Lisa's Method: A New Arrangement For Calculating Slope

Lisa’s Method: A New Arrangement For Finding Slope

The National Council of Teachers of Mathematics continues to advocate for more student-centered mathematics teaching (NCTM, 2000). One of my dreams as a mathematics teacher and tutor is to have a student come up with an innovative approach to doing or thinking about some mathematics. And one of my nightmares is hearing of or seeing a student do something creative and be ignored or slapped down by an insensitive or ignorant instructor who fails or fears to consider that a student’s ideas about something might be of value. One of the worst aspects of “sage on the stage” mathematics teaching is its propensity for minimizing opportunities for students to do any thinking in class that strays from the teacher’s directed and generally very beaten path.

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. My student, Lisa, had been working her way through a computer-based first semester geometry course and came to me unsure about how to find slope. As I anticipated, her problem centered the question of where to put the x- and y-coordinates in the slope equation. She was given the points (3, 2) and (-2, -4) with a picture of the graph, so I began by asking if she could tell from the graph whether the slope would be positive or negative. Once she was clear that the slope of a line that went up from left to right had to be positive, which was indeed the case with the line in question, I modeled for her how to draw a right triangle by extending a vertical segment down from (3,2) and a horizontal one to meet it from (-2, -4). I then asked her to find the lengths of the two legs of this triangle (5, and 6, respectively): an advantage for many students of this graphical approach is that lengths are positive regardless of the presence of negative numbers amongst the coordinates, and it is relatively easy for them to see or be convinced that adding the distances left and right or above and below the axes, respectively, as all positive numbers will get the correct lengths.

We then turned to the question of which of these numbers was the numerator of the slope. I offered the idea that slope represents a ratio of change between the vertical change of the line, positive, negative or zero, to its horizontal change from left to right, using the notation Dy/Dx, which Lisa responded to as something she knew from her physical science courses, stating, “Yes, that was how Ms. Acree [her science instructor] showed me.” I wasn’t sure, however, if Lisa saw that this was just a compact way of showing what we had looked at graphically, as well as indicating what calculations had to be done. When I asked if she knew what to do next, she said, “I’m not really certain.” I wrote Dy/Dx = (y2 - y1)/(x2 - x1) and asked her if that made sense. At that point, she said, “Oh, now I see why I keep getting the problems wrong: I’m putting the x’s on the top.”

Of course, this error is one all-too-familiar to mathematics instructors around the world. Countless students have lost untold numbers of points on exams due to this simple and understandable error. After all, x comes before y in the alphabet. And the x-coordinate precedes its partner y-coordinate in every ordered pair. Why wouldn’t it “come first” when writing the ratio for slope?

While we can all sagely agree that students “shouldn’t” make this error based on the underlying meaning of slope, as I pointed out to Lisa before going to notation, it is useless to speak about what is supposed to happen given what so often does. Much as it is important to try to get students to think clearly about the meaning of the mathematics they write and calculate with, it is difficult to ensure that this particular error, one I sometimes start to make myself if I’m not careful, does not occur.

To check her understanding, I posed a second problem to Lisa, giving her the points (3, 9) and (1, 1). She solved the problem graphically first, but was still a little unsure about doing it algebraically, so I did the usual set-up, writing m = (9 - 1)/(3 - 1) = 8/2 = 4.

Lisa looked at what I’d written down and suddenly took the paper and wrote something down, saying, “Couldn’t you do it like this?” (See Fig. 1)

Fig 1: Lisa’s Method

After looking at this for a few seconds, I thought that I understood what Lisa’s intentions were. However, I could see problems with her notation. To check, I asked what the symbols between the numbers in parentheses meant and she made clear to me that they were dashes, not subtraction signs. I asked about the slash and she said, “Oh, that means that you divide the two into the eight to get four, which would be the slope.”

This was what I thought she meant, but I asked her if it seemed possible that most people would be confused by what seemed like indicated subtractions, and if she felt that math teachers would be very distressed by the idea that what she’d written at the bottom should be read the way she intended. She said, “Oh, I see what you mean about the dashes. But I don’t see what’s wrong with the bottom part.”

I said that in fact she appeared to be claiming that two divided by eight equaled four rather than one-fourth. But I believed that I saw an easy fix to what she had done. I then wrote below her figure (see Fig. 2)

Fig 2:Teacher’s Version

This seemed more consistent with what was intended and didn’t use any non-standard or misleading notation. I asked Lisa if what I had done made clear what she wanted; she agreed that it did. The logical continuation would then be to perform the indicated division to get the slope (See Fig. 3)

Fig. 3 Teacher’s Version Completed

This correctly completed the solution for finding the slope of the line that contained the two given points.

Lisa said she agreed with the changes I’d made. At this point, I said: “I have never seen anyone find slope like this. Did another teacher show you this?” She answered, “No, but when I saw how you did it, it just seemed to me that this would work. It seemed natural to me.”

I asked her explain why this seemed like a good way to her. She said, “Well, we always did subtraction problems like that: it seems easier to me when the numbers are lined up this way.”
I told her that I thought she might have invented an original approach that made sense, and that I was going to write up what she’d done for publication so that other teachers would have a chance to share it with their students. I think she believed I was pulling her leg, and though I tried to make clear that I was serious, even when I told her again at the end of class that I was definitely impressed by her approach and was going to write it up, she appeared skeptical.

