The article that launched a thousand posts...

I can't believe I'm getting sucked into this one, but I'm not the only one. It comes down to, "Is it okay to teach kids that multiplication is repeated addition or not."



Devlin: It Ain't No Repeated Addition
Devlin: It Still Ain't Repeated Addition
And Devlin Is Still Wrong, see Neiderbergers comments to Let's Play Math

I don't know what else multiplication could be. Doesn't my computer base everything it does on repeated addition of ones and zeros? If Devlin comes up with a new paradigm wouldn't that be the greatest revolution in math in the latest 150 years?

And finally,

Text Savvy: here, here, here, here, and here.


And in my inbox this morning this email by Adrian. Who knew that after a marathon discussion on this from yesterday, (that I took notes on!) there was still something to be said?

Devlin has made a horrible gaffe. In the case of number fields, at any rate, you don't even have to make it to Peano's Axioms. Just the integers is far enough. Anything that produces the integers is forced to define multiplication as repeated addition. It follows from the distributive law and the fact that integers as an additive group are generated by the multiplicative identity. I wonder if there isn't a more general theorem in there about all fields reducing to the second operation of the field being repeated applications of the first operation of the field.

It is true that multiplication is not *just* repeated addition in the general case. But, the process of teaching it as repeated addition in the case of integers and then generalizing and extending it to other cases is not only not "just false", but it matches up exactly with the true intellectual development of the numbers. This isn't a technicality. This is the underlying truth of arithmetic.

Now, I do understand exactly where the notion of "multiplication isn't repeated addition" is coming from and do kind of appreciate the sentiment. But, if anything is "just false" it is the view that it is just false to say that it is repeated addition. Repeated addtion is the basis of multiplication and without it, there would be no extensions of it to the rational numbers, the real numbers or the complex numbers. It is not "mathematically correct" to teach it otherwise. It is more like a math gimmick. And, maybe it is a "harmless fallacy", itself, and one worth employing. But, it is not the "real" notion of multiplication.

Certainly solving problems like 5x=17 rely heavily on using the extension of multiplication to the rational numbers and the definition of multiplicative inverses, in particular (and/or the notion of division). But, that doesn't diminish one bit the fact that 5x is understood to be x+x+x+x+x or, for that matter, that 17 is understood to be 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1. Without this understanding, we don't even know what 5x=17 even *means* let alone have any means to solve for x.

The only plausible alternative to this is to abandon analytic geometry and algebra altogether and return to synthetic geometry. You could define multiplication as area. And, you would have to not start thinking of area by decomposing a region into a collection of unit squares and counting up the unit squares (since that is repeated addtion all over again). Something like that is far from more accessible or generalizable than the standard algebraic approach using repeated addition.

Now excuse me while I look up the phrases that I wrote down in my notes last night, because my lecture didn't sound like that email at all, but involved phrases such as: principle ideal domain, unique factorization domain, integral domain, ring, and field theory, matrix addition, functional analytic blah blah blah.


Since Frege, Peano, and Dedekind on the foundation of Arithmetic looked too expensive, I downloaded Dedekind's Theory of Numbers in pdf and I'll be able to print that out a few pages at a time and figure something out.

My appointment with the neuro is Aug 1st. I'll know more then.

Why not use Singapore's NEM?

This was the topic of a recent thread on the WTM message forums. The reasons given by many parents had nothing to do with the content of the book, but addressed student and parental factors. One person finally addressed content:

NEM does not provide lists of theorems and postulates for algebra and do "proofs or show tha this equals this" which I think is really improtant in algebra.

For that matter, the only American algebra text still in print that I have seen that does so in Foerster's Algebra. [I have no idea about Lial's, Chalkdust, Teaching Textbooks, or any other book you may be using. I do have an idea about Jacob's: It's not there. If you go flipping through your book to look for it also make a mental note, or even better a written list of all theorems proved in the book and make a note if the student is then required to prove theorems in the exercises] It lists out the field axioms in a list at the back of the book--something like getting assigned a basement office--but that at least is better than being completely absent.

Now that I've had time to think about it, I think what was missing from the ensuing conversation (monologue actually on my part) that these properties of the numbers, when considered together are very, very, very important in math. They aren't important because they help a 14 year old become a better rune manipulator and solver for x , they are important because they create a new universe of math called group theory. Asked one person puzzled about the need of such a list of properties at such a late date in the student's math career,

But aren't those properties all taught it most elementary mathematics programs? Or am I being naive? Certainly they've been featured and reviewed constantly in the program we've used. ... I'm not saying they're immaterial -- just that they're basic enough principles that I would expect students to know them well before they reach algebra (if not expressed in exactly the way that they are in your link). Am I wrong?

And so this is how I responded:

They aren't "basic" in the sense of being simple. They are more like foundational. Mathematicians have made careers talking about the differences between things (groups) that follow some of these rules and things that follow all of the rules. Essentially all of these various properties and what happens when they are or aren't true is the sum total of what one learns in Abstract Algebra so this isn't something that can be covered in its entirety in a K-8 arithmetic program. Usually what you might see in arithmetic is the appearance of the distributive, commutive, and associative properties and the book states it and then has the kid apply it in the simplest way possible or identify the use of it in a very simple way.

