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Abstractmath.org after four years

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I have been working on the abstractmath website for about four years now (with time off for three major operations). Much has been written, but there are still lots of stubs that need to be filled in. Also much of it needs editing for stylistic uniformity, and for filling in details and providing more examples in some hastily written sections that read like outlines. Not to mention correcting errors, which seem to multiply when I am not looking. The website consists of four main parts and some ancillary chapters. I will go into more detail about some of the parts in later articles.

The languages of math. This is a description of mathematical English and the symbolic language of math (which are two different languages!) with an emphasis on the problems they cause people new to abstract math (roughly, math after calculus). At this point, I have completed a fairly thorough edit of the whole chapter that makes it almost presentable. Start with the Introduction.

Proofs. Mathematical proofs are a central problem for abstract math newbies. People interested in abstract math must learn to read and understand proofs. A proof is narrated in mathematical English. A proof has a logical structure. The reader must extract the logical structure from the narrative form. The chapter on proofs gives examples of proofs and discusses the logical structure and its relationship with the narration. The introduction to the chapter on proofs tells more about it.

Understanding math. There are certain barriers to understanding math that are difficult to get over. Mathematicians, math educators and philosophers work on various aspects of these problems and this chapter draws on their work and my own observations as a mathematician and a teacher.

All true statements about a math object must follow from the definition. That sounds clear enough. But in fact there are subtleties about definitions teachers may not tell students about because they are not aware of them themselves. For example, a definition can really mislead you about how to think about a math object.

The section on math objects breaks new ground (in my opinion) about how to think about them. I also discuss representations and models and images and metaphors (which I think is especially important), and in shorter articles about other topics such as abstraction and pattern recognition.

Doing math. This chapter points out useful behaviors and dysfunctional behaviors in doing math, with concrete examples. Beginners need to be told that when proving an elementary theorem they need to rewrite what is to be proved according to the definitions. Were you ever told that? (If you went to a Jesuit high school, you probably were.) Beginners need to be told that they should not try the same computational trick over and over even though it doesn’t work. That they need to look at examples. That they need to zoom in and out, looking at a detail and then the big picture. We need someone to make movies illustrating these things.

These other articles are outside the main organization:
Topic articles. Sets, real numbers, functions, and so on. In each case I talk just a bit about the topic to get the newbie over the initial hump.
Diagnostic examples. Examples chosen to evoke a misunderstanding, with a link to where it is explained. This needs to be greatly expanded.
Attitudes. This explains my point of view in doing abstractmath.org. I expect to rewrite it.

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Constructivism and Platonism — a Third Way

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This is my response to Phil Wilson's article on Constructivism in a recent Plus Magazine, in particular to the following paragraph (but please read the whole article!) where he talks about

'…how firmly entrenched is the realist view that mathematical objects exist independently of the human mind, "out there" somewhere just waiting to be discovered, even in our every day conception of objects as fundamental as the real numbers. Intuitionism is radically antirealist: antirealist in that it claims mathematical objects only come into existence once they are constructed by a human mind (a sad quirk of language that this is called "anti"realist), and radical since it seeks to recast all of mathematics in this light.'

We don't have to choose between the view that mathematical objects exist independently of the human mind and the view that they only come into existence once they are constructed by a human mind. There is a third approach: We think about mathematical objects as if they exist independently of the human mind. In particular, mathematicians have gotten away with pretending that all the digits of a real number exist all at once and proving theorems such as trichotomy based on that view, without running into contradictions. The justification is just that: it works.

This approach has the advantage that our brain has a whole system of thinking about physical objects. We use this system to think about other things such as Sherlock Holmes and pi and appointment schedules and it works quite well. It doesn't work perfectly: for example, physical objects change over time and affect each other, whereas we must think of mathematical objects as eternal and inert if our proof techniques are going to work properly. Indeed, it is thinking of the decimal digits of pi (for example) as "going toward infinity" that gets students into trouble with limits.

Even so, objectification, if that is the right word, has worked very well for mathematicians and we don't need to give it up, nor do we need to be Platonists — we need only act as if we are Platonists.

I wrote about this in several places:

A scientific view of mathematics

Rigor and rigor mortis

Rich and rigorous

 

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Naming real numbers

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I am considering an Astounding Math story about how you can’t name an arbitrary real number. In this blog I will describe some of the math technicalities, teaching problems and writing problems that arise in writing the story. It would be great if other math popularizers would blog about the problems they faced in their writing and what decisions they made about them.

