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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.

MPGCube

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.

grid1

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:

50_times

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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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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New Florida Standards

It looks like we're seeing the first effects of the new Focal Points. Check out Florida's new standards:

  • The number of math concepts for grades K–8 has been significantly reduced. The purpose behind this is to allow K–8 teachers more time to teach mathematical concepts in depth and to mastery rather than quickly "covering" concepts. With the previous Grade Level Expectations (GLE), K–8 math teachers had 2 to 3 days per GLE. With the new math content standards, K–8 teachers will have 10 to 14 days per benchmark.
  • The extended amount of time with the new math content standards will allow teachers to help students move from concrete understanding, to several different models (with diagrams and manipulatives), to abstract representations with the new benchmarks. The intent is NOT to have students just do MORE practice of the same skills, but to have time to really explore the same skill in different contexts and to increase rigor over time to gain depth and mastery of the smaller number of math concepts in their grade level.
  • The K–8 benchmarks are built around 3 big ideas and 3 to 5 supporting ideas at each grade level. These provide a framework that allows the instructor to keep coming back to over-arching concepts that the benchmarks will help students see a consistent, unifying theme in their learning for mathematics at a specific grade-level. All benchmarks are of equal importance.
  • The power of the new K–8 math content standards is to allow teachers in these grade levels to FOCUS on the smaller number of mathematical concepts and teach them to mastery. Instructional materials should reflect this focus and not include material that is outside the scope of the concepts for a particular grade level. Although students are expected to build on math skills that were mastered in early grades, the new math materials are not intended to repeat full instruction of an early math concept (for example, adding and subtracting fractions, a Big Idea in Grade 5) all over again in the next higher grade (Grade 6, for example, is where students focus on multiplying and dividing fractions).
  • The math benchmarks in Grade 8 are intended to be a strong focus on pre-Algebra materials that will prepare 8th grade students who are not taking Algebra I to be fully prepared to successfully complete Algebra I in high school.


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Area of a Trapezoid

The formula for the area of a trapezoid is breathtakingly easy to understand when we slice and dice a picture to show why the formula works.

But most basal texts, when presenting this topic—and a number of other geometry topics involving formulas—use just one picture or maybe just a few different pictures to get the idea across. And doing so leaves open the question of whether the presentation indeed covers all examples of trapezoids or cylinders or parallelograms or what have you.

This is, in some sense, understandable. There is a limited amount of real estate in any basal text, and a more general (more accurate) explanation may make the topic much more difficult to understand. If basals try at all to explore an "all cases" type of explanation, they tend to do so with hands-on activities—a strategy that can be useful (especially with the presentation below) but is more often just an expansion of the number of cases that can be explored.

Let's take a crack at explaining—to adults like you—the formula for the area of all trapezoids: 1/2 h(b1 + b2), and maybe someone out there can improve upon it and/or develop a nifty hands-on complement to it.

We'll use the most typical definition for trapezoid: A quadrilateral that has exactly one pair of parallel sides. You can see the parallel sides and the non-parallel sides on the trapezoid below.

trap1

There are three important parts of a trapezoid: the bottom base, labeled b1, the top base, labeled b2, and the height, labeled h. You can see all of these in the figure.

The two dotted line segments—which include the non-parallel sides of the trapezoid—are fixed to the bottom horizontal line. You can imagine that both of these line segments can slide left or right and/or rotate on the points shown, stretching or shrinking the top and/or bottom of the trapezoid as they move and changing the angle(s) of one or both of the non-parallel sides. For a little warm-up you can mentally play with this idea in the figure above, sliding and swiveling the non-parallel sides. We don't mess with the parallel sides, though, because "exactly one pair of parallel sides" is essential for "trapezoid-ness." If you like, you can imagine rotating the parallel lines together, so they are not horizontal. As long as they are parallel, this won't affect our explanation.