What seems useful about Lisa’s method is that it solves the typical problem of forcing students to put the y-coordinates “first,” out of alphabetical order, when they may still be struggling to simply get a handle on what slope is and how to interpret it. So many students learn (and are often taught) mathematics in a mechanical, procedural way, and in the case of calculating slope, there is something counterintuitive about how we set up the ratio, even though we would like it to make sense. Clearly, solving the problem using her approach avoids putting students in a position where many will lose points on exams and become frustrated with this vital part of dealing with linear equations and their graphs. In the long run, I would expect that when they have a firmer understanding of slopes as ratios of change and of what it is that is changing relative to what (dependent variable relative to unit change in independent variable), they will become more comfortable and effective with the traditional approach. And of course, some students will be perfectly fine with the usual way of finding slope and will likely eschew Lisa’s alternative. However, as another possible way to set up this important calculation, it seems to have promise for struggling students.

Of course, I don’t mean to suggest that her method (or any approach) is a panacea for all possible student difficulties. Naturally, students still need to know how correctly subtract signed numbers. While my work with Lisa suggested that she was comparatively clear on integer arithmetic (based on work we did previously), most of my students are typical in struggling greatly with both the procedures and concepts in this regard. That is one reason I would strongly suggest that teachers have students find slopes of lines with points in the first quadrant and/or on the axes defining that part of the Cartesian plane. Further, it seems an aid to reviewing the requisite arithmetic to then have students solve problems graphically (using right triangles as I demonstrated to Lisa) before doing the symbolic arithmetic when including points in the other three quadrants.

For Lisa, the question of finishing the problem with dividing a smaller positive integer into a larger one was unproblematic. Further experimentation with numbers of opposite signs or with two negative numbers caused her no difficulty either, and she successfully concluded in cases that resulted in both proper and improper fractions that as long as she reduced to lowest terms, no further adjustment was needed to the results she calculated with her method. However, it is not at all obvious that this would be the case for all students, so further investigation along these lines will be important.

If nothing else, what Lisa has done can be shown to students as an example of creative solutions to simple but annoying sticking places for many teachers and students. I’ve personally never been happy with the “rise over run” idea, though I offer it to students as a mnemonic, because the word “run” doesn’t communicate horizontal motion to me clearly, even though it does to some students. I also worry sometimes that it locks students into thinking of slope too literally as restricted to a graphical interpretation, rather than keeping open the idea of slope as a ratio of change between dependent and independent variables.

Lisa seems to have spontaneously created something that for me might solve a nagging pedagogical problem: how to teach students to calculate the slope of a line from two given points without having to do something that for many does not come naturally. The key here for me (and I hope for teachers reading this article) is that I was open to seeing Lisa’s ideas and that we had established the kind of relationship where she felt comfortable sharing what seemed like a spontaneous burst of creativity with me. While it is far easier to be receptive to student thinking in the one-on-one teaching I often do as a consultant, coach, guest teacher, or in full-time work as a Title I mathematics teacher, I believe it is vital that teachers in more typical group and whole-class teaching situations remain open to opportunities to explore student ideas. As the work of teachers like Ball (1993) and Lampert (1985) shows, teachers must make dozens of decisions about such moments, and it is easy to lose them under the pressure to “cover the curriculum.” But if we are serious about teaching mathematics in ways that value and promote independent student thinking and active participation in a mathematical learning community, we cannot afford to miss the unexpected. At worst, we need to note when students have provocative, promising mathematical ideas so that we can revisit them later with their authors and other students. It’s not always possible in the heat of a lesson to see whether an idea warrants taking a detour from the official planned route. Even if something needs to be put on the back burner, it is important that students know that their ideas are valued, and vital that the teacher get back to the student to discuss and explore such insights and speculations further. And of course, it is necessary that teachers have adequate mathematical knowledge to recognize the potential significance of their students’ ideas (Ball, et al, 2005). But the more teachers take their students seriously as sources of rich mathematical ideas, the more likely it is that students will engage more fully in mathematics and indeed produce surprising and fruitful ideas.

References

Ball, D. L., Hill, H.C, & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator.

Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. Elementary School Journal, 93 (4), 373-397.

Lampert, M. (1985). How do teachers manage to teach?: Perspectives on problems in practice. Harvard Educational Review , 55 (2), 178-194.

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000.

Pinning The Blame On The Wrong Donkeys: A Response to "The Problem With Math Education"

The latest post on the blog "The Math Mojo Chronicles" states in part:

The Problem with Math Education

Filed under: math education, public schools; Author: Brian; Posted: December 8, 2008 at 8:18 pm;

Well, of course it’s not the problem, just one of many, but here it goes…

Somehow along the way, people got the feeling that math is supposed to always be right, and that math teachers are supposed to know all the answers.

Math has gotten the reputation of being an authoritarian science. I don’t think this is the fault of mathematicians, I think it is the fault of many math educators who have tried to turn mathematics from an art and science into a “subject.”

Math education is all too often about “standards” and “curricula” that students take “tests” about that they are “graded” on.

If Archimedes was to take a high-school math test today, he would be unfamiliar with the jargon, and would find little value in the trite little multiple-choice and partial credit nonsense that passes for assessment.

On the other hand, he could run rings around the math teachers with his knowledge of actual mathematics, and could twist their pedagogical dogma into moebius bands.

Trying to shove math into the education industry’s rubric is one of the worst educational crimes I can think of. School math seldom has anything to do with actual math, except for the very rare cases where an inspired person is doing the teaching. And when that happens, that person is invariably in trouble with the administration....

Click here to read the entire blog post.

A Mathematics Educator's Reply

I posted the following response to Brian's blog entry this morning:

As usual, I find your use of language problematic in a variety of ways. First, let’s take:

“Math has gotten the reputation of being an authoritarian science. I don’t think this is the fault of mathematicians, I think it is the fault of many math educators who have tried to turn mathematics from an art and science into a ’subject.’”

My gripes here are first as an ex-English teacher: do you mean “authoritarian” or, more likely, “authoritative”? The latter makes more sense in the context you’re speaking about, but the former is possible if you mean to imply that mathematics is somehow about bossing people around. Of course, mathematics can do no such thing, though mathematicians and those who use mathematics certainly try to do that a great deal of the time.

