Text Savvy

Repost: Adding and Subtracting Integers

2010-09-01T20:07:00.002-05:00

This is, without a doubt, THE most popular post at Text Savvy. I re-post it now with this request: Can we please introduce vectors in late-elementary/early middle-school math? Please?

One of the reasons we get these migraines over integers is that, at least up to the point that we as students were actually introduced to operations with negative numbers, we had been taught (correctly) that addition is an operation that describes combination and subtraction describes extraction. We know, for instance, that adding values is like combining collections of objects, and subtracting values is like removing a collection of objects from another collection.

Then we get to integer math, at which point we are asked, judging by present-day treatments in textbooks, to understand the idea that we should be able to, for example, add a "negative collection" to another "negative collection." Or we must throw away and disregard as ridiculous all that "collection" talk.

Mathematics is always described as a beautifully and rigorously universal subject in every detail--when an idea is laid down and proven in mathematics, it applies everywhere and always. But, to my mind, this is not the way mathematics works, and people should really stop spreading this "universal" rumor.

When you add or subtract with integers, you are NOT combining collections or extracting from collections; you are "moving" in certain directions.

As an example, below is a (poorly drawn) submarine at a depth of -5 whatevers. This doesn't mean that depth can be negative (how could it?); it simply means that the submarine's position, in relation to a number line, can be described as -5.

If we add 3, we are adding buoyancy. Because 3 is positive, we are adding positives. Thus, the sub goes up.

If the sub starts at -5, and we subtract 3, however, we are "getting rid of" positives. We are "getting rid" of When we get rid of positives, the sub must sink:

What about if we add negatives? You know what this would do. If we add -3, the sub must sink:

Finally, let's find -5 - (-3). Here we are "getting rid of negatives." When we get rid of negatives, the sub must go up.

"Interesting" Is not Subjective

2010-09-01T18:06:00.006-05:00

How could it be? Anyone with half a brain can see that this is a really poor attempt to interest kids in math.

Far more interesting are videos of water tanks slowly filling up. Or maybe paint drying.

18- and 72-Degree Rotations

2010-08-26T14:08:00.022-05:00

And now, finally onto 18° and 72° rotations. Once again we can get some inspiration from working with triangles.

The 36-72-72 triangle is an interesting one. Take a look at Triangle ABC at the right. Angle A is a 36° angle, angle B is a 72° angle (36 + 36 = 72), and angle C is a 72° angle, so this triangle is a 36-72-72 triangle. And it's an isosceles triangle, with AB congruent to AC.

When we draw line segment BD from point B to side AC, bisecting angle B, we create two similar triangles: Triangle BCD is also a 36-72-72 triangle, so it is similar to Triangle ABC.

This, of course, means that the two triangles' corresponding sides are proportional.

Since Triangle BCD has two congruent angles, it is an isosceles triangle, and the sides opposite these congruent angles are also congruent. Thus, side BC and side BD are congruent, as shown in the diagram at the left.

Interestingly, side AD is also congruent to sides BC and BD because Triangle ADB is also an isosceles triangle (a 36-36-108 triangle). Since BD is congruent to BC, and BD is also congruent to AD, then AD is congruent to BC.

Now we get to do something that mathematicians do all the time--we get to make stuff up and watch people wonder why.

So, let's just make up some values for two of the segments in this diagram: let's say that the length of DC is 1 and the length of BC is x. This of course means that BD and AD each have a length of x as well.

The values that we assign to these lengths is not terribly important, so long as we don't assign values that can't be true. (So, we can't say that both DC and BC are x, for example, because we know that these segments are not congruent.)

Since we know that Triangle BCD and Triangle ABC are similar triangles, and thus their corresponding sides are proportional, we can write a proportion to make the relationship between the sides a little clearer.

So, AB = x². Not very interesting. But look at side AC. We can describe its length as x + 1, whereas side AB has a length of x². Remember that Triangle ABC is an isosceles triangle with side AB congruent to side AC. So, because these two lengths are congruent, this equation must be true:

x² = x + 1
Now that is interesting. The value of x is such that when you square it, you obtain the same value as when you add 1 to it. What is this value? You can see how it is determined at the right.

(If you don't remember, or you haven't seen it before, the third equation at right comes from a technique for solving quadratic equations called completing the square.)

More importantly, though, this number is called the golden ratio. It is approximately equivalent to 1.618034, and it has its own special symbol, &Phi, which is spelled out as "phi" and pronounced "fee."

Okay, so back to our triangle. We can draw a segment which bisects angle A and bisects side BC at a 90° angle. This gives us an 18-72-90 right triangle, which we can work with to determine the formulas for 18° and 72° rotations. (We just have to imagine that the triangle is rotated left, with the 18° angle sitting at the origin of the coordinate plane.)

Recall that x is equal to this special number we are now writing as &Phi, so our hypotenuse, side AC, has a length of &Phi + 1 (which is the same as &Phi²!), and our vertical side has a length of &Phi/2.

We can make these values a little easier on the eyes by dividing both lengths by &Phi/2. This of course gives us 1 for the vertical side, and for the hypotenuse (remember that &Phi + 1 = &Phi² and that dividing a fraction is the same as multiplying its reciprocal):

All that's left is the horizontal side. The Pythagorean Theorem makes quick work of it (remember again, &Phi + 1 = &Phi²).

Now we have the triangle and formulas we were after for our 18° and 72° rotations:

As always, there's the helpful more that has come before: here, here, here, here, here, and here.

Slopes of Perpendicular Lines

2010-08-23T16:11:00.001-05:00

Why do the equations for perpendicular lines have slope values that are negative reciprocals of each other?

Easy. The formula for the 90° rotation of any point (x, y) about the origin can be written as (-y, x):

When we rotate an entire line 90°, we rotate all of the points 90°. The original line has a slope that can be described using any two of its points (x₁, y₁) and (x₂, y₂) with this formula:

When we rotate those points 90°, we have to replace every x with a -y and every y with an x, because, as we mentioned, the formula for a 90° rotation is (-y, x), which gives us

or, rewritten:

which is the reciprocal of the original slope formula multiplied by negative 1.