Let's draw a line inside the trapezoid above that will separate it into two triangles.

trap2

You can see that both of the triangles have the same perpendicular height, h. What's interesting is that one of the triangles has b1 as its base, and the other has b2 as its base. And this is key: We want to show that ANY trapezoid can be separated into two triangles—one with a base of b1, and the other with a base of b2--that both have the same height.

Okay, so let's swivel the dotted line on the right outward and leave the dotted line on the left alone.

trap3

By doing this, we stretch just the top of the trapezoid (b2). We also stretch that red line that divides the figure into two triangles. Yet, as you can see, we still have two triangles with the same height, and one triangle has a base of b1, while the other has a base of b2.

So far, so good. Now let's swivel and slide both of the dotted lines outward.

trap4

Here we've stretched the top and bottom of the trapezoid along with the red dividing line, but we still have two triangles with the same height, one with a base of b1 and the other with a base of b2.

It is possible to swivel and slide our way (inward) into a triangle, a figure that obviously does not meet our definition of trapezoid. Similarly, when we swivel both dotted lines out far enough, we create a parallelogram (in this case, a rectangle), which also does not meet our definition of trapezoid, because then the figure would have two pairs of parallel sides. (The formula still works for parallelograms. Since the two bases are congruent, the formula becomes 1/2h(2b), which simplifies to bh.)

trap5

And, in fact, we can stop there—we have shown (with the help of your imagination) that every possible trapezoid can be divided into two triangles with the same height, one with a base of b1 and the other with a base of b2.

But, you might ask, what about continuing the outward swiveling of our dotted lines? We can continue to rotate them outward, creating different trapezoids.

The reason we don't need to deal with these examples is because they are all inversions—basically, upside-down cases—of all the possibilities we have already seen. The image below is, for example, an inversion of the second image I showed above.

trap6

And, of course, as I mentioned, it doesn't matter how we rotate our parallel lines. The same explanation applies.

So, every trapezoid can be separated into two triangles with the same height—one with a base of b1 and the other with a base of b2.

To find the area of a trapezoid, then, we add the areas of the triangles. The area of one of the triangles is 1/2b1h, and the area of the other is 1/2b2h. If we add these together, we get 1/2b1h + 1/2b2h.

And a little fancy Distributive Property pencilwork reveals 1/2h(b1 + b2).

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Analogies Are Like Boxes of Chocolates

I mentioned here that I would come back to analogies, which I talked about here and here.

The reason for my revisiting this idea is to clarify to some extent where I'm coming from when I write something like this (the analogy idea is highlighted):

Hiding all the "mathy" stuff might very well be described as the central principle of modern elementary mathematics education, and those who promote mathless math education—consciously or unconsciously—have actually managed to convince people that they are the ones best concerned for the future of our children's brains. It's insane from any angle.

Operating inside this "frame" inevitably leads one to hold secretly fast to the assumption that students already know everything—that it is impossible to give them new concepts, new ways of thinking, and new knowledge. Instead we must "connect" everything we teach them with something we presume they already know, or with a way of thinking we presume they have.

For example, after reading the above and other examples like it, it may seem like I would disagree with the proudly un-pseudonymous Michael Paul Goldenberg when he writes, approvingly (emphasis is mine):

In their book MAKING SENSE: Teaching and Learning Mathematics With Understanding, Hiebert, et al., propose three features for appropriate mathematical tasks: 1) the tasks make the subject problematic for students; 2) the tasks connect with where the students are; and 3) the tasks engage students in thinking about important mathematics.

Or with Jenny D., who responds here to some criticism of a fifth grade math lesson presented by her dean (again, emphasis is mine):

As adults who are math fluent, we might not be able to know what it's like NOT to know math, and what it would take to get someone to learn math.

This is why expert mathematicians might not be the best teachers of fifth graders. It's why great athletes are not always great coaches. Because taking that expertise and turning it into building blocks for teaching is sometimes difficult for some who [have] achieved mastery and beyond.

