Frank Allen has a nice motivating speech about teaching proofs and logic in high school math here and it was useful for me to read it.
To back up a bit, we've been nicely plodding through the two column proofs. I've been having him use the same format that Euclid used 25 centuries ago. His cookie cutter proof format is to consist of these five parts:
1. Statement in prose of the thing to be proved.
2. Declaring the variables to be used.
3. Restating the theorem in symbols.
4. Proof in two columns
5. Conclusion.
The most pedagogically useful has been 1 & 3 since it's led to several discussions about how concepts can be expressed by both by natural language and by symbols and how it is that one can come up with logically meaningful conclusions when all one begins with is imprecise and vague concepts.
To undigress, when my son came across the following in his text he didn't even recognize it as a proof. I'll give you the exact wording of the original paragraph and then tell you how it was rendered meaningful.
Okay, so the first problem that the kid had was that it all looked like "blah blah blah." Admittedly, stating the theorem in bold before proving it would have served as a nice sign that what was about to follow was a proof so it's no surprise that my kid didn't have any idea what had happened when instead the "to be proven" came after the proof itself.An even integer is defined to be an integer which is a multiple of 2. Thus if n is an even integer, then n = 2k, where k is an integer. An odd integer is an integer which is not even.
If t is an odd integer, then t ≠ 2k, where k is an integer. By the multiplication property of inequality, it follows that t/2 ≠ k, where k is an integer. Since t/2 is not an integer, it must be between two consecutive integers which we shall represent by k and k + 1. Therefore, k < t/2 < k+1, and by multiplication property of inequality, we have 2k < t < 2k + 2. Since 2k +1 is the only integer between 2k and 2k + 2, we conclude that t = 2k + 1. Thus we have proved the following theorem: If t is an odd integer, then t = 2k + 1, where k is an integer.
The first thing I had him do was rewrite the above proof as a list of sentences rather than as a paragraph. The second step was to make distinctions with the individual assertions being made. "Is this a definition? Is this a justification or a conclusion? Why does he keep saying "where k is an integer?"
My next idea was to have him convert all of it into the friendly recognizable two column proof, but before I could do that he said he wanted to write it all out for himself in a way that was meaningful to him. He also wanted to include number lines so that he could understand the greater than and less than relationships and I requested that he make the number lines in a different color to make sure that he ultimately didn't confuse illustration with justification. So here is his chicken scratch:
In a nutshell, that is what it took to understand the paragraph on page 326 and those activities occupied the entire time we had set aside that day for math. The next day the assignment was, simply stated, "The proof of the converse is left to you as an exercise."
Grabbing his chicken scratch he decided to list all the step backwards (that paper is not shown here), convert the symbols to prose, and list them out as sentences and turn them in to Dad for grading. Dad innocently toiling away in his office at home had no idea what the assignment on his desk was even about and made a few comments before giving up and telling me it required too much effort to sort through:
There was a second draft followed by a final draft. I'm really pleased that the kid did it mostly on his own (the motivational speech to get him to do it included references to Spartans) it somehow clicked with him that he needed to write this out as if he were explaining this to someone rather than listing steps like you do in a two column proof. I told him that explicitly, but I didn't have to resort to dictating the words to him or any such thing.
My complaint with the book is that after this surprise attack of the proof in paragraph form, it looks like we will be returning to typical ninth grade algebra activities such as dividing polynomials and by the time we encounter another proof the kid will have forgotten how to write out proofs in this new way. I'm thinking I'll have him go back once a week and rewrite some of the two column proofs from prior chapters as paragraphs to keep on top of this skill.
12 comments:
I think this is a great illustration of the learning process in how to write a good paragraph proof.
However, the theorem being proved is a little bit too trivial, in my opinion. That is, most kids (though apparently not yours!) would think it's silly to be proving something so obvious, and the rest of the kids (again not yours!) would not understand what it means well enough to figure out the proof at all.
So, I would look for slightly harder things to prove. The infinitude of primes is a good example, or the irrationality of square root of 2: these are surprising results with interesting proofs!
Another good choice, but more in geometry than algebra, is that an angle inscribed in a semicircle is a right angle.
Oh, another potential algebra idea: Prove that if you subtract a two-digit number from its digit reversal, the result is divisible by 9. Then generalize ...
Yeah, the more time we spend in Frank Allen, the more impressed we are with him. It was Allen's "mindless formalism" that really taught our oldest to be able to be rigorous like that -- Allen and the way we teach it, I suppose. Myrtle knows she could be doing harder to solve problems -- like really hard Singapore word problems, say. But, proving something is the best kind of hard word problem. And, as you say, it is particularly difficult, actually, when the result is more obvious because it becomes less clear just what you would say to justify it.