This of course works for 270° rotations (y, -x) as well, because we just end up moving the negative sign from the denominator to the numerator.

Cinemathematics, the Sequel

2010-08-20T13:51:00.008-05:00

It becomes increasingly difficult over time to offer unique, novel responses to the same old stories that we hear over and over in education.

Here's Cocktail Party Physics with one such story:

A study by Capraro, et al in Psychological Reports (106(1), 49-53 (2010)), which draws on their previous data in Li, et al. (Cognition and Instruction, 26, 195-217 (2008) compares 6th grade students from different countries. . . .

The results from the first two questions I posed from their study (6 + 9 = __ + 4 and __ + 8 = 12 + 5) were surprising/appalling. Only 28.6% of American students got these questions right. The Chinese and Korean rates were in the 90+% range and the Turkish rates were 61% and 79% respectively. . . .

For the "Type C" problem __ + 3 = 5 + 7 = __, American students got the first blank right 23.8% of the time, while the rates for other students were 98.6% (Chinese), 86.5% (Korean) and 60.2% (Turkish). Interestingly, the correct rates for the second blank were much more comparable: 86.7% (American), 97.9% (Chinese), 93.3% (Korean) and 86.0% (Turkish).

And now, unfettered as we are at the moment by what's in the actual paper in question, let's say something a little different from what we normally say in response:

The sixth-grade American students who got these questions wrong were or are mathematical morons.

The little gomers did not get that way all by themselves.

Our failure as educators that these results highlight ultimately has nothing to do with our lack of knowledge about math or our lack of experience teaching. We "know" lots of math, and we impart this knowledge relatively well; we just impart the wrong math.

Traditionalist ideologues will argue that these results are to be expected, since students nowadays are taught to just guess and check their answers and have no experience solving basic problems like these. These ideologues are wrong.

Reform ideologues will argue that these results are to be expected, since students are still taught rote procedures without meaning or motivation. These ideologues are also wrong.

Chris Granade, whose post is referenced in the original, summarizes just what it is these American students are missing:

Fundamental to mathematics is the idea of a relation, which is a formal way of stating that two objects are related in some specific way. For instance, the object "2 + 3" is related to the object "5" by the equality. This notion, however, can hide that something very important has occurred. We have taken a conceptual process, addition, and restated it in terms of a statement about static relations. No matter what I do, I cannot break the relation "2 + 3 = 5." By contrast, if I constrain myself to thinking about the addition process, then it is harder to separate that statement about Platonic ideals from the perhaps imperfect implementation of the addition process. The equality relation, then, tells us about what is.

This is the problem that the study identifies. I've called it cinemathematics.

The American students got these questions wrong because they learn from us throughout their schooling that something like 6 + 9 means "9 happens to 6 to make 15." So, their understanding of the addition looks like this:

This representation may look a little odd, but, as far as our teaching is concerned, all the important parts are there. That is, 9 happened to 6, and the result is 15. The expression 6 + 9 is just a means to get the answer we're after. Once we've got that result, the other numbers become part of the past. The addends are just flat rocks spaced out nicely across a river. Students jump to the first one, then to the second one, and then, taking their cue from the equals sign, jump to the opposite bank of the river with their answer.

Imagine how frightening it must be for students who are educated in this manner to see 6 + 9 = __ + 4! Some students will see the equals sign and just jump, because they don't know any better, so they'll write 15 in the blank. Others may be able to tell that the opposite bank is farther away than normal but won't have a clue what to do next, so they'll just guess. Still others may see the "+ 4" sign in the distance and deduce that someone put it there as a hint for what to do next, so they'll write 19 in the blank.

And we haven't even discussed those students who don't know how to jump from rock to rock accurately in the first place. All of the errors described above come from students (who are taught by teachers) who KNOW HOW TO ADD!

I suppose I should go read the paper now.

Update: Pffh. I don't even need to read this stuff anymore to know what's in it. But I will anyway.

Falkner, Levi, and Carpenter (1999) reported from their investigation in a single school of the problem "8 + 4 = a + 5", all 145 sixth-graders filled the box with 12 or 17. . . . According to Carpenter, Franke, and Levi (2003), students may have three different misconceptions of the equal sign: equal sign may mean "the answer comes next" ignoring the rest of the problem (p. 10), that is, 8 + 4 = [12]; students may "use all the numbers" (p. 11) such as 8 + 4 + 5 = 17, arbitrarily restructuring the sentence; or they may put 12 in the box and "extend the problem" (p. 11) as 8 + 4 = [12] + 5 = 17.

75- and 15-Degree Rotations

2010-08-19T09:23:00.004-05:00

In this post, we get to what is close to the end of how far our initial perspectives can take us with regard to determining rotation formulas.

To recap, we have derived formulas for 90° (180°, 270°), 45°, and 30° rotations (Click on the degree measures to go to the relevant post. Click on the pics to enlarge):

90°: (-y, x)

45°:

30°:

The pattern to note above--the one I asked about in the previous post--is that the hypotenuses of the triangles can be found in the denominators in the formulas, and the other side lengths of the triangles can be found in the coefficients of x and y in the formulas.

In fact, for both 30° and 45° angles, the rotation formulas can be written as follows:

where, for the time being, opposite side means "the side that is directly opposite the rotation angle in question" and adjacent side means "the side that is not the hypotenuse or the opposite side."

Though we don't really know it yet, this is the formula for any rotation about the origin. That is, if we replace the coefficients above with their somewhat more abstract and useful equivalents, sine (sin) and cosine (cos), then we have the formula for any rotation (cos&thetax is read as "cosine of the angle times x"):

(cos&theta(x) – sin&theta(y), sin&theta(x) + cos&theta(y))
But that's getting ahead of ourselves. Let's wrap up this basic series by finding the formula for a 75° rotation:

Since we know the formula for any 45° rotation and any 30° rotation, we can just substitute the "x"s and "y"s of one formula for the x's and y's of the other formula to find the formula for a 75° rotation. Then, as we saw before, we can just flip around the coefficients to find the complement's rotation (15°):

Old Fraction Division Art

2010-08-15T13:17:00.001-05:00

A Pattern Emerges

2010-08-12T15:42:00.004-05:00

So, there seems to be a nifty little pattern emerging here. In fact, it's so nifty that it's not only a pattern. It's the pattern we're looking for:

Can you spot it? You can catch up here, here, and then here.

