From Mikusinski (and Adrian) With Love

You can have no end of fun poking little holes in my explanation of how multiplication is defined using induction.

I explained in a post before that, that I left out a number of steps, and I certainly rounded off some edges in the interest of trying to make it somewhat realistic. In general, though, I think the explanation was sound.

If you are interested, this is what I was working with, from Mikusinski (emphasis in original; also, I'm using the phrase "is an element of," rather than the symbol, because that symbol really messes with my layout):

The result of multiplication of two natural numbers x and y is called the product of x and y and is denoted by xy (or sometimes by x &bull y). The product of x and y is defined by induction as follows:

(&bull) x &bull 1 = x;
(&bull &bull) x(y + 1) = (xy) + x.

Let X be the set of all natural numbers y such that the product xy is uniquely defined for every natural number x. By (&bull) we see that 1 is an element of X. Consider some y such that y is an element of X. Then xy is uniquely defined and so is (xy) + x because of the uniqueness of addition. Thus, by (&bull &bull), the product x(y + 1) is uniquely defined, which proves that y + 1 is an element of X. By the Induction Principle the set X contains all natural numbers, so the product xy is uniquely defined for all pairs of natural numbers.

I do have at least one more post up my sleeve about all of this. And the reason I do is because I love talking and writing about this kind of stuff.

Hey, I even enjoy a good snark, even when I disagree with it (especially when it employs an analogy; I love analogies). This is from Adrian, defending the sacred unity of repeated addition and multiplication:

There is nothing more exactly identical in all of mathematics other than perhaps something like "0=0". Saying otherwise would be like saying that

"A triangle is not a three sided polygon -- a triangle is a triangle. That it has three sides quickly follows and 'it works', but it isn't part of the definition, and it is damaging and just false to say that 'a triangle has three sides'."

Preposterous.

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Devlin's Right Angle, Part VI

Let me quickly close out the definition of multiplication I was building to in Part V.

I fell far short of my goal of making myself completely clear at the end of that post, so I want to see if I can rescue the clarity of the presentation somewhat—for myself, if no one else.

Anyway, here it is, again:

When you're dealing with operations, like addition and multiplication, it's both different and the same. Different because you can't just throw an operation, pure and unadulterated, at the Induction Principle and ask, "Hey, does it work?" And the reason you can't do that is not because it's a stupid question—it's not. The reason you can't do it is because there is no such thing as an operation, pure and unadulterated. You have to define it first. Same because you still have to deal with the least element (1) and some x + 1 when you're dealing with the natural numbers.

So, let's make the following assumptions about the product of a and b (Note: These are assumptions. And they are not assumptions about "multiplying;" they are assumptions about the result of multiplying. And they are assumptions one would have to make, given the definition of the natural numbers.):

(1) a × 1 = a
(2) a(b + 1) = (ab) + a

Now, here's the freaking crazy part. We assume that (1) and (2) above are true. Then we simply imagine that we already have a jar that contains all the natural numbers, scooped up by the operation of a × b, with b being a natural number. All we have to do in order to show that what we have scooped up is indeed all the natural numbers is to find the number 1 and to find some b + 1.

According to our assumption, then, and (1), the number 1 is in our jar. That is, since we assume that a × b scoops up all natural numbers, and we assume that a × 1 = a, then 1 (substituted for b) is a natural number and must be in our jar.

It is very important to remember that we're not proving anything here, so to speak. As I mentioned in my previous post, we can use the Induction Principle to find out whether a statement like 1 + 2 + 3 + . . . n = n(n + 1) / 2 works for all natural numbers. We check that it works for 1, then we check whether or not it works for some b + 1. If the statement passes those tests, then we know that it works for all natural numbers.

With an operation like multiplication, the same idea applies. However, with multiplication (or addition), we first have to have some guess about how that operation works on 1 and some b + 1. You can see this in (1) and (2) above. The first statement, (1), imagines how the operation applies to the number 1, and (2) imagines how the operation applies to some b + 1. But after we send those statements through the tests of the Induction Principle, the only relevant conclusion that will come out is whether or not they describe an operation that applies to all natural numbers. That's it.

So how do we take those hair-brained statements, imagining some operation called multiplication in the natural numbers, and show that they do indeed work for all natural numbers?

Here's how: First, we just imagine that we're right on the big idea here and that the operation of multiplication (a × b, where both a and b are natural numbers) does indeed apply to all the natural numbers—or, as I said before, the operation of multiplication "scoops up" all natural numbers. Second, we assume that the statements we made in (1) and (2) are true. And these statements tell us how multiplication applies to both the number 1 and some b + 1. Third, we need to show that, given these assumptions, our "multiplication jar," as it were, could contain the number 1 and some b + 1, which would prove that our jar contained all of the natural numbers.

(1) says that a × 1 = a. And since we've assumed that that is true and that a × b scoops up all natural numbers, which we assume are in our jar, then 1 must be a natural number (because we can just substitute b for 1), and, therefore, must be—or, rather, could be—in our jar.

(2) says that a(b + 1) = (ab) + a. Since we've already assumed that a × b scoops up all the natural numbers, obviously (b + 1) can be substituted for b, which means that b + 1 is a natural number and could be in the jar. The only problem is that one must show that a(b + 1) is "uniquely defined," which means that you must show, using axioms of addition already settled upon, that (ab) + a is uniquely defined, which it is, because of the uniqueness of addition. We do, indeed, need to rely upon addition when we define multiplication for the natural numbers.

However, there is no reason to believe that repeated addition is criterial. It should be noted that nowhere--nowhere--in the "definition" above was it necessary to make use of repeated addition to define multiplication. Nowhere.

The Thorn

From what I've seen, what most people get stuck by is the statement in (2) above--a(b + 1) = (ab) + a--what is known as the distributivity of multiplication over addition. So, let's say a = 4, and b = 5. We know that the product of 4 and 5 is 20, right? What the statement above says—if you read it cinemathematically--is that if you want to multiply 4, or a, by the next natural number after 5 (b + 1, or 6), you can multiply 4 and 5 and then just add another 4 [(ab) + a]. In other words, to find 4 × (5 + 1), first find 4 × 5, then just add 4.