But, as I say, on the surface Frank Allen's text may seem a little too much like "mindless formalism" as the critics like to characterize it, or too much of the "proving of the obvious" that only technical experts are ever really expected to be able to do (and perhaps "need" to be able to do). But, on closer examination, the book isn't really like that. The fact that it doesn't teach axiomatic set theory, to me, is a big tip off. But, more generally, it isn't always the full monty on rigor at all. It is pretty rigorous -- probably almost completely so. But, it doesn't start with Peano's axioms, for instance, or anything like that which the up and coming experts would absolutely have to do. And, he doesn't just dwell on "proving the obvious", even though he doesn't avoid it at all costs, either.
It really does get a lot more rigorous than most of these New Math texts ever get. They are really just introducing some amount of rigor -- some amount of appreciation for "proving the obvious". And, that, alone, was the cardinal sin that got them dumped almost the minute they were created. I just think it is worth noting just what the issue really is and has always been with books like these. It isn't that they are hyper-rigorous but that they try to foster any appreciation for rigor at all.
You know, it's one thing to defend applied math and make sure there is a place in the world for it. It is another to try and stamp out pure math.
Hi Joshua,
I remember you had an excellent website up on your experiences with the Mathematical Circles in Russia that was very intriguing. I tried to find a similar group in my city but it seems that none exist. I did get Dmitri Fomin's book of problems, "Mathematical Circles" and had my eye on that, but found it very difficult for me since I have no background in math at all. My problemwhen using that book as a source of problems was not knowing what constituted a proof or not.
This algebra book does spend a lot of time on proving the obvious, but in my case, since I'm not quite so clever, it's helped me to see when assumptions are made and when they need to be justified. So, My son and I are learning this together. More specifically, for example, in the particular proof above the issue came up of, "How do we know that K + 1 is necessarily another integer?" We spent some time wondering if there wasn't something that we missed. Nowhere were we told that that this had to be the case. We knew that the reals were closed, but there was nothing that ever told us that the integers were closed. We made note that we were assuming it and moved on.
These exercises in proving the obvious have resulted in both of us being trained to pay more attention to definitions and prior theorems in a way that we hadn't before.
Another benefit of the book that we are using is that the problems are carefully introduced so that each one builds on the one before it and that there are none too diffult to get. So, while it perhaps looks difficult, it is because the problem is taken out of context of the entire book. It is not so difficult when the kid has been doing similar such obvious problems in preceding chapters. There were plenty where the trick was to "reverse the steps", so this was already in our bag of tricks. There are a few more proofs on divisibility in this chapter, "If a is a factor of b and b is a factor of c..." type of thing, but perhaps I will take out that Mathematical Circles book again and select a few on the topic of divisibility so that we have some more interesting problems to work on.
what adrian said.
also:
surprising results with interesting
proofs are all well and good.
but good luck every discovering
one if you haven't written
proofs of a few *obvious* results
with *routine* proofs:
this, as myrtle's kid is
(inspiringly) finding out,
is where you learn the *nature*
of proof -- what they are &
what they're for.
you're doing good work, myrtle.
oh, and hey adrian ...
i've probably looked up your blog
half-a-dozen times by now.
how about turning on comments?
i meant joshua, not adrian.
oops. v.
daggone it. one more time.
i meant joshua's *blog*.
(by "what adrian said"
i meant "what adrian said").
going away now.
Myrtle,
I was interested by your comments about Allen returning to typical types of algebraic activities. I had a similar response to some of the book in that much of the theory did not seem to get worked out in practice. I wrote to Ralph Raimi about Allen and he basically said the same thing. I would like to do parts of Allen with my eldest but I don't know...
I liked all your comments about converting statements to two column proofs. I have been thinking for a while that the best thing to start with for Geometry is Euclid's Elements.
Cheers...
Hi Andrew,
I can see many ways in which Allen's book could do a better job, but I also haven't found another book that does it better. If one exists, I'd certainly be interested in it. My husband's even considered the idea of writing such a book himself, but the amount of time and work that would take is horrifying.
What I have done so far is have my son memorize the proofs and regurgitate them. They are not always memorized rote though, we spend a lot of time discussing why each step is necessary and in some cases, where the notation is ambiguous, we change notation. In many cases I've had him begin with the conclusion and work his way to the beginning as I said above, working backward is in his bag of tricks. I tell him, "One way to being your proof is to be omniscient and know exactly what step you should begin with, another way is to figure that first step by seeing what you can do with the conclusion." And I will also have him attempt the proof before he reads it in the book. I give him clues and coach him through it if need be.
In the previous chapters I would have my son apply the steps of the proof to the solution of the problems rather than directly applying the results, if that makes any sense.