30-Degree Rotations, Part II

2010-08-11T12:36:00.003-05:00

Okay. So now that we've got the right perspective, we can work on a formula for 30° rotations about the origin.

Let's take a look very quickly at two of the important representations we used to determine the formula for a 45° rotation about the origin:

From our 45-45-90 triangle, we were able to construct a function table, which took our diagonal side length as an input and returned a horizontal and vertical location (the height and width of the triangle, which had the same length) as an output.

And now that we have our 30-60-90 triangle, we can construct a similar function table.

Once we know the length of the diagonal segment (the length of the x-arm of our "claw"), we can simply divide by 2 to find the vertical location (the y-coordinate) of the end of the arm. To find the horizontal location (the x-coordinate) of the end of the arm, we multiply by the square root of 3 and divide by 2.

So, once again, we take our point at (3, 2) attached to our "claw" and rotate it--this time 30°--about the origin. Using our function table, we can determine the coordinates of the end of the x-arm:

When we turn our attention to the y-arm of our claw, we run into a bit of a snag: the y-arm actually goes up and to the left at a 60° angle, not a 30° angle.

When we rotated our point 45°, we could use the same function table to convert the x-arm length and the y-arm length to coordinate points, because both the x-arm and the y-arm were oriented at 45° angles to the grid. Now it seems that we have to come up with another function table--one that deals with 60° angles--to work with the y-arm of this claw.

Ah, but that is not the case! The triangle we are working with now has a 60° angle. All we need to do to get the right perspective here is to flip this triangle around so that the 60° angle is sitting at the origin:

You'll notice, I think, that the diagonal segment remains in the same position, while the height and width of the triangle have been transposed. This means that the columns "Horizontal Location" and "Vertical Location" in the function table for the 30° angle can just be transposed to get the function table for the 60° angle.

Using this information and the process we used before, we can determine the final location of the rotated point:

And, of course, the formula for any 30° rotation about the origin:

30-Degree Rotations, Part I

2010-08-09T12:01:00.005-05:00

So, back to rotations. Last time, we found a formula for determining the coordinates of the image of any point on the coordinate plane rotated 45° (counterclockwise) about the origin. That formula is

where x and y are the coordinates of the original point.

What helped us out tremendously in coming up with this formula was the fact that there is a functional relationship between the length of a line segment oriented at a 45° angle and the horizontal or vertical location of one of its endpoints on the coordinate grid. Once we know the length of any 45° diagonal line segment, in order to find the horizontal or vertical location of a specific endpoint, we can simply divide the length by the square root of 2.

In other words, we were able to come up with a formula because of this triangle . . .

. . . and because we know that all similar triangles have the same ratio of side lengths. For this triangle (a 45-45-90 triangle) and all similar triangles, if we divide the vertical side length by the horizontal side length, we should always get a ratio (a slope) of 1. From there, all we need is the Pythagorean equation to come up with the square root of 2.

It is natural to wonder whether we can, likewise, identify a constant ratio (a slope) for 30° angles. And of course, with 30° angles, it gets trickier:

As you can see, a line oriented at a 30° angle does not hit gridline intersections as neatly as does the 45° angle line. But we can get some sense of the slope we are looking for by noting where this line does cross intersections. For instance, it comes very close to crossing (14, 8), which gives us a slope of 8/14, or about 0.5714. The point (24, 14) is not as good, but it is close, and it gives us a slope of 0.58333... And (38, 22) seems almost spot-on. At that point, we see a slope of about 0.5789—a number that is, in fact, only about 15 ten-thousandths off from the number we will soon figure out.

As accurate as this sloppy method is, however, it doesn't help us figure out the numbers (the vertical and horizontal side lengths) we can use every time to arrive at it. And, of course, without my having said so earlier, we shouldn't know at this point how accurate this ratio is anyway.

So, how can we figure out exactly what these side lengths should be every time? Let's take a look first at how we might have done it for the 45° angle.

We could have started by noticing that what we were dealing with was a square, whose side lengths are all equal, divided into two congruent triangles by its diagonal. So, we could narrow the three variables of the Pythagorean equation down to two:

The Pythagorean equation then gives us b² + b² = c², or 2b² = c².

Solving for c, we get c = b × sqrt(2).

Plugging in that all-important number 1 for b (because we like to think of the axes on the coordinate grid as being labeled in intervals of 1), we get that 45-45-90 triangle mentioned above:

For 30° angles, let's start by taking an equilateral triangle--a triangle with three congruent sides and three congruent angles (each 60°)--turning it on its side by rotating it 90° counterclockwise, and drawing a line through its center as shown below.

The line through the center divides the "top" 60° angle into two 30° angles, and it also divides the base into two congruent segments, each of which is half the length of each of the original sides. If we ignore that bottom right triangle and call these new half-side-lengths b, then we have

Once again, we've narrowed it down to two variables instead of three. Now we just need to figure out what a is in terms of b:

And once again, we can plug in the all-important number 1 for b to get our side lengths:

This means that the slope for any 30° angle will be the same ratio as 1/sqrt(3), or 0.577350269 . . ., which you'll recall is very close to any of the ratios we were getting when we determined this slope from the grid. But now it's exact!

Celebrate the Exposition

2010-08-04T13:52:00.002-05:00

Matt has a wonderful video here, which I've also embedded below, that shows a simple way to visualize the imaginary number line.

It is a brief yet superb exposition of a not-so-intuitive concept in mathematics, requiring nothing more than the presenter's hands and speaking voice.

Note also the very real difference between addition and multiplication that becomes clear in this presentation.

Repost: 45-Degree Rotations

2010-07-27T15:35:00.001-05:00

I'm hoping to pick up this topic again soon. But it's been a while since I've posted about it. So here's a little reminder:

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 the horizontal or vertical location of one of its endpoints 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 the horizontal or vertical location of a specific endpoint, 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's Complicated

2010-07-25T16:37:00.001-05:00

Kate's post here is brilliant, to me anyway.