Now, of course, this "works" for all natural numbers. And, if you think about it, you could just continue this process repeatedly, using the second part of the definition for multiplication I showed above. In other words, you could find 4 × (5 + 2) by, again, first finding 4 × 5, and then adding 4 and then another 4. The product is 28, bravo!

Hey, you know what we could do? We could basically extend this definition—that is, we can say that "all anyone means when they say" a(b + 1) = (ab) + a is just (a + a + a . . . )_btimes! That way, we can say that multiplication is just, by definition, repeated addition, right?

NO. NOPE. NYET. NOPEY-NOPE-NOPE.

Of course, the idea of repeated addition follows very directly and immediately—as a way to calculate multiplication problems—from a definition of multiplication on the natural numbers, but IT. IS. NOT. THE. DEFINITION. OF. MULTIPLICATION.

As Nunes and Bryant (1996!) argued (I'll get the source up soon):

Although there is a conceptual discontinuity between multiplication and addition, there is a procedural connection between these operations. Because multiplication is distributive with respect to addition, repeated addition can be used as a procedure to solve multiplication sums.

What people (including now someone named Adrian with his/her one-person press release to bloggers who write on the subject) constantly refer to is the "procedural connection" while ignoring the "conceptual discontinuity."

Here's another good quote from the same source (again, I'll put it up soon; keep in mind that this is from '01):

The English National Numeracy Strategy (DfEE, 1999, p. 14) suggests that pupils should be taught to understand multiplication as repeated addition. In contrast, the Japanese Association of Mathematical Instruction proposes that "repeated addition is a way to calculate multiplication, not a meaning of it" (Yamonoshita & Matsushita, 1996, p. 291).

Alas, another post shall follow.

Reference:Park, J. & Nunes, T. (2001) "The development of the concept of multiplication." Cognitive Development 16 (2001) 763-773

Part I | Part II | Part III | Part IV | Part V | Part VI

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Devlin's Right Angle, Part V

I think Mr. Devlin, in his follow-up article on the subject, provides the best transition from my previous post to this one (emphasis mine):

Over a century ago, mathematicians finally learned to sidestep that unanswerable "What is it?" question by adopting the axiomatic approach, where you simply specify the properties of numbers and the arithmetical operations, and concentrate on manipulating them according to those rules. . . .

At the turn of the twentieth century, an Italian mathematician called Peano did formulate what are often called the Peano axioms, but their purpose is to show how the positive whole numbers can be defined from first-order logic; they are not a descriptive axiom system that tells you how to work in the system.

Now, what Joe and Myrtle refer to when they call upon Peano and Edmund Landau to defend multiplication as repeated addition is probably almost exactly like what I will present below. Yet, I do not see the necessity of the concept of repeated addition in this definition—yes, it's still basically just a definition. And, after reading my previous post over again, I'm not convinced that Joe and I disagree all that much--at least mathematically--as he puts the word repeated in parentheses and describes repeated addition to represent multiplication as "informal."

So, let's start with a definition, taken from Mikusinski, but which I will translate for the uninitiated:

Denote by 1 the least element of N. If x is an element of N, then the least element of the set of all natural numbers greater than x is denoted by x + 1.

Translation: Let's just say that 1 is the least number of all the natural numbers. If some number (let's call it x) is a natural number, then we'll use the expression x + 1 to mean the natural number that comes right after x. So, essentially, the natural numbers start with 1 and go up by 1's forever, and there is no natural number between x and x + 1.

All right. Now on to the big show, the Induction Principle Theorem, for which I won't be showing a proof because we skipped over a lot to get here:

Let X be a subset of N such that

(a) 1 is an element of X;

(b) If x is an element of X, then x + 1 is an element of X.

Then X = N.

Translation: We have essentially defined the natural number system as starting with 1 and going up by 1's forever. So, imagine you could scoop up a bunch of natural numbers in a jar—you don't know how many. (We use N to refer to all the natural numbers and X to refer to what you scooped up.) What the Induction Principle Theorem says is that if you can prove that

(a) The number 1 is in your jar, and that

(b) No matter what natural number you name, both it and the natural number that comes right after it are in the jar,

Then your jar contains all the natural numbers.

I know that, in some sense, this all seems ridiculous—hey, and in some sense, it is. But I think the example shown here is a good example of how the Induction Principle can be used to check whether or not a statement applies to all natural numbers. In brief, you must check that the statement is true for the least element (1), then, given some natural number m, you must check that the statement works out for m + 1. Presto. It works for all natural numbers—certainly better than guessing.

When you're dealing with operations, like addition and multiplication, it's both different and the same. Different because you can't just throw an operation, pure and unadulterated, at the Induction Principle and ask, "Hey, does it work?" And the reason you can't do that is not because it's a stupid question—it's not. The reason you can't do it is because there is no such thing as an operation, pure and unadulterated. You have to define it first. Same because you still have to deal with the least element (1) and some x + 1 when you're dealing with the natural numbers.

So, let's make the following assumptions about the product of a and b (Note: These are assumptions. And they are not assumptions about "multiplying;" they are assumptions about the result of multiplying. And they are assumptions one would have to make, given the definition of the natural numbers.):

(1) a × 1 = a
(2) a(b + 1) = (ab) + a

Now, here's the freaking crazy part. We assume that (1) and (2) above are true. Then we simply imagine that we already have a jar that contains all the natural numbers, scooped up by the operation of a × b, with b being a natural number. All we have to do in order to show that what we have scooped up is indeed all the natural numbers is to find the number 1 and to find some b + 1.

According to our assumption, then, and (1), the number 1 is in our jar. That is, since we assume that a × b scoops up all natural numbers, and we assume that a × 1 = a, then 1 (substituted for b) is a natural number and must be in our jar.

Update: I'm deleting the part of the post that went on to describe how we find the b + 1, because it sucked.

Oy. More to come.