But, in fact, I don't disagree with any of these thoughts.

It is obviously extremely important and useful for teachers and for curricula to try to "connect with where the students are" and to turn expertise "into building blocks for teaching." A sad sort of hilarity ensues, however, when "connection" and "building blocks" consume all thinking about mathematics education—as they have—to the extent that we completely disregard both the need for students to have a deep understanding of mathematics and the need for students to gain formal knowledge in the discipline of mathematics.

One can see this, I think, when one takes a look at just about any topic in elementary mathematics, but let's take a look again at division.

The need for "connection" to students' everyday thinking is so powerful in mathematics education in the United States that division as a topic is introduced and discussed almost exclusively as a process of "separating [dividend] objects into [divisor] equal groups and then counting the number in 1 group." We choose this perspective because students can almost instantly relate to dividing "things" into equal groups, and, more generally, they can relate to doing something to a group of objects in order to create a result.

This perspective is helpful, in my opinion, when explaining the long division algorithm, but relying on it solely may present serious problems.

First, it is much more difficult, under this perspective, to visualize the relationship between multiplication and division. We typically present multiplication as "starting with an amount and replicating that amount a certain number of times"—all activity, occurring from left to right. Then we talk about division in a separate perspective, still occurring from left to right. Each operation has a different "starting" point and a different "ending" point. Accessing one puts a student on a certain path that has nothing to do with the other.

Second, when students encounter division with fractions (say, 8 ÷ 1/2), they naturally become confused when they think, "Separate 8 objects into 1/2 group and then count the number in 1 group." What the hell does that mean? And when they see the solution, it is close to unbelievable—how can one divide a group of objects into equal groups, count the number in 1 group, and come up with an answer greater than the original number? We train students to think about division as a process whereby a group is shattered into so many equal groups. We shouldn't be surprised when they have difficulties with fraction division.

But what if we introduced and modeled the meaning of multiplication and division as follows, assuming an understanding that multiplication and division deal with equal groups:

Multiplication: If there are [first factor] in 1 group, how many are there in [second factor] groups? Division: If there are [dividend] in [divisor] groups, how many are there in 1 group?

The models I presented here can then take on several different meanings that (a) relate multiplication and division and (b) make fraction division easier to grasp.


6 ÷ 3: "If there are 6 in 3 groups, how many are there in 1 group?" This is the same as 6 × 1/3: "If there are 6 in 1 group, how many are there in 1/3 group?"

2 ÷ 1/3: "If there are 2 in 1/3 group, how many are there in 1 group?" This is the same as 2 × 3: "If there are 2 in 1 group, how many are there in 3 groups?"


8 ÷ 1/2: "If there are 8 in 1/2 group, how many are there in 1 group?" This is the same as 8 × 2: "If there are 8 in 1 group, how many are there in 2 groups?"

16 ÷ 2: "If there are 16 in 2 groups, how many are there in 1 group?" This is the same as 16 × 1/2: "If there are 16 in 1 group, how many are there in 1/2 group?"

I for one would be willing to sacrifice a little more time in the early stages presenting this understanding of multiplication and division if I knew that it would make more difficult topics easier for students down the road.

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Translating Division

Despite appearances, I am not opposed to analogies, so long as they are "narrowly tailored to further a compelling interest."

I'll come back to that in a future post. I begin with the idea here because what I thought was a neat little analogy about division popped into my head recently. It's not really an analogy as much as it is a "translation" of the basic idea of elementary division. In essence, every division problem that students run across in the elementary grades can be translated as . . .

If there are [dividend] in [divisor] groups, how many are there in 1 group?

Now, obviously, for purists like myself, there are gigantic problems with that question. First, it's not always the case that division problems "ask" for the number in 1 group. Sometimes they ask for the number of groups, given that an amount is divided into groups of equal size (and, perhaps "each" should be used instead of 1). Second, the question mysteriously leaves out the very important word equal. Amounts are divided into so many equal groups.