There are more than a few patterns in the quotes she provides, but certainly Kate identified the most visible one in the comments:

Education is messy and there are no easy answers.

It's true! We're dealing with human beings. So it's not going to be much like programming. If education were even close to a science today, it would still be a social science, where a correlation hovering around 0.6 is cause for champagne popping.

But how we--and that's we as in not just teachers, not just administrators, not just publishers, not just high school, not just elementary school, and so on--how we deal with this ambiguity is more important than the simple fact of its existence.

Luckily, the writers that Kate links to seem to be dealing with it smartly. Here's an example from Math Be Brave (italic in original, boldface is mine):

I feel renewed in my enthusiasm for doing this job knowing that when I feel like there aren't clean-cut answers, it's because there really aren't any. Now I feel free to simply enjoy asking the questions and trying to find answers, without feeling like there's something wrong because things aren't already nice and tied up. When I feel like the job is too hard, it's because it really is. When I get confused about a problem I face in my curriculum development or my pedagogy, I can just sink my teeth into the discovery filled process of searching out a solution. Like doing mathematics.

But, distressingly, we don't seem to know how to deal with it all that well in education. Too often we turn to what sounds like a god-of-the-gaps–type argument:

There is a gap in understanding of some aspect of the natural world.

Therefore the cause must be supernatural.

So . . .

Education is messy and there are no easy answers.

Therefore everyone should be doing it this way.

or . . .

Education is messy and there are no easy answers.

Therefore no one way is fundamentally any better or worse than any other way.

or . . .

Education is messy and there are no easy answers.

Therefore suggesting that there are some general answers is ridiculous.

or . . .

Education is messy and there are no easy answers.

Therefore clean, tidy, expository teaching is to be avoided.

The problem with the god-of-the-gaps argument is not the conclusion; it's that nothing follows from that single premise. You can't jump from total uncertainty to some kind of knowledge. If you don't know, you don't know. The end. If you want to know, you have to try to find out.

Another problem is that, often in my experience, the person making a god-of-the-gaps argument for education sneaks in a hidden premise:

Education is messy and there are no easy answers.

(And that's the way it will always be!)

This makes any conclusion that follows sound a little more reasonable. After all, if we presume that education's messiness is inevitable, then it seems like we have no choice but to move on, ignore or reject criteria for knowing, and just make stuff up and try stuff out.

But that's still a poor way to deal with the messiness of education. Even if there's little hope of pinning anything down, we--a professional community devoted to improving mathematics education for everyone--should be working to reduce ambiguity, to reduce uncertainty, not using ambiguity and uncertainty to bludgeon each other.

Make Arguments, Not War

Listen, no one likes to be told things like what to drive or how to raise your kids, in general. Sure, you can hand out suggestions just for me about what kind of car I should buy. You can use what you know about me to kindly suggest a vehicle with, say, a lot of seating capacity or power windows, or good fuel efficiency, etc. Or, you can be an agency which simply lists features of cars, and I can look at your information and decide for myself *what I can do with that.* But don't tell me I should be picking out a vehicle that gets good gas mileage by making abstract, reasoned arguments about oil or terrorists or global warming. That's annoying, right?

And, sure, you can come over to my house and make polite suggestions about a kind of crib to buy or a discipline style that worked for you. Or, you can be a parenting magazine which simply lists all kinds of ideas for raising children, and I can look at your information and decide for myself *what I can do with that.* But don't tell me I should avoid disciplining my children with spankings by making abstract, reasoned arguments about encouraging children to become hitters or distracting children from learning correct behavior. That's just annoying.

As a result of this stubborn annoyance with abstract, reasoned, prescriptive arguments, two things happen: (1) more people drive guzzlers and more people hit their kids than otherwise would and (2) we stop caring about and making abstract, reasoned, prescriptive arguments to the people in our community. Thus, argument-free incentives and *celebrity* public-service announcements become some of the only ways we can influence people's behavior.

In education, too, general arguments about actual teaching (not ideology) are hard to come by, not only because of our embrace of messiness as a reason; but, let's face it, it's also because, "Screw you. Don't tell me what I should or should not do. You don't know me or my kids." So nothing really changes. And we stop making those arguments. And nothing changes even more.

Instead we talk about my teaching, what worked for you, how this affects me, what worked for my kids--it's a kind of Facebook-style discourse where we only get to press the "Like" button, make brief, positive comments, and talk about what we are doing now.

That's not to say that those kinds of discussions aren't helpful for some people. But we need something more.

Draw an Overbar for Repeating Digits

2010-07-18T19:36:00.003-05:00

Surani asks if I know of other kinds of shortcuts like the one I presented here.

The answer is that I'm not sure because I don't know specifically what folks are looking for. If you want to play around with more field codes, you can google "field codes, Word." Also, Word has a complete list. Just go to the Insert menu, click on Quick Parts, and select Field...

Another field code I use creates an overbar for a single repeating decimal digit, though it requires a little tweaking to make it look right:

Step 1: Type 0.1 and then type Ctrl + F9 to insert a field code. Type EQ \O(6, (that O is an "oh," not a zero). Six will be the digit that repeats.

Step 2: Click on the Insert tab at the top of your screen. Then click on Symbol all the way to the right. Select More Symbols...

Step 3: In the box next to Character code: type in 00AF (those 0's are zeros). This should take you to the macron symbol. (The name of the symbol is shown right above the AutoCorrect... button.) Click on the Insert button at the bottom.

Step 4: Finally, you're going to want to make that macron superscript and 2 points larger than the type size you're using. To do this, click and drag to select the macron in the field code. Type Ctrl + Shift + = (this is the shortcut key for superscript). Then click on the Home tab and adjust the type size up 2 points. If you're in Times New Roman 12, change it to 14 and press Enter. Type Alt + F9 to toggle between the code and the finished product. (You may have to type that sequence more than once when you first do it.)

Assigning a shortcut key to the macron symbol makes this whole process a little quicker and easier.

There are ways to extend this process to make overbars for lengthier chains of repeating digits and also for indicating line segments and the like. And, of course, macros make all of this really easy. But, as I said, I'm not sure what people are interested in, specifically.