Part I | Part II | Part III | Part IV | Part V | Part VI

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Devlin's Right Angle, Part IV

Oh, gosh, I may have to go back on my promise to make this my last post on the topic.

A gentleman by the name of Joe Niederberger has been dominating the comment thread at Let's Play Math. His first comment is uninspiring:

Devlin unfortunately makes the mistake of thinking of multiplication as one “thing.” It’s true multiplication of any two real numbers cannot be simply reduced to repeated addition, however, the multiplication of any two integers *can* always be reduced (or thought of, or defined by) repeated addition.

Very well, then, Mr. Niederberger. Solve for a (as an integer):

a + a + a + a = 6
If it is indeed true that the "multiplication of any two integers *can* always be reduced [to] (or thought of, or defined by) repeated addition, then it should be a snap to solve this repeated addition problem without recourse to multiplication (or its inverse, division). I certainly could be wrong, but it seems to me there are two choices if you actually want to solve this problem within the restrictions given (and without guessing or modeling) rather than simply define it: (a) you can subtract a from 6 four times, or (b) you can subtract 4 from 6 a times, which wouldn't make any sense, given that we don't know about multiplication. Either way, you're up a creek without a Peano (or with one).

Here's Joe again, on the comment thread:

I’d like to repeat a key point that I make: even Peano in his axiomatic defintion needs to define multiplication of whole numbers with a recursive (repeated) addition definition.

That’s just the way it is and Devlin has not offered an alternative.

LOL! Niederberger ignores a great deal here. As I alluded to before, just because you can define something in terms of something else, does not make it mathematically useful, which is why I presented my algebra challenge above. I think it's wonderful that Joe can call upon Peano to define multiplication:

You can refer to a previous most of mine that gives the Peano recursive definition of multiplication that uses addition. You can do it more informally by saying that MxN (M,N integers) denotes a function whose definition is given as MxN = M+…+M (N times). That’s all anybody means when they say that.

But if the only way Joe can use that definition is to rewrite multiplication expressions as repeated-addition expressions, then it's not very useful in our present discussion. And it certainly isn't very useful in solving mathematical problems. So why should we think of it as something necessary to plant into our kids' brains?

By the way, there IS a way to define the operation of multiplication on natural numbers that makes no use of the concept of repeated addition. Next time. Got some pacing to do.

Part I | Part II | Part III | Part IV | Part V | Part VI

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Devlin's Right Angle, Part III

So I'll start to wrap up here—just one more post after this, I promise—by making a long story short and a short story a bit longer.

Devlin has the right angle on this topic. Multiplication is not repeated addition, and we should really stop telling kids that it is.

However, as I tried to relate in my first post on the topic, Devlin is wrong to ask classroom teachers (or homeschoolers) to make this adjustment. Even if teachers are not, in effect, forced by state standards or their current mathematics curricula to introduce multiplication as repeated addition, separating the two operations from each other requires a fundamental change in how most elementary mathematics teachers go about the business of teaching multiplication. For evidence of that fact, you need go no further than Homeschool Math Blog, run by Maria Miller, who has an actual financial stake in these kinds of arguments. So I agree with Goldenberg when he says that "i[t] does not suffice merely to assert that the two [multiplication and repeated addition] are, for the most part, not the same."

Yet, as I tried to get at in my second post, it does not similarly suffice for teachers—or textbook publishers or curriculum experts—to simply throw up their hands and declare that, well, the nice NPR math guy might be right, but he didn't spell out for me word for word how I am to change my practice, so I'll leave well-enough alone.

Of course, that doesn't mean that educators should just agree with Devlin and move on to implementation. I thought this, from Maria, for example, was a pretty good rejoinder:

Even our word "multiply" refers to multiple copies of the same... people and animals "multiply", we talk about multiples, etc. We use the word "times" referring to doing the same thing over and over, such as "I opened the door three times."

And consider these graphs from a book by Jan Gullberg:

Multiplication was thus defined by Robert Recorde (c. 1542) [spelling edited slightly for clarity]:

Multiplication is such an operacion that by two sumes producyth the thyrde, whiche thyrde sume so manye times shall containe the fyrst, as there are unites in the second.

[. . . .]

The earliest form of multiplication known is the Egyptian method of duplation, which reduces multiplication to a form of continued addition. The method . . . was frequently copied by other peoples, and is commonly found in textbooks from the Renaissance.

But it also doesn't mean that educators should simply reject Devlin's idea out of hand. After all, we're not ancient Egyptians. And it's never going to be 1542 again.

Part I | Part II | Part III | Part IV | Part V | Part VI

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Devlin's Right Angle, Part II

Let me continue here by addressing some of the counterarguments to Keith Devlin's assertion that multiplication is not repeated addition.

Michael Paul Goldenberg declares that Devlin's complete statement

Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not.

is both right and wrong.

Goldenberg believes that the first clause of Devlin's statement is right because "multiplication simply is not repeated addition" is true for some number systems. And the reason he thinks Devlin is wrong?

So now I need to go out on a limb and suggest that while Devlin isn't wrong, he also isn't quite right, and the problem lies with the nature of school mathematics and its teaching, as well as issues of mathematical maturity.

I agree with the first part, and the second argument is as old as the hills, and it alone is why I started this blog. I have at times referred to it as the Stork Argument, the 170 Hearts Argument, and the Columbus Syndrome. Basically, the argument goes like this:

The truth is unteachable.

I'll come back to this, maybe. Here's Goldenberg again:

One thing I find lacking in his piece is a solid example that would communicate well and clearly to K-5 mathematics teachers (based on the ones I've known and worked with) how multiplication differs in some deep way from addition. I[t] does not suffice merely to assert that the two are, for the most part, not the same.

One has to wonder, given that Goldenberg's essential disagreement with Devlin has to do with Goldenberg's role as shepherd of his local K-5 mathematics flock, why he can't come up with an example "that would communicate well and clearly . . . how multiplication differs in some deep way from addition" himself. I would bet good money on the fact that the reason he doesn't is because he hasn't the slightest clue what he's agreeing with.