Second, I would be awfully careful about how you use the term “math educators” these days. In the “Math Wars” era, that’s come to mean almost exclusively people with advanced degrees, generally doctorates, in mathematics education, an academic field focused on research on math teaching and learning, as well as upon teacher education. And I seriously doubt that it’s those folks who are centrally responsible for the problem you mention.

You need to look at who tends to hold major sway these days (since the late 1990s) in many states and on a national basis regarding state curriculum standards. It isn’t for the most part mathematics educators, though occasionally they get to have some say. Look, for example, at the Presidential National Math Panel. Exactly how many “mathematics educators” were on it? I count three. Liping Ma, Deborah Ball, and Skip Fennell. The rest were mathematicians, cognitive psychologists, one middle school math teacher with a long track record of hostility to what we can loosely call “reform mathematics education,” and a bunch of other folks whose presence on such a panel would be puzzling were it not one called under the watch of George W. Bush, not exactly known for giving much of a damn about actual educational or scientific expertise (see THE REPUBLICAN WAR ON SCIENCE for starters). The folks who undid two decades of reform work in California were, to a man (and I use that word advisedly) mathematicians who got in through a political back door to substitute their own state standards for the ones the legitimately appointed panel had submitted (see The Politics of California School Mathematics: The Anti-Reform of 1997-99 by Jerry P. Becker and Bill Jacob).

You then go on to talk about tests as if the folks promoting them are, once again, math educators rather than a narrow band of anti-reformers who are virtually never math ed researchers or teacher educators, but rather politically-motivated activists, some with little or no established knowledge of mathematics or its applications (but VERY loud voices), some mathematicians with a conservative educational viewpoint, and all of them openly dismissive of “mathematics education” and “mathematics educators.”

You really seem in this entry to be terribly confused about whom is to blame for the reduction mathematics teaching and learning to many things that have very little to do with what mathematics as a living field, as an activity, as a “doing” are about. It’s not mathematics educators.

Of course, many of the points you make about curriculum, assessment, etc., are a function of a system that is not really very much in the control of the folks you are inadvertently or purposely attacking here. The education “industry” is another loaded term that seems to get used to mean different things to different folks. There are of course people in education because it’s a job, a way to make a living (however limited) with what they may find minimal risk or effort. There are increasingly people in it as an explicit business (see, for example, major publishers, obviously, but less obviously perhaps those companies run by folks with no interest or training in education who are coming in through the charter and voucher routes to grab up as many education dollars as they can, often without even a passing glance at kids as any more than the income they represent. I speak from bitter personal experience in this regard, having worked for and with enough of these companies at the bottom rung (know vaguely as “teacher”), but with eyes open to the corporate memos that speak of the “business we’re in,” focusing upon enrollments and count days, but with never a mention of curriculum or, heaven forfend, learning or what students are actually getting from being at their schools. It’s uglier than you can imagine unless you’ve looked at such management companies with real scrutiny, and from the outside it’s damned hard to get access to what they’re up to. (Of course, I can provide quotations from REAL corporate memos to back my assertions).

Many on-the-job math teachers and mathematics researchers and teacher educators are not of this ilk. They’re hard-working people with a passion for mathematics and for kids. They actually interested in seeing kids become effective with and excited about math and its applications, beauty, and power. And they are trying to find math curricula (read “books, lessons, materials, tools”) that are effective in engaging and educating kids. They’re also interested in teaching: how to better get key aspects, concepts, techniques, procedures, applications, implications, and pleasures of mathematics across to kids.

Your penultimate paragraph talks about questions. And of course that’s a huge part of what math and education are about. But questions in math and in math class are pedagogical issues, not purely mathematical ones. They’re the stuff many math educators but too few mathematicians and virtually no business people or politicians are concerned with in this thing we’re all fighting about. You have identified some of the problems, but your loose language and lack of focus once again appears to be pinning blame on the wrong donkeys more often than not.

What Can Division Tell Us About Multiplication? (Part 2)

Preface

This post has been sitting in the hopper since the beginning of November. Due to a number of conflicting writing projects and a major presentation I've been preparing to give on Dec. 1., I'm at a point where I need to either fish or cut bait with this now, pending a lot of reading that I don't have time to do immediately. My decision today is to post this "as is," with the major investigation of existing research left hanging fire until I have time to do it real justice. So for some readers, this second post may disappoint, not getting to what I personally think is the nitty-gritty that must be explored to determine whether Keith Devlin's heuristic columns on multiplication and repeated addition are simply provocative pieces that fail to lead to urgent rethinking of how we present multiplication to students in K-12, or if in fact he has put his finger on something mathematics teacher-educators and elementary and middle school teachers need to take very seriously indeed. I continue to try to occupy a semi-moderate middle ground, not happy with some of the inflammatory language that has been used by various participants in the debate (including Devlin himself, of course, though I believe he did so with the intention of stirring up things; regardless of intention, he clearly succeeded in provoking a lot of discussion, some of it actually quite productive), but unwilling to buy the simplistic viewpoint that there's nothing underpinning Devlin's ideas that need to be looked at carefully from the perspective of what we teach kids and how regarding multiplication.

Part 2

Previously, I explored two major interpretations of whole number division, the partitive (sharing) and quotative (measurement) "models." My purpose was to see how well these interpretations work for whole numbers (though they do not exhaust possible interpretations), and also how these interpretations are almost, but not quite adequate for integers. The problem comes when one tries to interpret dividing a positive integer by a negative integer using either of these ideas: there's simply no way to make sense of them. Of course, looking at division as the inverse of multiplication solves that problem readily. And there are other interpretations of multiplication and division that might do equally well.

