact, you could program a package for lists using pointers (a process taught in class!) and then use your package to write a C program that looks like the  "multiply the numbers from $1$ to $n$" approach.  I don't know much about C; a reader could probably tell me other better ways to do it.

So notice what happened:

  • You can translate the "multiply the numbers from $1$ to $n$" directly into Mathematica.
  •  For C, you have to write a program that implements multiplying the numbers from $1$ to $n$. Implementation in this sense doesn't seem to come up when we think about instruction sets for putting furniture together.  It is sort of like: Build a robot to insert & tighten all the screws.

Thus the concept of program in computing science comes with the idea of translating the program instruction set into another instruction set.

  • The translation provided above for Mathematica resembles translating the instruction set into another language. 
  • The two translations I suggested for C (the program and the definition of a list package to be used in the translation) are not like translating from English to another language.  They involve a conceptual reconstruction of the set of instructions.

Similarly, a compiler translates a program in a computer language into machine code, which involves automated conceptual reconstruction on a vast scale.

Other metaphors

In writing about this, I have brought in other metaphors, for example:

  • C or Mathematica as like a natural language in some ways 
  • Compiling (or interpreting) as translation

Computing science has used other VIM's (Very Important Metaphors) that I need to write about later:

  • Semantics (metaphor: meaning)
  • Program as text — this allows you to treat the program as a mathematical object
  • Program as machine, with states and actions like automata and Turing machines.
  • Specification of a program.  You can regard  "the product of the numbers from $1$ to $n$" as a specification.  Notice that saying "the product" instead of "multiply" changes the metaphor from "instruction" to "specification".

References

Conceptual metaphors (Wikipedia)

Images and Metaphors (article in abstractmath)

Images and Metaphors for Sets (article in abstractmath)

Images and Metaphors for Functions (incomplete article in abstractmath)

 

 
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Improved clouds

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome.

The source code for these demos is Animated Riemann.nb at my Mathematica Site. The notebook is is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

The animated clouds show two hundred precalculated clouds for each picture, so you get the same clouds each time you run the animation.  It would have taken too long to generate the random clouds on the fly.  Each list of two hundred took about seven minutes to create on my computer.

In my post Riemann Clouds Improved I showed examples of clouds of values of Riemann sums in such a way that you could see the convergence to the value, the efficiency of the midpoint rule, and other things.  Here I include two Riemann sums that are shown

  • as manipulable graphs,
  • in clouds in an animated form.

Each manipulable graph (see Elaborate Riemann Sums Demo) has a slider to choose the mesh (1/n) of the partitions.  The small plus sign besides the slider gives you additional options. The buttons allow you to choose the type of partition and the type of evaluation points.

Each cloud shows a collection of values of random Riemann sums of the function, plotted by size of mesh (an upper bound on the width of the largest subdivision) and the value of the sum.  The cloud shows how the sums converge to the value of the integral. 

Every dot represents a random partition.  The sums with blue dots have random valuation points, the green dots use the left side of the subdivision, the brown dots the right side, and the red dots the midpoint.  The clouds may be suitable for students to study.  Some possible questions they could be asked to do are listed at the end.

Pressing the starter shows many clouds in rapid succession.  I don't know how much educational value that has but I think it is fun, and fun is worthwhile in itself.

Quarter Circle

Manipulable graph:

Animated cloud

 

Sine wave

Manipulable graph:

Animated cloud

Questions

I am not sure of the answers to some of these myself.

  • Why is the accuracy generally better for the sine wave than for the quarter circle?  
  • Why are the green dots above all the others and the brown dots below all the others in the quarter circle?
  • Why are they mixed in with the others for the sine curve?  In fact why do they tend upward? (Going from right to left, in other words in the direction of more accuracy).
  • Why are the midpoint sums so much more accurate?
  • Why do they tend downward for the sine wave?
  • Is it an optical illusion or do they also tend downward for the quarter circle? 

Notes:

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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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Offloading abstraction

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The interactive examples in this post require installing Wolfram CDF Player., which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Tangent Line.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.