So let's help him out. Below is a cube (I should have used a cone). Each of its six faces are squares. The volume of such a figure can usually be found by employing the formula l × w × h. However, given that multiplication is the same as repeated addition, we should be able to describe the volume of the figure below using only the operation of addition, repeated.

Go.

Part I | Part II | Part III | Part IV | Part V | Part VI

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Devlin's Right Angle, Part I

Keith Devlin has written a wonderfully rebellious piece titled It Ain't No Repeated Addition, and I can't pass up the opportunity to comment.

The ideas in the article—and from reactions by MPG, Maria Miller, and Denise--touch on a number of fundamental issues that I like to write about here.

Let me start by quoting the concluding statement of Devlin's article:

In the meantime, teachers, please stop telling your pupils that multiplication is repeated addition.

And then I think it is important to point out that the article in question is, as Devlin notes, an extension of a September 2007 article on conceptual understanding. The key graphs from the earlier article are these (I hope this isn't lifting too much):

One of the math ed folks explained to me that teachers often explain whole number equations by asking the pupils to imagine objects placed on either side of a balance. Add equal numbers to both sides of an already balanced pairing and the balance is maintained, she explained. The problem then is how do you handle subtraction, including cases where the result is negative? I jumped in with what I thought was an amusing quip. "Well," I said with a huge grin, "you could always ask the children to imagine helium balloons attached to either side!" At which point my math ed colleagues told me the awful truth. "That's exactly how many elementary school textbooks do it," one said. . . .

Sure, I can see how the helium balloon metaphor can work for the immediate task in hand of explaining how subtraction is the opposite of addition. But talk about a brittle metaphor! It not only breaks down at the very next step, it actually establishes a mental concept that simply has to be unlearned. This is surely a perfect example of using a metaphor that is not consistent with the true concept, and hence very definitely does not lead to anything that can be called conceptual understanding.

I use these two quotes to make three points: (1) Teachers are real-time education delivery agents. They must—MUST—at the very least, adhere to state standards:

California (2005), Grade 2, Number Sense 3.1: Use repeated addition, arrays, and counting by multiples to do multiplication.

Florida (2007), Grade 3, Big Idea 1, MA.3.A.1.1: Model multiplication and division including problems presented in context: repeated addition, multiplicative comparison, array, how many combinations, measurement, and partitioning.

New York (2005), Grade 2, Number Sense and Operations, 2.N.20: Develop readiness for multiplication by using repeated addition.

Ohio (2001), Number, Number Sense, and Operations: By the end of the K–2 program students should "I. Model, represent, and explain multiplication as repeated addition, rectangular arrays, and skip counting."

Michigan (2005), Grade 3, Number and Operations, N.ME.03.04: Count orally by 6's, 7's, 8's, and 9's starting with 0, making the connection between repeated addition and multiplication.

So, even if teachers—the ones who are in real-live classrooms every day—were adamantly opposed to the idea of linking multiplication with repeated addition (and they have, by and large, neither the time nor the resources to be adamantly opposed to, or proponents for, academic issues), it doesn't matter. They're hands are effectively tied by state law. Gone are the days (as of now) when an elementary mathematics teacher could be successful in teaching her students to memorize the multiplication table up to 10 × 10. Now her students MUST show the same result with repeated addition.

And, therefore, (2) Devlin is wrong in calling out "teachers"—if what he means by that term are the actual classroom education delivery agents—to make this correction. They have, effectively, no say, no power, when state standards are so prescriptive.

However, (3) the inaccuracy of saying that all multiplication IS repeated addition belongs to an entire class of inaccuracies, or lies, perpetrated by modern mathematics education. One who considers the problem of the alleged mismatch between repeated addition and multiplication as an isolated incident of inaccuracy in mathematics education is someone who has his or her eyes firmly shut against reality.

More on this tomorrow.

Part I | Part II | Part III | Part IV | Part V | Part VI

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Transformations in the Coordinate Plane, II

When I started my previous post, I wanted to use the subtopic of teaching transformations to address loftier concerns with education.

And then I took a look around online and saw that most of the presentations about transformations in the coordinate plane—like so many presentations of mathematics online—were composed with, seemingly, no audience in mind. Several hundred thousand people, all giving you (or me, or someone, who knows) bad directions.

So, why not add to the confusion?

Transformations

The word transformation is a rather unfortunate choice on the part of mathematicians (or whoever came up with the name)—at least for school children—because what we're talking about when we talk about transformations has nothing to do with "transforming" anything, if by "transform" we mean "to change in structure or form." Indeed, when elementary and middle-school texts introduce transformations, they define them more or less (but more less) correctly as "movements of plane figures that do not alter the shape or size of the figures"—the exact opposite of what novices would likely think if we were to tell them that we were going to "transform" a triangle. Transform it into what? they might ask.

This is one reason I don't allow the use of the verb transform when talking about transformations in text. (The other reason is that it sounds really stupid.)

When we talk about transformations at the elementary and middle-school levels, we're talking about the END-RESULT of movements of geometric figures (it's not really the movements themselves, and the figures don't have to be plane figures like squares and rectangles; they can be points, or, later, 3-D objects).

That's it. The end-results of movements of figures.

However, in some small defense of the word transformation, I should note that the transformation (i.e., the end-result of a movement) of a geometric object is, at least mathematically speaking, a different object, called an image. For example, the transformation (i.e., the end-result of the movement) of, say point D is treated as a different figure—the image of point D. Blah, blah, we'll get there.

Stay tuned.

Summary: (1) most presentations about mathematical transformations online are written with no audience in mind, (2) the definition and use of the word transformation in elementary and middle-school mathematics texts is at odds in one way or another BOTH with students' intuitive understanding of the word and with mathematics knowledge, (3) mathematical transformations at the elementary and middle-school levels are simply the end-results of movements of geometric figures.

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Transformations in the Coordinate Plane, I

I have mentioned here a couple of times how lower-level representations can be valuable both in and of themselves and as teaching aids.