Consider first that we can name every rational number. You can describe rational numbers in terms of equivalence classes of expressions of the form “m/n” where m and n are names of integers, n not zero. Then any expression of the form “m/n” names a specific rational number and every rational number can be named in this way. Yes, the naming can be made unique by using expressions in lowest terms, but that is not the point here, which is that in theory you really can name every rational number.

Every real number has a decimal expansion. The expansion is nearly unique: some rational numbers have two different decimal expansions, but that is all (see (2) below). We may define a decimal expansion precisely using the “regular” expression [-]?[0..9]*[.][0..9]^(infinity). This means:

One minus sign or nothing
followed by
A string of any finite length of decimal digits
followed by
A decimal point
followed by
An infinite string of decimal digits.

This is not really a regular expression since regex’s don’t allow you to specify an infinite string. But it is a precise definition of decimal expansion, and every decimal expansion refers to a specific real number. You can then define the real numbers as equivalence classes of regular expressions of decimal digits, with each equivalence class containing one or two members.

The Astounding thing is that as a result of this construction

a) You can describe precisely the set of real numbers.

b) Each real number has a description as an infinitely long decimal expansion.
c) You cannot give a name to every real number. That’s because the description is an infinite sequence and you cannot give every infinite sequence even in theory (see (3).)

d) So when mathematicians deal with real numbers, they are dealing with things that in most cases they cannot refer to.

Complications

My purpose in writing Astounding Math Stories is to get people who are already somewhat familiar with math to have their consciousness raised about all the fascinating things that go on in math. This requires a delicate balancing act when I write them.

1) I get comments from readers like this one: “That is not astounding. I already knew about it.” This is probably inevitable. In the case of the names of real numbers, however, I’ll bet there are practicing mathematicians who understand item (d) implicitly but have never heard it said out loud.

2) You have to say precisely which two infinite sequences are in the same class. When I was teaching discrete math in the eighties and nineties, I realized that I had never seen this written out explicitly. Every description depended heavily on pattern recognition, as in the description “I am referring to the phenomenon that for example 0.9999… denotes the same number as 1.0000…” (See remark (6).) I included a nearly explicit description in my discrete math class notes (page 12).

Perhaps this problem should be slurred over. Really every real number has one decimal expansion. That thing with the 9’s is just a technicality. (This makes me a heretic. Mathematicians don’t usually say things like that.)

3) You cannot give the name of every real number because the set of linguistic expressions is countable and the set of infinitely long decimal expansions is uncountable. Do I just quote this fact? Do I write another Astounding Math Story about infinite cardinality? Probably.

Some would object that you can’t give the name of every rational number either. But there is a name (a finitely long linguistic expression) for every rational number. You can’t in physical fact “give” the humongous ones but that is a practical problem. In contrast, most real numbers have no linguistic expression naming them.

4) I need to keep the demands on the reader as low as is reasonable, but not lower. A minor example in this case is that I express everything in terms of decimal expansion instead of binary expansion or Cauchy sequences (more abstract) or Dedekind cuts (even more abstract). In theory, binary expansions are not any more abstract than decimal expansions and require less data, but in fact most of the people I am trying to attract are less familiar with binary than decimal, and that drags on their understanding.

5) When I say “You can describe a real number as an infinitely long decimal expansion” you run into the ubiquitous difficulties math-newbies have with infinite sequences. Namely, they think of then as progressing through time, so you never get to “the end”. In fact, experienced mathematicians think of an infinite sequence as existing all at once: every entry is there now.

Students complain that they can’t “visualize” the entries all at once, but that is not the point: You are not suppose to visualize the whole sequence at once, you are suppose to think and talk about the entries as if they are all there. That is, assent to the concept that the whole thing is “there”, which is not the same thing as visualizing it. (I also wrote about this phenomenon in abstractmath and in a previous blog.)

So when I write about the infinitely long decimal expansions I know many readers’ understanding will falter right there; they will not be able to take in the rest of what I write. What do I do about this? Well, I suppose I could include the last paragraph!

Note: This discussion is not about what infinite sequences “really are”, but about how you think about them. This way of thinking about them has been around for a couple of centuries and have produced many useful theorems and no known contradictions. Philosophers may have a problem with this point of view, but mathematicians don’t.