But, if we put the first objection aside for the moment and assume that we all know we're talking about equal groups (another idea I'll come back to), we can see some advantages of using this "translation."

First up: whole numbers. Consider the division problem 6 ÷ 3 = ? This can translate to "If there are 6 in 3 groups, how many are there in 1 group?":

div1

(Update: The image above should be separated into 3 separate parts, not connected. Apologies.) Certainly one advantage of this is that it avoids cinemathematics--the dangerous tendency to present mathematical operations as "things that happen." When presented "statically," not only is the problem more accurate with regard to what division really is, it is also easier for students to see how division and multiplication are related.

Next up: fractions. This is where you'll see why I left off the word "equal" in the translation. Consider the division problem 8 ÷ 1/2 = ? Translation: If there are 8 in 1/2 of a group, how many are there in 1 group?:

div2

With fractions, one might also ask the question this way: "If there are 8 in one half of a group, how many are there in two halves?" The translation here makes the idea of multiplying by the reciprocal seem obvious—or at least, more obvious—doesn't it?

What about something like 3/8 ÷ 1/8 = ? Translation: If there are 3/8 in one eighth of a group, how many are there in eight eighths? Easy, right? At the very least, the translation makes it easier to see that one must multiply to find the answer. And, of course, that makes it easier to see that the quotient will be greater than the dividend.

I thought it was kind of nifty, anyway. I'll have more to say about it in another post.

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Decimal Multiplication

A number of basal math texts I have seen explain how to place the decimal point in a product but fail to explain why it is done the way it is.

The following is a typical presentation:

decmult

This is usually accompanied by a brief paragraph explaining that students should ignore the decimal points and multiply decimals in the same way they multiply whole numbers. The sum of the decimal places (the places shown after the decimal point) in the factors is the number of decimal places in the product.

In the example above, the product of the whole numbers 86 and 115 is 9,890. Since each factor has 2 decimal places, the product will have 4 decimal places, yielding 0.9890.

The explanation as to why is remarkably simple. First, write the decimals as fractions and/or mixed numbers. Then multiply:

fracmult

Multiplying the numerators gives us the product of the whole numbers, and multiplying the denominators gives us the number of decimal places (a decimal to tenths goes to one decimal place, a decimal to ten thousandths goes to four decimal places, etc.).

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Hands-On, Brains-Off

Methodologies and philosophies that facilitate classroom management are almost always preferred over those that facilitate actual learning.

That's not to say that classroom management and learning can't coexist, nor is it to say that most practitioners don't look to improve learning in their students. However, given the choice between a programmatic methodology or philosophy and a pedagogical one, educators seem almost always to be more attracted to the former. The environments in which they work make it nearly impossible not to have such preferences.

One such programmatic methodology—although it has pedagogical potential—is the nefarious hands-on activity, which is so beloved by elementary math teachers that entire programs can be rejected for not including it as a feature. Is it because educators see hands-on activities as vital to the academic success of students? Nope:

Patricia S. Moyer-Packenham, a researcher from George Mason University, in Fairfax, Va., interviewed and observed 10 middle-grades teachers using manipulatives to teach math. In a paper published in 2001, she noted that many of the teachers saw the classroom toys as a "fun" reward for students, rather than as a way to enhance their learning.

This is a quite shocking, though not entirely unexpected, admission from even such a small sample of teachers. But it is consistent with my experience:

Most hands-on activities aren't written to help kids "move from the concrete to the abstract." They are written to keep kids firmly in the concrete, playing with fraction strips; counters; ones, tens, and hundreds blocks; etc. . . Hands-on activities are preferred because they are seen as effective classroom management tools.

In this article from Education Week, which I also linked to above, we find research that calls into serious question the current use of manipulatives in mathematics classrooms:

Researchers found that children taught to do two-digit subtraction by the traditional written method performed just as well as children who used a commercially available set of manipulatives made up of individual blocks that could be interlocked to form units of 10.