However, what you don't see (but if your arithmetic book does this I'd be interested) is go in depth into distinguishing between properties of numbers, axioms, definitions, theorems, and notation. It's all thrown in there and all mixed up. It's all ad hoc. [I should have said here is that I'm talking about algebra books, not arithmetic] And the students aren't required to use them as axioms, but instead they are taught "facts." [Axioms play the role of the premise in a logical syllogism. Think of algebra as having 11 premises, you will get the bigger picture in the long run if in algebra you start thinking of them as assumptions rather than as "facts." What follows from accepting these assumptions, if you do your derivations and proofs correctly, are all things that you use to do algebra arithmetic such as the algorithm you use to find equivalent fractions, multiplying double digit numbers together, etc). The sceptic then asks, "What happens if I reject one of these premises and instead only use these five? What sort of a math world would that create? You ask the question that open ups a worm hole and you leave the 9th grade algebra universe and go straight into group theory. Another sceptic might say, how can those 11 properties be proven? And that takes you to a different area of math as well, Back to the original post...]

That 2 X 3 = 3 X 2 is true is not "learning the commutative law". In fact, even learning that such a property is called "the commutative law" is not "learning the commutative law.""Learning the field axioms" entails being able to write down the general statement of the principle involved, not merely as a fact about the integers, but as a property that any set of elements together with any kind of operater [+ - x ÷ √] on that set may or may not satisfy.

"Learning the field axioms" also entails being able to use the list of axioms/properties that some given set and its operators might satisfy and actually derive significant mathematical results from that list through an unbroken chain of completely valid logical deductions.

What seems to happen in most algebra books (Saxon, Singapore not excluded) is that the student never is expected to make a distinction between the field axioms and other "facts" in algebra. They don't understand that multiplication distibutes over addition is axiomatic, but that it distributes over subtraction is not. Or, that the natural numbers are closed under addition is axiomatic but that they are closed under subtraction must be proved. And, they would "know" and use the rules of order of operations but they wouldn't know how those fit in with the other rules they've memorized. The rules about order of operations are not on "the list" of field axioms, for example. So, unless there is a master list ,the student is likely to go through his algebra book lumping it all together because he's never been given an indication that there is some sort of structure or hierarchy to all these facts that he's memorizing. [But then again, unless you have theorems for the student to prove or something else to say about the field axioms, nothing magically will come of simply memorizing it. ]

Once this list is memorized then there can follow discussion about what happens if one of the properties is missing. So even beginning in ninth grade algebra, with a text like Allen or Dolciani the student is getting an introductory taste of groups, rings, and field theory along with the skills it takes to prove theorems. This maturity of proving theorems, to gauge the appropriate amount of rigor for a given problem, constitute "true" math skills that cannot be acquired over night, skills that are absolutely necessary in doing in higher math (not to be understood as engineering calculus), For instance, after a diet of Allen, Gelfand, Oakley and Allendoerfer even an average student has practiced the skills needed for a Calculus text such as Spivak. Even a really good program like Singapore would leave a student unprepared for Spivak.

That's our story and we're sticking to it.

Always Look on the Bright Side of Life

I finally got around to seeing a neurologist who thinks I may have multiple sclerosis and I had a battery of MRIs last week done to look for lesions in my brain or problems with my spinal cord that account for those pesky symptoms I've been having.

I don't have a diagnosis yet and the anxiety and stress of waiting for that phone call alone is causing another flare up of whatever it is that I have.

The optimism of the people around me makes me cranky, "You know there's a lot of people with MS that live to old age." Yeah, and there's a lot that don't. As far as I can tell if this were a game of roulette there is one empty chamber, four chambers with rubber bullets, and one chamber with a lead bullet. I am not amused.

I was going to post how I am just not that interested in math any more, or really much of anything since I'm having a lot of difficulty concentrating in general, and more specifically with a mystery diagnosis hanging over my head the anxiety alone is disruptive, but at any rate, a couple of days ago I did read a chapter in David Berlinski's book "Infinite Ascent" about how Galois Theory explains why the quintic is unsolvable and it got me motivated to want to learn (not to be confused with putting out the effort to actually learn) abstract algebra again.

Related to that, there is an interesting proof in Frank Allen's Algebra that I was dismissive of the first time I saw it and now, after finding out what the formal defintion of a normal subgroup is, I wonder if there might not be more to it. Allen proves that "d-x+x=d" for the real numbers. The whole thing led to a neat discussion, at my house not in Allen, about graphing the solutions to the "roots of unity". I just don't remember discussing that all in any class I ever took and it's neat stuff. I'm pretty sure that it never came up because to graph it properly requires knowing polar coordinates and de Moivre numbers which I never got around to. And this is all a round-about way of saying that I now have a more personal reason to work through Gelfand's Trig than I did before. Although I just need to see how things are going to go in the next few weeks.