For example, repeated addition (a lower-level representation of multiplication) can be used to find a product when a benchmark fact is known. And even something as simple as knowing how to count on your fingers (a lower-level representation of addition) is not scrubbed away suddenly when you learn how to add. Such knowledge can still have some value.

Intelligent adults still maintain the ability to count on their fingers—counting time intervals that cross over the hour (or the month of January), counting inclusive ranges (e.g., the range "8-12 years old" covers 5 ages, not 4), teaching their children addition and subtraction, etc.—when they lack a practiced algorithm to do otherwise; when they forget, under pressure, a method they have learned; or when they need to communicate using the less-efficient method.

One kind of lower-level representation that is pervasive in upper-elementary and middle-school teaching is the presentation of transformations in the coordinate plane as a kind of visuo-spatial training.

Take a look at the grid below. Students in sixth, seventh, and then eighth grade might be asked to name the coordinates of the point shown after it is rotated 90 degrees counterclockwise about the origin.

However, no "math" is used. Students are, as far as I've seen, expected to arrive at the correct answer by rotating the image mentally—or by drawing. I've yet to see a lesson in a purely middle-school text explaining how this mental rotation is to be done in any kind of systematic way.


But there is a "higher-level," mathematical way of figuring out these kinds of transformations that, because of education's obsessive fixation on—well, I'll leave that for my next post.

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Experimental Probability, Part I

Experimental probability is usually introduced to students somewhere between the fifth and seventh grades and could be defined as follows:

In the context of a probability experiment, the experimental probability of an event is the ratio of the observed outcomes in the event to the total number of trials in the experiment.

Suppose a person rolls a number cube (a die) 50 times. In that case, rolling the die is the probability experiment, and the total number of trials in the experiment is 50. The table below shows possible results of that probability experiment:


So, consider the event "rolling an even number." According to the probability experiment results shown above, there were 24 outcomes observed in this event—the person rolled a two 7 times, a four 8 times, and a six 9 times: 7 + 8 + 9 = 24. Thus, one could say that the experimental probability of the event "rolling an even number" is 24/50, or 12/25, or 0.48, or 48%.

Now consider the event "rolling a 2." According to the probability experiment results shown above, there were 7 outcomes observed in this event—the person rolled a two 7 times. Thus, one could say that the experimental probability of the event "rolling a 2" is 7/50, or 0.14, or 14%.

When it is introduced, experimental probability is usually distinguished from theoretical probability, which could be defined as follows:

In the context of a probability experiment, the theoretical probability of an event is the ratio of the number of possible outcomes in the event to the total number of possible outcomes in the experiment.

Consider again the probability experiment described above—rolling a number cube, or die—and consider the first event we discussed above, "rolling an even number." The number of possible outcomes in that event is 3—there are 3 ways to roll an even number: by rolling a 2, 4, or 6. And the total number of possible outcomes in the experiment is 6—the person could roll a 1, 2, 3, 4, 5, or 6. Thus, one could say that the theoretical probability of the event "rolling an even number" is 3/6, or ½, or 0.50, or 50%.

Now consider again the event "rolling a 2." The number of possible outcomes in that event is 1—there is 1 way to roll a 2—and the total number of possible outcomes in the experiment is 6. Thus, one could say that the theoretical probability of the event "rolling a 2" is 1/6, or about 16.7%.

The key differences to note, at least for this post, between the concepts of experimental and theoretical probability as they are introduced to students are that where experimental probability employs "number of observed outcomes in an event," theoretical probability employs "number of possible outcomes in an event," and where experimental probability uses "total number of trials in an experiment," theoretical probability uses "total number of possible outcomes in an experiment." Obviously, these different definitions give us different probabilities for the same events. In the context of the probability experiment described above, the experimental probability of rolling an even number is 48%, whereas the theoretical probability of the same event is 50%. The experimental probability of rolling a 2 is 14%, whereas the theoretical probability of the same event is about 16.7%.

The Argument

Given that background information, consider the following argument put to me a few months ago, in reference to a word problem like the following:

A bag contains ten tiles numbered 1–10. Megan chooses a tile from the bag, records the number on it, and then replaces it seven times. She chooses the number 4 twice, the number 6 once, the number 2 three times, and the number 5 once. Based on these results, what is the experimental probability that the next number Megan chooses will be a 2?

Quite simply, the argument was this: Experimental probability can not be used to discuss the probability of a future event such as "the next number Megan chooses" because experimental probability uses "total number of trials in an experiment," and, for future events, there are no trials to consider.

Well, such an idea contradicts at least one published source:

Megan plays on the high school's varsity softball team. She has been at bat 35 times this season. She gets a hit 9 times. What is the experimental probability that she gets a hit her next time at bat?

So, what do you think?

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A History-Making Campaign

Excellent . . .

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Dichotomies Are the Reason Daddy Drinks

This happens to me way too often. I stumble across some education "research," read the abstract, print it, and quickly become bored with it.

Then I feel guilty for wasting paper, struggle through the entire piece (PDF), and, usually, find some ideas that seem to be worth further exploration:

There appears to be an inclination within the education community to dichotomise and an associated tendency to (i) ignore the connectedness of the dichotomous categories, and (ii) on occasion, to privilege one category while denigrating the other . . . .

This paper addresses five of these dichotomies: Teaching and Learning; Abstract and Contextualized mathematical activity; Teacher-Centred and Student-Centred classrooms; the teacher's contemporary dilemma: to Tell or Not to Tell; and the related issue for students: to Listen or to Speak. (376)

[The Teaching and Learning dichotomy] is a particularly insidious consequence of the constraints that language (and the English language, in particular) imposes on our theorizing . . . . this is particularly evident in the various translations of Vygotsky, in which the Russian word 'obuchenie' has been represented as either teaching or learning in different translations (Clarke, 2001). The integration of teaching and learning as components of a jointly enacted single activity also occurs in several other languages, including Chinese, Japanese and Dutch. (378)

Differences in the use of abstract and contextualised tasks seem strongly connected to a perceived need in Western classrooms to present mathematics as relevant to students. (379)

In the Swedish classroom, the students demanded that the teacher justify the relevance of what was being taught . . . . Despite the teacher's efforts, students were outspoken in their lack of belief in the relevance of the mathematics they were studying. . . .