But my initial purpose for looking at division and interpretations of it was to see what light, if any, it could shed on meanings for multiplication. In particular, my concern was to see whether "repeated addition" sufficed as a definition for multiplication, given Keith Devlin (and other's) insistence to the contrary. Minimally, he asserted in his most recent column on this continuing controversy that teachers should not leave students with the impression that multiplication is "nothing more than repeated addition."

One hint that multiplication can't simply be repeated addition comes from the above-mentioned breakdown of the two interpretations of division of whole numbers when applied to integers, particularly the failure of the measurement model, which can also be viewed as "repeated subtraction." Though I'm very fond of using that viewpoint to help convince people that division by zero cannot result in a real number, it's clear that repeated subtraction will not result in any meaningful interpretation for 6 / (-2) = -3. There are no -2's to subtract from 6. The colored chip model, which works so well for the same examples that partitive and quotative interpretations work for with integer or whole number division, will not make any sense for this problem. Adding zero pairs and then subtracting (removing) pairs of negative (red) chips gives the wrong result. It seems that division "ain't no repeated subtraction" after all, or it least it isn't JUST repeated subtraction.

I'm not sure that I mind all that much that the repeated addition and repeated subtraction ideas don't work all the time; ditto for the chip model and the partitive/quotative interpretations. After all, these aren't definitions so much as they are interpretations or physical models: the map is not the territory. And it's pretty clear that while it's possible to stretch the repeated addition/subtraction ideas to positive rational numbers, the same difficulties will arise for signed rationals, and the entire thing will collapse when one looks at irrational numbers. I won't try to address complex numbers here, as they are well beyond the scope of K-5/6 arithmetic.

By saying that I don't mind, what I mean is not that I would abandon these notions or not use them in elementary mathematics teaching, but rather that I would insist that kids need to see OTHER ideas about multiplication and division, other sorts of models, with every reasonable effort made to ensure that: A) elementary teachers understand that "repeated addition/ subtraction" does NOT suffice for discussing multiplication/division with students; and B) that students will NOT be left with the mistaken belief that these operations are nothing more than repetitions of addition and subtraction.

So What The Heck IS Multiplication?

Don't continue if you're expecting a definitive or single answer to the above question. For one thing, I'm not a mathematician, professional or otherwise, and I would never claim to have a sufficiently deep background in mathematics to offer 'the' answer. For another, a major part of my point of view is that there are rarely, if ever, single definitions of that type, even in mathematics. Perhaps Keith Devlin is right in that regard to ask:

Why not say that there are (at least) two basic things you can do to numbers: you can add them and you can multiply them. (I am discounting subtraction and division here, since they are simply the inverses to addition and multiplication, and thus not "basic" operations. This does not mean that teaching them is not difficult; it is.) Adding and multiplying are just things you do to numbers - they come with the package. We include them because there are lots of useful things we can do when we can add and multiply numbers. For example, adding numbers tells you how many things (or parts of things) you have when you combine collections. Multiplication is useful if you want to know the result of scaling some quantity.

But then again, this quotation might just be Professor Devlin's cry in the wilderness, or perhaps it's an example of much ado about nothing. Is there really such a difference between multiplication and repeated addition?

Consider the following, offered at "TheMathPage.com," a page maintained by a former colleague of mine at Borough of Manhattan Community College, Lawrence Spector:

Here is the most general definition of multiplication:

Whatever ratio the multiplier has to 1
the product shall have to the multiplicand.
Consider this multiplication of whole numbers:

3 × 8 = 24

The multiplier 3 is three times 1; therefore the product will be three times the multiplicand; it will be three times 8.

Similarly, to make sense out of

½ × 8,

the multiplier ½ is half of 1. therefore the product will be half of 8.

Proportionally,

As the Multiplier is to 1, so the Product is
to the Multiplicand.

Or inversely,

As 1 is to the Multiplier, so the Multiplicand is
to the Product.

The Product, then, is the fourth proportional to 1, the Multiplier, and the Multiplicand.

To be perfectly fair, Spector's earlier discussion of multiplication (of whole numbers) is grounded in precisely the "multiplication is repeated addition" interpretation that Devlin decries. But in his "most general definition," he clearly sees multiplication as deeply related to proportional reasoning, not repeated addition. And this is exactly what Devlin and others see as a serious matter for concern and inquiry.

Given the critical importance of proportional reasoning in the US middle school math curriculum (or so the NCTM Standards and many mathematics educators and state curriculum frameworks suggest), and given that there is disturbing evidence that American students as a group do less and less well in mathematics as they get older, it's important to consider their performance on proportional reasoning in particular. Here, I'm not just talking about by comparison with students in other countries, since there are reasons to question whether such comparisons are accurately comparing similar groups; some critics of TIMSS results have claimed that some of the higher-scoring countries are testing a narrower band of students than does the United States. I'm talking about the general sense that the percentage of mathematically proficient students declines in America as students continue through middle and high school and into post-secondary education.

It seems reasonable to suppose that if such a decline is real, that one reason that middle school students underperform compared with elementary students is that they are not well-positioned to tackle the central curriculum focus for their grade band: proportional reasoning. (I would suggest that a continuing weakness in integer arithmetic goes along with and exacerbates this shortcoming). And I suspect that part of that weakness in handling proportional reasoning can be found in how multiplication is taught and (mis)understood.

What remains to be done, then, is to look carefully at the research that is already out there on children's mathematical thinking, as well as that of older students (middle school, high school, and college students in, say, freshman calculus) and see if and how their thinking about multiplicative situations, proportional reasoning, etc., is impacted, for good or ill, by their introduction to whole number multiplication. Is there in fact evidence that some/many/all students who are introduced to multiplication as repeated addition and left continuing to think of it solely or primarily as repeated addition are hamstrung when it comes to doing and/or understanding multiplication as they move to other kinds of numbers?