Finally, for those with a classical education bent, you may enjoy "Lapses in Mathematical Reasoning". It's an English translation of a Russian book from the 1950's. It is a collection of 80 false proofs that are at the high school level and you read through the solution to some problem and spot the fallacy. Some of them just reflect run of the mill mistakes and some of them lead to interesting discussions about deeper issues, long-winded answers are at the end of every chapter. At any rate, as you can see by the table of contents the authors of the book chose to extend Aristotle's refutations to sophisms of a mathematical nature which gives it a very philosophical feel that I had a lot of fun with. The chapter on arithmetical errors was not as good as the others since they relied on pecularities of algorithms that we don't use any more.

I'm not paranoid. They really are watching me.

My neighbor's camera pointing into the room where I homeschool our children:



And it's all perfectly legal since he's doing it from his property so I keep the blinds shut.

If it stays this way I guess we'll take it to the HOA.

Other ideas:
A) Put a sign in my window that says, "If you can read this you're invading my privacy."

B) Put a sign in my front yard that says, "I don't think my neighbor is a pedophile just because he has a camera pointed at our home and videotapes the children."

The background to this story is that my neighbor had his home broken into a couple of weeks ago so this weekend he was putting up security cameras and they all were pointed to catch anyone approaching the side of the house. I'm hoping the camera somehow got knocked out of place by accident and he'll notice this and fix it. Still, it's kind of creepy.

Solutions to Gelfand's Trig

Is anyone interested in a solutions guide to Gelfand's Trig? Did you benefit from the algebra solutions?

Paragraph Form of Proof

My sixth grader is progressing extremely slowly through Frank Allen's Algebra. A couple of days ago we ended up on page 326 which ostensibly is on the topic of odd and even numbers and divisibility tests, however, nearly overlooked by me was the first proof in paragraph form. All prior proofs were presented in two columns. And as I flip forward through the book it looks like more and more of these will be in the form of a paragraph.

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.

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.

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.

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.

How Math Doesn't Get Done

Say I have a theorem I want to prove because its given in an exercise set. In this case show that,

│x₁ + x₂+...+x_n │ ≤ │x₁ │+ │x₂ │+...+ │x_n │

and my next thought is that I need a “good” definition. How do I decide which definition to use? I find a bunch and I think they ought to be good definitions because I looked them up in a book written by someone with a degree in math.

Definition 1: │a│ = { a, -a } ⁺
Definition 2: │a│ = max{ a, -a }
Definition 3: │a│= a sgn a
Definition 4: │a│ = sqr root of a²
Definition 5: The absolute value of a real number a is defined to be a if a is positive or zero, and to be -a if a is negative.

Day Two:

But on the other hand, what if I don't have a book and I'm online or I get a bunch of trash from wiki. How am I supposed to know a good definition when I see one?

And say I get in the middle of a logic book to figure this out and I start reading the chapter in the middle of the book on “Theory of Definitions” and it makes a lots of references to Aristotelian logic, then out comes the Aristotle and see that I have at least 350 pages to read if I want to make any sense of it. I get back online and look up what I can about the original theorem. Why is it given in the exercise set? What makes it important? It doesn't look like the rest of the problems in the exercise set so it must be special.

Day Three:

So I go back to wiki and there is a discussion about the “properties” of absolute value and one of its properties is “subadditivity” which, lo and behold, looks exactly like the problem that I have in my problem set. So is there something important about subadditivity? By the dogs, it seems there is something to this subadditivity thing. All sorts of things can have this subaddtivity property, but most importantly, it seems a distance function must have this property....(What if it doesn't?) What are the other properties that a distance function must have? Is a distance function the same thing as a metric? Is a metric the same thing as a metric space? No. it seems not. A metric space includes a distance function. What sorts of distance functions are there other than absolute value?

Day Four:
So while I'm thinking about what makes a good defintion good, it occurs to me that perhaps illustrations are not definitions. Are illustrations actual objects? In geometry an illustration of a circle is not the actual circle we are discussing, it's an idealization of some circle, but wait a second, there aren't any actual Platonic circles floating around anywhere, so what can an illustration of a circle really represent?

Day Five:

This inequality is like the triangle inequality but for a triangle with an infinite number of sides...er sort of...like a circle? No. Must stop thinking in pictures. Maybe Euclid could be done without any pictures at all, maybe circles aren't really circles but sets of points, lots of symbols, what would Euclid look like if it was written in set notation? (Answer handed down from above: Don't worry, Euclid defined a circle, you're safe, they aren't just illustrations).


Day Six:
And when I look at a kids textbook and it says that absolute value is "the" distance, is it really the distance or is it a way of measuring that distance? I mean, the actual number yielded is the distance, but the thing that got you that number, the absolute value thingy itself does not seem to be "the" distance. See what I mean? So maybe, since there are other distance measuring thingies like absolute value,which I have no idea what they are but wiki or some other math site said it so it must be true, maybe I should say, "Absolute value is one way of measuring distance." I'm having issues with the idea of measuring distances on non-physical objects.

Six days later, I'm finally finished with my little proof, but if I'm going so slow because I can't really focus on the actual problem then it's going to be forever before I learn anything.

But before I go on to problem #26 let me just check this one little thing out about this Fekete's Lemma. Wonder what that's good for? And if I could find out would I even understand? It's not in any books around the house but it looks interesting.