By contrast, in the classroom in Shanghai, mathematics tasks tended to be very abstract in character and the teacher made no effort to demonstrate or argue for the real world applicability of the mathematics being studied. . . . However, in the post-lesson interviews, the Chinese students consistently expressed strong beliefs in the utility of mathematics in general and in relation to the specific mathematics they were studying [ed: There is a bit of apples-and-oranges at work here, so be warned. "Demanding" relevance of what you're studying and "believing" that what you're studying has relevance are two completely different things.]. . . . . Svan has christened this the "Expanded Relevance Paradox" (Svan & Clarke, in preparation) and means, by this term, to refer to the paradoxical character of application-oriented mathematics teaching associated with subjective irrelevance and pure mathematics-oriented mathematics teaching associated with subjective relevance . . . . Mathematical tasks are a constituent element of the social activity in which students engage. Attempts to increase the 'relevance' of these tasks through a figurative contextualisation may be counter-productive if these efforts are perceived by students to be artificial and are interpreted as reifying the very distinctions they seek to dissolve. (380)

This one was really good. Hell, it's been "good" for a long time. (emphasis mine):

One common interpretation of the constructivist manifesto (i.e., that "knowledge is the result of a learner's activity rather than of the passive reception of information or instruction," von Glasersfeld, 1991, p. xiv) has been that it became no longer legitimate for teachers to "tell" students anything. This position is not a logical consequence of adherence to constructivist learning theory, which suggests that students inevitably construct their own mathematics, whatever the classroom situation (Cobb, 1995). However, Telling or Not-Telling have been constructed oppositionally with such success that publications on contemporary pedagogy, . . . while usefully discussing many pedagogical strategies, see no need to address any strategies that might be construed as analogous to "telling" and even articles that purport to address the issue (such as Chazan and Ball, 1999) offer teachers little insight into how (and, as importantly, when) their mathematical knowledge might be articulated explicitly to the benefit of their students.

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Wu and I

Allison pointed me to some notes from a presentation by Hung-Hsi Wu at NCTM's 2007 meeting, and I've been meaning to write about them.

I gave a presentation at that conference, so during Wu's talk I was, if I remember correctly, holed up in my hotel room frantically reviewing my notes—a shame, considering that what appears to be the central point of Wu's presentation was, with fewer differences than similarities, the idea that I tried to flesh out for textbooks when I first started writing this blog three years ago. (A further shame is that many of my posts which directly addressed that idea are gone forever.)

Wu's idea below, minus the brilliant parallel to engineering, is essentially what I have referred to as the Boundary Principle (emphasis in original):

Engineering cannot cater to any human need no mater [sic] how scientifically absurd, any more than it should produce any product that is useless though scientifically correct.

Engineering must mediate between two extremes:

(1) inviolable scientific principles.
(2) user-friendliness of the final product.

Mathematics education, as Wu explains, is constrained by the same impossibility on the one hand and impracticality on the other. That is, it is not possible for mathematics education to serve every student desire, nor practical for it to attempt to do so. Rather, mathematics education (what Wu refers to as "mathematical engineering") must strike a principled compromise between its often conflicting fealties to both mathematics students and to mathematics itself.

Wu goes on to list five "inviolable scientific principles" for mathematical engineering, four of which fit reasonably well under my three for textbooks—accuracy, coherence, and language:

Precision: Mathematical statements are clear and unambiguous. At any moment, it is clear what is known and what is not known.

Definitions: Bedrock of mathematical structure (no definitions, no mathematics).

Reasoning: Lifeblood of mathematics; core of problem solving.

Coherence: Every concept and skill builds on previous knowledge and is part of an unfolding story.

Purposefulness: Mathematics is goal-oriented. It solves specific problems.

I would disagree that the last of these rises to the importance of being a principle, and Wu's contention that the damaging split (visit the links in Allison's post [linked above] for more details) is between mathematicians on the one hand and mathematics educators on the other is, I think, a bit simplistic. Nevertheless, I was heartened to see so many similarities between Wu's ideas and my own.

And here's just a fantastic section from Wu's article "How mathematicians can contribute to K-12 mathematics education" (again, see Allison), which, to my mind, connects with some ideas here, here, here, definitely here, and certainly here and here:

At present, what most children get from their classroom instruction on fractions is a fragmented picture of a fraction with all these different "personalities" lurking around and coming forward seemingly randomly. What a large part of this research does is to address this fragmentation by emphasizing the cognitive connections between these "personalities". It does so by helping children construct their intuitive knowledge of the different "personalities" of a fraction through the use of problems, hands-on activities, and contextual presentations.

This is a good first step, and yet, if we think through students' mathematical needs beyond grade 7, then we may come to the conclusion that establishing cognitive connections does not go far enough. What students need is an unambiguous definition of a fraction which tells them what a fraction really is. They also need to be exposed to direct, mathematical, connections between this definition and the other "personalities" of a fraction. They have to learn that mathematics is simple and understandable, in the sense that if they can hold onto one clear meaning of a fraction and can reason for themselves, then they can learn all about fractions without ever being surprised by any of these other "personalities".

Update: I promised to post this great quote from Allison on the Wu presentation/paper. It's a fantastic quote, and it's like a bottom line for people who read this post (or read this blog) and say, "Harumphh, he's so abstract":

The point is to simplify without lying, misleading, or otherwise undermining future growth, so that the presentation true for the 4th grader is still true for the 12th, but the 12th can handle more, and can see the connections to the stuff already in his mind. Mastery is then possible.

Update II: One reason I like Allison's statement so much--aside from the fact that it's pretty much spot on in interpretation--is that she uses the word lying. Exactly. I can go into detail later, but that's exactly what a lot of mathematics education is today--straight-up lying to kids' faces.

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Ordinals and Denominators

At the moment, what is fascinating about the book are the little interesting nuggets.