Math: Your Backstage Pass

2010-07-14T19:56:00.002-05:00

Download the files for this blog post here (for Word '07; downloads on click) or here (for Words '97-'03; downloads on click).

I really like the idea of using MS Word to do some math activities. I'm at a stage where I don't know how much I can do with it, or whether it would be interesting to my kids.

But I'm intrigued by the idea. First of all, a Word document is probably one of the least "mathy" interfaces one could think of even though there's a lot of math behind it. That's not only good for taming the wolverine, but it also hits on what I think is probably the most important lesson for students trying to learn mathematics: math is a backstage pass. Sure, you can be satisfied to go to the concert, hear the music, have the fun--but how much better is it to tell your friends that you talked to the band, found out how they party, heard stories in person about how they met, figured out something about how they do what they do . . .

Math--and, you know, really anything worth studying--is almost always under the surface, the background music, something you have to seek out and find--and once you get it, you're a VIP.

In Word, math is like that. It's so there if you want to look for it--or if someone will show it to you--but it's not really there if you just show up, just like the "real world."

I could, and will, go on about it later. But just take a look at this first, very simple task--not really the simplest I could think of, but one that will give you a sense of some of the possibilities.

Here again is the Word 2007 link (downloads on click). And here is the Word 97-'03 link (downloads on click).

Of Wolverines and Lambs

2010-07-12T21:22:00.003-05:00

I should just admit that this Virtual Conference on Soft Skills is producing some thought-provoking posts.

Here is Dan's and then a follow-up. And this is from Kate.

I wanted to discuss for a moment this quote from Dan, and then I think I have to get back to some mathy posts:

Most educators, I think, understand instinctively the tension between access and correctness, the difficulty of extending one while insisting on the other.

There is a demographic, though, that feels little tension along that line. Call them "wolverine wranglers." These people handle dangerous animals like you and I can't believe. They're gifted and there aren't a lot of them. Their most striking feature, though, is their conviction that wolverines are dangerous and you are not taking that seriously enough. Work up the nerve to approach a wolverine and the nearest wrangler will remind you of all the ways that could go wrong.

First, I know a lot of "wranglers" who feel this tension between "access and correctness": Eric Mazur, not really a friend but, hey, a Facebook friend at least : ); Professor Devlin, and many other editors, curriculum coaches, university professors, business leaders, computer science folks, and mathematicians. But, as I said before, I haven't known nearly as many educators who I would characterize as feeling this tension at all.

And that actually makes sense, I think, for people in general. If you're really into the most efficient way, the most proper way, the best way to tie your shoes, it's going to be a struggle for you to provide access to that knowledge--that very personal knowledge--to a small child. It's not easy to know something really well and explain it to someone who doesn't. On the other hand, if you know something about tying shoes, and you know that you know this, then access is mostly what you've got. There is little tension at all.

A Fields Medal winner with no experience in front of children (who's that?) trying to explain fractions to a fourth-grade classroom has about a billion factors to work with (considering they're human) to determine if she is providing sufficient "access" for the children in that context. A fourth-grade teacher who could make any material accessible to anybody but does not know fractions very deeply has far, far, far fewer ways of knowing whether what he's saying is correct or not in that context.

Just because we're human, social animals, the person who knows nothing but correctness always starts off better at the access than the person who knows nothing but access starts off at the correctness.

So I don't really buy the idea that educators en masse feel this tension and some "demographic" doesn't. I think it's probably actually the reverse--that most folks who aren't in teaching but really know the content have a much better sense of this tension than do most educators.

Second, look at this from the comments in Dan's follow-up post linked above:

Devlin is a deep and interesting dude, but I think he’s got some wrangler tendencies that subvert his overall demystification program. Do you remember how he kept pulling mathematical rank in that flap about multiplication and repeated addition?

Wow, "pulling mathematical rank." What exactly is that rank up against when it comes to teaching mathematics? We do remember what we're teaching, don't we? Am I on The Daily Show and this is like a joke where someone complains that the top economists in the country are "pulling economic rank" and tellin' us somethin' we don't want to believe in the first place?

How on earth does someone complain about "pulling mathematical rank" unless he's been lead to believe that actual mathematics is completely at odds with teaching mathematics?

Oh, I know how.

We're Taking the Plunge

2010-07-10T19:17:00.003-05:00

We're taking our kids out of the public schools and homeschooling them. At least for this year. We'll see how it goes.

Very little of this has to do with my disaffection with current educational practices, although, I have been consistently ticked off with the invented spelling baloney that my third-grader going into fourth grade had to deal with at his school. (Just a brief survey of online material about invented spelling brings up a surfeit of naturalistic fallacies--descriptivism to prescriptivism fallacies--that are the coin of the education realm. My son's DS game Scribblenauts seemed to have more effect on his spelling ability than did an entire year of third-grade spelling instruction. See, cuz there you have to spell a word correctly in order to get what you want; he seems to have suffered no trauma as a result.)

That said, I'm kind of relieved that my kids will, at least for a time, not be subject to this obsession with access above content--something that is fully represented by invented spelling. Despite what the author of that post says, I don't see any tension for teachers between access and correctness (or, in my terms, inclusion and precision.) It's ALL about access. It's ALL about inclusion. It has precious little to do with correctness or precision.

I mean, seriously, F--k the Exposition: Can anyone do any better than that to describe the silliness of this "inclusion ethic"?

To continue my baseball analogy, these educators who practice "access" are great middle relievers. When you're down 0 - 6 in the bottom of the 6th, they're the folks you want to bring in to make all those "access" pitches. They've got kids who are terrified of math. They're not going to win it for you every time, but those are the kind of pitchers you want for that specific situation. But, as an assistant coach of Team USA, I don't want those middle relievers--especially those who use TV-show-creator quotes rather than actual evidence to make their case--to be telling my starting pitchers or closers how to pitch, because their jobs are to WIN the game, not to try to make up for losing.

I don't want my kids or any kids to be down 0 - 6 at any point in their educational careers. So I don't like educational philosophies that start with that score.

One Person's Intuition . . .

2010-07-06T21:44:00.006-05:00

Best example ever (emphasis in original):

And the incompetent apologist of a teacher, who is sympathetic to creationism himself, isn't doing his job, which is to explain to them exactly how biology explains these phenomena. Instead, he makes excuses: "How could I say to a student, 'your ideas are trash'?"

