I said in this post that what I am interested in is the "text" of education itself, however that text presents itself.

And I've been thinking lately that where I need to go first in explaining this orientation toward education is deeper down into my assumptions and my experience—closer to the trunk of the tree, if you will, and further away from the various branches that I have explored.

So let me start with experience. I would characterize the bulk of my experience in education as simply writing and editing math lessons--on specific topics for specific grade levels. And to each writing and editing task, I bring four important resources: (1) my own knowledge about the lesson that I am writing, (2) a list of related lessons that have likely already been presented, (3) a list of related lessons that will likely be presented in the future, and, most importantly, (4) a memory not only for the content of each of the lessons mentioned above, but also a memory of all the different ways those lessons have been presented.

And that's pretty much it.

Now, I say "that's it," but that's a lot. For myself and for the many thousands of people who do the same kind of work that I do, believe me (or us, rather), that's a lot. I say "that's it" because this kind of work does not involve--or, rather, does not have to involve--thinking about students, which is also, to use the same phrase above, a lot.

Yet, and again in my experience, even if intelligent and/or experienced discussants were to restrict themselves to a specific lesson at a specific grade level and bring only those four resources mentioned above into a debate, one could write ahem, a lot, about--and education could gain a lot from--the ensuing conversation. (Try it yourselves. Pick a math topic and a grade level. Follow 1–4 above.)

Bollocks

What capsizes this happy floaty boat more often than not is an attitude from bad apples in all kinds of different groups in the U.S.--constructivists, traditionalists, teachers, non-teachers, administrators, etc.--those who seem to believe that personality and power can rescue and/or improve mathematics education in the United States.

Well, "believe" is probably the wrong word. These folks, by and large, just want to get paid.

But that's neither here nor there. What's important, I think, to keep in mind is that the Western approach to education is much more concerned with personalities and power centers than it is with getting anything done. Just copy everything in this and the above two paragraphs and post it on an education forum somewhere. The probability that you will encounter a reaction from some idiot who thinks she channels all the education gods and can prescribe a remedy for what ails us is pretty close to 1.

The point I'm trying to make is not that this hypothetical person doesn't have the answers (she doesn't); it's that all the answers that can be got in the West are drawn from one bag. The solutions that we consider in the U.S. all have a certain flavor to them. One would think that after 60-100+ years of comparative education failure, we would wise up and try a different flavor.

I t seems funny to me to say that I am taking a break from blogging. Funny because I don't post every day or even every other day. At worst, it's about once a month.

But that's what's going to happen. I'm going to be taking a break from blogging. It might be for the summer only; the plan right now, though, is that it will be for much longer than that.

I'm still going to be posting links and jabbering on Twitter, and I may pop in with a word or two about Mr. Dyer's upcoming post(s) or other posts if I can organize my thoughts pretty quickly, but that's about it.

The biggest reason for this break is that the majority of my time blogging is spent thinking, pacing, reading, analyzing, etc., and I want all of that time back so that I can apply it to a new project--a math book, which I've already started. The other reason I'd like to take some time off is so that I can get reacquainted with the ideas that motivated me to start this blog in the first place.

I have always seen education--in particular, K-8 mathematics education--as a very large written work (what would you expect from a textbook editor!). There are people who have created the language used to write this work (mathematicians), people who write it (educators), people who distribute it (teachers), and people who read it (students). Of course, many of these people play multiple roles, and there is seemingly an infinite variety of ways of creating, writing, distributing, and reading this work. But I'm not really all that interested in the people or the various ways they fulfill their roles in education. What I am interested in is the writing itself, in whatever form it presents itself at any given time or in any given place.

Take a look at this, from Bert at Learning and Unlearning Math (bookmark this fella), which I've edited for layout purposes only:

Problem: Simplify 3x + 5 + 7; Jessica's response: 15x. . . . What is Jessica thinking so that her response follows from that thinking? . . . . Underlying my question is an assumption that Jessica isn’t just guessing randomly, but does indeed think from some kind of model, a model that has probably worked for her in prior situations, a model that she now applies in a situation where it is insufficient.

If you remove both the "Jessica" construct and the notion that there is "thinking" involved here, then you would be pretty close to my general orientation toward education.

And why not? If, for purposes of analysis, it is useful to assume that "Jessica isn't just guessing randomly" and that her thinking proceeds "from some kind of model," then it would be reasonable to assume that the model we're talking about is what we taught Jessica in the first place. We can go on to ask, What did we teach Jessica that resulted in her error? and come up with all kinds of possible answers, but, given the above assumptions, we really don't need to speculate about Jessica or what's going on in her brain to come up with the answers that I'm interested in. We can look at the writing itself, the "text" of the teaching.

This is just one example that illustrates (not perfectly) the kind of thinking I want to get back to--the kind of "world," if you will, that I want to go back to for at least a little while. My sense, at least for now, is that after having immersed myself in that world to my satisfaction, I will want to write about it in a book, not in a blog.

But, until then, this blog has ended. Go in peace to love and serve your students.

--Josh

Time seems to move faster as we get older. But every time an experience reminds us of this, we're surprised (and probably a little disappointed) to find that it is still true.

For me, last night was a case in point: I sat down to write something about this post from Michael, imagining at the time that I was coming back to it after a month or two-month hiatus. Yet, with an admixture of surprise and disappointment, I found that nearly six months had passed since I had read it.

Of course, age isn't the only factor here. Another reason Michael's post seems much younger than six months to me is that one of the points of disagreement in the discussion was that of the value of student errors--a "theme," if you will, that can be found in one way or another almost daily in discussions about education.

Still, even though student error became a sticking point for myself and others, the main idea of the post was not the value of student error, but the value of student creativity in the mathematics classroom.

Recently, I had one of those much-desired opportunities to see a student spontaneously come up with what was, to me at least, an original approach to something that is easy for and familiar to many, but distressingly hard for a significant number of students: calculating the slope of a straight line given two points.

The "original approach" to finding slope that Michael describes in his post was the result of a collaboration between Michael and a student named Lisa and can be summarized with the following formula:


In order to see why this approach is both novel and interesting, let's briefly review the traditional, or textbook, way students are taught to find the slope of a line given two points.

The Basics

Look at the figure below. Line f is a straight line that passes through points A, B, and C. Point C is located at 3 on the vertical axis, or y-axis. Point B is at 2 on the vertical axis. So, we can describe the vertical distance between points C and B as 3 &minus 2, or 1. There is a vertical distance of 1 between points B and C.


Now let's look at horizontal distance. Point C is located at 6 on the horizontal axis, or x-axis, and point B is at 4 on the horizontal axis. So, we can describe the horizontal distance between points C and B as 6 &minus 4, or 2. There is a horizontal distance of 2 between points B and C.

The slope of line f can be described using a ratio, as shown below. The first term (top number) of the ratio is the vertical distance between any two points on the line, and the second term (bottom number) of the ratio is the horizontal distance between those points. Between points B and C, which are points on the line, there is a vertical distance of 1 and a horizontal distance of 2. So, line f has a slope of 1/2.


As I mentioned, though, the slope of a line can be found using any two points on the line. If we use points A and C instead of B and C, we get the same ratio. Look at the graph and table below.


The vertical distance between points A and C is 2 (3 &minus 1), and the horizontal distance between the points is 4 (6 &minus 2). Using points A and C, then, we would write the slope ratio as 2/4 (vertical distance over horizontal distance). And, of course, this ratio simplifies to 1/2.

To write the formula for the slope of a line given two points, we actually write two expressions--one for the vertical distance between the two points and one for the horizontal distance between them. Then we combine these two expressions together into one ratio.

To find the vertical distance between the points in our examples above, we subtracted the vertical, or y-axis, location of the first point from the vertical, or y-axis, location of the second point. So, we can write y₂ &minus y₁ to describe the vertical distance between points.

To find the horizontal distance between the points in our examples, we subtracted the horizontal, or x-axis, location of the first point from the horizontal, or x-axis, location of the second point. So we can write x₂ &minus x₁ to describe the horizontal distance between points.

When we combine these expressions together into one ratio, recalling that vertical distance (y₂ &minus y₁) is the first term and horizontal distance (x₂ &minus x₁) is the second term, we get the following:


Now, it should be noted that the order of the terms in each expression is not important, so long as they are consistent with each other. That is, the vertical distance could be described using the expression y₁ &minus y₂ instead of y₂ &minus y₁. So long as the second term of the ratio uses the same order, the ratio will accurately describe the slope of a line.


Typically, we teach students to find the slope of a line given two points in two different ways--using a graph and using ordered pairs. The difference between these two approaches has to do with how the locations of the points are presented, either as points on a line or as ordered pairs of numbers. In our example above, we found the slope of a line using points graphed on that line, but typically students would also be expected to find the slope of the line when given the ordered pairs describing the locations of any two of the points on the line. The ordered pairs for the points in the example above are as follows: A (2, 1), B (4, 2), C (6, 3).

Potential Problems

There are at least a few issues that immediately present themselves when considering the typical slope formula and the teaching that has (again, typically) occurred prior to the introduction of the formula.

First, students are taught (correctly) that, in ordered pairs--those pairs of numbers in the form (x, y) that can locate points on a coordinate grid--the x-coordinate comes first and the y-coordinate second. For slope it is, in some way, the opposite. The y-coordinates are listed on top (first), and the x-coordinates are listed on bottom (second). Second, the slope formula does not keep the ordered pairs intact, though students work with them as such. The formula strips out the y's and places them on top while it does the same for the x's on the bottom. The ordered pairs no longer exist as independent entities in the formula.

Another issue worth considering--an issue that is independent of prior teaching--is that simple tracking errors might occur when students (or anyone) attempt to substitute information written in one form (e.g., horizontally as ordered pairs) into a very different form (e.g., vertically as a ratio).

Lisa's Method

The approach that Michael dubbed "Lisa's Method" eliminates at least the first two problems listed above. (Though, as I will point out in a future post, this is not a satisfactory criterion on which we can base an evaluation of Lisa's method.) Let's briefly compare the two "approaches":


As you can see, in "Lisa's Method," for each horizontal line, the x's always come first and the y's second, and the ordered pairs remain intact: the top line shows the ordered pair of the first point, and the second line shows the ordered pair of the second point.

The last line in "Lisa's Method" is, in fact, the complete slope formula shown above on the right. Each triangle symbol can be read as "change in," and, in this case, "change in" simply refers to the distance between the points. So, the last line in "Lisa's Method" shows the distance between the y-values divided by the distance between the x-values. The fraction bar in the traditional formula is also a division symbol, so the meaning is (pretty much) the same.

Scientific Thinking

Given that background, there are an extraordinary number of questions, arguments, hypotheses, etc., that could be generated, all starting with the simple comparison between two approaches used to find the slope of a line given two points. Each of these could, in turn, lead to other questions, other hypotheses, and other areas of inquiry. For example, one might question the conceptual equivalence between the long division symbol used in Lisa's method and the fraction bar used in the formula. Do they really mean the same thing?

Let's take that up next time.

My son participated tonight in an annual fair at his school where students involved in the Quest program show off their projects.

I was flipping through some paperwork next to one of the projects--a research project about space created by a fifth grader--when I noticed this familiar picture:


Although the photo was lo-res and black-and-white, it was still nice to see, and I thought that any parents/grandparents and students who had not yet seen that particular picture might find it to be a novel take--as I did when I first saw it--on the relative sizes of our solar system planets (with the exception now of poor Pluto, of course).

The caption beneath that picture was what you might expect--here are the planets of our solar system, different sizes, etc.