6) Studies show that most math students do not believe that 0.999… is the same number as 1.000… Some mathematicians I know say things like: “Why are you writing for people like that? They are too stupid to understand anything about abstract math.” But it is time mathematicians stopped insisting that there is no point in getting people who are not especially talented in math interested in math, or trying to explain anything to them. In fact, it appears to me that this elitist attitude is in the process of dying out. It had better be dying out.

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Unconstructed existence

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Astounding Math Stories is intended to attract the interest of bright, technically oriented non-mathematicians or beginning math students by telling them about some of the phenomena in math that are more surprising than, say, Venn Diagrams. This is turning out to be hard to do. You get responses from sophisticated people who say, in effect, “Well I knew that!” or “That’s not surprising!”. But sometimes they uncover real difficulties. Example:

Last August, Pubkeybreaker posted this message on sci.math about Finding Factors or Not:

I’m not sure why being able to show that an integer is composite without finding a factor is “astounding”.
Even at a beginning, first year algebra level we know that linear equations (with real coefficients) have a real solution, without knowing what the solution is. Similarly for any odd degree polynomial. We know that positive real numbers have a positive square root without knowing its value…. etc. etc.
Why should it be astounding that we know something
exists yet not know its value?

This brings up a sticky point. I was trying to find a simpler example of the sort of nonconstructive proof of existence that Hilbert used for the finite basis theorem. In that proof there is an inductive construction where at each step you find a polynomial of minimal degree in a certain ideal. But the ideal is infinite, so how do you know when you have found a minimal degree member? That is why it is nonconstructive.

Now Fermat’s Little Theorem gives you a way (if it works — it doesn’t always) of finding a proper factor of a composite number without showing how to calculate it. But finding a proper factor of a composite number has an obvious algorithm: try every number up to its square root. This is a finite but slow process; it is not nonconstructive.

What FLT does is give you a fast way of showing the existence of the proper factor without give you a fast way to find it. (See the appendix below.) That is simply not the same situation as in the finite basis theorem.

Clearly I need to rewrite the FLT story in such a way as to emphasize speed-of-algorithm difference instead of the nonconstructive part. Then I need to write another Astounding Math Story that describes a truly nonconstructive proof directly. The Hilbert Basis Theorem itself might be a candidate; the ideas involved are not too hard, although they are more advanced than I was hoping for for Astounding Math Stories.

Appendix

Here are some more detailed comments on Pubkeybreaker’s specific examples:

The proof that odd degree real polynomials have a real root is that such a function goes to positive, resp. negative infinity in the positive, resp. negative direction, so must cross the x axis. This immediately suggests an algorithm for finding the solution: choose larger and larger inputs in both directions until you find one that gives positive output and one that gives negative output. Now you have trapped the solution between two real numbers. Now use bisection. The square root can be found in a similar way, although of course there are faster algorithms.

The proof that a number is composite using the contrapositive of Fermat’s Little Theorem does not obviously (or even non-obviously) suggest a way to find a proper factor of the number. Not only that, but people have fancier ways (only partly based on Fermat’s Little Theorem) to test for primality that always work, and work faster than any known method of producing an explicit proper factor. This is a real contrast to the situation with cubic polynomials.

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"Mathematical objects" rewritten

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Vaughan Pratt has rewritten the Wikipedia article on mathematical object, and the result is a great improvement.

I still think that a suitable approach would be to pick out mathematical objects among all abstract objects by specific properties they have.

  • A mathematical object is always an abstract object that satisfies certain axioms specific to the object.
  • A mathematical object is inert and unchanging.
  • A mathematical object is defined crisply, no fuzzy allowed.
  • And so on.

I wrote about some of those points here, but not the part about axioms. Watch this space!

These ideas are not settled. As one commenter said, Wikipedia articles should not be the product of current research. What Vaughan has written is about right for now.

Specific comments:

1) Above, I reworded my comment about mathematical objects satisfying axioms in response to an objection by a reader. For example, a model of untyped lambda calculus is an object S for which the function space S -> S is isomorphic to S. Such a thing does not exist in the category of sets, but it does in the category of topological semigroups and also in the realizability topos.

2) In a note to the category mailing list, Vaughan also said:

“It might also be worth mentioning coalgebras, and perhaps more importantly dually defined structures such as locales which are understood better in terms of the morphisms from them to a cogenerator rather than those to them from a generator, i.e. dual elements rather than elements. Also toposes as a more general codomain of the forgetful functor than the particular topos Set.”