It's not hard. One student at the end [of the video] says this: "How can like an African-American person evolve from a white person? We're different skin."

Hey, student! Your ideas are trash!

It's funny how when the focus is on "real" biology or math, every student is different and we are being unfair to them by asking them to subscribe to one viewpoint. But when the focus is on students' intuition, we assume that everyone's intuition is the same, and there's just no reason to treat any of those intuitions as freaking crackerjack crazy.

You must be able to tell your students when they are wrong if you're going to teach at all.

Yup.

Ooooh. Here's more:

I have a very bad brain for the purposes I want to use it for. It's pretty good, but prone to awkward mistakes, for deciphering behavioral cues and inferring intent in my conspecifics, which is still a useful skill, but other functions, like the ability to search out fruit and tubers, or to coordinate a hunting party, or to detect predators lying in wait, I've let slide out of a lack of utility. I'd like a brain that could hold more than half a dozen numbers at once in my head, or that wasn't prone to perceptual errors, or that could process written information a bit more efficiently than this linear, one-word-or-phrase-at-a-time parsing. I wish I had a memory that could accurately record events and scenes, rather than storing a few key hints and reconstructing the rest. I'd like a brain that was actually evolved for doing mathematics naturally, rather than requiring years of discipline and training to acquire the skill artificially.

We really do have very untrustworthy brains. The capacity for abstract, rational thought is a byproduct of general cognitive capacity, and doesn't come easily to any of us. We have to work at it.

Yeah, "work at it." Sounds like work.

Show and Tell

2010-07-02T13:36:00.001-05:00

This right here provides a nice outline to and analogy for the prickly discussions that have been happening here lately:

The mantra "Show, don't tell" has become stock advice for fiction-writers. Janet Evanovich considers it to be one of the most important principles of fiction: "Instead of stating a situation flat out, you want to let the reader discover what you're trying to say by watching a character in action and by listening to his dialogue. Showing brings your characters to life." "It is the difference between actors acting out an event, and the lone playwright standing on a bare stage recounting the event to the audience."

Here's an example:

[Wrong:] I'll never forget how I felt after Fido died. I was miserable.

[Right:] Whenever puppies in the pet store window distracted me from the serious business of taking him for his walk, Fido growled, his little ears flattened against his scruffy head. Yet he always forgave me. Even after his hearing and sight faded, when he felt the leash click on his collar and smelled fresh air, he still tried to caper. He's been dead for three months now. This morning I filled his water bowl all the way to the top --just the way he likes it -- before I remembered.

No doubt, one could likely find a few other reasons for preferring showing to telling, but those would be outnumbered 100 to 1 by the number of times advice-givers cite reader engagement as the reason.

Of course, there's the other side too:

"Show, don't tell", like all rules, has exceptions. According to James Scott Bell, "Sometimes a writer tells as a shortcut, to move quickly to the meaty part of the story or scene. Showing is essentially about making scenes vivid. If you try to do it constantly, the parts that are supposed to stand out won't, and your readers will get exhausted." Showing requires more words; telling may cover a greater span of time more concisely. A novel that contains only showing would be incredibly long; therefore, a narrative can contain some legitimate telling.

Note that the reasons given in this last paragraph have mostly to do with the story, rather than the reader—moving to the "meaty part of the story," thinking about the "parts that are supposed to stand out," concision, etc.—though the fact that reader exhaustion was mentioned should not be overlooked.

The clear implication here is that too much telling—or telling in the wrong places—may damage reader engagement, while too much showing—or showing in the wrong places—may hurt the story you're trying to convey.

This debate in fiction writing maps pretty easily onto mathematics education. But there we seem to have less of a concern for the story than we do for the students, so the exceptions to the "show, don't tell" rule have to fight harder to get noticed or are considered relatively unimportant or are ignored altogether.

In Defense of Problem 11

2010-06-27T17:08:00.001-05:00

After criticizing Dan Meyer's TED presentation and follow-up post on more than one occasion, Mr Meyer provided some reasoned and measured responses (in short comments at my posts linked above), which dealt primarily with two key issues that I had not before considered important to this discussion: (1) my blog etiquette and (2) Meyer's improvement of one single problem in one single textbook.

In my defense, the reason I had not before considered these issues important is because they're not. To frame them as though they are is to turn Mr Meyer's presentation into nothing but show-and-tell (just an azza argument) and I and a few other miscreants in the back of the room into a-holes making fun of some poor kid's new stuffed animal.

So, for both our sakes, I choose to frame the discussion differently--as a serious discussion with serious arguments for education reform. Unfortunately, in that light, Meyer's presentation doesn't appear so wonderful.

Let's start with Meyer's water-tank problem:

How long will it take you to fill the tank? [picture of water tank]

And now the problem that Meyer criticizes in a textbook published by Key Curriculum Press:

11. A water tank is in the form of a regular octagonal prism. The base octagon has side length 11.9 cm. The lateral edge of the water tank is 36 cm.

a) What is the surface area of the base?

b) What is the volume of the water tank?

c) If you pour water into the tank at a rate of 1.8 oz/sec, how long will it take you to fill the tank?

The criticism here is that problems like these set up students to be blank fillers rather than "patient problem solvers" by giving them not only all the information they need up front to solve a problem but also the substeps (the "breadcrumbing") laid out in order.

Note where this problem falls in the overall sequence of the lesson. It is Problem 11. This is a practice problem set in the context of an application. It is presumably not the goal to have students try to figure out what information they need and do not need in order to calculate how long it will take to fill a certain volume. Rather, it is to assess their understanding of what they just learned--either how to calculate volume, how to work with volume and rates, or both. Plunk Mr Meyer's problem down in the same location in the sequence, and it becomes a practice problem as well, having as little to do with intuition as the textbook problem. Students will certainly be required to assemble more information and possibly follow a set of steps of their own design in order to solve the problem, but as Problem 11, this will be a matter of measuring and remembering (or looking back to the instruction page(s) to remember how to calculate volume), not one of actively constructing anything.