On the next page of said paperwork, I saw this picture:


I couldn't find the original site where these pictures were posted if my life depended on it, but I do remember that both pictures were part of the same "project," so I wasn't surprised to see this one following the "scale picture" of the planets above.

What did surprise/horrify me was the caption under this second picture. I don't remember exactly how it read, but I do know that something like this sentence was there:

Here are some other planets we have discovered.

Now, I can certainly tolerate a student project titled "The Conspiracy Theories of John F. Kennedy." But when a student who is (a) in fifth grade, (b) in a gifted-and-talented program, and (c) given weeks to put together a project on a topic of his/her own choosing manages to confuse stars and planets, then there is really something terribly wrong somewhere.

Okay, so we're going to pick up the topic of transformations where we left it in this post.

You'll all remember from that post (nearly a year ago) that we "derived" from a few examples a simple rule we could use to determine the coordinates of the image of a point rotated 90 degrees (counterclockwise) about the origin: the x-coordinate becomes the new y-coordinate, and the negative y-coordinate becomes the new x-coordinate.


As a general rule, if we take our starting coordinates to be (x, y), then there are really only two sets of coordinates to remember for degree rotations about the origin that are multiples of 90 . . .

90-degree (counterclockwise) rotation: (^-y, x)
270-degree (counterclockwise) rotation: (y, ^-x)

. . . assuming one needs no help remembering that the coordinates of a point under a 180-degree rotation simply take opposite signs (^-x, ^-y).

For clockwise rotations, obviously we just transpose these sets of coordinates.

90-degree (clockwise) rotation: (y, ^-x)
270-degree (clockwise) rotation: (^-y, x)

45-Degree Rotations

We can use the same kind of thinking to determine a general rule for 45-degree rotations as well. But first--and again--we need to get the right perspective (or, rather, a useful perspective).

Take a look at the line graphed below (in red). The measures of the angles formed in the first and third quadrants by this line and the x-axis of the coordinate grid are each 45^o.

It is important for you to see that for each point on this red line, the x- and y-coordinates are equal. That is, the coordinates for each point on the red line can be represented as (x, x) or (y, y). This is fairly obvious, since we can see that the line passes through the points (1, 1), (2, 2), etc. But keep in mind that (0.0015, 0.0015), (4,501.2, 4,501.2), etc., are also points on the line.


Something else that is extremely important to see is that, because of the all-powerful Pythagorean equation, the diagonal (d) formed by the red line inside each grid square has a length equal to the square root of 2, which is approximately 1.414.


Okay, so that was the easy part.

As I mentioned, we will approach 45^o rotations in the same way that we approached 90^o rotations. So let's rotate the point at (3, 2) 45^o about the origin.


We take our "claw," which is connected to the point located at (3, 2) and rotate it 45^o. Then we take a look at each "arm" of the claw.

Right away, we find the situation to be a bit trickier than the one we encountered with 90^o rotations. Where are the endpoints of the rotated arms? We can see that the end of the rotated x-arm is pretty close to (2, 2), but not exactly there, and it is even harder to guess where exactly the end of the rotated y-arm is. So, just like that, we need to once again pause and enjoy a tall, frosty mug of fresh perspective.

Take a look again at the red line and consider what we know:


(1) no matter where on the line we draw a point, the x- and y-coordinates of that point will be equal, and (2) the diagonal formed inside each grid square has a length equal to the square root of 2. What (1) and (2) indicate is that there is a functional relationship between the length of any diagonal line segment and its horizontal or vertical location on the coordinate grid.

How might we describe this relationship? Here's how (at least for the first quadrant):

Once we know the length of any (45^o) diagonal line segment, in order to find its horizontal or vertical location, we can simply divide the length by the square root of 2.

You can see this in the split table below. Dividing each of the segment lengths in the first column by the square root of 2 gives the value in the last column.


So, let's look again at our claw, attached to the point at (3, 2) and rotated 45^o about the origin. We know the length of the diagonal line segment formed by this rotation. It's simply the x-arm of our claw, which has a length of 3.

Therefore, the location (both the x- and y-coordinates) of the end of the x-arm can be found by dividing the length of the arm, 3, by the square root of two:


To make the next section a little clearer, we can write the coordinates of the end of the rotated x-arm this way:


When we write the coordinates in this way, we can more easily see them as vectors--that is, they are numbers that describe not only magnitude, but direction as well. On the coordinate grid, movement to the right on the x-axis can be represented by a positive number, and movement to the left can be represented by a negative number. Similarly, on the y-axis, movement up can be represented by a positive number, and movement down can be represented by a negative number. If we begin at the origin (0, 0), then to find the location of the end of the rotated x-arm, we follow the x-axis to the right (positive) and follow the y-axis up (positive). This is why both of the coordinates that describe the location of the end of the x-arm are positive numbers.

Anyway, obviously the location of the end of the x-arm is not what we want. We want to know the location of the end of the y-arm of our "claw." This is the location of the point (3, 2) rotated 45 degrees about the origin.

As it turns out, all the hard work is behind us now, because the line that contains the rotated y-arm segment also forms a 45-degree angle with the x-axis. (There are probably dozens, if not hundreds of ways to show this, and I'll leave it up to curious readers to tackle it themselves.)

Since the rotated y-arm is also oriented at a 45-degree angle, we can treat it the same way we treated the x-arm. To make this as clear as possible, let's first detach the rotated y-arm from the rotated x-arm and make its "starting point" (0, 0):


If we take the length of this arm, 2, and divide by the square root of 2, we will come up with this location for the end of the detached y-arm:


But, of course, this isn't right, because we did not take into account direction. We still follow the y-axis up to find the end of this arm, so the y-value should be positive but, in this case, we follow the x-axis to the left to find the end of the arm. This means that the x-value should be negative:


So, take a look again at the rotation. We see that the x-arm moves up and to the right, while the y-arm moves up and to the left.


To find the location of the end of the y-arm, we use the end of the x-arm as a starting point and simply add the coordinates:


All that's left is to notice that in the numerators of the fractions above, we see the coordinates of the original point (3, 2). Play around with the grid below to see that the rule is true. The coordinates of a point (x, y) rotated 45 degrees (counterclockwise) about the origin are:



What is not immediately obvious is why this rule should hold no matter what quadrants are involved. For example, the 45-degree rotation in the second quadrant has a down y-movement, which is not reflected in the rule. It's pretty simple to figure out how this works after you play with the rotations a bit (or, of course, you can figure it out algebraically too). Enjoy.

It always makes me laugh a little when I hear or read the question, How do children learn mathematics?

Regardless whether the question is explicit or implicit, my first reaction is to ask back (usually to a piece of paper or to my computer screen), How do children learn how to drive?

Actually, it is this reaction--this thought--that makes me laugh, because what seems to be meant by the first question is, How do children naturally learn mathematics, without the aid (or with the very minimal aid) of teaching or culture or environment? So when I replace mathematics in the question with how to drive, then that question--the real question--sounds, appropriately, ridiculous: How do children learn how to drive without the aid (or with the very minimal aid) of teaching or culture or environment? They don't.

Now, obviously there are a thousand points one could quibble with here. One might argue that I have misrepresented the "real" question--that teaching or environment or culture is taken into account but what is truly sought after is how those factors fit with students' potential (or proclivities or "reasoning," whatever). Or one might laugh back and call my comparison ridiculous because learning mathematics is simply nothing like learning how to drive. Or one might argue that transplanting the phrase how to drive into the first question doesn't make it ridiculous at all--that there is, indeed, a set of skills/abilities/proclivities/etc. that makes us naturally capable of driving. Et cetera, et cetera.

But even though those considerations crowd my brain, I still laugh whenever I hear that question: How do children learn mathematics?

Maybe it's just a dumb question. I mean, really, what could we do fundamentally differently in education if we ever got an answer to that question--the real question?

Okay, so back to soup. Did you read the recipe I included in my previous post? If so, what did you think about it?

For point-making purposes, it is my sincere hope that your answers to those questions are something close to no and nothing. But, regardless, my assumption (which I think is a safe one) is that, of the percent of people who might bother to read every word of the recipe, a relative few, not including myself, would have any kind of comment on it, per se--especially given such an open-ended question as the one above.

The Setup

However, and again for point-making purposes, it's worth examining this "content" more closely, especially in light of some principles that will be familiar to readers of this blog:


Let's start by very briefly contrasting the original recipe (the "original" that I followed for cooking anyway) with the recipe that you read, which was a modification of the original. (We'll pursue this analysis in more detail later.) I made only one substantive change to the original recipe. By taking the ingredients (and preparation suggestions [e.g., "diced," "sliced"])--which were tidily chunked together under one section in the original--and spreading these out among the cooking instructions, I changed the recipe's coherence, or, more specifically, the "packaging" of the content--how related material "hangs" together (and, of course, how unrelated material doesn't):


It is important to note here that I did not destroy the original recipe's coherence, nor did I create a coherence where none existed previously. I changed how the content was packaged, giving the rewritten recipe a different kind of coherence. The original recipe contains two larger bundles in which the content objects are related by a certain characteristic--ingredients/preparation or cooking instructions. The rewritten recipe has seven smaller bundles in which the content objects are related by point of use. One can make various arguments in support of either type of coherence, drawing on the other standards (order, clarity, and precision) or on relevant ideas outside of the principles.

It is also important to note that although the recipes above are written down, each can represent both spoken and written instructional content. That is, each recipe can be a representation of a lesson that is taught to students.

The Question

So here's the million dollar question--and the pivot to math education: Is it reasonable to hypothesize that the difference in coherence between the two lessons would effect some kind of difference in students' learning?

The "Answer"

You can guess what my answer to that question would be. But what does research in mathematics education have to say about a question like this?

Flaw #1: Research in mathematics education devotes relatively little time and effort to the mechanics of school mathematics.

To illustrate education research's relative silence on the subject of 'pedagogical mechanics,' so to speak, I report to you first about a recent visit I made to ERIC--Education Resources Information Center--which is billed (by Google at least) as "the world's largest digital library of education literature." I conducted two searches. For the first search, I entered--in quotation marks--the phrase teacher salaries. Result: Your search found 4945 results. (It was 6,011 without the quotation marks.) For the second search, I entered the phrase fractions--a topic which is correctly described by the first sentence of the second abstract that was actually pulled in my search as "one of the most problematic areas in primary school mathematics." Result: Your search found 2115 results.

Then I tried a favorite of mine, ScienceDirect. I came up with 822 results for the search term "teacher salaries." For "teaching fractions," 34.

Of course, this is a bit of cherry picking to try to illustrate a point. One might likely easily find examples of searches that would suggest a very different conclusion. Still, the relative lack of attention given to issues "inside education" on the part of the mathematics education research community is real and has not gone unnoticed even by members of that very community:

The panorama of work represented at professional education meetings or in publications is vast and not highly defined. . . Research that is ostensibly "in education" frequently focuses not inside the dynamics of education but on phenomena related to education—racial identity, for example, young children's conceptions of fairness, or the history of the rise of secondary schools [Josh: Or "teacher salaries"]. These topics and others like them are important. Research that focuses on them, however, often does not probe inside the educational process. (emphasis mine)

There can be legitimate disagreement as to where exactly "inside the educational process" is (or what "pedagogical mechanics" are)--and, of course, there will almost certainly be a sometimes dissonant, sometimes harmonious overlap of various personal interests within the study of "the educational process." (My own [and others'] area of interest, for example, is content--the "text" of teaching--from whence my cooking example comes.) But it should be clear that, for example, investigating different ways of teaching long division is of far more direct utility to education than is investigating, say, how children "reason"--especially when those who purport to study "reasoning" refer to it as "something one feels interacting with people" (like nausea?).

