This post has been rewritten and posted on the abstractmath.org Articles Page as Elaborate Riemann Sum Demo.
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Every post that talks about representation of mathematical objects in the most general sense.
Send to KindleMore about the definition of function
Maya Incaand commented on my post Definition of "function":
Why did you decide against "two inequivalent descriptions in common use"? Is it no longer true?
This question concerns [1], which is a draft article. I have not promoted it to the standard article in abstractmath because I am not satisfied with some things in it.
More specifically, there really are two inequivalent descriptions in common use. This is stated by the article, buried in the text, but if you read the beginning, you get the impression that there is only one specification. I waffled, in other words, and I expect to rewrite the beginning to make things clearer.
Below are the two main definitions you see in university courses taken by math majors and grad students. A functional relation has the property that no two distinct ordered pairs have the same first element.
Strict definition: A function consists of a functional relation with specified codomain (the domain is then defined to be the set of first elements of pairs in the relation). Thus if $A$ and $B$ are sets and $A\subseteq B$, then the identity function $1_A:A\to A$ and the inclusion function $i:A\to B$ are two different functions.
Relational definition: A function is a functional relation. Then the identity and inclusion functions are the same function. This means that a function and its graph are the same thing (discussed in the draft article).
These definitions are subject to variations:
Variations in the strict definition: Some authors use "range" for "codomain" in the definition, and some don't make it clear that two functions with the same functional relation but different codomains are different functions.
Variations in the relational definition: Most such definitions state explicitly that the domain and range are determined by the relation (the set of first coordinates and the set of second coordinates).
Formalism
There are many other variations in the formalism used in the definition. For example, the strict definition can be formalized (as in Wikipedia) as an ordered triple $(A, B, f)$ where $A$ and $B$ are sets and $f$ is a functional relation with the property thar every element of $A$ is the first element of an ordered pair in the relation.
You could of course talk about an ordered triple $(A,f,B)$ blah blah. Such definitions introduce arbitrary constructions that have properties irrelevant to the concept of function. Would you ever say that the second element of the function $f(x)=x+1$ on the reals is the set of real numbers? (Of course, if you used the formalism $(A,f,B)$ you would have to say the second element of the function is its graph! )
It is that kind of thing that led me to use a specification instead of a definition. If you pay attention to such irrelevant formalism there seems to be many definitions of function. In fact, at the university level there are only two, the strict definition and the relational definition. The usage varies by discipline and age. Younger mathematicians are more likely to use the strict definition. Topologists use the strict definition more often than analysts (I think).
Usage
There is also variation in usage.
- Most authors don't tell you which definition they use, and it often doesn't matter anyway.
- If an author defines a function using a formula, there is commonly an implicit assumption that the domain includes everything for which the formula is well-defined. (The "everything" may be modified by referring to it as an integer, real, or complex function.)
Definitions of function on the web
Below are some definitions of function that appear on the web. I have excluded most definitions aimed at calculus students or below; they often assume you are talking about numbers and formulas. I have not surveyed textbooks and research papers. That would have to be done for a proper scholarly article about mathematical usage of "function". But most younger people get their knowledge from the web anyway.
- Abstractmath draft article: Functions: Specification and Definition. (Note: Right now you can't get to this from the Table of Contents; you have to click the preceding link.)
- Gyre&Gimble post: Definition of "function"
- Intmath discussion of function Function as functional relation between numbers, with induced domain and range.
- Mathworld definition of function Functional-relation definition. Defines $F:A\to B$ in a way that requires $B$ to be the image.
- Planet Math definition of function Strict definition.
- Prime Encyclopedia of Mathematics Functional-relation definition.
- Springer Encyclopedia of Math definition of function Strict definition, except not clear if different codomains mean different functions.
- Wikipedia definition of function Discusses both definitions.
- Wisconsin Department of Public Instruction Definition of function Function as functional relation.
Send to KindleOffloading chunking
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
- Abstractmath article on chunking.
- Wikipedia on chunking
Send to KindleOffloading abstraction
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.
Send to KindleAbstract objects
Some thoughts toward revising my article on mathematical objects.
Mathematical objects are a kind of abstract object. There are lots of abstract objects that are not mathematical objects, For example, if you keep a calendar or schedule for appointments, that calendar is an abstract object. (This example comes from [2]).