Now, I don't think Mr Meyer made his water-tank problem the eleventh problem following instruction. It seems unlikely that one could talk about the "level playing field of intuition" or "guesses" as he did unless one made this problem the opening problem to introduce the instruction. In that case, students would have to make guesses, use their intuition, discuss, and actively construct some of the mathematics to help them get closer to an answer--because they wouldn't have any other way of solving the problem.

But, see, from that perspective, the comparison seems a bit unfair. I might just as well find a problem from an EOG exam and complain that problems like that don't promote "patient problem solving." Well, yeah, they're not supposed to. Where those problems fall in the sequence of instruction, not just their content, is important to consider when evaluating their usefulness or effectiveness.

Recall, also, this quote:

Most of my teaching strays only four or five degrees from the path beaten by my textbooks. We do a full WCYDWT [what can you do with this] unit — the kind of home run that I post here — perhaps once every two weeks.

That's what . . . say, as high as 25% of the school year? Do any of the textbooks mentioned in the presentation spend 25% of their print real estate on open-ended, performance assessment-type problems? Are there any suggestions for those kinds of activities in the teacher's guide? in their ancillary material? Do they make explicit suggestions in their respective forewords that teachers are forbidden from changing the problems in any way or from straying away from the order laid out in the pages?

It seems like it would only be fair to answer those questions before throwing all textbooks under the bus with "the way we teach math in this country virtually guarantees that they won't retain it" or "such a calamity for society."

What about those nasty substeps that turn our kids into impatient problem solvers? Have you read Sweller's work on his cognitive load theory? Check out some past issues of the journal The Educational Psychologist for some work around this theory. One strong suggestion from that research is that substeps such as the ones in Problem 11 above can help reduce intrinsic cognitive load, which makes it easier for students to form and strengthen schemas (i.e., learn) at the beginning.

Have you seen this obliquely related video from ~~TV show creator~~ cognitive psychologist Daniel Willingham which knocks down the oft-repeated claim that students today need to multi-task in order to be engaged?

There's also this research which I posted about a few years ago that suggests that students' integration of new learning has little to do with how much they like math or enjoy math or fear math.

It would have been a welcome change to see any one or all of these "counterarguments" brought up (at least considered) and then knocked down in the original presentation or any one of the follow-up posts by anybody.

But that is just not how we roll in education.

In Case You Wanted to Know . . .

2010-06-25T19:43:00.004-05:00

Open a Word document or start a new one. Type "The probability of rolling a 1 on a fair number cube is about". Then hit the spacebar. Then do the following:

Type Ctrl + F9. You should see some curly brackets. Then press Backspace just once. Type this exactly as you see it:

EQ \F(1,6)

The only space should be between "EQ" and "\F".

Hit the right arrow key once and then Backspace once. Right click on the little inscrutable phrase you just typed and select Toggle Field Codes.

Finally, insert a period at the end of your sentence, because you're not a caveman.

P.S.: Play around with this one too if you're doing some square root work:

EQ \R(,9)

If you want it to be a root other than a square root, you just type in the number in front of that comma.

Repost: Framing: Curriculum by Analogy

2010-06-14T08:05:00.000-05:00

I'm going to give this a shot, even though I have much more to say about it than I care to write in one post. The topic will certainly pop up again.

I mentioned here that the theme I have been building up to in my recent posts has to do with framing. Mark C. explains this concept nicely:

"Framing" is not spinning. And even the most vocal opponents of framing are doing framing in their arguments. It's unavoidable. Whether you like it or not, framing is an inescapable part of communication. Framing is, quite simply, a term for describing the way in which you present information or arguments. If you're communicating, that communication takes place in a frame. The people who advocate framing are simply saying that it's important to consider how you frame your arguments: that the way in which arguments and information are presented affects how they're going to be received.

To draw the obvious parallel, similarly, all mathematics education takes place in frames of our own design, and the way we present mathematics affects how it is received by our students.

Okay, granted. However, it is one thing to consider one's audience and the framing of one's argument. It is quite another to set aside important truths in one's argument and call that framing. This version of framing is what PZ and others have been arguing against:

Now we all know that we have to dole out the technical details appropriately—I've misgauged an audience a few times myself—but our possession of the data is one of our greatest strengths—if we're going to start equating explaining the evidence to puppy-strangling, we might as well hang it up and go home right now. . . . It's like suggesting that we could do a better job of promoting science if we could only hide that sciencey stuff.

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

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

Very few seem to have considered the idea that the reason kids are in school is because they don't know anything and their ways of thinking need improvement.

Dan Meyer Is Ambitious

2010-06-13T20:22:00.005-05:00

From his March, 2010 TED talk:

Be less helpful. Because the textbook is helping you in all the wrong ways. It's helping you--it's buying you out of your obligation for patient problem solving and math reasoning, to be less helpful. . . . Whatever your stake is in education--whether you're a student, parent, teacher, policy maker, whatever, insist on better math curricula. We need more patient problem solvers. Thank you.

And now from a blog entry dated June 7, 2010:

Most of my teaching strays only four or five degrees from the path beaten by my textbooks. We do a full WCYDWT [what can you do with this] unit — the kind of home run that I post here — perhaps once every two weeks.

Ambitious. And I think he was being dishonest in his TED talk.

Repost +: Cause and Purpose in Text

2010-06-12T14:26:00.004-05:00

A neat study in Educational Studies in Mathematics (Link) points to a familiar yet disturbing characteristic of mathematics textbooks.

In the study, samples from eighteen different elementary mathematics texts used in the UK were analyzed. Researchers were interested in how often the texts provided "reasons" for the mathematics they presented—that is, how often the texts explained a mathematical idea (or solicited an explanation from students) in terms of purposes and causes:

There is evidence that the strength and number of cause and purpose connections determine the probability of comprehension and the recall of information read (Britton and Graesser, 1996) and can indicate a teacher's or writer's concern for reasons (Newton and Newton, 2000). Even when writers withhold reasons and provide activities to help children construct them, they cannot assume that this will happen. In books, a concern for reasons, therefore, is often indicated by their presence. Clauses of cause and purpose can, within limits, serve as indicators of this concern (Britton and Graesser, 1996; Newton and Newton, 2000). . . . Clauses are commonly used as units of textual analysis (Weber, 1990). Amongst these clauses, clauses of cause (typically signalled by words like as, because, since) and purpose (typically signalled by in order to, to, so that) were noted.