The Consequence

Flaw #2: When mechanics are not considered in education research, they can become confounds.

In research, a confound is something that a researcher fails to look at, or doesn't "control for," in an experiment but which could very well explain his or her results.

Suppose, for example, that you study the produce departments of two different grocery stores in the same city. You rate both departments' cleanliness, the friendliness of their employees, the variety they each offer, and how much (in dollars) each store sells in produce. After your experiment ends and you analyze these data, you see that Store A sold much more produce than Store B did. You also observe that Store A scored significantly higher in cleanliness than did Store B, but that each store had about the same rating for friendliness and variety. You conclude that the cleanliness of a store's produce department matters more than friendliness and variety in its ability to make money in the short term.

Now suppose that you collected data from both stores for exactly four hours on the same (non-holiday) Friday, but that for Store A, data were collected between 2:00 p.m. and 6:00 p.m. and, for Store B, data were collected from 7:00 p.m. to 11:00 p.m. In that case, time of day is certainly a confound, because it could explain not only the superior sales of Store A (most grocery stores are much more likely to be busier on a [non-holiday] Friday between 2:00 p.m. and 6:00 p.m. than they are between 7:00 p.m. and 11:00 p.m.), but it could also explain the difference in cleanliness between the two stores. Just considering the possibility of a confound is enough to make your result go down in flames.

How about our recipes? Suppose we want to know whether or not it makes a significant difference in learning to have a teacher with 8-10 years versus 1-4 years of cooking instruction under her/his belt. We scour the literature on the subject and conclude, perhaps, that it doesn't matter. We look at 32 studies (all focused on the issue of teacher competence [operationalized as 'years experience']), and eighteen of those studies tell us that it does matter one way or the other and the other 14 say the opposite; it's a veritable tie.

But what if none of those studies looked at the specific recipes used? If the type of lesson used might likely sway the results, then ignoring it compromises those results.

Stay tuned.

A question that I asked in Part II of this series was the following:

If we can indeed restrict ourselves to considering a limited set of approaches to teaching mathematics, then what restrictions do we use?

The author of the paper we are examining draws a distinction between two different possible answers to that question with regard to funding (i.e., value): (1) the approaches most worthy of funding are those that are connected to published research which show positive effects, and (2) the approaches most worthy of funding are those that are connected to precedent and meet student needs. As I mentioned in my previous post, I think both of these answers are seriously flawed. So, I'll use this almost-penultimate post in the series to set up my explanation. Hang in there!

Soup Sequitur

Several months ago, I developed an interest in what you might call cooking--though you should know that it's difficult for me to describe looking up easy-to-follow recipes on the Internet, buying the ingredients, and following the instructions verbatim as "cooking."

Anyway, one of the first recipes I looked up turned out to be a simple one--perfect for an eternal "beginner" like myself--and one that I've always loved: Zuppa Toscana (sp?) à la Olive Garden.

Instructions

Sauté 1 pound of ground Italian sausage and 1 and 1/2 teaspoons of crushed red pepper in a large pot. Drain excess fat and refrigerate while you prepare other ingredients.

Dice a large white onion and tear 4 tablespoons of bacon strips into pieces. In the same pot, sauté bacon, onions, and 2 teaspoons of garlic puree for approximately 15 minutes or until the onions are soft.

Mix together 5 chicken bouillon cubes and 10 cups of water, then add it to the onions, bacon, and garlic. Cook until boiling.

Thinly slice 3 large baking potatoes. Add potatoes to the pot and cook until soft, about half an hour.

Add 1 cup of heavy cream and cook until thoroughly heated.

Stir in the sausage.

Add 1/4 of a bunch of kale just before serving.

It's certainly one of a very small number of "keeper" recipes for me (and, more importantly, my family). I would suggest trying it with some fresh parmesan sprinkled on top and a good bread to lap it up with. If you don't like spicy, I think you can skip the crushed red pepper without dramatically affecting the taste.

"And a Spaceship Lands."

Suppose we want to teach this recipe to people who have no idea about cooking; or, if you like, we want people who have no idea about cooking to learn this recipe. For this specific content (the recipe), that might mean that we want people to be able to recite it from memory and be able to prepare the soup to a certain level of quality unaided.

If this is what we want, then over time it would be reasonable--even important--at some point to ask, What are some better ways to teach this recipe? or, again if you like, How do people best learn this recipe? especially if we observe that (a) either the teaching or learning (or both) is, for some reason, not natural or easy, and (b) despite all the time, energy, and money we put into this hypothetical endeavor, our students consistently get their butts handed to them at international food competitions.

Asking ourselves how we can better teach a topic or subject (or better learn a topic or subject) is indeed important. But the question that I mentioned at the beginning of this post is one that we can ask first: What restrictions should we use to narrow our search for the answers?

We know there must be some restrictions. While all of us draw circles--for various reasons--that contain different ideas and methods and even questions within education that are thereby endowed with value, excluding others either partly or completely, for a short time or for a long time, there are certainly areas where we can see widespread agreement. For example, as open-minded as we are (or think we are), none of us would spend much time considering my shin-kicking proposal from the previous post. And, as iconoclastic as each of us might be, almost all of us would agree that one or both of the restrictions Mr Garfunkel mentions in his paper, research and precedent--what one might call the "what works" restrictions--are necessary.

In other words, given the universal set U (below) of ideas/methods/questions/etc., that could possibly pertain to better teaching and/or better learning, there is widespread agreement that elements like s, shin-kicking, should be excluded from consideration (i.e., they are not valuable).


And despite the fact that there may be no agreement whatsoever regarding the value of elements a, b, or c (or, perhaps, unanimity among the population that all three are worthless), there is nearly universal agreement that either Set P (the set of all elements that are connected to precedent and meet needs) or Set R (the set of all elements that are connected to published research and show positive effects) or both (either P &cap R or P &cup R) are necessary circumscriptions--even if we believe those sets are empty.

Lest ye believe that the paper I introduced in my previous post was authored by some kind of anti-research nutter, let me share this from the conclusion:

I wish to be clear. I recognize that Faffufnik [i.e., any mathematics education researcher] has done important research. I recognize that Chaim Yankel's [i.e., any statistician's] protocols can help quantify our results. We have to learn from the past and theoretical frameworks are important for future work. But we also have to recognize that quoting Faffufnik and Chaim Yankel is not a substitute for imagination, creativity, and the application of common sense.

This is, I think, a conclusion that one can easily agree with, mostly because it is a weaker version of the perspective that holds throughout the balance of the paper--that judgments about the quality of mathematics education projects should not be anchored in any way to research.

I agree with the weak version of this conclusion and, in part, with the stronger perspective. But I disagree somewhat with how the author gets there. So, for the remainder of this post, I will focus on those disagreements--or rather, one disagreement.

From page 3 of the paper:

It is simply not possible to prove that an approach to teaching and learning will be effective before the fact. . . . It is in the nature of the enterprise that we cannot discover what works before we create the what.

What a relief! Now I can dust off that shin-kicking proposal that's been gathering dust on my bookshelf. To be brief, it's a proposal--chock-full of "imagination" and "creativity"--to teach mathematics to second and third graders by kicking them each in their shinbones every ten minutes during math time.

I know what you're thinking, and you can laugh at my idea all you want, but you really can't possibly know if this method will work until it's tested. Right? Well, of course you can. And, yes, I realize that that's not what Mr. Garfunkel is suggesting. He almost certainly has a limited set of "approaches" in mind.

But, see. That's just it. If we can indeed restrict ourselves to considering a limited set of approaches to teaching mathematics--an idea that contradicts the two sentences quoted above--then, for crying out loud, what restrictions do we use? The following is, in part, Mr. Garfunkel's answer to that question:

Curriculum development, in particular, is best related to an engineering paradigm. In order to test the efficacy of an approach, we must analyze needs, examine existing programs, build an improved model program, and test it--in the same way we build scale models to design a better bridge or building.

So, according to Garfunkel, in order to judge the quality of a mathematics education project, one should not have to tie it to the research literature, but rather one should prove that it's (a) connected to something that's worked before (and/or an improvement on something that hasn't worked before) and (b) something that meets "needs."

He contrasts this approach with the one that produces the Faffufnik-Chaim Yankel Effect:

There appears to be an underlying assumption here that mathematics education projects must proceed in the following way. First, they must be based upon research. Therefore, we heavily quote the results of prior research (See the papers of Faffufnik). Then based upon that research we make a new research hypothesis and test it with a small number of students. If at all possible we make this experiment as close to a 'gold standard' double-blind medical approach as possible. Then using certain statistical protocols (See the work of Chaim Yankel) we conclude that there is some measurable effect and write a new proposal to test this effect on a larger population. This process is then iterated. This is now a necessary condition for funding--independent of the content and strength of the ideas being considered.

In other words, quoting research and using statistical protocols to determine effects: bad; quoting predecessor projects and using the criterion "meets needs" to determine effects: good.

The truth is that both Garfunkel's method and the current research method are seriously flawed.

Next time.

An accessible, and admittedly basic, introduction to the concept of natural selection can be read at Wikipedia:

The basic concept of natural selection is that the physical and biological environment ("nature") selects those variant traits of individual organisms that contribute better to the survival and reproduction of their carriers: Individuals carrying unfavorable variants might die before reproducing and/or produce fewer or lower-quality progeny, while individuals carrying favorable variants are more likely to survive until reproduction and/or to produce more and/or higher-quality progeny.

It doesn't take a great deal of gumption to admit that natural selection is, as envisioned by those who work with the theory, an "undirected process." But "undirected process" cannot reasonably be read as equivalent to "total randomness," so the oft-implied claim that intelligent design proposes a kind of common-sense alternative to the wacky idea of speciation and adaptation as being slapped together by an unbelievable coincidence of cosmic chance is, at the very least, disingenuous. Indeed, there is a fair amount of assumed regularity in what has been proposed as the mechanism of natural selection—this, in part, leads some to question, with good reason, whether regularity, chance, and design really are mutually exclusive explanations for events.

The essential difference between natural selection and design detection as explanatory tools is one akin to the difference between scientific and mathematical truth.

Mathematics defines a triangle more or less as a closed figure with three straight sides (line segments). Nothing that anyone has ever held in her hand or viewed from afar (or closely) conforms to this definition in the real world. Nothing. A triangle's sides are perfectly straight and are one-dimensional. The usefulness of the strict definitions of mathematics are derived by the benefits they provide in further deduction, not at all by any one-to-one clarity they provide in regard to the real world.

Science is different. The twin rovers Spirit and Opportunity, which were so successful on Mars, each carried a thermal emission spectrometer (TES [or, rather, mini-TES]). These devices (tests, really) operate on the empirical reality that rocks of different material composition reflect thermal infrared light differently, making it possible to identify different materials in matter by looking at the thermal infrared signatures. This was the way scientists hoped to identify the mineral hematite on the surface (or below the surface) of Mars, which would, in turn, lend credibility to the existence—past or present—of water on the planet. The constructed test was based on real-world observations, and the hopefulness behind its potential was, in part, based on the idea that Mars and Earth are both subject to the same physical laws.

Design detection is a construct that has no empirical inputs that would give us confidence that what was tested for was indeed what we were looking for. If by some impossibility we actually detected something in the real world that conformed perfectly to a strict definition of a triangle, we would have no reason to claim that our test works at finding "real" triangles. The only evidence we would have that we found a triangle would be the test itself. Intelligent design may well indeed find design 100% of the time. But we would have no objective reason to believe this is true, because the only evidence for the findings would be the test itself.