And, in due time, I hope to explore some of that research here. More to come. Perhaps much more.

What Can Division Tell Us About Multiplication? (Part I)

In my tireless pursuit of the question raised this past summer by Keith Devlin in a series of columns in the MAA MONTHLY ("It Ain't No Repeated Addition" (June 2008); "It's Still Not Repeated Addition" (July-August 2008); and "Multiplication and Those Pesky English Spellings" (September 2008)) regarding the potential harm of leaving students with the belief that multiplication is nothing more than repeated addition, I am doing extensive reading about how children learn about, think about, and are taught to think about mathematics and the connection amongst and differences between the operations. One leg of this journey brings me to consider division. There are a number of ways to think about division, but the two "big ideas" looked at in school mathematics are the partitive or "fair sharing" model and the quotative or "measurement" model.

Partitive Division With Whole Numbers

As a quick overview or review, let's take some simple examples of both. Consider, for example, a simple sharing situation:

"Alexa has 30 cupcakes and wants to share them equally amongst six people. How many cupcakes does each person get?"

Note that this situation has several key elements, perhaps most important being the idea of "sharing equally" or "fair sharing." Without that stipulation, there's no clear-cut mathematical solution and instead any distribution one chooses that results in 30 cupcakes being shared in any manner whatsoever is "correct." But in this situation, there is only one right answer, which of course is that each person gets five cupcakes.

The other things to notice are that we have a fixed total (30), a known number of equal shares (six), and an unknown SIZE of each fair share which is to be determined.

How might a lower middle school student solve this problem? One way would be to model the problem with 30 counters of some sort. The student could keep placing one counter into each of six piles until all the counters are exhausted. She would then count how many there were in one pile and thus determine the size of the equal shares.

A similar approach would be for the student to count "one" during the placement of each counter on the first round, "two" during the second round, and so on through the fifth and final round, whereby the student would know that the last number that is repeated, "five," is the desired answer.

Students might all try subdividing strategies in which the original pile of 30 is subdivided into three equal groups of ten, and then each of those groups could be split into two equal groups of five, resulting in the final answer of five (in six groups). There are several variations on this sort of scheme depending on the factors of 30.

A more seat-of-the-pants approach would be to take a rough estimate of the share size, say "four," give that many to each of the six piles, and note that there are enough left to give one more to each resulting in a share of five. Or if one guessed too high, some chips could be redistributed until the piles were equal. Clearly, this sort of approach becomes less and less viable as the number of items to be shared and/or the number of shares goes up.

Quotative Division With Whole Numbers

Turning to the quotative or measurement model, let's take an example similar to the previous one, with appropriate differences:

"Zane has 30 cupcakes and wants to give six each to a number of his friends. How many friends can he do this for?"

Note that here we still have a known total (30) but now the size of a fair share is given (six). What is not known is the NUMBER of such shares possible of the given size with the original total. Though the solution here (five) is the same as that of the previous problem and the numbers given are the same, they do not represent the same things (in the case of the two smaller numbers). In the partitive case, we know the number of fair shares and seek to find the size of one fair share given the total; in the quotative case, we know the size of a desired share and need to find how many such shares can be made from the given total.

Possible solution strategies include taking a pile of 30 counters and subtracting six at a time until the pile is exhausted, then counting the number of piles to get the desired answer. Thus, it may be clear why the quotative or measurement model relates to the idea of division as repeated subtraction: one can repeatedly subtract a given number (the divisor) from the dividend until either zero or a number smaller than the divisor is reached. The number of such subtractions is the quotient, and any left over is the remainder. This relates the quotative model to the so-called division algorithm (more accurately a theorem):

given two integers a and d, with d ≠ 0

There exist unique integers q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes the absolute value of d. The integer q is called the quotient, r is called the remainder, d is called the divisor and a is called the dividend.

The relationship and difference between partitive and quotative models of division is not always made clear in elementary mathematics education, and often their existence comes as news to elementary school teachers and education students. When asked to create division word problems, teachers and students often find it difficult to do so for the quotative model, even though the natural approach to solving such problems, repeated subtraction, would appear to be "obvious" and to link to the widely-held belief that multiplication is "nothing more than" repeated addition. The imbalance between ease of constructing partitive examples of real-world problems and difficulty of construction quotative examples becomes exacerbated for many teachers and students when dealing with rational numbers in fractional form. There is also an interesting conceptual break-down of the partitive and quotative models when dealing with division of integers, as we shall see shortly.

A Side Trip To Division of Fractions

One notable point about division of rational numbers in fractional form must be made before turning to integer divisions. It has become increasingly well-known to those interested in elementary mathematics learning and mathematics teacher education that many Americans have difficulty with the concept of division of fractions. While fraction multiplication is a fairly straight-forward proposition (the product of a/b and c/d is equal to ac/bd, so that a standard error in fraction addition and subtraction, which is to add/subtract numerators and denominators from left to right, is correct when it comes to computing the product of two fractions: simply multiply numerators and denominators, respectively, from left to right.

When it comes to dividing fractions, however, there is often both procedural and conceptual difficulty (not that the simplicity of computing products with fractions is an indication that conceptual understanding is common): the old standby rhyme, "Ours is not to reason why, just invert and multiply," not only indicates that many teachers and learners either give up worrying about students' understanding of the mathematics or never cared about it to begin with, but also isn't even a guarantee that students will compute correctly: after all, it matters enormously WHICH fraction is inverted before doing the multiplication.