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Mathematical objects in Wikipedia

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The Wikipedia article on “mathematical object” needs to be rewritten. It begins: “In mathematics any subject of mathematical research that can be expressed in terms of set theory is a mathematical object”. It also says, “Outside mathematics, a mathematical object is an abstract object that is referred to or occurs in mathematics”.

That is defining mathematical object in terms of a historical detail. Yes, it has been shown that most mathematical objects can be constructed from sets in one way or another, often in strange or unintuitive ways. However, that is a theorem, not a main property of mathematical objects. Besides, there are objects that exist in other categories but not in sets, such as models of untyped lambda calculus. Those categories involve proper classes of objects, not sets of them.

I would prefer some definition such as this: “A mathematical object is an abstract object defined by axioms”, together with explanations of abstract object and axiom. Mathematical objects should be distinguished from abstract objects such as “schedule” that change over time and also from objects in narrative fiction.

The article should describe the different points of view taken by philosophers and mathematicians who have written about the idea. It should refer to some of these articles and books:

Davis and Hersh, The Mathematical Experience (Mariner Books, 1999), sections on Mathematical Objects and Structures: Existence and on True Facts about Imaginary Objects.

Goodman’s article in New Directions in the Philosophy of Science, Princeton, 1998.

Hersh, What is Mathematics, Really?, Oxford University Press, 1997.

Stanford Encyclopedia of philosophy article on abstract object.

I have written about mathematical objects at length on abstractmath.org. It pulls together many of the ideas in the articles listed above.

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Representations IV

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Mark Meckes recently commented on my post on writing Astounding Math Stories that students “usually think of decimal expansions as formal expressions”. His point is well taken, but I would go further and say that they think of real numbers as decimal expansions, hence as formal expressions.

For example, 1/3 is approximately 0.333… However, 1/3 is exactly 1/3. The expression “1/3” is more exact than the decimal expansion. Similarly \sqrt 2 is defined exactly as the positive real number whose square is 2. And of course there are still other representations of some or all real numbers, for example binary notation, representations as limits, as solutions of equations, and so on.

The main thing to understand is that every interesting mathematical object has several representations, each representation coming from a different system of metaphors. And if you are going to understand math you have to be aware of various representations of the same object and hold (some of) the details of several of them in your head at once. Even “2 + 3 = 5″ is talking about two different representations of a number simultaneously. William Thurston once said that it was a revelation to him as a child that when you divide 127 by 23 you get 127/23. That notation”127/23” tells you two things: exactly what the number is and one way it is related to other numbers. That kind of phenomenon is what makes math work.

I went on and on about this stuff in abstractmath.org here and here.

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What I learned in school is the Only Truth

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Michael Barr recently commented on another post about a Dutch student who insisted that the words “long” and “short” in English referred to qualitative differences such as that between “ride” and “rid”, whereas linguists use the words to refer to temporal length, such as the different in the vowels between “hid” and “hit”.

I assume the student acquired the qualitative meaning from English courses in school; that meaning is very still widely used in English classes in the USA and Britain, so that (I’ll bet) nearly any person on the street in the USA would expect the meaning of “long” and “short” to be the difference between “ride” and “rid”.

This is an example of a phenomenon mathematicians have to put up with too. We know that the same word or symbol can have many different meanings in math, but people who know a little math assume that all meanings that they learned in whatever math courses they took are universal and set in granite. They are startled that “pi” can mean anything other that what they think it means. Someone recently started talking to me about “phi” as if I should know what it means, but I recovered fairly quickly, since I had become vaguely aware that it means the so-called golden ratio to laymen. In my experience mathematicians mostly use phi to denote some function.

When I taught, I was constantly in trouble with students who told me that 0 was not a natural number if the textbook said it was, or was a natural number if the textbook said it wasn’t, because that was the definition in some previous course they had taken.

I have written about this phenomenon here and here.

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A Scientific View of Mathematics

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Over most of the history of human thinking, both philosophers and theologians have come up with explanations of some natural phenomenon, only to be faced with scientific investigations that give a successful evidence-based explanation of the phenomenon they have written about.