Design detection proposes an unreality--and I don't think intelligent design theorists would object to this description--that specified events of extremely small probability must have a designed causality. It is not an "unreality" because it is not objectively true; it is such because it is not derived from any reality.

As scientific truth, well, it simply isn't. As a kind of philosophical/mathematical truth, intelligent design has yet to show its usefulness.

You can't possibly have the right experience with the paper I will refer to below without reading it yourself first. So go here (PDF), do the Google searches, and come back.

The paper, titled "The Faffufnik-Chaim Yankel Effect: A Cautionary Tale," was meant to address the "evolution of theoretical frameworks in mathematics education" and was submitted for a conference for the International Commission on Mathematical Instruction (ICMI) in March of 2008. The paper's author is Solomon Garfunkel, who is listed as the executive director of an organization called COMAP, the Consortium for Mathematics and Its Applications.

The main idea of the paper seems to be that, although the process by which math education projects were approved in the past was certainly flawed, the process that exists today--one which leads to what the author calls the Faffufnik-Chaim Yankel Effect (FCE)--is much worse:

Years ago a typical review of a COMAP proposal would read, "This is an excellent idea with an excellent staff with an excellent track record, we recommend this project for funding." The FCE refers to more typical current reviews that read, "This is an excellent idea with an excellent staff with an excellent track record. However, we have to recommend against funding because they don't make any reference to the seminal research papers of Faffufnik, nor do they plan to use the statistical protocols of Chaim Yankel." The reviewers may very well be students of Faffufnik and/or Chaim Yankel.

Now, I assume I'm like most people in that (a) I would think it a pitiable state of affairs that mathematics education funding would be so tightly tethered to one researcher's line(s) of inquiry, and (b) I would find it extremely difficult, if not impossible, to pick out the fake names in a list of names of alleged mathematics education researchers (unless they were Madonna or Jon Bon Jovi, or something like that), and (c) sometimes things go right over my head that I later think should not have gone right over my head.

So, after reading Garfunkel's paper, I looked up Faffufnik and Chaim Yankel. For the former, the search results simply led me back to the paper in question. And the top result for the latter was this definition from a "Yiddish dictionary":

Chaim Yankel (khai-yam Yonk-l) a country bumpkin; the guy who just fell off the turnip truck; an ineffectual nobody; a fool. Also, Mr. Anyone; the man on the street; any Tom, Dick or Heshie.

Only then did I fully realize that Faffufnik and Chaim Yankel were made-up names.

Of course, this realization unfortunately could not change (b) and (c) above. But it did change (a). That is, upon realizing that Faffufnik and Chaim Yankel were merely placeholder names, I also realized that I was not being asked anymore to despair alongside the author--and, I assume, alongside most others--that mathematics education funding might be held hostage by the work of one researcher, but rather to despair that it might be held hostage by any research. All research.

And that's a completely different question. One that I'll take up in the next post.

These lessons have already had a pretty thorough review from some tough critics, and they seem to have survived all right.

So, here you go. Merry Christmas! Don't say I never gave you nothin':

Multiplication Lessons Grade 3 (Draft)

Publish at Scribd or explore others: Primary and High Sch Education lessons multiplication

All other things being equal, students from more individualistic cultures/societies are harder to educate than their counterparts.

So, let's end this series of posts by first reviewing the previous two posts discussing the research.

In the first post, we looked at how Ainsworth and Burcham defined the concepts they were working with--text coherence and self-explanation. Text coherence was defined, in general, as "the extent to which the relationships between the ideas in the text are made explicit" and self-explanation was defined as "additional knowledge generated by learners that states something beyond the information they are given to study." In addition, we learned that both text coherence and self-explanation can produce gains in students' learning from text. But they seemed to do so in opposing ways: Whereas text coherence advantages learners by "repairing" (i.e., removing) inferences, self-explanation often produces gains even when—and perhaps especially when—text remains minimally coherent.

For their experiment, four groups of undergraduates were created: (1) a group who was assigned "maximally coherent" text and received self-explanation training, (2) a group who was assigned maximally coherent text and received no self-explanation training, (3) a group who was assigned "minimally coherent" text and received self-explanation training, and (4) a group who was assigned minimally coherent text and received no self-explanation training. Each group was given a three-section pretest and a matching three-section posttest, which assessed students' knowledge of the "explicit propositions in the text," or textbase. Participants were also given (at posttest only) "implicit questions" and "knowledge inference questions" (so, a two-section additional posttest), both of which assessed students' situation models.

Two hypotheses were proposed:

(a) The minimal text condition when accompanied by self-explanation training will present the optimal conditions for learning. Minimal text is hypothesized to increase self-explaining, and self-explanation is known to improve learning. Consequently, low knowledge learners who self-explain will not only be able to overcome the limitations of less coherence but will actively benefit from it as they will have a greater chance to engage in an effective learning strategy.

(b) Maximally coherence [sic] text accompanied by self-explanation will present the optimal condition for learning. Although maximal text is hypothesized to result in less self-explanation than minimal text, when learners do self-explain they will achieve the benefits of both text coherence and self-explanation.

What Ainsworth and Burcham found was a main effect for self-explanation on four out of the five sections of the posttest (the multiple-choice section of the textbase assessment was the only exception). Self-explainers scored significantly better than did non self-explainers on all but one of the sections of the posttest. In contrast, a signficant main effect for text coherence was observed on only one section of the posttest--inference questions. Those who read maximally coherent text scored significantly better than those who read minimally coherent text on only one of the five sections of the posttest.

If we stop here, we would be justified in concluding that (a) was the winning hypothesis here. It would seem that self-explanation has a more robust positive effect on learning outcomes than does text coherence. And since the literature tells us that minimally coherent text produces a greater number of self-explanations than does maximally coherent text, minimizing text coherence is desirable for improving learning.

Luckily, Ainsworth and Burcham went further. They coded the types of self-explanations made by participants and analyzed each as it correlated with posttest scores. While they did find that students who read minimally coherent text produced significantly more self-explanations, they also noted this:

Whilst using a self-explanation strategy resulted in an increase in post-test scores for the self-explanations conditions compared to non self-explanation controls, there was no signficant correlation within the self-explanation groups between overall amount of self-explanation and subsequent post-test performance. Rather, results suggest that it is specific types of self-explanations that better predict subsequent test scores.

In particular, for this study, "principle-based explanations" ("[making] reference to the underlying domain principles in an elaborated way"), positive monitoring ("statements indicating that a student . . . understood the material"), and paraphrasing ("reiterating the information presented in the text") were all significantly positively related to total posttest scores, though only the first of those was considered a real "self-explanation."

Now, each of those correlations seems pretty ridiculous. They all seem to point in one way or another to the completely unsurprising conclusion that understanding a text pretty well correlates highly with doing well on assessments about the text.

What is interesting, however, is the researchers' observation that the surplus of self-explanations in the "minimal" groups could be accounted for primarily by three other types of self-explanation, none of which, in and of themselves, showed a signficant positive correlation with total posttest scores: (1) goal-driven explanations ("an explanation that inferred a goal to a particular structure or action"), (2) elaborative explanations ("inferr[ing] information from the sentence in an elaborated manner"), and (3) false self-explanations (self-explanations that were inaccurate).

To put this in perspective, there were only two other types of "self-explanation" coded that I did not mention here. Out of the remaining six, three showed no significant positive correlations with posttest scores (or, in the case of false self-explanations, a significant negative correlation), yet those were the self-explanations that primarily accounted for the significant difference between the minimal and maximal groups.

Or, to put it much more simply, the minimal groups had significantly more self-explanations, but those self-explanations were, in general, either ineffective at raising posttest scores or actually harmful to those scores. It is possible that the significant positive main effect for self-explanation in the study could, in fact, have been greatly helped along by the better self-explanations present in the maximal groups. All of this leads to this conclusion from the researchers:

This study suggests that rather than designing material, which, by its poverty of coherence, will drive novice learners to engage in sense-making activities in order to achieve understanding, we should design well-structured, coherent material and then encourage learners to actively engage with the material by using an effective learning strategy.

Reference: S AINSWORTH, S BURCHAM (2007). The impact of text coherence on learning by self-explanation Learning and Instruction, 17 (3), 286-303 DOI: 10.1016/j.learninstruc.2007.02.004

Folks are still kicking around the repeated-addition-vs.-multiplication stuff, and clearly there are more than two sides to this debate, which, to the extent that it represents a development at all, is a positive development.

Deserving of more scrutiny, though, is this widespread notion among commenters--a keystone belief supporting many different arguments--that students' collective intuitions about mathematics serve as a kind of useful, or even necessary, foundation on which curricula should be built. This is, essentially, a warped version of the notion (or perhaps mantra) of what's known as "teaching from the known to the unknown."

Known to Unknown

At any point in a child's life or schooling, he or she presents with a number of things he or she can do and a number--which could be 0--of things he or she knows. We can refer to these collectively as the "knowns." And, of course, the "unknowns" are all those things a child does not know or cannot do at any of the same points. The problem of teaching from the known to the unknown involves making some kind of connection from a student's knowns to a very restricted set of unknowns, which, taken together at any point, form a kind of immediate curriculum.

Now, of course, it is impossible to teach without going from the known to the unknown in some way. On the one hand, a student can't learn anything if s/he has absolutely no knowledge or skills (because then s/he wouldn't exist), and on the other hand, nothing can be described purely in terms of itself (although God takes a good crack at it in Exodus when he says to Moses, "I am that am"). The inevitable connection from known to unknown itself is not at issue. What is at issue is the way this connection is made. What knowns are connected to what unknowns?

The Best "Known"

Over a wide variety of topics, educators will often argue about the quality of the knowns to be connected to specific unknowns. The ongoing debate about whether to teach fractions first or decimals first is an area where this argument pops up, with some making the case that place value is the better "known" to be connected to the unknown of rational numbers (decimals first) while others argue that equal shares is the better known (fractions first). Similarly, one can argue that, for the unknown of improper fractions, proper fractions serve as the best "known," whereas another can argue that, because improper and proper fractions are used in such diverse situations (e.g., "no one says that they have 8/5 dollars"), we must scrap the use of proper fractions as the "known" in introducing improper fractions and come back to the connection later.

Proponents of the equivalence between repeated addition and multiplication--for brevity's sake, I'll just call them "repeaters"--often argue that repeated addition is the best "known" (or, in most cases, the only known) that can be connected to the unknown of multiplication. And there are several problems with these arguments, given the various ways they are presented and the premises on which they rest. One of those is called the appeal to common practice.

Appeal to Common Practice

This is a fallacy. And it works like this: Such and such an action is justified because it is what everyone else is doing or what we've always done. Now, virtually no repeaters that I've read actually commit this fallacy so nakedly. But it does creep up somewhat, um, "un-nakedly." Here's Mr. Mark, falling into the fallacy with repeated multiplication:

Try to give me a simple definition of exponentiation, which is understandable by a fifth or sixth grader, which doesn't at least start by talking about repeated multiplication. Find me a beginners textbook or teachers class plans that explains exponentiation to kids without at least starting with something like "5² = 5 × 5, 5³ = 5 × 5 × 5."