It may be represented as a physical object or you may keep it entirely in your head. I am not going to talk about the latter possibility, because I don't know what to say.
- If it is a paper calendar, that physical object represents the information that is contained in your calendar.
- Same for a calendar on a computer, but that is stored as magnetic bits on a disk or in flash memory. A computer program (part of the operating system) is required to present it on the screen in such a way that you can read it. Each time you open it, you get a new physical representation of the calendar.
Your brain contains a module (see [5], [7]) that interprets the representation in (1) or (2) and which has connections with other modules in your brain for dates, times, locations and whether the appointment is for a committee, a medical exam, or whatever.
The calendar-interpreter module in your brain is necessary for the physical object to be a calendar. The physical object is not in itself your calendar. The calendar in this sense does not exist in the physical world. It is abstract. Since we think of it as a thing, it is an abstract object.
The abstract object "my calendar" affects the physical world (it causes you to go to the dentist next Tuesday). The relation of the abstract object to the physical world is mediated by whatever physical object you call your calendar along with the modules in the brain that relate to it. The modules in the brain are actions by physical objects, so this point of view does not involve Cartesian style dualism.
Note: A module is a meme. Are all memes modules? This needs to be investigated. Whatever they are, they exist as physical objects in people's brains.
Mathematical objects
A rigorous proof of a theorem about a mathematical object tends to refer to the object as if it were absolutely static and did not affect anything in the physical world. I talked about this in [10], where I called it the dry bones representation of a mathematical object. Mathematical objects don't have to be thought of this way, but (I suggest) what makes them mathematical objects is that they can be thought of in dry bones mode.
If you use calculus to figure out how much fuel to use in a rocket to make it go a mile high, then actually use that amount in the rocket and send it off, your calculations have affected your physical actions, so you were thinking of the calculations as an abstract object. But if you sit down to check your calculations, you concentrate on the steps one by one with the rules of algebra and calculus in mind. You are looking at them as inert objects, like you would look at a bone of a dinosaur to see what species it belongs to. From that point of view your calculations form a mathematical object, because you are using the dry-bones approach.
Caveat
All this blather is about how you should think about mathematical objects. It can be read as philosophy, but I have no intention of defending it as philosophy. People learning abstract math at college level have a lot of trouble thinking about mathematical objects as objects, and my intention is to start clarifying some aspects of how you think about them in different circumstances. (The operative word is "start" — there is a lot more to be said.)
About the exposition of this post (a commercial)
You will notice that I gave examples of abstract objects but did not define the word "abstract object". I did the same with mathematical objects. In both cases, I put the word "abstract object" or "mathematical object" in boldface at a suitable place in the exposition.
That is not the way it is done in math, where you usually make the definition of a word in a formal way, marking it as Definition, putting the word in bold or italics, and listing the attributes it must have. I want to point out two things:
- For the most part, that behavior is peculiar to mathematics.
- This post is not a presentation of mathematical ideas.
This gives me an opportunity for a commercial: Read what we have written about definitions in References [1], [3] and [4].
References
- Atish Bagchi and Charles Wells, Varieties of Mathematical Prose, 1998.
- Reuben Hersh, What is mathematics, really?
- Charles Wells, Handbook of Mathematical Discourse.
- Charles Wells, Mathematical objects in abstractmath.org
- Math and modules of the mind (previous post)
- Mathematical Concepts (previous post)
- Thinking about abstract math (previous post)
- Terrence W. Deacon, Incomplete Nature. W. W. Norton, 2012. [I have read only a little of this book so far, but I think he is talking about abstract objects in the sense I have described above.]
- Gideon Rosen, Abstract Objects. Stanford Encyclopedia of Philosophy.
- Representations II: Dry Bones (previous post)
http://plato.stanford.edu/entries/abstract-objects/
Send to KindleWhole numbers
Sue Van Hattum wrote in response to a recent post:
I’d like to know what you think of my ‘abuse of terminology’. I teach at a community college, and I sometimes use incorrect terms (and tell the students I’m doing so), because they feel more aligned with common sense.
To me, and to most students, the phrase “whole numbers” sounds like it refers to anything that doesn’t need fractions to represent it, and should include negative numbers. (It then, of course, would mean the same thing that the word integers does.) So I try to avoid the phrase, mostly. But I sometimes say we’ll use it with the common sense meaning, not the official math meaning.