The diagram above shows you the tangent line to the curve $y=x^3-x$ at a specific point.  The slider allows you to move the point around, and the tangent line moves with it. You can click on one of the plus signs for options about things you can do with the slider.  (Note: This is not new.  Many other people have produced diagrams like this one.)

I have some comments to make about this very simple diagram. I hope they raise your consciousness about what is going on when you use a manipulable demonstration.

Farming out your abstraction load

A diagram showing a tangent line drawn on the board or in a paper book requires you visualize how the tangent line would look at other points.  This imposes a burden of visualization on you.  Even if you are a new student you won't find that terribly hard (am I wrong?) but you might miss some things at first:

  • There are places where the tangent line is horizontal.
  • There are places where some of the tangent lines cross the curve at another point. Many calculus students believe in the myth that the tangent line crosses the curve at only one point.  (It is not really a myth, it is a lie.  Any decent myth contains illuminating stories and metaphors.)
  • You may not envision (until you have some experience anyway) how when you move the tangent line around it sort of rocks like a seesaw.

You see these things immediately when you manipulate the slider.

Manipulating the slider reduces the load of abstract thinking in your learning process.     You have less to keep in your memory; some of the abstract thinking is offloaded onto the diagram.  This could be described as contracting out (from your head to the picture) part of the visualization process.  (Visualizing something in your head is a form of abstraction.)

Of course, reading and writing does that, too.  And even a static graph of a function lowers your visualization load.  What interactive diagrams give the student is a new tool for offloading abstraction.

You can also think of it as providing external chunking.  (I'll have to think about that more…)

Simple manipulative diagrams vs. complicated ones

The diagram above is very simple with no bells and whistles.  People have come up with much more complicated diagrams to illustrate a mathematical point.  Such diagrams:

  • May give you buttons that give you a choice of several curves that show the tangent line.
  • May give a numerical table that shows things like the slope or intercept of the current tangent line.
  • May also show the graph of the derivative, enabling you to see that it is in fact giving the value of the slope.

Such complicated diagrams are better suited for the student to play with at home, or to play with in class with a partner (much better than doing it by yourself).  When the teacher first explains a concept, the diagrams ought to be simple.

Examples

  • The Definition of derivative demo (from the Wolfram Demonstration Project) is an example that provides a table that shows the current values of some parameters that depend on the position of the slider.
  • The Wolfram demo Graphs of Taylor Polynomials is a good example of a demo to take home and experiment extensively with.  It gives buttons to choose different functions, a slider to choose the expansion point, another one to choose the number of Taylor polynomials, and other things.
  • On the other hand, the Wolfram demo Tangent to a Curve is very simple and differs from the one above in one respect: It shows only a finite piece of the tangent line.  That actually has a very different philosophical basis: it is representing for you the stalk of the tangent space at that point (the infinitesimal vector that contains the essence of the tangent line).
  • Brian Hayes wrote an article in American Scientist containing a moving graph (it moves only  on the website, not in the paper version!) that shows the changes of the population of the world by bars representing age groups.  This makes it much easier to visualize what happens over time.  Each age group moves up the graph — and shrinks until it disappears around age 100 — step by step.  If you have only the printed version, you have to imagine that happening.  The printed version requires more abstract visualization than the moving version.
  • Evaluating an algebraic expression requires seeing the abstract structure of the expression, which can be shown as a tree.  I would expect that if the students could automatically generate the tree (as you can in Mathematica)  they would retain the picture when working with an expression.  In my post computable algebraic expressions in tree form I show how you could turn the tree into an evaluation aid.  See also my post Syntax trees.

This blog has a category "Mathematica" which contains all the graphs (many of the interactive) that are designed as an aid to offloading abstraction.

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Bugs in English and in math

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Everyone knows that computer programs have bugs.  In fact, languages have bugs, too, although we don't usually call them that.  