From a purely symbolic perspective, one approach is to write a/b divided by c/d as a complex fraction - a fraction that has one or more rational expressions in the numerator and/or denominator. One then shows how to simply this fraction by multiplying both numerator and denominator by the reciprocal of the latter. Thus, (a/b) / (c/d) = (a/b)(d/c) / (c/d)(d/c) =
(a/b)(d/c) / (cd/cd) = (a/b)(d/c) / 1 = (a/b)(d/c). Thus, we show a justification, though informally, for the "invert and multiply" procedure for fraction division.

A second commonplace is to ask students to construct real world situations that would be correctly addressed by solving something like 2/3 divided by 1/2, and/or to simply compute the correct quotient. A very significant percentage of both elementary students and elementary teacher education students will produce problems that represent 2/3 * 1/2 and give 1/3 as the desired answer. These failures have been cited as proof that the groups in question lack procedural knowledge of rational number division, conceptual knowledge of that process, or both.

I suggest that while it is true that many people have difficulty with rational number division, it is important to look at how the above sorts of questions are posed before drawing conclusions about any given individual or group. It is particularly risky to draw to strong of a conclusion if the question is posed orally with 1/2 as the divisor. It seems plausible and even likely that a significant number of people are so used to being asked to find 1/2 OF a number that they will mishear/misconstrue a question asking them to divide by 1/2. I think it would be important for formal research, classroom assessment, or instruction to be sure that students have the opportunity to respond to written as well as oral questions, and that unit fractions, particularly 1/2, are not the only divisors they are asked to work with. I would be more confident that there is a significant conceptual and/or procedural problem if someone were asked to construct a problem that would be solved by dividing 3/4 by 2/3 for which they gave something like, "Dominique has 3/4ths of a pizza and wants to give 2/3 of it to his brother. How much should he give him?" or gave the answer for the computation as 1/2, than if s/he gave similar responses to the previously mentioned problem, particularly if the first one were only given orally.

Division of Integers

It is heuristic to have in-service and pre-service elementary teachers, as well as their students, to try to apply on their own the partitive and quotative interpretations of division that they employed successfully with whole numbers to division of integers.

Take the simple example of dividing the four combinations of +/- 6 by +/- 2. First, consider dividing +6 by +2. It is to be hoped that individuals will conclude that both the partitive and quotative interpretations can be used sensibly. Six balloons can be shared equally by two friends, with each getting three balloons. Or six cups of flour can make three batches of cookies if it takes two cups of flour per batch.

Consider next -6 divided by +2. It does not make sense to use the quotative interpretation here. There are no positive 2's to be taken from -6. However, the partitive interpretation can be used sensibly, as there is no difficulty dividing -6 into two equal groups of -3. So, for example, if two friends wish to share equally a deficit of 6 dollars (-$6), each gets a share of -$3.

Next, consider dividing -6 by -2. Here, the partitive interpretation fails because it does not make sense to talk about -2 groups or sharing equally among -2 friends, etc. However, the quotative interpretation can be used sensibly: If you want to share a deficit of six dollars (-6) so that each person gets a share of -$2, then 3 people are needed.

Finally, we come to +6 divided by -2. If one attempts to use the partitive interpretation of division, the same problem mentioned in the previous example arises. If one tries the quotative model, we run into the problem raised in the second example: there are no -2's to remove from +6. So how does one interpret 6 / -2 intelligibly? One obvious solution is to talk about division as the inverse of multiplication. Thus, x = 6 / -2 asks the question, "What number times -2 equals 6?" and the answer is -3, which for the moment we will assume is something already established with students. There is no problem finding the correct answer.

Does this mean that partitive and quotative interpretations of division are wrong and should not be taught? Well, it's clear that they make perfect sense in whole number arithmetic. And they cover 3 of the 4 possibilities for integer division where the dividend is an integer multiple of the divisor. It seems reasonable to believe that by the time students reach integer arithmetic (typically 5th or 6th grade), most will be ready for some of the necessary algebraic concepts needed to make sense of all cases, with notions of inverses, identity elements, and so forth coming into play. It doesn't seem like too much of a stretch to think that many will be comfortable with the question given in the previous paragraph as a way to think about six divided by negative two.

No model or interpretation of a mathematical idea is likely to be perfect. But that should not be shocking. The model/interpretation is NOT the mathematics. It's a way of helping students to think about the mathematics. A model or interpretation may in fact be precisely what a given student will carry away with him from his education, though this is not necessarily "the" goal of mathematics teaching: certainly most educators hope that every student will have the opportunity to experience the beauty and power of mathematical abstraction for its own sake, as well as a way of thinking that can deepen as one encounters increasingly complex and challenging mathematical ideas. We should not leave students with the mistaken impression that the map is the territory, that the model or application or interpretation IS the mathematics. But absent evidence that exploring models and interpretations is harmful to student learning or that it inevitably limits their long-term mathematical growth, it seems reasonable for teachers to continue to use them thoughtfully.

What next?

The title of this piece suggests that there is something in all this talk about division, interpretations, models, and applications that can inform us about multiplication and whether it suffices to leave students thinking of it as "no more than repeated addition." In the next part of this piece, I hope to help shed some light on just that question.