Having these data, researchers then compiled the "reason-giving" statements into seven different categories based on their "explanatory purpose." The results from the study are shown below. The labels used are my own.

The first four categories (working counterclockwise from the largest section) were considered non-mathematical. Forty percent of the clauses in the sample provided "the purpose of and instructions for games and other activities intended to provide experience of a topic"; 22% provided "reasons in stories, real-world examples and applications and in descriptions of the basis of analogies"; 1.3% provided "the purpose of text in terms of its learning aims and objectives and could be described as metadiscourse"; and another 1.3% of the clauses "justified assertions of a non-mathematical nature." Nearly 65% of "reason-giving" in the texts was non-mathematical.

The next two categories were considered mathematical. Just over 13% of the clauses in the sample provided "the intentions of procedures, operations and algorithms for producing a particular mathematical end"; and just over 9% "attempted to justify [mathematical] assertions (e.g., 'It is a square number because 5 × 5 = 25').

Clauses in the final category (symbols) were considered mathematical or non-mathematical, depending on whether or not the symbols in question were mathematical ones. These clauses provided "the purpose of certain words, units, signs, abbreviations, conventions and non-verbal representations."

This is not to say that writers explain only through clauses of cause and purpose. They may use other devices to the same end and this analysis does not detect them. There is also what the teacher and the child do with the textbook to support understanding, perhaps through practical activity (Entwistle and Smith, 2003). This approach does not detect these directly. The aim of the study, however, is to consider the potential of the children's text to direct a teacher's attention to reasons.

It is important to remember that the results do not tell us that, for example, 40% of the clauses in the sample were instructions. They tell us that 40% of the "reason-giving" clauses were used in instructions. The graph above shows how "reason-giving" statements were used in the textbooks.

Although these results are generally supportive of the conclusions drawn in the study, they also provide further support, especially in light of these values . . .

Clauses of cause ranged from nil to 3.96% of text (using clauses as the unit) with a mean of 0.68% (s.d. 1.08). Clauses of purpose ranged from nil to 8.03% of text with a mean of 4.77% (s.d. 2.08).

. . . for the long-standing contention that contemporary elementary mathematics textbooks are, primarily, classroom management tools.

---------------------------------------------------------------------------------------------------

And it seems that most of our debates about mathematics education have to do with classroom management or behavior management, rather than with how to teach. How do we explain to children, for example, (or, if you like, how do we get children to come to know) why inverse operations are used to solve one-step equations? It seems to me that we need to talk about how to tell the truth (or, if you like, how to get students to come to know the truth), how to make it clear, where that explanation (or discovery) comes in the sequence, and how it should be grouped with other concepts. That's--once again--precision, clarity, order, and cohesion.

But that's not what we talk about.

We talk about "compelling and real". Or, we talk about motivation. Or we talk about entertainment and the TV show Deadwood. Or we talk about mastery. Or basic facts. Or conceptual understanding. All of these discussions are, at their root, about how students should orient themselves to the study of mathematics--how students should behave, how we manage the behavior of students when they learn about mathematics, not about teaching and learning mathematics.

I really couldn't care less whether we never ever tell kids about inverse operations and let them discover the concept and reason for themselves or whether we tell them everything in detail straight up and then drill them with fifty problems and call it a day. Do students have the truth about inverse operations/one-step equations? Is it clear to them? Was the teaching timed right, such that it can support both current learning and future learning? Did we group the concept of inverse operations appropriately with other topics so that students can make the best connections?

No classroom-management orientation to education reform can really answer these questions. Your Hot vs. Crazy graph doesn't really answer these questions. Your vociferous insistence on mastery doesn't really answer these questions. And "I did it this way" answers don't really answer these questions.

Reference: Douglas P. Newton, Lynn D. Newton (2006). Could Elementary Mathematics Textbooks Help Give Attention to Reasons in the Classroom? Educational Studies in Mathematics, 64 (1), 69-84 DOI: 10.1007/s10649-005-9015-z

Meyer, Mazur, and the Azzas

2010-05-10T17:09:00.003-05:00

Whenever I watch political debates or read newspaper editorials or blog comments, I cringe when I encounter "azza" arguments. They start like this:

"As a veteran, I . . ."

"As someone who lives there, I . . ."

"As a teacher, I . . ."

"As an engineer, I . . ."

And the reason they make me cringe is that this particular opening phrase seems to invariably release the author or speaker from the hard work of anticipating and then dealing with disagreement with what follows and from wading more deeply into an issue.

Abortion? No problem: "As a doctor, I . . ." or "As a Christian, I . . ." How about capital punishment? Ha! Piece of cake: "As a police officer, I . . ." or "As the mother of a death-row inmate, I . . ." Math education? Wow, that's easy: "As a teacher, I . . ." or "As the co-writer of such-and-such textbook, I . . ."

So long as we include azza, we don't have to think about all the other azzas that have a stake in what we're talking about. We are merely and humbly inserting our opinion, and we can continue blabbering until the sun comes down, unconstricted by counterargument or criteria other than our tiny stake in the larger issue. We can sleep easily; we said what we wanted to say, and let whoever or whatever is in charge sort out the right way to go, if there is one; who the hell knows.

And azza is a perfect defense against criticism too: got a problem with what I said? Hey, I was just giving *my* perspective, right?

So, take a look at the videos below and tell me, are these just azza arguments?

Dr. Mazur's talk at least has some science behind it. I don't think anyone can say as much for Mr Meyer's presentation. And that's probably good enough to get on CNN, but it really shouldn't be good enough for a community of educators committed to doing what is right to teach children mathematics.

Here's Dan in response to a comment I left on his blog:

I can imagine a lot of teachers, editors, and publishers conceding the point but rebutting it with those practical considerations. I can’t imagine the ideological argument for the textbook’s water tank problem, unless we’re operating under very different assumptions about the point of math applications.

So he has a supposed airtight ideological argument that is completely impractical. Good for him. Bad for the rest of us.