Later on, though, the children who used the toys had trouble transferring their knowledge to paper-and-pencil representations. Mr. Uttal and his colleagues also found that the hands-on lessons took three times as long as the traditional teaching methods.

Three times as long to teach kids to have a somewhat inflexible knowledge of two-digit subtraction. Certainly not a glowing endorsement of hands-on activities. Yet, the problem isn't in the manipulatives themselves (though that can't be discounted when you're talking about mathematics); it's in how they are used in instruction:

One problem is that children . . . sometimes fail to grasp the symbolic value of the objects they're using, according to a panel of experts who presented research on the topic during a national meeting of the Society for Research in Child Development held in Boston last month.

Students might correctly perform the classroom procedure, connecting 10 blocks here, for instance, or taking away blocks from another pile, without thinking about what the objects are meant to represent. Younger children, in particular, also can get lost in play with the toys or become distracted by superficial features of the toys, such as realistic details or bright colors, that have nothing to do with the academic concept being taught.

And textbook publishers, though they spend a lot of time washing their hands of what actually goes on in classrooms, also share some of the blame for this.

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The Point

The responses I received to my vocabulary comment were not contradicted by the research I presented . In fact, they could be supported by it.

And there are other reasons to call out vocabulary in a text. Here's Robert with one of them:

If my calculus students encountered a problem involving relative maximum values of a function, the first thing they need is to be fluent with the meaning of the term "relative maximum." If they're not, then they need to go look it up--and the cues show them where to look (by flagging the point at which the term is first introduced).

I agree that highlighting important vocabulary words helps signal their importance and can help with navigating a search for their meaning. But, to me, the idea should be to also help students remember the meaning of the words. In truth, publishers don't highlight vocabulary words because doing so facilitates recall of those words or because it signals their importance or because it helps with textbook navigation. They highlight vocabulary words because it signals to customers that the publisher cares about vocabulary. The point is that if we truly cared about learning, we could do more with text to facilitate learning.

Consider the following series of experiments conducted by Bruce Britton (whose work we have looked at before) and Robert Sorrells out of the University of Georgia. They began with questions that are rarely, if ever, asked in educational publishing:

Our development of the method began when we encountered Colomb and Williams' (1987) claim that all expository texts have a point, which is a central tenet around which the text is organized. From this idea, our three hypotheses emerged. First, do expert readers generally agree about the point of a text, whereas less skilled readers differ in their judgments of the point? Second, do readers who have the point of a text signaled to them recall more of that text than readers who do not have the point signaled? Finally, do readers who have the point signaled to them construct a different structure of the text, incorporating more of the important information than those who do not? If these hypotheses are supported, they suggest that identifying the point of a text and informing readers that it is the point would improve the learnability of that text.


Experiment 1

To answer the first question, the researchers assembled five faculty with Ph.D.s in English as the "expert readers" and 12 introductory psychology students as the "less skilled readers." Each individual was given three brief expository texts, without titles, and asked to simply underline the point of each after a careful read. The texts were provided by Colomb, who identified the point of each to the researchers (point units) along with the "themes related to the point." The "themes" (theme units) were ideas that were not as informative as the point, but were more important than other ideas (other idea units) in the texts. Each text was a different length, and "the point" in each text occurred in a different location.

The researchers found that the "expert readers" identified 86.4% of the points, whereas the "less skilled readers" succeeded only in identifying 49.5% of the points, a significant difference (p < 0.05). Less skilled readers also showed greater variance in their individual results.

The importance of this result is that it points to the possibility of experts signaling the point for less skilled readers. The next two experiments focus on whether or not such signaling can improve recall and/or comprehension. In these experiments, Britton and Sorrells set out to test this general hypothesis:

If the point of a text telegraphs the information in that text, as Colomb and Williams suggested, then cuing the point should affect the recall of other information.