Her comments brought up a couple of things I want to blather about.
Official meaning
There is no such thing as an "official math meaning". Mathematical notation has no governing authority and research mathematicians are too ornery to go along with one anyway. There is a good reason for that attitude: Mathematical research constantly causes us to rethink the relationship among different mathematical ideas, which can make us want to use names that show our new view of the ideas. An excellent example of that is the evolution of the concept of "function" over the past 150 years, traced in the Wikipedia article.
What some "authorities" say about "whole number":
- MathWorld says that "whole number" is used to mean any of these: Any positive integer, any nonnegative integer or any integer.
- Wikipedia also allows all three meanings.
- Webster's New World dictionary (of which I have been a consultant, but they didn't ask me about whole numbers!) gives "any integer" as a second meaning.
- American Heritage Dictionary give "any integer" as the only meaning.
- Someone stole my copy of Merriam Webster.
Common Sense Meaning
Mathematicians think about and talk any particular kind of math object using images and metaphors. Sometimes (not very often) the name they give to a math object embodies a metaphor. Examples:
- A complex number is usually notated using two real parameters, so it looks more complicated than a real number.
- "Rings" were originally called that because the first examples were integers (mod n) for some positive integer, and you can think of them as going around a clock showing n hours.
Unfortunately, much of the time the name of a kind of object contains a suggestive metaphor that is bad, meaning that it suggests an erroneous picture or idea of what the object is like.
- A "group" ought to be a bunch of things. In other words, the word ought to mean "set".
- The word "line" suggests that it ought to be a row of points. That suggests that each point on a line ought to have one next to it. But that's not true on the "real line"!
Sue's idea that the "common sense" meaning of "whole number" is "integer" refers, I think, to the built-in metaphor of the phrase "whole number" (unbroken number).
I urge math teachers to do these things:
- Explain to your students that the same math word or phrase can mean different things in different books.
- Convince your students to avoid being fooled by the common-sense (metaphorical meaning) of a mathematical phrase.
Send to KindleMathematical usage
Comments about mathematical usage, extending those in my post on abuse of notation.
Geoffrey Pullum, in his post Dogma vs. Evidence: Singular They, makes some good points about usage that I want to write about in connection with mathematical usage. There are two different attitudes toward language usage abroad in the English-speaking world. (See Note [1])
- What matters is what people actually write and say. Usage in this sense may often be reported with reference to particular dialects or registers, but in any case it is based on evidence, for example citations of quotations or a linguistic corpus. (Note [2].) This approach is scientific.
- What matters is what a particular writer (of usage or style books) believes about standards for speaking or writing English. Pullum calls this "faith-based grammar". (People who think in this way often use the word "grammar" for usage.) This approach is unscientific.
People who write about mathematical usage fluctuate between these two camps.
My writings in the Handbook of Mathematical Discourse and abstractmath.org are mostly evidence based, with some comments here and there deprecating certain usages because they are confusing to students. I think that is about the right approach. Students need to know what is actual mathematical usage, even usage that many mathematicians deprecate.
Most math usage that is deprecated (by me and others) is deprecated for a reason. This reason should be explained, and that is enough to stop it being faith-based. To make it really scientific you ought to cite evidence that students have been confused by the usage. Math education people have done some work of this sort. Most of it is at the K-12 level, but some have worked with college students observing the way the solve problems or how they understand some concepts, and this work often cites examples.
Examples of usage to be deprecated
Powers of functions
In
Set "in" another set. This is discussed in the Handbook. My impression is that for students the problem is that they confuse "element of" with "subset of", and the fact that "in" is used for both meanings is not the primary culprit. That's because most sets in practice don't have both sets and non-sets as elements. So the problem is a Big Deal when students first meet with the concept of set, but the notational confusion with "in" is only a Small Deal.
Two
This is not a Big Deal. But I have personally witnessed students (in upper level undergrad courses) that were confused by this.
Parentheses
The many uses of parentheses, discussed in abstractmath. (The Handbook article on parentheses gives citations, including one in which the notation "
More
The examples given above are only a sampling of troubles caused by mathematical notation. Many others are mentioned in the Handbook and in Abstractmath, but they are scattered. I welcome suggestions for other examples, particularly at the college and graduate level. Abstractmath will probably have a separate article listing the examples someday…
Notes
[1] The situation Pullum describes for English is probably different in languages such as Spanish, German and French, which have Academies that dictate usage for the language. On the other hand, from what I know about them most speakers of those languages ignore their dictates.