The theologians and philosophers eventually lose the argument. The word “eventually” means that (1) the scientific investigation has to produce a pretty solid theory that explains a lot of the evidence and (2) the older theologians and philosophers have to die, since they rarely change their minds about stuff after the age of 50. (Some theologians and philosophers still continue to argue for things such as flat earth, intelligent design and dualism, but they are not really engaging in the intellectual world’s ongoing conversation.)

It is now possible to investigate the theory and practice of mathematics using evidence-based scientific reasoning. In particular, recent findings in neuroscience and cognitive theory make it plausible to provide a description of mathematics that is based on the interaction between brain, body and culture. By continuing to study mathematics and its practice scientifically we can hope to come up with a theory of mathematics that will be a part of cognitive theory.

I have been writing about bits and pieces of this idea for a long time, and so have many others. What I am going to give here is a lacunary sketch of my current thoughts with references. Most of these ideas originated with other people!

Math is an activity of our brains.
Our brains contain ideas. These ideas are real physical structures, organizations of neurons or something. [MO], [TaBa2002] I will call them PSB’s (physical structures in the brain). This is early days in neuroscience and exactly how the ideas exist physically in the brain is still controversial. I assert only that ideas are physical, nature in part yet to be determined.
Among our ideas are representations of objects and lists of rules.

Objects are represented in our brains.
The objects represented in our brains may be physical objects, fictional objects or abstract objects, including mathematical objects [AbMO], [Her97].

There is presumably a PSB that is triggered when you think about any kind of object. It is triggered if you think about the Parthenon, Sherlock Holmes, or the function f that takes a real number x to x^2. (If you are not an experienced mathematician, you might in fact not think about f as an object, but rather as rule or procedure. This can cause serious difficulties for students in calculus classes who are faced with such concepts as “the derivative of f”.)

There is no doubt another PSB’s that recognize that the Parthenon is a physical object in contrast to the squaring function, which is an abstract object. But our brain clearly recognizes both as an object because we talk about physical, fictional and abstract objects using the same grammatical structures and we think about them using similar mental operations. [AbLM]

The fact that we think and talk about the set of all real numbers (for example) as an object is explained by this PSB. It does not imply that the set of all real numbers exists anywhere, physically or ideally.

Our brains are organized to follow rules.
We apparently have a PSB that implements a rule-following mode. (This has been studied but I don’t have a good reference for it.) We learn and follow rules very easily when we play any game, baseball, chess or whatever. We also learn rules for algebraic manipulation and for mathematical reasoning using the rule-following mode. People who are good at math seem to engage the rule-following mode easily in these situations.

Math is communicated among people using the languages of math.
Math has several languages. Mathematical English is a special dialect of English with some disconcertingly different rules. Other major languages have a similar special dialect. The symbolic language of math is a special purpose system that is largely independent of any particular natural language. Graphs, geometric drawings and diagrams form a system for communication as well. The various systems are intertwined with each other in conversation, in lectures and in written math. [AbLM], [O’H], [Wel2003].

When we do math we think about math objects as if they were things.
Conceiving of math as talking about (abstract) objects enables us to think about it using the machinery in our brain we use to think about physical objects. This machinery is highly developed and uses metaphors and physical reasoning (maybe using mirror neurons). We could not do math without it. [LakNun], [WMN], [AbImMet].

Useful mathematical ideas tend to come from our physical experiences with our body and the world.
This is a thesis of [LakNun]. This understanding of the origins of mathematical objects might be developable into an explanation of the “unreasonable effectiveness of mathematics”. Note that you have to explain the effectiveness of mathematical reasoning as well as the usefulness of the objects we talk about.

Mathematical computation and reasoning lead to consistent results.
When we find inconsistent results using math we expect to find a mistake somewhere, and we usually do. This claim is about both numerical and algebraic computation and also formal mathematical reasoning. This phenomenon gives us confidence that mathematical processing is dependable.

This is what was behind my point about actual infinity in [AI]. When we envision the real numbers (for example) as an infinite set that exists all at once, and follow the correct rules of mathematical reasoning, it all works.

It’s also true that we now know of genuine limitations to what we can know, because of incompleteness results as well as cardinality results that say, for example, that there is an uncountably infinite number of real numbers that we cannot refer to individually.

Since in particular we can’t prove the consistency of a system within the system, our experience of consistent results is the only evidence we have that mathematics really
works and can be applied to the world.