Experiment 2

This experiment was conducted to answer the second and third of the researchers' specific questions:

Do readers who have the point of a text signaled to them recall more of that text than readers who do not have the point signaled? [And], do readers who have the point signaled to them construct a different structure of the text, incorporating more of the important information than those who do not?

Each of twenty-nine introductory psychology students was given the three texts used in Experiment 1. (Recall that each text was broken down by the researchers into "point units," "theme units," and "other idea units.") Each participant was asked to read the text and write down what he or she remembered (free recall). For participants in the experimental group, the point of each text was underlined, and written instructions identified the underlined sentence as the point of the text. Participants in the control group received the texts without modification of any kind. Reading time and recall time were measured.

[Note: There is no indication in the write-up as to whether participants were asked to read all three texts before recall or whether they were asked to recall after each text. However, in this experiment, the mean reading times ranged from 7.77 min to 9.08 min and the mean recall times ranged from 10.53 min to 12.04 min. The longest text in the experiment was 559 words. It seems unlikely that it would take about 8 to 9 minutes for college students to read such a brief passage and then spend 11 to 12 minutes writing down everything they could remember about it. But, hey, I've heard crazier things.]

Not only was overall recall significantly improved in the experimental condition--in which the point was underlined--but recall in every subgroup—point units, theme units, and idea units—was significantly improved:

First, the experimental group recalled significantly more of the points than the control group . . . Second, a subset of data was analyzed for differences between the groups for recall of the important themes (as identified by Colomb). The experimental group also recalled significantly more of these theme-related units than the control group . . . Also, the differences between the groups in recall for the unsignaled content (the themes and the other units) were assessed. The experimental group recalled more of these nonpoint units than the control group. [(p < 0.05)]

Importantly, no significant differences were found between the experimental and control groups for either reading time or recall time, which indicates that the underlining and extra instructions for the experimental group did not cause that group to spend more time processing the text or to take the task more seriously.

The results from the first two experiments suggest that not only is identifying the point possible, it is also beneficial for recall.

Experiment 3

This experiment replicated Experiment 2 and added another condition in which a random sentence was underlined and designated as the point. If the mere underlining of a sentence was responsible for the effects found in Experiment 2, then the random underlining condition would exhibit recall patterns similar to those found for the experimental group in Experiment 2. However, if the pattern of recall for the random group was similar to that for the control group of Experiment 2, then evidence would be provided that it is the semantic importance or informative content of the point that, when signaled, facilitated recall.

This experiment, conducted on forty-one introductory psychology students, found the same results as those in Experiment 2, while also finding no difference between the control group and the random group, indicating that signaling the actual point—not just signaling a random sentence and calling it the point—was responsible for the improved recall. Furthermore,

It is also interesting to note that in Experiment 3, the control group recalled more of the actual points, as well as more of the related themes than the random group (although these differences were not significant). If this particular finding is replicated, then an argument could be made that signaling random content can be detrimental to recall.

Of course, these experiments were conducted on college students, and, while the texts were expository, none of them were about mathematics. Still, the results suggest that, should we take seriously the idea of vocabulary instruction in mathematics (among other things), we should make new vocabulary--the words and their definitions--the explicit "point" of instruction and signal this point to students. These results also suggest that authors should rethink how they present content in textbooks. Rather than finding ever more clever ways to distract and "engage" students, they could spend some thought on what the "point" of the instruction is and build from there.


Reference:Learning from Text across Conceptual Domains. Contributors: Cynthia R. Hynd - editor. Publisher: Lawrence Erlbaum Associates. Mahwah, NJ (1998)

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Getting to the Point

I can remember a meeting back in '01 or '02 in which I and other editors had the opportunity to comment on prototype pages for a basal revision.

I remember the meeting, because my comment got a very icy, if not downright hostile, reaction. The pages we were looking at were from a lesson opener, and they had a number of new vocabulary words, all boldfaced and highlighted in yellow. This was my comment:

I don't understand why we boldface and highlight vocabulary words. What purpose does that serve? Kids don't care. It's just a distraction.