[2] Actually, they may use more than one corpus, but I didn't want to write "corpuses" or "corpora" because in either way I would get sharp comments from faith-based usage people.
References on mathematical usage
Bagchi, A. and C. Wells (1997), Communicating Logical Reasoning.
Bagchi, A. and C. Wells (1998) Varieties of Mathematical Prose.
Bullock, J. O. (1994), ‘Literacy in the language of mathematics’. American Mathematical Monthly, volume 101, pages 735743.
de Bruijn, N. G. (1994), ‘The mathematical vernacular, a language for mathematics with typed sets’. In Selected Papers on Automath, Nederpelt, R. P., J. H. Geuvers, and R. C. de Vrijer, editors, volume 133 of Studies in Logic and the Foundations of Mathematics, pages 865 935.
Epp, S. S. (1999), ‘The language of quantification in mathematics instruction’. In Developing Mathematical Reasoning in Grades K-12. Stiff, L. V., editor (1999), NCTM Publications. Pages 188197.
Gillman, L. (1987), Writing Mathematics Well. Mathematical Association of America
Higham, N. J. (1993), Handbook of Writing for the Mathematical Sciences. Society for Industrial and Applied Mathematics.
Knuth, D. E., T. Larrabee, and P. M. Roberts (1989), Mathematical Writing, volume 14 of MAA Notes. Mathematical Association of America.
Krantz, S. G. (1997), A Primer of Mathematical Writing. American Mathematical Society.
O'Halloran, K. L. (2005), Mathematical Discourse: Language, Symbolism And Visual Images. Continuum International Publishing Group.
Pimm, D. (1987), Speaking Mathematically: Communications in Mathematics Classrooms. Routledge & Kegan Paul.
Schweiger, F. (1994b), ‘Mathematics is a language’. In Selected Lectures from the 7th International Congress on Mathematical Education, Robitaille, D. F., D. H. Wheeler, and C. Kieran, editors. Sainte-Foy: Presses de l’Université Laval.
Steenrod, N. E., P. R. Halmos, M. M. Schiffer, and J. A. Dieudonné (1975), How to Write Mathematics. American Mathematical Society.
Wells, C. (1995), Communicating Mathematics: Useful Ideas from Computer Science.
Wells, C. (2003), Handbook of Mathematical Discourse
Wells, C. (ongoing), Abstractmath.org.
Send to KindleThinking about abstract math
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
- Math and modules of the mind (previous post)
- Mathematical Concepts (previous post)
- Mental, physical and mathematical representations (previous post)
- Images and Metaphors (abstractmath.org)
- 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.
- Conceptual metaphor (Wikipedia article).
- What is mathematics, really? by Reuben Hersh, Oxford University Press, 1999. Read online at Questia.
- 18 Unconventional Essays on the Nature of Mathematics, by Reuben Hersh. Springer, 2005.
- Mathematical objects (abstractmath.org).
Send to KindleTurning definitions into mathematical objects
When G&G was moved to this current location, most of the links were trashed, so I have been repairing them a bit at a time. There are still some broken links from 2009 and before but I am working on them. Honest.
G&G contained a series of posts about turning definitions into mathematical objects, mostly written in 2009. Not only were their links broken (and they used many links to each other), but two of the articles were trashed. I have now removed them from this website. They are all still at the old website: http://sixwingedseraph.wordpress.com/ and as far as I know all the links to each other work.
When I have time I will combine them into one long article. Until then, the old website will remain.
Send to KindlePicturing derivatives
The CDF files in G&G posts no longer work. I have been unable to find out why.I expect to produce another document on abstractmath.org that will include this example and others. A link willl be posted here when it is done.
This is my first experiment at posting an active Mathematica CDF document on my blog. To manipulate the graph below, you must have Wolfram CDF Player installed on your computer. It is available free from their website.
This is a new presentation of old work. It is a graph of a certain fifth degree polynomial and its first four derivatives.
The buttons allow you to choose how many derivatives to show and the slider allows you to show the graphs from 
How graphs like this could be used for teaching purposes
You could show this in class, but the best way to learn from it would be to make it part of a discussion in which each student had access to a private copy of the graph. (But you may have other ideas about how to use a graph like this. Share them!)