Bugs in English 

  

Right

Q: "Should I turn left at the next corner?" A: "Right".  Probably most Americans who drive now know this bug.  The answer could mean "yes" or "turn right".  So we have to stop and think how to answer this question.  That makes it a bug.  

Too, two

Comment: " We will take Route 30".  Answer: "We will take Route 30 too".  This bug is probably responsible for the survival of the word "also".  

Note that unlike the case of "right", this is a bug only of spoken English.

Subject and predicate

In Comma rule found dysfunctional, I wrote about the problem that in formal English writing there is no way to indicate where the subject ends and the predicate begins.  This causes a problem reading complicated sentences with many clauses such as academic writing often uses.  Of course, one way around this is to write short, simple sentences!  (That sounds like the subject of a future blog…) 

Bugs in the symbolic language of math

  

Fractions

In both Excel and Mathematica, "1/2*3" means 3/2. Now, I would think "1/2a" means "1/(2a)", but younger mathematicians are taught PEMDAS (see Purplemath), which says that division and multiplication have the same precedence and operations are evaluated from left to right.  

 If in Mathematica you define a function f[a_] := 1/2a, f[3] evaluates to 3/2, so Mathematica (and most other computer languages) agree with PEMDAS. (Note: When you write 1/2a in a Mathematica notebook, it automatically puts a space between the 2 and the a, and space in Mathematica means times, so it does warn you.)

Nevertheless, my ancient education would lead me to write (1/2)a for that meaning.  This means I must learn to write 1/(2a) for the other meaning instead of 1/2a.  

Questions:

  • Did the language really change or was I always "doing it wrong"?  I would like to hear from other ancient mathematicians.  (But I don't know very many who would read blogs or Purplemath.)
  • Should such a phenomenon be called a bug? 

Repeated exponentiation

In Excel, "2^2^3" means $(2^2)^3$, in other words, 64.  In Mathematica, it means $2^{(2^3)}=2^8=256$.  My impression is that most mathematicians expect it to mean $2^{(2^3)}$.  

References: This post in Walking Randomly, my post Mathematical Usage, Wikipedia's article.  

Exponentiation on functions is ambiguous

If $f:\mathbb{R}\to\mathbb{R}$ is a function, $f^2(x)$ can mean either $f(f(x))$ or $f(x)f(x)$, and both usages are common.  You should tell your students about this because no one is ever going to make one of the usages go away.

A far worse catastrophe is the fact that in calculus books, $\sin^2x=(\sin\,x)(\sin\,x)$ but $\sin^{-1}x=\text{arcsin}\,x$.  I betcha (lived in Minnesota four years now) we could succeed with a campaign to convince calc book publishers to always write $(\sin\,x)^2$ and $\arcsin\,x$.  

Bugs in the Mathematical Dialect of English

The mathematical dialect of English is what I call Mathematical English in the abstractmath website.  It is a different language from the symbolic language, which is not a dialect of English.

I have written about the problems with Mathematical English in a ridiculous number of places.  (See references in The Handbook of Mathematical Discourse).  It is normal for a dialect of a language to use words and grammatical structures that in the original language mean different things.  (See Dialects below).

Words with different meanings

  • A set is a group in standard English, but not in math English.  
  • The number 2+3i is a real number in standard English, but not in math English.  
  • And so on.

Use of adjectives and prefixes

  • A "noncommutative ring" has commutative addition.
  • A "semigroup" has a fully defined binary operation.

If, then

The bug that grabs math newbies by the throat and won't let go is the meaning of "If P, then Q".  

  • "If a number is divisible by 4, then it is even" in math dialect means a number not divisible by 4 might be even anyway.
  • "If you eat your broccoli you will get your dessert" in standard American Parental English does not mean you might get your dessert if you don't eat your broccoli.

And then there is the phenomenon of Vacuous Implication, which leaves students gasping and writhing.

About "dialects"

Most Americans are not familiar with dialects in the sense I am using the word here, since the only really different dialects we have are Gullah and Hawaiian Pidgin, both of which are very hard to understand; although for example Appalachian English and African-American urban vernacular [1] are dialects of a milder sort.  I grew up in Savannah and heard diluted Gullah sometimes on the street (didn't understand much).  I am also rather familiar with Züritüütsch since we lived in Zürich for a year.   