The second of those sentences is pretty clearly the fallacy of appealing to common practice, to the extent that it is used in any way to justify or excuse the teaching of exponentiation as repeated multiplication. But notice what is said in the first sentence: "Try to give me a simple definition of exponentiation, which is understandable by a fifth or sixth grader, which doesn't at least start by talking about repeated multiplication." This, too, is an appeal to common practice, but the practice in this case is not necessarily the teaching of exponentiation as repeated multiplication to fifth or sixth graders but, rather, the teaching of everything before that. The argument is that repeated multiplication is the best "known" because currently the 8 to 10 years of schooling prior to teaching the "unknown" of exponentiation don't prepare students for learning exponentiation any other way (or any better way).

But these circumstances do not make repeated multiplication the best "known," just the most expedient "known." The same goes for repeated addition as a "known" connected to the unknown of multiplication. Expedience is, of course, criterial when one is a classroom teacher forced to get Johnny to grok multiplication any way she possibly can, but it is not a convincing justification for repeated addition as the best foundation for learning multiplication.

The idea that multiplication and repeated addition are not the same has implications for the reorganization of our mathematics curricula. It cannot be judged convincingly one way or the other by how well or how badly it fits into those curricula.

As always, more later.

During the summers when I was going to school, I worked as a tour guide and teacher for groups of Korean students who came to Los Angeles to see the sights and learn some English.

My job was fairly straightforward: Help keep all 100 students safe and happy during our trips to Disneyland and Universal Studios and then work with them for about two hours a week to help them master the English language.

We were all put up in a nearby hotel, so my work for those five weeks every summer was pretty much 24/7. And during the off hours or on bus rides, I had several conversations with students, covering just about everything a 10-15 year-old could talk about with her or his teacher--from homework problems (they had homework from their Korean schools to complete over the summer while they were in the States) to can you please go buy me some tampons (at 1 a.m.).

Anyway, although I didn't have a lot of conversations with students about math, there were a handful, and I was thinking about those conversations today. And what strikes me most, in retrospect, about my conversations with those students was their ability to detect whether an answer was reasonable or unreasonable.

I chuckle when I write that last phrase, because I know that there has been a concerted effort in the U.S. in the past several years to try to teach kids how to distinguish reasonable from unreasonable answers. (You can see it in most elementary basal mathematics texts, where it pops up occasionally as a problem-solving strategy.)

But, see, I'm not talking about that quintessential American "algorithmic" mathematical ability, which is still sadly algorithmic even when it is described as "constructivist" or "understanding." The ability these Korean students displayed--ahem, about 15 years ago--was one of recognition of the basic relationships in mathematics.

So here's a lesson my young Korean tutors taught me way back when for simplifying basic fractions. (Well, they never really got to this point verbally. But I assumed at the time that this was how they were thinking based on hand gestures and some incoherently phrased English.) [And, by the way, this is yet another reason to teach children fractions as division expressions.]

[begin lesson]
To simplify fractions, most students are taught that you first have to find a common divisor--a divisor common to both the numerator and denominator. If the only common divisor you can find is 1, then the fraction is in simplest form.

This is true, of course. But that algorithm is designed for "worst-case scenarios"--like 3/102--when typical elementary-school students don't know (a) whether the fraction can be simplified and (b) how far they have to go to get the fraction in simplest form.

A lot of basic fraction simplifying is a lot simpler. All you need to know are basic multiplication/division facts. Oh, and one other box-of-rocks-simple idea (besides the concept of fractions as division expressions)--how to turn something upside down.

Let's simplify the fraction 3/18 the "algorithmic" way:


Okay, and now let's just think about it. What's 18 ÷ 3? (Um, 6.) How would y'all write that as a fraction? (Uhhhh, 18/3.) So 18/3 is 6. If you just flip the fraction 18/3 over, what division expression do you have? (Zuhhhhh, 3/18.) Is there anyone here who thinks that the answer, the quotient, is NOT going to be the upside-down version of 6/1? Anyone? Anyone?
[end lesson]

Now, if you're a third-, fourth-, fifth-, or sixth-grade student, and you know your basic multiplication/division facts, don't you save valuable time on those high-stakes tests by just knowing this stuff?

As always, more on this later.

This one's going to be fun:

OK - in what *mathematical* manner of speaking is multiplication not a function? I've probably always taken it for granted that multiplication is a function. This could be my big awakening moment now - the big Aha! That I can't just assume that multiplication is a function for just everyone --- I'm holding my breath!

I'm going to print this big, so that even if you squint at a computer screen all day long, you can see it:


I'm not sure how to make this clearer. Dare I open it for comment?

The study (PDF) I introduced in my previous post lays out what seems like a provocative dilemma in its introduction.

On the one hand, a comprehensive—though shallow—read of the text coherence literature tells us that improved text comprehension can be achieved by "repairing" text incoherence—by closing informational gaps in text. On the other hand, research shows that significant improvements in learning from text can come from employing a strategy of self-explanation during reading—a method that practically feeds off textual incoherence.

Ainsworth and Burcham tease out a good question from this apparent dilemma:

Is maximally or minimally coherent text more beneficial to learning when accompanied by self-explanations? Two alternative hypotheses are proposed:

(a) The minimal text condition when accompanied by self-explanation training will present the optimal conditions for learning. Minimal text is hypothesized to increase self-explaining, and self-explanation is known to improve learning. Consequently, low knowledge learners who self-explain will not only be able to overcome the limitations of less coherence but will actively benefit from it as they will have a greater chance to engage in an effective learning strategy.

(b) Maximally coherence [sic] text accompanied by self-explanation will present the optimal condition for learning. Although maximal text is hypothesized to result in less self-explanation than minimal text, when learners do self-explain they will achieve the benefits of both text coherence and self-explanation.

Which of these hypotheses do you think will win out?

The Experiment

Forty-eight undergraduate students were randomly separated into four groups, each of which was assigned either a maximally coherent text (Max) or a minimally coherent text (Min) about the human circulatory system. Each group was also given either self-explanation training (T) or no training at all (NT):


All forty-eight students completed a pretest on the subject matter, read their assigned text using self-explanation or not, and then completed a posttest, which was identical to the pretest. The results for each of the four groups are shown below (the posttest results have been represented using bars, and the pretest results have been represented using line segments):


The pretest and matching posttest each had three sections, as shown above. Each of these sections comprised different kinds of questions, but all of the questions assessed knowledge of the textbase:

The textbase contains explicit propositions in the text in a stripped-down form that captures the semantic meaning.

As you can see, each of the four groups improved dramatically from pretest to posttest, and those subjects who read maximally coherent text (Max) performed slightly better overall than those who read minimally coherent text (Min), no matter whether they used self-explanation during reading (T) or not (NT). However, the effect of text coherence was not statistically significant for any of the three sections of the tests. Self-explanation, on the other hand, did produce significant results, with self-explainers scoring significantly higher on two of the three sections than non–self-explainers.

In addition to the posttest, subjects also completed a test comprised of "implicit questions" and one comprised of "knowledge inference questions" at posttest only. The results for the four groups on these two tests are shown below.


Each of these two tests assessed students' situation models:

The situation model (sometimes called the mental model) is the referential mental world of what the text is about.

The researchers found that self-explainers significantly outperformed non–self-explainers on both tests. Those who read maximally coherent text also outperformed their counterparts (readers given minimally coherent text) on both tests. However, this effect was significant for only one of the tests, and approached significance for the other test (p < 0.08).

More on this later.

Reference: S AINSWORTH, S BURCHAM (2007). The impact of text coherence on learning by self-explanation Learning and Instruction, 17 (3), 286-303 DOI: 10.1016/j.learninstruc.2007.02.004

When I write about text coherence, I usually refer to what I call the Coherence Principle—a principle that involves two standards for content: order and packaging.*

These two standards are somewhat similar to the two standards of the Accuracy Principle--clarity and precision—in that the first is a consumer-level (student-level) standard, and the second is a distributor-level (teacher-level) and/or producer-level (content-level) standard.


From a research standpoint, however, text coherence is something very different and much simpler. The authors of the paper (PDF)¹ I will discuss in this and the next post, Ainsworth and Burcham, follow the lead of many researchers, including Danielle McNamara (2001), in conceiving of text coherence as "the extent to which the relationships between the ideas in the text are made explicit." In addition to this conceptualization, the authors also adopt guidelines from McNamara, et al. (1996)² to improve the coherence of the text used in their experiment—a text about the human circulatory system. These guidelines essentially operationalize the meaning of text coherence as understood by many of the researchers examining it:

(1) Replacing a pronoun with a noun when the referent was potentially ambiguous (e.g., replacing 'it' with 'the valves'). (2) Adding descriptive elaborations to link unfamiliar concepts with familiar ones and to provide links with previous information presented in the text (e.g., replacing 'the ventricles contract' with 'the ventricles (the lower chambers of the heart) contract'). (3) Adding connectives to specify the relation between sentences (e.g., therefore, this is because, however, etc.).

Maximal coherence at a global level was achieved by adding topic headers that summarised the content of the text that followed (e.g., 'The flow of the blood to the body: arteries, arterioles and capillaries') as well as by adding macropropositions which linked each paragraph to the overall topic (e.g., 'a similar process occurs from the ventricles to the vessels that carry blood away from the heart').

Many studies have found that improving text coherence (i.e., improving the "extent to which the relationships between the ideas in the text are made explicit") can improve readers' memory for the text. Ainsworth and Burcham mention several in their paper, including studies by Kintsch and McKeown and even a study by Britton and Gulgoz, which I discussed a couple of years ago here.

What Britton and Gulgoz find is that when "inference calls"—locations in text that demand some kind of inference from the reader—are "repaired," subjects' recall of a text is significantly improved over that of a control group. These results may sum up the advantages seen across research studies in improving text coherence: in general, although there are certainly very few if any simple, straightforward, unimpeachable results available in the small collection of text-coherence studies, researchers consistently find that "making the learner's job easier" in reading a text by making the text more coherent provides for significant improvement in readers' learning from that text.

Self-Explanation

In some sense, the literature on self-explanation tells a different story from the one that emerges from the text-coherence research. Ainsworth and Burcham define self-explanation in this way:

A self-explanation (shorthand for self-explanation inference) is additional knowledge generated by learners that states something beyond the information they are given to study.

The authors then go on to describe some advantages offered to readers by the self-explanation strategy, according to research:

Self-explanation can help learners actively construct understanding in two ways; it can help learners generate appropriate inferences and it can support their knowledge revision (Chi, 2000). If a text is in someway [sic] incomplete . . . then learners generate inferences to compensate for the inadequacy of the text and to fill gaps in the mental models they are generating. Readers can fill gaps by integrating information across sentences, by relating new knowledge to prior knowledge or by focusing on the meaning of words. Self-explaining can also help in the process of knowledge revision by providing a mechanism by which learners can compare their imperfect mental models to those being presented in the text.

Thus, whereas text coherence advantages learners by "repairing" (i.e., removing) inferences, self-explanation often produces gains even when—and perhaps especially when—text remains minimally coherent.

What shall we make of this? Which is more important—text coherence or self-explanation? And how do they (or can they) interact, if at all? These are the questions Ainsworth and Burcham attempt to address in their experiment, which I will discuss in my next post.


* I'm putting this seemingly irrelevant information at the top of this post, so that I can refer to it later in the context of the research I write about here.

1: S AINSWORTH, S BURCHAM (2007). The impact of text coherence on learning by self-explanation Learning and Instruction, 17 (3), 286-303 DOI: 10.1016/j.learninstruc.2007.02.004

2: Danielle McNamara, Eileen Kintsch, Nancy Butler Songer, Walter Kintsch (1996). Are Good Texts Always Better? Interactions of Text Coherence, Background Knowledge, and Levels of Understanding in Learning From Text Cognition and Instruction, 14 (1), 1-43 DOI: 10.1207/s1532690xci1401_1

Let’s take a look at some further criticism of education research from the Ball and Forzani paper (PDF) I discussed briefly in my first and second posts.