References

[AbImMet] Images and metaphors
[AbLM] The languages of mathematics
[AbMO] Mathematical objects
[AI] Actual infinity
[MO] Gyre&Gimble (2007), Mathematical objects are “out there” ?
[LM] Gyre&Gimble (2007), The languages of mathematics
[NM] Gyre&Gimble (2007), More about neurons and math
[RIII] Gyre&Gimble (2008), Representations III: Rigor and rigor mortis.
[Her97] Hersh, R. What is Mathematics, Really? Oxford University Press, 1997. ISBN 978-0195113686
[LakNun] Lakoff, G. and R. E. Nüñez (2000). Where Mathematics Comes From. Basic Books. ISBN 978-0465037711.
[O’H] O’Halloran, K. L. (2005), Mathematical Discourse: Language, Symbolism And Visual Images. Continuum International Publishing Group. ISBN 978-0826468574
[TaBa2002] Tall, David and Barnard, Tony (2002). Cognitive units, connections and compression in mathematical thinking
[Ta2001] Tall, David (2001). Natural and formal infinities.
[Wel2003] Wells, C. (2003). The Handbook of Mathematical Discourse.
[WMN] Wikipedia on mirror neurons.

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Actual Infinity

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The infinite sequence 1, 1/2, 1/3, 1/4, … is commonly described as “getting closer and closer to 0”. This fits with the mental representation (metaphor) students have of the sequence as something whose successive entries are revealed or created over time.

People who are used to abstract math have another handle on that sequence: It is the function whose value at each positive integer n is 1/n. Stated that way, the sequence is being pictured as existing all at once (another metaphor). It is a completed infinity or actual infinity. Completed infinities occur all over modern math and mathematicians rarely remark on them. But some people outside math, in particular philosophers, get all hot and bothered about it.

I have not found a good reference on the internet to the idea of completed infinity. The article on actual infinity in Wikipedia gives lots of reasons against the idea but few reasons in favor of it. Many other articles on the internet also mostly discuss the opposition to the idea.

The phrase “the function whose value at each positive integer n is 1/n” is a definite and clear description of the sequence. Mathematicians have proved all sorts of statement about the sequence using classical logic; it is a Cauchy sequence converging to 0, for example. These statements all seem to be consistent with each other and with other parts of math. To put this in another way, we have a clear syntax for talking about this sequence and others, and we have built up a fund of statements about it that all hang together.

As always when talking about anything, we use metaphors and images when talking about this sequence. The image of the sequence as getting closer and closer to zero is one, which we could call the Xeno image. The image of the sequence as existing all at once is another, let’s say the actual-infinity image.

Neither image says anything about physical reality. The sequence is of course represented physically in our brains by arrangements of neurons or some such thing, but in no way does that imply that every entry is represented in our brain. What is represented in the brain are properties of the sequence, the metaphors and images we have mentioned (and others), relationships with similar sequences, and so on, what the math ed people call our schema of the concept. (See [1] and [2].)

I propose that the trouble philosophers and students have with the actual-infinity image occurs because there is a physical arrangement in a brain that serves to recognize big bunches of individual things. (Compare this post.) It may be triggered, for example, if you look out of a high window and see a big crowd of people standing in the street. You have a person-recognizer in your brain and you also have a big-bunch recognizer in your brain, which works with the person-recognizer to identify a crowd of people. (I talked about a similar idea, the I-saw-this-before recognizer, here).

A smart person soon realizes that there is a difference between a crowd of people and an infinite sequence, namely that the crowd is large but finite whereas the sequence is infinite. Well, that is true. So what? The crowd may be finite but you still can’t hold an image of each individual in the crowd in your head. The finite-infinite difference is indeed a difference, just as the idea that one is composed of physical objects (people) and the other is composed of mathematical objects (numbers). You can reason about both the crowd and the sequence, make discoveries about them, and so on.

Why is there a difficulty? Perhaps it is because a device in your brain that is usually used for bunches of physical objects is now being triggered by a bunch of abstract objects. That can be disconcerting.

But: You don’t have to get all upset about something just because it disconcerts you.

Now, then, no one ever needs to worry abou
t actual infinity again. (Fat chance).

REFERENCES

[1] Ed Dubinsky and Michael A. McDonald, APOS: A Constructivist Theory of Learning

in Undergraduate Mathematics Education Research

[2] David Tall, Reflections on APOS theory in Elementary and Advanced Mathematical Thinking.

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