What the rest of the world call dialects have many distinctive properties:

  • They have nonstandard pronunciation to the point where they are difficult to understand. 
  • They have differences in grammar.  (Both Gullah and especially Hawaiian Creole have differences in grammar from Standard English.) 
  • They have differences in vocabulary, enough sometimes to cause misunderstanding.

I grew up speaking an Atlanta dialect, which really did have differences in all those parameters.  But what people today call a Southern accent is really just an accent (minor variations in pronunciation), not a dialect.  

Hawaiian Creole, and possibly Gullah, but not the other dialects I mentioned, are singled out by linguists as creoles because they been modified heavy influence from another language.  Züritüütsch is not a creole, but it is quite difficult for native German-speakers to understand.  The Swiss situation particularly emphasizes the distinction between "dialect" and "accent".  The typical native of Zürich speaks Züritüütsch and also speaks standard German with a Swiss accent.  

Reference

[1] What Language Is (And What It Isn't and What It Could Be) by John H. McWhorter. Gotham, 2011.

 

 

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Freezing a family of functions

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook algebra1.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

Some background

  • Generally, I have advocated using all sorts of images and metaphors to enable people to think about particular mathematical objects more easily.
  • In previous posts I have illustrated many ways (some old, some new, many recently using Mathematica CDF files) that you can provide such images and metaphors, to help university math majors get over the abstraction cliff.
  • When you have to prove something you find yourself throwing out the images and metaphors (usually a bit at a time rather than all at once) to get down to the rigorous view of math [1], [2], [3], to the point where you think of all the mathematical objects you are dealing with as unchanging and inert (not reacting to anything else).  In other words, dead.
  • The simple example of a family of functions in this post is intended to give people a way of thinking about getting into the rigorous view of the family.  So this post uses image-and-metaphor technology to illustrate a way of thinking about one of the basic proof techniques in math (representing the object in rigor mortis so you can dissect it).  I suppose this is meta-math-ed.  But I don’t want to think about that too much…
  • This example also illustrates the difference between parameters and variables. The bottom line is that the difference is entirely in how we think about them. I will write more about that later.

 A family of functions

This graph shows individual members of the family of functions \( y=a\sin\,x\) for various values of a. Let’s look at some of the ways you can think about this.

  • Each choice of  “shows the function for that value of the parameter a“.  But really, it shows the graph of the function, in fact only the part between x=-4 and x= 4.
  • You can also think of it as showing the function changing shape as a changes over time (as you slide the controller back and forth).

Well, you can graph something changing over time by introducing another axis for time.  When you graph vertical motion of a particle over time you use a two-dimensional picture, one axis representing time and the other the height of the particle. Our representation of the function y=a\sin\,x is a two-dimensional object (using its graph) so we represent the function in 3-space, as in this picture, where the slider not only shows the current (graph of the) function for parameter value a but also locates it over a on the z axis.

The picture below shows the surface given by y=a\sin\,x as a function of both variables a and x. Note that this graph is static: it does not change over time (no slide bar!). This is the family of functions represented as a rigorous (dead!) mathematical object.

If you click the “Show Curves” button, you will see a selection of the curves in middle diagram above drawn as functions of x for certain values of a. Each blue curve is thus a sine wave of amplitude a. Pushing that button illustrates the process going on in your mind when you concentrate on one aspect of the surface, namely its cross-sections in the x direction.

Reference [4] gives the code for the diagrams in this post, as well as a couple of others that may add more insight to the idea. Reference [5] gives similar constructions for a different family of functions.

References

  1. Rigorous view in abstractmath.org 
  2. Representations II: Dry Bones (post)
  3. Representations III: Rigor and Rigor Mortis (post)
  4. FamiliesFrozen.nb.
  5. AnotherFamiliesFrozen.nb (Mathematica file showing another family of functions)
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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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