The explanation I was given was that teachers liked the highlighting. Sigh.

I was also told that the highlighting at least helped kids remember what the important vocabulary of the lesson was, which is absolutely hilarious, given that—wait, where did I put that . . . is it . . . over . . . no . . . come on . . . ah! Here we go:

The literature supports the idea that typographical cues, such as underlining, facilitate memory for a text, but for the cued text only (Lorch, 1989). These studies manipulate location of, amount of, and type of typographical cue (Hartley, Bartlett, & Branthwaite, 1980; Lorch, Lorch, & Klusewitz, 1995), as well as subject underlining (Johnson, 1988). These manipulations generally produced better memory for cued information, whereas uncued information was unaffected.

More later.


Reference:Learning from Text across Conceptual Domains. Contributors: Cynthia R. Hynd - editor. Publisher: Lawrence Erlbaum Associates. Mahwah, NJ (1998), p. 96.

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Repost: The Text Effect

Teacher guides are almost always written by teachers for teachers, so they provide a bit of insight as to what might be going on in classrooms.

Here is the first part of some teacher guide material I looked at today (from the very old Math Steps program:

Learning to use mathematical terms precisely is an important part of learning mathematics. Doing so, however, requires a full understanding of the sometimes complex ideas behind these terms. Most students only achieve this precision over an extended period of time. Leading questions and frequent but gentle reminders are probably the best approach.

Teacher: I plant a garden that is 9 yards in length and 5 yards in width. What is the area of my garden? Student: 45. Teacher: 45 what? Student: I don't understand. Teacher: Let's suppose you're asking me a question. You ask me how large my garden is and I say "45." Would you know whether I had a large garden or a small garden? Student: No, I wouldn't know. Teacher: Why not? What piece of information did I forget to give you? Student: Well, you didn't tell me whether you're talking about inches, feet, or whatever. Teacher: That's right. My garden is 5 yards by 9 yards. How should I describe the area of my garden? Student: You could say that the area of the garden is 45 yards. Teacher: When I picture 45 yards, I usually think of a distance. Something might be 45 yards wide or 45 yards long. But we're talking about area here. Student: Oh, yes, I guess I should say "45 square yards."

This is a bit more prescriptive than the language in other, more recent guides I am familiar with, but it contains something that all teacher guides seem to have—an assumption that all topics are difficult, require a lot of time and understanding, and cannot be approached directly.

These characteristics do not necessarily reflect a certain philosophical orientation toward education either. It is much, much more difficult (probably impossible) for a publisher to get in trouble today with customers when they treat all topics laboriously. Teachers who need that material will like it; the ones who don't will just ignore it. If, on the other hand, teacher guides were to prioritize topics in one way or another, publishers would run the risk of producing in their customers the pedagogical equivalent of the feeling many new mothers have when they read this book--my kids are all stupid, and I'm a bad teacher.

The indirect approach to teaching is certainly grounded in a philosophy, but this too may at least be supported by the practicalities of writing teacher guides. You have to have something to fill up the page, after all. No one's going to buy the teacher guide I've been shopping around:

Lesson 1: Add and Subtract Fractions
Teach them to add and subtract fractions.

Lesson 2: Add and Subtract Mixed Numbers
Teach them to add and subtract mixed numbers.

Is it possible that textbooks, as cultural tools that embody and operationalize the intangibles of education, can have an effect on our perception of and orientation to education? I think so. Isn't it possible that publishers' need to create laborious, long-winded, indirect content for teachers at least reinforces the idea among teachers that instruction should be laborious, long-winded, and indirect? Food for thought. In my view, as a medium for presenting content, text can at least narrow, channel, and sculpt practice as strongly as it can shape the ideas it presents.

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More Invisibility

At the end of my most recent post I quoted the following idea from a paper on the cognitive analysis of mathematics com