Education research frequently focuses not on the interactions among teachers, learners, and content—or among elements that can be viewed as such—but on a particular corner of this dynamic triangle. Researchers investigate teachers' perceptions of their job or their workplace, for example, or the culture in a particular school or classroom. Many excellent studies focus on students and their attitudes toward school or their beliefs about a particular subject area. Scholars analyze the relationships between school funding and student outcomes, investigate who enrolls in private schools, or conduct international comparisons of secondary school graduation requirements. Such studies can produce insights and information about factors that influence and contribute to education and its improvement, but they do not, on their own, produce knowledge about the dynamic transactions central to the process we call education.

Here Ball and Forzani expand their criticism of education research, pointing to a lack of good research that not only looks at teachers, students, or content but also at the interactions among these three. Their critique of the now-famous Tennessee classroom-size study illustrates clearly this further refinement of the authors' concept of research "inside education":

Finn and Achilles (1990) investigated whether smaller classes positively affected student achievement in comparison with larger classes. . . . The results suggest that reducing class size affected the instructional dynamic in ways that were productive of improved student learning. The study did not, however, explain how this worked. Improvement might have occurred because teachers were able to pay more attention to individual students. Would the same have been true if the teachers had not known the material adequately? Would reduced class size work better for students at some ages than at others, or better in some subjects than in others?

I'll leave the summary of Ball and Forzani's paper there, though it is by no means complete. I refer interested readers to the paper linked above and of course to my two earlier posts.

Next time, I'd like to take a look at some ideas that I think are missing from Ball and Forzani's paper.

Ref: Ball, D., & Forzani, F. (2007). 2007 Wallace Foundation Distinguished Lecture--What Makes Education "Research Educational"? Educational Researcher, 36 (9), 529-540 DOI: 10.3102/0013189X07312896

Below are 50--yes, 50--theories regarding education/instruction. Go here to read about them.

ACT* (J. Anderson)
Adult Learning Theory (P. Cross)
Algo-Heuristic Theory (L. Landa)
Andragogy (M. Knowles)
Anchored Instruction (J. Bransford & the CTGV)
Aptitude-Treatment Interaction (L. Cronbach & R. Snow)
Attribution Theory (B. Weiner)
Cognitive Dissonance Theory (L. Festinger)
Cognitive Flexibility Theory (R. Spiro)
Cognitive Load Theory (J. Sweller)
Component Display Theory (M.D. Merrill)
Conditions of Learning (R. Gagne)
Connectionism (E. Thorndike)
Constructivist Theory (J. Bruner)
Contiguity Theory (E. Guthrie)
Conversation Theory (G. Pask)
Criterion Referenced Instruction (R. Mager)
Double Loop Learning (C. Argyris)
Drive Reduction Theory (C. Hull)
Dual Coding Theory (A. Paivio)
Elaboration Theory (C. Reigeluth)
Experiential Learning (C. Rogers)
Functional Context Theory (T. Sticht)
Genetic Epistemology (J. Piaget)
Gestalt Theory (M. Wertheimer)
GOMS (Card, Moran & Newell)
GPS (A. Newell & H. Simon)
Information Pickup Theory (J.J. Gibson)
Information Processing Theory (G.A. Miller)
Lateral Thinking (E. DeBono)
Levels of Processing (Craik & Lockhart)
Mathematical Learning Theory (R.C. Atkinson)
Mathematical Problem Solving (A. Schoenfeld)
Minimalism (J. M. Carroll)
Model Centered Instruction and Design Layering (A.Gibbons)
Modes of Learning (D. Rumelhart & D. Norman)
Multiple Intelligences (H. Gardner)
Operant Conditioning (B.F. Skinner)
Originality (I. Maltzman)
Phenomenonography (F. Marton & N. Entwistle)
Repair Theory (K. VanLehn)
Script Theory (R. Schank)
Sign Theory (E. Tolman)
Situated Learning (J. Lave)
Soar (A. Newell et al.)
Social Development (L. Vygotsky)
Social Learning Theory (A. Bandura)
Stimulus Sampling Theory (W. Estes)
Structural Learning Theory (J. Scandura)
Structure of Intellect (J. Guilford)
Subsumption Theory (D. Ausubel)
Symbol Systems (G. Salomon)
Triarchic Theory (R. Sternberg)

Which ones do you like? Put me down for none.

Well, this is actually Orwell on the English language, but just replace "English language" or "English" with "mathematics education."

Oh, and replace "writers" with "mathematicians":

A man may take to drink because he feels himself to be a failure, and then fail all the more completely because he drinks. It is rather the same thing that is happening to the English language. It becomes ugly and inaccurate because our thoughts are foolish, but the slovenliness of our language makes it easier for us to have foolish thoughts. The point is that the process is reversible. Modern English, especially written English, is full of bad habits which spread by imitation and which can be avoided if one is willing to take the necessary trouble. If one gets rid of these habits one can think more clearly, and to think clearly is a necessary first step toward political regeneration: so that the fight against bad English is not frivolous and is not the exclusive concern of professional writers.

Via Clicked.

I'm reposting this because someone from JSTOR spent some time reading it, and I'd like to take this opportunity to beg said person for free access.

That and, of course, it continues a train of thought that has been rolling across this blog lately.

At the end of what seems like a long chain of events, I asked, and answered yes to, this question about professionalizing teacher practice:

Is there one or more cultural "teaching scripts" that might tend to stymie the practice of collecting and critically analyzing specific best-practice knowledge linked to academic outcomes?

The question was, at the time, the end result of my thinking about Jenny D.'s terrific post, Chris Correa's input on the subject, and ideas presented by different commenters.

Regardless whether yes is the correct answer to that question or not, I'd like to follow up and suggest one script that I think may be a significant culprit. Of course, in doing so, I will be making a solid leap away from firm ground, because cultural scripts are constructs that one can observe only indirectly, if at all:

[Cultural scripts] are not proposed as rules of behaviour but as rules of interpretation and evaluation. It is open to individuals in concrete situations whether to follow (or appear to follow) culturally endorsed principles, and if so, to what extent; or whether to manipulate them, defy them, subvert them, rebel against them, play creatively with them, etc. Whether or not cultural scripts are being followed in behavioural terms, however, the claim is that they constitute a kind of shared interpretive "background."

One part of an "interpretive background" that I would suggest we share with regard to the idea of teaching is this:

Teaching is "ethotic" and "pathotic" persuasion.
That is, the input of teaching is gauged in terms of the character of teachers (ethos) and their ability to navigate and control the emotional, cognitive-psychological, and interpersonal dynamics of learning (pathos). "Logetic" persuasion (logos)—which involves consideration of the presentation and organization of content in isolation--is really not part of the script for teaching or is, at best, completely overshadowed.

Consider these ethotic/pathotic selection criteria for the National Teacher of the Year award as a bit of indirect evidence for the existence of this script:

Inspire students of all backgrounds and abilities to learn.

Have the respect and admiration of students, parents, and colleagues.

Play an active and useful role in the community as well as in the school.

Be poised, articulate, and possess the energy to withstand a taxing schedule.

In short, we tend to view better teaching exclusively as a function of better people, and almost never as a function of better information--a script which makes the compilation and dispensation of best-practice knowledge nearly unimaginable.

I would also suggest that this script plays a role in answering a question I asked here, about Direct Instruction:

It has consistently fascinated me over the last five or six years to see veteran educators' reactions to even a mention of DI—snarling, eye-rolling, etc.—because these reactions are often followed by admissions that "the program achieves remarkable results." How can it be that educators are so dismissive of something that so clearly improves student achievement?

In that post, it was my (obvious) intention to answer the question by suggesting that deciding on Direct Instruction might feel to educators like choosing the "fake" experiences of the Experience Machine--a hypothetical device that can provide users with manipulable, yet absolutely real, experiences--over "authentic" experiences, because Direct Instruction simply doesn't match educators' views of what "real" teaching is.

But the script mentioned above, although related to my answer, may be a more powerful way of explaining the phenomenon. Look at part of an exercise in this sample (PDF) from a Direct Instruction reading program (the words in italics are instructions to the teacher):

Everybody, find part A in your textbook. Wait. Touch column 1. Check. The words in column 1 are hard words that will be in your reading stories.

Touch under the first word. Check. The first word is difficult. What word? Signal. Difficult. Spell difficult. Signal. D-i-f-f-i-c-u-l-t. What word did you spell? Signal. Difficult.

Next word. Pause. That word is exhausted. What word? Signal. Exhausted. Spell exhausted. Signal. E-x-h-a-u-s-t-e-d. What word did you spell? Signal. Exhausted.

Where is our ethos? Where is our pathos?

I was going to write something about this shortly after the most recent "controversy" in the edusphere died down, but . . .

This captures my feelings about it almost perfectly.

Via Pharyngula.

Last time, we took a look at a paper that called education research to task for too often being "related" to education, rather than "inside" it.

In future posts, I'd like to briefly summarize some of the authors' further criticisms of education research and then describe and comment on their suggestions for fixing the problem.

In this post, however, I'd like to highlight a key point that the authors, Ball and Forzani, make in the introduction to their paper. Although they do not explicitly link this point to their criticism of education research, I see no reason to consider the two to be unrelated:

One impediment is that solving educational problems is not thought to demand special expertise. Despite persistent problems of quality, equity, and scale, many Americans seem to believe that work in education requires common sense more than it does the sort of disciplined knowledge and skill that enable work in other fields. Few people would think they could treat a cancer patient, design a safer automobile, or repair a bridge, for these obviously require special skill and expertise. Whether the challenge is recruiting teachers, motivating students to read, or improving the math curriculum, however, many smart people think they know what it takes. Because schooling is a common experience, familiarity masks its complexity. Powell (1980), for example, referred to education as a "fundamentally uncertain profession" about which the perception exists that ingenuity and art matter more than professional knowledge. Yet the fact that educational problems endure despite repeated efforts to solve them suggests the fallacy of this reliance on common sense.

Ball and Forzani here accurately describe the environment in which many of our discussions of and debates about education take place. Instruction itself is shielded from our view by ideas—some of which may indeed be correct—that are too often based on common-sense notions about education. As a result, good questions and reasoned arguments that challenge fundamental assumptions about instruction are brushed aside without consideration.

I certainly have experience with this. Not too long ago I brought up a question with my (now former) colleagues--one that I discuss here: If we expect students to be able to represent data accurately and efficiently, then why do we present students with data representations that are neither accurate nor efficient? On that specific occasion, I was talking about bar graphs that were purposely designed to make the data values hard to ascertain and/or graphs that wasted space, as there were only three or four data values to consider. Both of these characteristics are characteristics of bad graphs. Yet there we were, teaching kids how to read and create bad graphs. Shouldn't we--can't we--do it differently? Nope.

One of the all-too-common assumptions underlying this practice and many others like it is the assumption that no matter what kind of content we present to students, they'll eventually get the right idea. Another assumption is that third graders can handle 4 or 5 bars on a graph, but if you push it to 10 or 20, they will spontaneously combust. (Research showing that 8- to 10-year-olds cannot selectively attend to different parts of a 10-bar graph without becoming distracted [such that it impairs learning] would be evidence for the need to move the introduction of bar graphs up to a later grade; it should not be a justification for presenting students with bad bar graphs.)