And, at the risk of being exposed to a collective raising of readers' eyebrows at its obviousness, I present below one of those nuggets—about fractions—that I came across tonight.

Every third lot wins, every fifth man is Chinese. It seems that this is the origin of the ordinal numbers as a means to indicate denominators of fractions: Counting 1, 2, 3, . . . , 10 to count out the tenth; all these "tenth" people or objects together form a (one, the) tenth of the whole. Thus the tenth part is in fact the last of all of them. In an obsolete terminology nine parts means 9/10, the remainder that is left if the tenth is counted out. "Decimate" originally meant counting out the tenth (to be shot).

At this point in his writing about fractions, Freudenthal is not making teaching suggestions (thankfully, given that last sentence), but simply conducting a survey—a thorough survey, it seems—of the different ways fractions (have) come up in informal contexts. Still, the paragraph above brings up an interesting (to me) perspective and mathematical connection that may be didactically useful.

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Not Choosing Sides

If I have not already forewarned readers and visitors that in the coming weeks I may annoy them with all things Freudenthal, well, let me do so now.

From Chapter 2, this paragraph gets at what is a core tension in debates about mathematics education:

In order to have some X conceived, one teaches, or tries to teach, the concept of X. In order to have numbers, groups, linear spaces, relations conceived, one instills the concepts of number, group, linear space, relation, or rather one tries to. It is quite obvious, indeed, that at the target ages where this is tried, it is not feasible. For this reason, then, one tries to materialise the bare concepts (in an "embodiment"). These concretisations, however, are usually false; they are much too rough to reflect the essentials of the concepts that are to be embodied, even if by a variety of embodiments one wishes to account for more than one facet. Their level is too low, far below that of the target concept. Didactically, it means the cart before the horse: teaching abstractions by concretising them.

Though I am extremely skeptical about Freudenthal's remedy for this tension, didactical phenomenology (more on that later), it is encouraging to see that he does not simply give up and choose a side.

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Meet Hans Freudenthal

I finally did get my book, though I must confess I haven't made it very far yet. Great first sentence, though: "Men die, systems last."

Okay, I have made it further than that. Anyway . . .

Freudenthal's thinking about mathematics education is interesting to me for two reasons: (1) it seems to deliberately avoid (or perhaps be unconscious of) what I have been thinking of lately as "didactic maternalism" and "didactic paternalism."

With apologies to women and men everywhere for such a gross generalization, I would link the former with the sanctification of error, an obsession with "behavior," and an unhealthy animosity to uncomfortable change; the latter (which will be much more familiar with "edusphere" readers) with stoic arrogance, intimidation, and an unhealthy obsession with individuality at the expense of everyone else (i.e., parochialism); (2) it seems to recognize the complexity of the subject of mathematics education, which is pretty rare on both of the made-up sides of our education debate in the U.S.

Here's a nice quote I wanted to share:

In order to write a phenomenology of mathematical structures, a knowledge of mathematics and its applications suffices; a didactical phenomenology asks in addition for a knowledge of instruction; a genetic phenomenology is a piece of psychology.

All the psychological investigations of this kind which I know about suffer from one fundamental deficiency: investigations on mathematical acquisitions (at certain ages) have involved the related mathematical structures in a naive way—that is, they lack any preceding phenomenological analysis—and as a consequence, are full of superficial and even wrong interpretations. The lack of a preceding didactical phenomenology, on the other hand, is the reason why such investigations are designed in almost all cases as isolated snapshots rather than as stages in a developmental process.

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Historical Phenomenology of Mathematics

One of these days, I'll have to pick up the book. For now, I'll write about the ideas, using incomplete information.

First, a definition from the man himself, the late Hans Freudenthal. This is his description of historical phenomenology as it concerns mathematics:

Our mathematical concepts, structures, ideas have been invented as tools to organise the phenomena of the physical, social and mental world. Phenomenology of a mathematical concept, structure, or idea means describing it in its relation to the phenomena for which it was created.

In a previous post, I shared an example of what seems to be an early conception of the median, quoted in a paper by Bakker and Gravemeijer titled An Historical Phenomenology of Mean and Median. The example comes from a book by Edward Wright, cartographer and mathematician, circa 1599:

Neither if there be disagreement betwixt observations, are they all by and by to be rejected; but as when many arrows are shot at a mark, and the mark afterwards away, he may be thought to work according to reason, who to find out the place where the mark stood, shall seek out the middle place amongst all the arrows: so amongst many different observations, the middlemost is likest to come nearest the truth.

For a more accessible example of "Wright's median," one can think of a guess-the-number contest at a fair in which participants are invited to publicly guess (write down) the number of, say, jellybeans in a jar to win a prize. Here, each guess is like an arrow, and the exact number of jellybeans like the missing bullseye. Assuming that every participant had access to the same information (they all saw the same jar of jellybeans) and that the distribution of their "errors" was relatively normal, the median (or mean) of all the guesses would likely fall close to the exact number of jellybeans in the jar:

Treynor, former editor of Financial Analysts Journal, told us that when he taught finance, he would pass a jar of beans among his students and have them guess the number. As I wrote: "The guesses would vary wildly, but always, when the number guessed in total was divided by the number of students guessing, the result was within 3% of the correct number, he said. As there were 52 of us assembled, and a bowl of peppermint candies on the table, we tried the experiment. A low guess of 32 was recorded, a high of 71. The median guess was 46, the mean was 45. The correct total was 46, a number only one of the 52 had guessed."

More later. Bye!

Reference: Bakker & Gravemeijer. 2006. An Historical Phenomenology of Mean and Median Educational Studies in Mathematics. 62: 149-168.

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The Wright Median

Historical and didactical phenomenology are both mouthfuls. Interesting, though. More on these later.