Does This Mean That The Phone Company Did Kill Kennedy?: Even More on Keith Devlin, Multiplication, and Repeated Addition

A recent post on the continuing conversations, fights, and debates inspired by Keith Devlin's recent trio of columns on problems he and some mathematics educators (in England, if not elsewhere) see with teaching multiplication-as repeated-addition (or, variously ONLY teaching it that way and never leading students to see that multiplication either is not repeated addition or is a lot else besides repeated addition) to elementary students, let me to write a response that gets at some of what I've been thinking about as I read those British researchers. Below is a re-edited version of my reply:

A crucial point I've tried to make previously in various places: there is a serious problem with the absolutist language a lot of folks in this debate insist upon using and which Prof. Devlin falls into at times (and hardly just at math-teach: check the comments at lets play math; good math, bad math, and any number of other blogs where this issue has been debated for several months. Some folks on those blogs make the argument on this thread at math-teach seem tame. See in particular comments by Joe Niederberger). However, as I've come to appreciate Prof. Devlin's style and intentions, I'm not so worried about his habit of overstating his case as I was initially. I think he wanted to draw attention to a number of important issues in the teaching of school mathematics, perhaps most broadly the likelihood (or at least the possibility) that sloppy teaching of elementary mathematics from a perspective that doesn't consider things from only slightly higher mathematics causes difficulties for a lot of kids as they progress into college mathematics courses. That would certainly be worth thinking about if true, and given the many posts we get on this list about how the sky is falling, too many kids aren't ready for college math, and so forth, one might expect that Prof. Devlin's columns would meet with approval from some of the educationally conservative thinkers here. Indeed, I was caught off guard at the anti-Devlin sentiments from Wayne Bishop and, later, Haim Pipik (though perhaps given that Devlin is on speaking terms with mathematics educators and doesn't find all math education theory or research to be garbage, I should have expected their knee-jerk dismissals).

I tried initially in my first blog post on Devlin's columns (before his third one appeared on this issue) to call into question his very strong assertion that there would be inevitable problems for students if they were taught that multiplication was repeated addition. But by now, anyone who is really paying attention to what he's said knows that he modified the claim near the end of his third column in a way I find acceptable:

Meanwhile, now you know why (or at least you know where to start finding out why) it is crucially important that we not teach children that multiplication is repeated addition. (Or, if you prefer, why we should not teach them in a way that leaves them believing this!)

That last parenthetical remark is precisely what I think most reasonable thinkers on this issue would likely conclude if they had taken the trouble to read the references. And the language in those articles and books, if you look at what Devlin himself quotes, is actually very modest. It's only Devlin who goes a bit heavy-handed for much of his three columns (though he gives plenty of explanation for both his perspective and his concerns), but ultimately I'm unable to fathom how anyone could argue vehemently and unrelentingly against the very last sentence in the quotation above.


For example (but many other examples are similar),

Research SUGGESTS that such teaching MAY BE at the root of later misconceptions. Alternative models for teaching have been shown more effective in experimental studies "(Clark & Nunes, 1998) but EVIDENCE IS STILL LIMITED AND MORE RESEARCH IS URGENTLY NEEDED. [emphasis added]

I'd add that this research would likely be most profitable if it were of the kind Ball describes when she and her colleague talk about research "inside education." The interaction between a number of factors needs to be considered in judging whether there really is a meaningful disadvantage in even including, let alone teaching exclusively, the multiplication-as-repeated-addition idea, or a meaningful advantage in the "alternative models" just mentioned. It's not merely a question of materials or merely of teacher moves or MERELY any one thing in isolation.


I think Prof. Devlin has gone on to point to a lot more than what he raised in his first two pieces. I doubt, for example, that he meant that we MERELY say that addition and subtraction are just two things we do with numbers, but rather that among other things we teach students, we make clear that two main things we do with numbers (or "to them") are to add them and multiply them and that the two are not identical. I think he's right that students should see that and continue to be exposed to the differences, just as we can't be satisfied with tell students that exponentiation IS repeated multiplication.

I would argue that students will, as they gain mathematical maturity, be increasingly able to see different ways in which there are deep differences between addition/subtraction and multiplication/addition, even if they ALSO consider some similarities. One issue Devlin raises that I think is worth noting is his concern that we're hobbling kids by teaching them that "multiplication makes things bigger," and I agree that it's a problem for a lot of students that arises repeatedly (one glaringly obvious example is on the typical SAT question where students are asked: for x > 0, which of the following is greatest - A) 1/x B) x C) x^2 D) sqr(x) or E) the relationship cannot be determined. Even when the problem is modified to exclude x = 1, many students are all-too-quick to pick "C" and I think this reflects the "multiplication (and exponentiation, which is "merely repeated multiplication") makes things bigger" idea. But I would point out that we see the same confusion about addition/subtraction: students get stuck at the whole number (or even natural number) assumption that addition always makes things bigger; subtraction always makes things smaller. So I believe we need to address a related though probably less complex problem with how we present addition and subtraction to students as if they are NEVER going to have to deal with integers. If no one else things any of this is important, the folks who write the SAT, ACT, GMAT, and similar test where math is at issue obviously do, because they exploit student misconceptions about all this relentlessly.

Returning to Devlin, he repeatedly states that one can arrive at the correct answer to many (though not all) multiplication problems through the act of doing repeated addition, but that doesn't mean that the former IS the latter or is "merely" the latter. I would have balked at the second notion, without ever having read Devlin, but I think he was trying to break through a lot of inertia amongst K-12 teachers, especially in the lower grades. And clearly he was right, given the reactions his columns engendered, as many teachers did (and do) seem to think that multiplication IS repeated addition and that's all ye know or need to know. Or at least all THEY want to know or teach. And that is really not acceptable.


Frank, the fellow whose post inspired this entry wrote,

The rest of the first article and the second article were mainly devoted to explaining why equating multiplication and repeated addition is mathematically incorrect (without providing any clear examples of why it was incorrect or important for K-12 education.) At the end of the second article, Devlin pleads:

Let me stress again that I am not suggesting we teach children arithmetic the way professional mathematicians view it. Rather, my point is that, however you teach it (and I defer to professional teachers in figuring out the how), don't do anything that is counter to the way the mathematicians do it .... some of your pupils may well end up in universities where they will HAVE to do it the right way.

In other words, don't create any misconceptions in K-12 that might make my job harder.