Devlin makes a point similar to that put forward by Ball and Forzani at the end of his September 2008 article:

While most of us would acknowledge that, while we may fly in airplanes, we are not qualified to pilot one, and while we occasionally seek medical treatment, we would not feel confident diagnosing and treating a sick patient, many people, from politicians to business leaders, and now to bloggers, feel they know best when it comes to providing education to our young, based on nothing more than their having themselves been the recipient of an education.

One may presume, given that Ball and Forzani and then Devlin ascribe this common-sense view of education to "many Americans," or to "politicians, business leaders, and bloggers," that these people consider, or are justified in considering, education researchers or teachers or other education professionals to be immune from similar assumptions and common-sense notions. Of course, they don't and aren't.

Thus, if education researchers are as susceptible as the rest of us to a "common sense-y" view of instruction impervious to reasoned probing, this may explain, in part, Ball and Forzani's criticism of education research as dealing with questions "related to" education rather than questions "inside" education. Many researchers may simply avoid questions inside education because their common sense has already answered them.


Ref: D. L. Ball, F. M. Forzani (2007). 2007 Wallace Foundation Distinguished Lecture--What Makes Education Research "Educational"? Educational Researcher, 36 (9), 529-540 DOI: 10.3102/0013189X07312896

Devlin's final article on the "multiplication = repeated addition" issue is up over here.

If you are new to the issue, start with his first article on the subject here, and then read his second article here. All together, they should make for some good back-to-school reading.

My posts on the subject start here.

I know that many people in education, including education bloggers, share my disappointment with what is published today as education research.

For some, it is primarily a disaffection with what seems to be a set of "constructivist" assumptions that unnecessarily undergird a lot of research. For others, it is frustration with the lack of experimental rigor in many of the research designs. For still others—and I would include myself in this group—it is mostly that a lot of research simply doesn't address what they care about.

Ironically, veteran education researcher Deborah Ball (along with co-author Francesca Forzani) provide some measure of validation for our frustrations, disappointments, and disaffections. In a paper (PDF) titled "What Makes Education Research 'Educational'?" published in December 2007, Ball and Forzani point to a less obvious but still visible problem with education research that is closely related to the problems we all see—its tendency to focus on "phenomena related to education," rather than "inside educational transactions":

In recent years, debates about method and evidence have swamped the discourse on education research to the exclusion of the fundamental question of what constitutes education research and what distinguishes it from other domains of scholarship. The panorama of work represented at professional education meetings or in publications is vast and not highly defined. . . Research that is ostensibly "in education" frequently focuses not inside the dynamics of education but on phenomena related to education—racial identity, for example, young children's conceptions of fairness, or the history of the rise of secondary schools. These topics and others like them are important. Research that focuses on them, however, often does not probe inside the educational process.

Certainly many of us have read terrible "studies" that are, in fact, "inside education," as we might intuitively understand that term—they are situated in classrooms, they focus on students or teachers or content, etc. Nevertheless, Ball and Forzani make an important point, and the consequences of ignoring problems "inside education" may already be playing out:

Until education researchers turn their attention to problems that exist primarily inside education and until they develop systematically a body of specialized knowledge, other scholars who study questions that bear on educational problems will propose solutions. Because such solutions typically are not based on explanatory analyses of the dynamics of education, the education problems that confront society are likely to remain unsolved.

Consider brain-based education. How many other "solutions" like this can we name?

I'll have more on these ideas, including Ball and Forzani's suggestions for better education research, in a future post.


Ref: D. L. Ball, F. M. Forzani (2007). 2007 Wallace Foundation Distinguished Lecture--What Makes Education "Research Educational"? Educational Researcher, 36 (9), 529-540 DOI: 10.3102/0013189X07312896

Note: If you've arrived at this site through a direct link to the interview below, you will need to scroll to the end of the interview to leave a comment.

After the recent repeated addition = multiplication kerfuffle, I asked Keith Devlin if I could interview him, and he graciously agreed.

Below is the full interview—just ten questions long. The interview was conducted via E-mail from August 10, 2008 to August 16, 2008. I have not edited either my questions as I originally wrote them or Mr. Devlin's responses in any way. (Update: With permission from Devlin, I have gone through and very lightly copyedited his responses--spelling, punctuation, etc.)

It's important for me to note that I am a newbie at interviewing. I think some of my questions are too complicated and confusing, and I think at times I appear to not understand an answer (though that was something I was trying to do to some degree).

I'll likely have some follow-up comments in a future post, and if you have any reactions, comments, or questions that I could include in that post, please send me an E-mail. For now, enjoy:

(1) I wanted to start by simply asking you about you. I think it's fair to say that you have a greater public profile than many other living, working mathematicians. Many people, including myself, know something about you through your books, your work at NPR, and your speeches and interviews over the years. Yet, in doing a bit of research for this interview, I was surprised to learn that you ride your bike tremendous distances every day, and you also enjoy playing a certain very popular online role-playing game. What else might one not know about Keith Devlin?

Devlin: The cycling is fairly recent. I was a long-distance runner for most of my adult life, including road races, fell running, and cross-country racing in the U.K. and then road running and trail running in the U.S. after I moved here in 1987. I managed to get my marathon time down to 2 hours 41 minutes, but no amount of effort could lower it further. I did best at 10 milers, usually coming in around 52 or 53 minutes. Then about five years ago my knees gave out – I’d run down too many mountain trails I guess – and I switched to cycling. I try to get out two or three days during the week for 15- to 25-mile rides (an hour to an hour-and-a-half), and longer rides at weekends, anything between 60 and 100 miles at a time. My longest, hardest ride was The Death Ride this last summer, a famous 129 miles in the California High Sierras that takes you over five high mountain passes. It took me 11 hours to get round. I’ve ridden in Europe a few times, including two ascents of the infamous Mont Ventoux. If I can’t get out of doors and exercise every few days I start to feel edgy. Mathematics is hard work, but not physically taxing. I took up cycling too late in life to be competitive, even within my age group, but I console myself that my marathon time is still considerably better than Lance Armstrong’s.

The World of Warcraft began as a desire to try to understand better the students I find myself teaching these days, but I soon noticed that such games have HUGE educational potential, especially for mathematics. It’s a new literacy, and teachers need to be familiar with it. Approached in the right way, they are great fun, and way more challenging than is popularly supposed. Since I have limited time to play, it took me 2 years to reach the top level of play in WoW. I’m working on a book about using videogames to teach mathematics.

Early in my career, I wrote a radio play based on mathematics that the BBC performed and broadcast, and worked on a novel that I finished in first draft, but mathematics proved too great a pull. Recently I’ve started to move some of my writing back to a more story-telling mode. My latest book, The Unfinished Game, is a mathematics history, and I am working on another one in the same genre.

Of course, I read all the time, but that’s surely true for everyone in education. We love learning.

Other than that, with a time-consuming job at Stanford, I don’t have time for much else!

(2) In your very first article for your monthly online column, "Devlin's Angle," at The American Mathematical Association's Web site (January, 1996), you wrote to debunk a myth about an alleged computer virus, saying, "There is, you see, no data-destroying "Good Times" virus out there. The warning message is itself the virus." And just before that you wrote, "To date, the only known defense to this virus—information—has failed to stem its spread." Since then, a number of your articles have dealt directly or indirectly with debunking mathematical myths, educating the general public about mathematics, stirring up thinking about innovation in mathematics education, etc.

Jumping twelve years into the future, you wrote the following just nine months ago in an article titled "American Mathematics in a Flat World," regarding the "outsourcing" of mathematics and the need for innovation in mathematics and mathematics education:

"I gave up on the country of my birth (the UK) twenty years ago when it told me it no longer had need for people such as myself (not far from an exact quote from the Vice Chancellor ("President") of the university where I taught, acting under government pressure to reduce its mathematics department by 50%). Having lived through the decline of my home country as a world powerhouse in innovation and economics, I am not about to give up on the country that welcomed me with open arms."

So, my question is, Can one draw a line connecting your commitment to educating the public about mathematics, debunking mathematical myths, provoking thinking about innovation in mathematics and mathematics education, and so on, back to your experience in the UK twenty years ago? Does the United States now have the same "virus" that attacked the UK—one spread by misinformation and misunderstanding about the importance of mathematics?

Devlin: I certainly see signs of the “cult of ignorance” in this country. Right now we are (hopefully) coming off one of the all-time low points in U.S. history, with a President who seems to take pride in his ignorance. Bush has to an extent legitimized ignorance and made it respectable. The U.S. that I immigrated to in the mid-1980s led the world in science, mathematics, technology, and business innovation, all built on a strong commitment to, in particular, higher education and pure research. We need to rekindle that “Can do, will do!” positive attitude that took us to the Moon. The strength of the U.S. is its huge diversity, and I’m hopeful the next few years will see an upsurge in the standards by which Americans conduct their public lives, both at home and abroad. The U.K. was and is far more homogeneous, and I doubt it will ever recover from the damage done to its education system during the 1980s. I had no alternative than to leave the U.K. (except to simply give up, which is what a lot of my colleagues did). So the two situations are very different. Here, we can recover. Still, my experience in the U.K. has perhaps raised my awareness of just how bad things can get, to an extent that I am not going to stop ringing alarm bells here. A country where a reported 50% plus of the population believes in Creationism clearly has problems.

(3) Yet, the countries that consistently outperform everyone else on measures of students' mathematics and science knowledge—Singapore, Japan, South Korea—are barely represented in lists of top universities for technology/mathematics. U.S. universities dominate in this category. Similarly, Singapore, Japan, and South Korea are barely represented, if at all, among Fields Medal winners. Since 1982, the U.S. and U.K. combined can claim about a third of all winners. Is all our worry in the United States about mathematics education somewhat misplaced, since we seem to still dominate at and above the university level?

Devlin: Two things. First, we dominate in part by immigration. I myself am just one of many university mathematicians who were trained abroad and came here because the working conditions are better than in my home country. If you want to find the world-class academics who were educated in those countries you list, just walk onto any major American university campus. There is nothing wrong with that strategy, which the U.S. has used effectively ever since it built its higher education system in the early twentieth century. But as I argued in my MAA “Flat World” column, that approach may not continue in the digitally connected new world. There is an irony here that our very own university system created the technologies that have given us today’s flat world.

Second, the current dominance we have in high-tech industries has its clear origins in massive government funding for “blue skies” university research in the 60s, 70s, and 80s. Without that level of support for basic scientific research, including mathematics, there will be no base for the next wave of innovation. That wave will come, the only question is where. I came to Silicon Valley in 1987 because that is where the action was, and still is, in my own area of interest. The talent will always go to where the opportunities are best. We can keep it here. But only if we have the vision and the will to make it happen.

(4) I understand that you have worked for the U.S. government, using mathematics to help better prevent terrorist attacks. Are you able to describe one or more fundamental problems that mathematicians are confronted with in this area? What might be covered in a Using Math to Prevent Terrorist Attacks 101?

Devlin: Countering terrorism is a two-pronged issue. Primarily, it’s a political problem. The only long-term solution is eliminate the causes of terrorism. In the interim, the only countermeasure is information gathering. It’s not very effective, but it’s the only thing we can do. My own work was in trying to develop protocols for taking massive amounts of information and reasoning about it. I approached it as the development of a “higher order logic.” That is how, and why, I got into that work; the first twenty years of my career I specialized in mathematical logic. In standard logic, we consider that case where a single individual reasons within mathematics. The reasoning considered is context free, ambiguity free, and linear, with limited information, a well defined starting point, and a clear goal. The problem we looked at (I was involved in a big, nationwide project) involved multiple, interacting agents, reasoning over effectively unlimited information, in a fashion that was generally non-linear, often holistic, and where the goal is not always clear. There is no Anti-Terrorism Math 101; it’s a horrendously difficult research area. I like that kind of challenge, but it’s not one you could build an academic career on.