The paragraph below seems to refer to the concept of the median, and was written by Edward Wright, cartographer and mathematician, in about 1599:

Neither if there be disagreement betwixt observations, are they all by and by to be rejected; but as when many arrows are shot at a mark, and the mark afterwards away, he may be thought to work according to reason, who to find out the place where the mark stood, shall seek out the middle place amongst all the arrows: so amongst many different observations, the middlemost is likest to come nearest the truth. (Eisenhart, 1974, p. 52, spelling modernized)

Reference: Eisenhart, C.: 1974, 'The development of the concept of the best mean of a set of measurements from antiquity to the present day', 1971 ASA Presidential Address. Unpublished manuscript.

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A Mathematical Magic Trick

I will soon get back into some serious writing (I think). In the meantime, learn this card trick and amaze your friends/students/children.

STEP 1: Shuffle a full deck of 52 cards (no Jokers). As you shuffle, ask your volunteer how many cards are in a full deck (52). Then ask, "What's half of 52?" (26)

STEP 2: Hold the deck in your hands with all cards face up. Deal 26 cards face up, one on top of the other. You can count while you do this, or you can have your volunteer count. The most important part of this step, however, is to remember the seventh card (face/number and suit). When you finish this step, you should have two sets of cards—the cards that are in your hand and the face-up pile that contains 26 cards, including that key card (the seventh card you dealt).

STEP 3: Without putting down the cards that are in your hand, take the face-up pile, turn it over so that all the cards are facedown, and set it aside. Turn the cards in your hand over so that they are all facedown in your hand.

You'll probably want to explain the next step to your volunteer before proceeding. However, such an explanation is not essential. What you will be doing is creating three "columns" of cards. Each column will have a face-up card at the top and anywhere from zero to nine facedown cards under it. Face cards are worth 10, Aces are worth 1, and the number cards are worth their face value (2s are each worth 2, 6s are each worth 6, etc.). For each column, you will be "making tens" using this equation:

Face-up Card Value + Number of Facedown Cards = 10
STEP 4: Deal the top card in your hand face up. The value of this card is the first addend in the equation above. This value will determine the number of facedown cards you deal beneath it. For example, if you deal an 8 face up, then you will deal 2 facedown cards beneath it (8 + 2 = 10). If you deal an Ace face up, then you will deal 9 facedown cards beneath it (1 + 9 = 10). And if you deal a 10 or a face card face up, then you will not deal any facedown cards beneath it (10 + 0 = 10). The image below shows an example of what your first column might look like. Note that it is not important to keep the facedown cards separate, but it is important to keep the face-up card visible.


STEP 5: Deal the next card in your hand face up to create a second column. Again, use the equation above to determine the number of facedown cards to deal beneath it. Repeat this process to create the third column. The image below shows what your three columns could look like after you've finished. To create the columns shown below, I dealt an 8 face up to start the first column, then 2 facedown cards beneath it, a King face up for the second column, an Ace face up to start the third column, then 9 facedown cards beneath it.


STEP 6: Take the remaining cards in your hand and place them facedown on top of the pile that you set aside in Step 3. Pick up this pile and keep it facedown in your hand. Ask your volunteer to add up the face-up values in your three columns. In the example above, the sum would be 19 (8 + Ace + King = 8 + 1 + 10 = 19).

STEP 7: Deal the cards in your hand facedown, one on top of the other, and count the cards until you reach the sum found in Step 6. The last card you deal is that magic seventh card that you remembered way back in Step 2. In the example above, the 19th card would be the card that you remembered. Announce to your volunteer what this card is before turning it over.

I'll let you enjoy explaining the math behind this trick. And bonus points for finding the big flaw in this trick. And bonus bonus points if you can figure out a way to overcome said flaw.

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Place Value and the Alphabet

I've been giving myself little writing projects lately—trying to just write what I want without thinking too hard about it. Here's an intro to place value:

Take a look at the sentence below. It is a very famous sentence in the English language:

"The quick brown fox jumps over the lazy dog."
This sentence is famous because it has all 26 letters of the alphabet in it. Yep, that's right. Every letter from A to Z is in that sentence at least once. Can you find all 26 letters of the alphabet in the sentence?

Of course, some of the letters in the sentence repeat. There are 2 T's, 2 H's, 3 E's, 2 U's, 2 R's, and 4 O's. But every letter of the alphabet is in that sentence at least once.

Now think about all the words you know. You might not think you know very many words. But actually, if you're reading this right now, you probably know thousands of different words! And to make all of those thousands of different words, you only need to use 26 different letters—the 26 letters of the alphabet!

Don't believe me? Well, just think of all the words you could make with the letters E, T, A, and B. Here are some of the words I can make using just those 4 letters:

A BE AT BAT TAB TEA BEE TEE
EAT ATE BET BEAT BEET EBB ABET
That's 15 words. And to make those 15 words I only needed 4 different letters. Think of how many words someone could make from 26 different letters. Well, it's a lot!

We can make just about all of the words in the English language using only the 26 letters of the alphabet. In mathematics, we can make just about all of the numbers using only 10 special numbers called digits. Here are the 10 digits we use in mathematics:

0 1 2 3 4 5 6 7 8 9
Using just those 10 digits, I can make all of these numbers:

4 152 99 3,678 8 521 2,222
215 654,301 70,953 540 123,456
And I could make a lot more! How many different numbers can you make in, say, 30 seconds?

Okay, so here's the good part. Take a look at the 3 words below. These are 3 of the words we made earlier:

TEA EAT ATE
Do you notice something special about these words? If you said that the words all have the same letters, but they each have a different meaning, you would be so right! All the words have an A, an E, and a T. But in each word the letters are in a different order. And each word has a different meaning.

Now, take a look at the numbers I made above. There are 3 of them that look alike. Here they are:

152 521 215
Do you notice something special about those numbers? The numbers all have the same digits, but they each have a different meaning. All the numbers have a 1, a 2, and a 5. But in each number the digits are in a different order. And each number has a different meaning.

When you don't know the meaning of a word, like ebb or abet, you can ask your mom or dad or your teacher what the word means. Or, even better, you can look up the word in a dictionary to help you figure out what the word means.

In mathematics, we use a special system to tell us what numbers mean. And that system is called . . . dah duh duh DAH . . .! Place value.

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