Some possible discussion questions:
- Click button 1. Now you see the function and the derivative. Move the slider all the way to the left and then slowly move it to the right. When the function goes up the derivative is positive. What other things do you notice when you do this?
- If you were told only that one of the functions is the derivative of the other, how would you rule out the wrong possibility?
- What can you tell about the zeroes of the function by looking at the derivative?
- Look at the interval between
and
. Does the function have one or two zeroes in that interval? On my screen it looks as if the curve just barely gets above the
axis in that interval. What does that say about it having one or two zeroes? How could you verify your answer?
- Click button 2. Now you have the function and first and second derivatives. What can you say about maxima, minima and concavity of the function?
- Find relationships between the first and second derivatives.
- Now click button 4. Evidently the 4th derivative is a straight line with positive slope. Assume that it is. What does that tell you about the graph of the third derivative?
- What characteristics of the graph of the function can you tell from knowing that the fourth derivative is a straight line of positive slope?
- What can you say about the formula for the function knowing that the fourth derivative is a straight line of positive slope?
- Suppose you were given this graph and told that it was a graph of a function and its first four derivatives and nothing else. Specifically, you do not know that the fourth derivative is a straight line. Give a detailed explanation of how to tell which curve is the function and which curve is each specific derivative.
and 
. Does the function have one or two zeroes in that interval? On my screen it looks as if the curve just barely gets above the Making this manipulable graph
I posted this graph and a lot of others several years ago on abstractmath.org. (It is the ninth graph down). I fiddled with this polynomial until I got the function and all four derivatives to be separated from each other. All the roots of the function and all its derivatives are real and all are shown. Isn’t this gorgeous?
To get it to show up properly on the abmath site I had to thicken the graph line. Otherwise it still showed up on the screen but when I printed it on my inkjet printer the curves disappeared. That seems to be unnecessary now.
Mathematica 8.0 has default colors for graphs, but I kept the old colors because they are easier to distinguish, for me anyway (and I am not color blind).
Inserting CDF documents into html
A Wolfram document explains how to do this. I used the CDF plugin for WordPress. WordPress requires that, to use the plugin, you operate your blog from your own server, not from WordPress.com. That is the main reason for the recent change of site.
The Mathematica files are New5thDegreePolynomial.nb and New5thDegreePolynomial.cdf on my public folder of Mathematica files. You may download the .cdf file directly and view it using CDF player if you have trouble with the embedded version. To see the code you need to download the .nb file and open all cells.
Here are some notes and questions on the process. When I find learn more about any of these points I will post the information.
- At the moment I don’t know how to get rid of the extra space at the top of the graph.
- I was surprised that I could not click on the picture and shrink or expand it.
- It might be annoying for a student to read the questions above and have to go up and down the screen to see the graph. I had envisioned that the teacher would ask the questions and have the students play with the graph and erupt with questions and opinions. But you could open two copies of the .cdf file (or this blog) and keep one window showing the graph while the other window showed the questions.
- Which raises a question: Could it be possible to program the graph with a button that when pushed would make the graph (only) appear in another window?
Other approaches
- I have experimented with Khan Academy type videos using CDF files. I made a screen shot and at a certain point I pressed a button and the graph appropriately changed. I expect to produce an example video which I can make appear on this blog (which supposedly can show videos, but I haven’t tried that yet.)
- It should be possible to have a CDF in which the student saw the graph with instructional text underneath it equipped with next and back buttons. The next button would trigger changes in the picture and replace the text with another sentence or two. This could be instead of spoken stuff or additional to it (which would be a lot of work). Has anyone tried this?
Note
My reaction to Khan Academy was mostly positive. One thing that struck me that no one seems to have commented on is that the lectures are short. They cover one aspect (one definition or one example or what one theorem says) in what felt to me like ten or fifteen minutes. This means that you can watch it and easily go back and forth using the controls on the video display. If it were a 50-minute lecture it would be much harder to find your way around.
I think most students are grasshoppers: When reading text, they jump back and forth, getting the gist of some idea, looking ahead to see where it goes, looking back to read something again, and so on. Short videos allow you to do this with spoken lectures. That seems to me remarkably useful.
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