For example,
Theorem 16 on pg 236 is the theorem on subtracting expressions which we phrased informally as "subtracting an expression is like adding its inverse." This is a derivation of 6 steps, but in the exercise set rather than apply the results of the theorem, I had him repeat the steps in the derivation with each problem. This is one reason why it's taking us so long. At some point I have to use my judgment and decide that he knows why it is that he's adding the inverse and let him take the short cut by skipping the steps in between.
I really did take a close look at "Mathematical Circles" the other night and there are just some really, really interesting problems in it which require proof. Of the many methodological remarks this one caught my eye, "A formal proof of this fact is not very difficult, though for beginners it may seem full of technicalities." I would say that it is those "technicalities" which is what Allen seems to cover.
We supplemented Allen's treatment of inequalities with Beckenbach's "Introduction to Inequalities" and somewhere online my husband reproduced a transcript of that conversation. So, we are not absolutely adhering only to what is printed in Allen but discuss it and expand on it when we need to.
All very interesting thoughts, Myrtle. I looked up that theorem in Allen and I have to admit that I'm not sure that I want to do this with my eldest. I'm not saying that I know I shouldn't, but rather that I don't know. What I want to avoid is getting further down the road with math and having my kids accept something like the operation of a differential without having any idea of how to prove it. Students need to understand why differentials work. Modern textbooks are generally focused on plug-and-chug utilitarianism that I want to avoid. But does this mean that I have to treat every or even most mathematical statements like theorems? I'm not convinced that I do and I wonder if Pearson and Allen haven't gone overboard here.
Raimi told me that Allen had come to him with a textbook some years back but Raimi could not remember if it was Modern Algebra. Much to Allen's chagrin Raimi would not write a favorable review. He thought that while axiomatics are important for the teacher to comprehend and utilize in his or her teaching methodology, it was not generally necessary for the student to have to go through all these axiomatic steps. He gave me an example or two which made sense to me. Also, Raimi pointed out that oftentimes in the New Math texts, logic theory did not manifest itself in practical problems that utilized this theory. I think he said that it was Richard Feynman who went systematically through a number of these texts and could not find the connection between theory and practice.
Raimi recommended one of older editions of Dolciani. I picked a copy up through ILL. I liked what I saw but did not spend enough time with it before I had to bring it back.
Cheers for now...
Well, I guess it all depends on what your issues are. I could definitely see coming at it from the perspective of "I just want my kid to be able to do an epsilon-delta proof," and otherwise being okay with "engineering" or applied, formula-manipulation-intensive math. But, the problem for me is that, for one thing, I think there is more to doing the proofs than just really knowing what you know when it comes to calculus. There really is a broader intellectual interest for that that does not exist for most other things, including the more general idea of heuristic "mathematical reasoning".
But, also, to really be able to wield the epsilon-type argument, it really takes some ability and mathematical maturity. You can kind of mechanically know a formula for doing extremely standard epsilon-delta proofs. but to really "know what you know" for calculus, you really need to do something like Baby Rudin -- a classical real analysis text. I guess what I am saying is that just "doing a little" is going to help you out just a little bit. You might have a better understanding of differential equations than "multiplying by dx" like it's an ordinary quantity, but I don't know that you really can do everything you want to do rigorously, much less extend anything rigorously.
In other words, just doing a little bit of rigor or just elaborating on that one, the weakest, concept you can think of from a standard undergraduate program is just going to push the whole issue back one concept. Then, the next generation goes, "People really need to understand what continuity is" and eventually you find yourself doing point-set topology. So, let's just cut to the chase and do that -- do it the right way and stop trying to figure out new ways to justify our less than completely correct way of doing things. And, by "correct", I don't just mean that the theorems are technically stated correctly, but that they are also taught correctly: proofs first and applications second. And, in particular, the proofs are really taught, as in the students are given homework and test questions that cover that. We know that it isn't correct the way we do it now, and we know that the correct way isn't pedantic. If it was, it really wouldn't exist -- we certainly don't do that when it's for real and students really have to know their math (e.g. when we are training the next generation of math professors). We really need to stop infantilizing our students.
I'm so tired of some applied math guy jumping in on how overly pedantic pure math is, saying just what everyone wants to hear and completely taking over undergraduate education when it really ought to be more like the other way around. The whole problem revolves around the fact that when we do things the right way, it is so hard and time consuming that it makes it really easy for someone else to gloss over some things and appear to make way more progress than we are. So, there is always some Jones out there that talks a real good game and ostensibly knows how to solve PDEs before we even know the fundamental theorem of calculus. And so, even if we recognize the Joneses as being a little bit of a scam, we nevertheless have to do something to mitigate what seems to be such a large gap. And so, we end up having to sacrifice some of the math in a compromise we have to make in order to close that gap to an acceptable level....
I think he's going to go slowly because it has many pages it could be boring for him, I think you should reading him something different.
Algebra is so hard I don't think so that student is slow, what I think is we should know the level the student is getting in order to help him, and not saying he's an slow student.
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