(5) Okay, now here's a tough one for many people, including myself. I'd like to ask you about Cataglyphis fortis. This desert ant wanders aimlessly away from its nest—for hundreds of meters--in search of food, and then, after finding food, it makes what is pretty much a straight line back to its nest. Not only does it know the exact straight-line direction to travel, but it knows the exact distance to travel as well. And, this is the most interesting part, when scientists have tested the ant by moving it after it finds its food, it follows the same "plan" of traveling back to the nest as though it had not been moved.

Now, you have said that Cataglyphis fortis actually uses mathematics to make this work: "This is not implicit mathematics. The mathematics is being done inside that creature. It isn't being done anywhere else . . . It's doing the math. It's been optimized through natural selection to solve that dead-reckoning problem and to solve it with great accuracy."

How can we say that this ant is "doing" math? Isn't it just a simple matter of keeping track of directions and distances?

Devlin: Yes, it’s a matter of keeping track of directions and distances. (“Simple” is a highly subjective term here so I’ll leave it out.) And what word do we have to describe the way humans do that? Mathematics. The point of my book The Math Instinct, where I describe that example and others like it, was to point out that “doing math” is a description we use for certain kinds of “mental” activities. We do this all the time. We talk about a handheld calculator or a supercomputer “computing” or “solving an equation” or whatever, but of course it isn’t. It’s an inanimate device that simply follows the laws of physics. We interpret its behavior as “doing math” because that’s what it looks like to us. Of course, it looks like that to use because we design those devices to appear precisely that way to us. It’s the same with living creatures. Natural selection works by optimization of behavior, and as a result, living creatures usually exhibit one or more behaviors that maximize their survival chances, and in general the natural way for humans to understand such optimized behavior is as “doing (some particular) math”, because that’s how we solve optimization problems. We don’t know exactly how Cataglyphis fortis determines the direction and distance to get home from any point. But then, we don’t know how humans actually solve math problems either. The explanations we give to our students and the arguments we publish in papers are after-the-fact rationalizations. Once you realize that “doing math” is a description of an activity from the outside, largely based on input-output behavior, the Desert Ant and the Fields Medal winner don’t look that different. (Note that I say “don’t look that different”, not “aren’t that different”.)

(6) This idea, that "the explanations we give to our students and the arguments we publish in papers are after-the-fact rationalizations" is extremely interesting to me. It's one I see eye to eye with, but I don't know that everyone in education feels the same way.

We might very well describe most of mathematics education as composed of "explanations we give to our students." Can we gain some insight into teaching by conceiving of explanations as "after-the-fact rationalizations"?

Devlin: First I should clarify the answer I gave to the previous question. In saying that we don’t know how people actually solve mathematics problems, I’m referring to the genuinely creative act of solving a novel problem, a process that inevitably involves the famous “Aha!” moment when we suddenly see the light. That is why I went on to compare the Fields Medal winner (the mathematicians’ equivalent of the Nobel Prize) with the Tunisian Desert Ant. In contrast, a lot of activity that goes under the name of “doing math” is simply the routine application of well established procedures. That kind of activity is entirely transparent both to the doer and to an observer, and it can be learned and practiced to the point of mastery. Creative mathematics usually involves a lot of the routine stuff, but there comes a point where something new is required. Once the key step has been made, we usually have no difficulty finding a rational explanation for what we have done, so that everything appears like a clean, linear, logical progression. But that is not how we got there!

I think many students give up on mathematics because they don’t see how they could possibly come up with the solutions to problems they see in their textbook, or which the teacher gives on the board, or which some of their classmates produce. What they don’t understand is that the clever argument they have just been presented with was not arrived at by deliberate, rational thought. It was constructed after-the-fact. And so the student misses the crucial lesson that the secret to doing mathematics is not an unusual brain but sheer persistence, trying one thing after another and failing each time until eventually the light comes on.

Of course, people will differ as to how successful they can be in solving mathematics problems and in how long it takes them. But that is a matter of degree, not category. Presenting mathematics as a logical progression gives the totally false impression that such is how it is actually DONE. It’s not. The logical progression is an after-the-fact account.

As it happens, I have just completed a book that illustrates this perfectly. It comes out this fall from Basic Books, and is an account of the origins of modern probability theory. Called “The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern”, it examines the detailed correspondence between those two great French mathematicians in which they developed a solution to a hitherto unsolved problem about computing probabilities in a game of dice. The solution they eventually came up with is so “logical and simple” that today it can be taught to middle school pupils. But as the correspondence makes clear, it took two of the greatest mathematical minds of all time a great deal of effort to find that solution. Moreover, it is clear from what they wrote that the solution seemed “logical” only after they had found it and agreed it was correct! I would like to see this book made compulsory reading for all teachers and students of mathematics. Not because I wrote it. I am just the reporter. But because of the story itself -- how it illustrates vividly the way mathematics is really DONE.

(7) Once Pascal and Fermat found their solution, what became part of our mathematics "vocabulary," in large part, was not the effort that got them there, the "doing," but the end-result of that effort, "the meaning." What we try to impart to students when we teach them mathematics is this vocabulary, this collection of end-results, the "meanings." And, of course, we (and in many cases, students) must "do" some mathematics to get to these meanings, by discussion or through clever arguments.

Is there a danger in confusing the "doing" of mathematics with the "meanings" in mathematics?

Devlin: I’m not sure there is a danger of generating confusion among our students all the time the classroom focus remains solely on the finished product. Rather, the loss is that we leave our students ignorant of mathematics as a living, growing discipline. Mathematics, like the natural sciences, history, art, etc. are about DOING, not knowing. Knowing is necessary in order to do, but the goal is the doing. Concentrating on “facts” to learn makes the teacher’s job easier, and makes measuring “progress” straightforward, but it misses what the subject is really about. So yes, there is confusion, but it is in the way the educational curriculum is designed. I think this confusion will persist until more professional mathematicians take an active interest in K-12 education.

(8) You have written before that teachers sometimes present mathematics using "brittle metaphors." Does this relate in some way to the distinction, if any, between the "doing" and "meaning" of mathematics? Doesn't using brittle metaphors help students avoid in many cases the "totally false impression," as you say, that mathematics is done in a logical progression?

Devlin: The problem with metaphors – and this has been noted many times – is that they are helpful only if the user is aware of their limitations from the outset. Otherwise, the metaphor persists and can cause immense problems later. There was a now famous case a few years ago where a TV crew went to Harvard on Commencement Day and interviewed the graduating seniors and some of their professors, asking them, among other things, why it was warmer in the summer than in the winter. Many of those interviewed (I forget the percentages) said this was because the earth was nearer to the sun in the summer months. This is clearly false (consider the southern hemisphere), so the question arises why so many well educated people got it wrong. All of the students had taken obligatory science courses, and all had at some point “learned” the real reason, namely the inclination of the Earth’s axis of rotation to the plane of its orbit. Why did they not recall this simple fact? Because it was over-ridden by a powerful “rule” we all learn when young children and we come close to open fires, hot stoves, and the like; namely, when there is a heat source, it gets stronger the closer we get. That’s a good thing to learn, since it prevents us from burning ourselves. But like all rules, it has its limitations. Of course, this example is about a rule-of-thumb rather than a metaphor, but the message is the same. You need to be aware of the limitations of the rule, metaphor, or whatever that you apply.

Using a brittle metaphor (multiplication is repeated addition, for example) inevitably leads to problems later, when the metaphor no longer holds but gets in the way of a better understanding of the concept. It’s hard enough grasping the abstractions of mathematics without compounding the issue with brittle metaphors. One problem is that metaphors inevitably lead to natural inferences. For instance, thinking of multiplication as repeated addition leads to the belief that multiplication makes things bigger. This false belief often persists throughout people’s lives. It’s particularly hard to eradicate since it is often something the child observes him or herself, and as we all know, knowledge we generate ourselves tends to stick like glue. When it comes to mathematics, I think it is probably always unwise to use metaphors as “interim definitions”, which is what often seems to be done, since they all break sooner or later. Rather we should present the student the same instances, but as motivational and illustrative examples of, not metaphors for. Mathematics is abstract. It does students no good in the long term to present it as something concrete. Moreover, there is no need to do so. There is plenty of evidence that children can handle abstraction, particularly when the learning is scaffolded by a range of concrete examples.

(9) How can educators and the mathematics community come together to improve mathematics education? It seems that, in many cases, we can't understand what mathematicians want, and, on the other hand, mathematicians don't understand the limitations we face in the classroom. Will it ever be possible to bridge these differences?

Devlin: I hope so. For many years, I think the problem was largely on the side of the professional mathematicians, who did not involve themselves in K-12 educational issues. That is changing, but as it does a new problem is arising, namely the huge resistance of some in the teaching profession who, for whatever reason, resent the involvement of mathematicians, who they think have nothing to offer.

I experienced this reaction first hand recently when I wrote a couple of MAA “Devlin’s Angle” columns on the issue of multiplication not being repeated addition. As you know, it generated a huge reaction on various math ed blogs, much of it negative. On a human level, I can understand the reaction. No matter how gently it is put, for many teachers it comes down to being told that they have been doing it wrong all of their careers. But it is only by examining what we do, reflecting on it, and then learning what needs to be changed and how to do it, that we can progress. None of us is perfect. What matters is how we react to learning that what we believed or have been doing was wrong, or could be done better.

Mathematicians are actually well prepared for this. It is notoriously difficult spotting one’s own mistakes in a mathematical argument of any length, so having one’s proofs shot down in flames by others, sometimes in full public view, is part of everyday life for the mathematician. The case of Andrew Wiles’ proof of Fermat’s Last Theorem is a spectacular illustration of this. He did eventually correct his proof, but it took him over a year after his first argument was found by others to be badly flawed. I suspect teachers are less used to this kind of thing, particularly in the U.S., where there is virtually no regular peer review of classroom teaching, no regular testing of teachers’ knowledge of the subject, and no regular in-service education about developments in educational methods — all of which are commonplace in other professions, including colleges and universities.

(10) You have mentioned that you know little about K-12 math education. Any plans in the future to dive more fully into our crazy little world?

Devlin: Well, it’s not true that I don’t know much about mathematics education. I have probably read more research in the math ed field than many practicing teachers, and I try to stay current. The research institute I direct at Stanford includes a world-famous educational research center, the Stanford Center for Innovations in Learning, so I get regular exposure to the latest research in the subject. I also served on MSEB for several years, and I learned a lot then. But having never taught in the K-12 mathematics system, my knowledge is all theoretical. In contrast, I have been active as a professional mathematician for forty years, and as a result have a wide and deep understanding of the subject from the inside. When it comes to mathematics, I am comfortable with being described as an “expert”. But when it comes to K-12 mathematics, I might know a lot, but I am in no way an expert. It’s possible I overplay my lack of experience in K-12 education more than is necessary, but I believe my record in mathematics (and my Stanford affiliation) gives considerable weight to anything I say or write about mathematics, and I think that to many outsiders that weight would extend (unjustifiably) to the K-12 mathematics education field. Math ed has its own (true) experts -- but I am not one of them.

Go check out Mathway. Pretty cool stuff over there.

Truthiness

This is already almost three years old. But it will likely live on for many years as a perfect metaphor for much of mathematics education.