That doesn't seem to be Devlin's point, but rather not to make those students' lives harder. I don't think he is worried about the difficulty of his job, but the intractability of some student conceptions.


Frank continues:

In his third article (Multiplication and Those Pesky British Spellings), Devlin finally gets around to explaining why K-12 teachers and students (as opposed to university mathematicians) should care whether or not multiplication is exactly the same process as repeated addition: It may make it difficult to separate the concepts of addition and multiplication when students do proportional reasoning. Devlin states that

"Arguably Nunes and Bryant know more about multiplication and how to teach it than any other researchers in the world", so he quotes them on the subject of repeated addition:

"The common-sense view that multiplication is NOTHING BUT [Frank's emphasis] repeated addition, and division is nothing but repeated subtraction, does not seem to be sustainable after a careful reflection about situations that involve multiplicative reasoning. There are certainly links between additive and multiplicative reasoning, and the actual calculation of multiplication and division sums can be done through repeated addition and subtraction. [DEVLIN NOTE: They are focusing on beginning math instruction, concentrating on arithmetic on small, positive whole numbers.] But several new concepts emerge in multiplicative reasoning, which are not needed in the understanding of additive situations.

After reading the portion of their book that is available on the Internet, Nunes and Bryant make a great deal of sense to me. Notice that Nunes and Bryant say that multiplication is MORE than repeated addition, while Devlin says that "It Ain't Not Repeated Addition". Nunes and Bryant refer to repeated addition a "common-sense view" (that is incomplete), while Devlin equates it with the level of ignorance represented by the phrase "ain't not".

Do you think Professor Devlin has done a good job of educating the mathematics community about this issue?

I think Devlin was citing sources earlier than in the third article, but I won't quibble. He clearly was informed by that research, and it's interesting that it's pretty much all BRITISH research. He pokes some fun at the fact that it seemed to be news to a lot of the blog respondents that this body of work existed. I'm not sure many of them would be any more aware of it had it been American research. We have a few members of this list who don't seem to much care what research is out there if it doesn't confirm their prior beliefs.

And yes, I think Prof. Devlin has done everyone a wonderful service with these columns. Was he absolutely right in every particular? Probably not, though I think all three columns bear rereading with great care: he answers a lot of questions that folks claim he does not, and he certainly provides support for his ideas, though as he's told me, that didn't stop some some folks from excoriating him for NOT providing support for his ideas. Go figure.

I know that there aren't many people in mathematics or mathematics education who can reach as many people as Keith Devlin can, and as the issue he's raised is clearly of great interest and likely importance, he deserves everyone's thanks and respect, though that hardly means that everything or anything he says should be accepted. He meant to provoke thought and has done so, though unfortunately he has also provoked a lot of thoughtlessness. That's the nature of things these days, of course, and I have no doubt he can handle the heat.


Postscript: for those of you puzzled about the title of this post and the accompanying photo: the actor Tom Bower played a character name Cecil Skell in a terrific movie with James Woods and Robert Downey, Jr. called TRUE BELIEVER. Skell is a schizophrenic who believes that the phone company killed JFK (because he wanted to break up the phone company). He happens to be a key eyewitness in a murder who is never called by the defense in the original trial of Woods' client, a Korean who has now killed an "Aryan Brother" in self-defense while in prison for the first crime. Skell insists that the murderer "wasn't Chinese, because Asians have a different aura." When Woods and Downey, Jr. discover proof that in fact Skell is right and the murderer was not Asian but merely looked like he might have been, Downey, Jr. says to Woods, "Does this mean that the phone company DID kill Kennedy?" To me, it's the all-time greatest movie line to use when you realize that something that initially struck you as a bit crazy is in fact probably true.

Plus ça change (plus c'est la même chose): language and The Math Wars

Jackson Turner Main

In recent posts (see both) at math-teach@mathforum.org, veteran "math warrior" Wayne Bishop has complained about "so-called Constructivists" and other alleged abuses of "jargon" by what I imagine he thinks of as "so-called progressive educators," and the like. He's very incensed about such language, all the while representing groups like Mathematically Correct and Honest Open Logical Debate (HOLD) whose very names are fascinating examples of propaganda and dissembling, and whose major tactic in the "Math Wars" is to find new epithets for the educators, programs, practices, and ideas they want to destroy.

These are folks who have been tormenting the English language into pretzel-like bits of obfuscation, perhaps the most outlandish being Bishop & Friends' attempt to paint those who support progressive mathematics education reform, affirmative action, or pretty much anything that doesn't fit the MC/HOLD religion, as "racists."

But there's not much new going on. Consider:

They are called the Antifederalists, but it should be made clear at once that they were not antifederal at all. In reality they were determined to preserve the Confederation, and the name, far from being their own choice, was imposed upon them by their opponents, the so-called "Federalists." The attachment to them of a word which denotes the reverse of their true beliefs, and which moreover implies that they were mere obstructionists, without any positive plan to offer, was part of the penalty of defeat. The victors took what name they chose, and fastened on the losers one which condemned them. Since the victory was a lasting one, the name and the stigma have endured.

It was a nice piece of misdirection by the Federalists. Originally, the word "federal" meant anyone who supported the Confederation. Several years before the Constitution was promulgated, the men who wanted a strong national government, who might more properly be called "nationalists," began to appropriate the term "federal" for themselves. To them, the man of "federal principles" approved of "federal measures," which meant those that increased the weight and authority or extended the influence of the Confederation Congress. The word "antifederal" by contrast implied hostility to Congress. According to this definition, the antifederal man was opposed to any effort to strengthen the government and was therefore unpatriotic. Eventually the term became a general word of opprobrium applied by the Nationalists to anyone who opposed their designs. [THE ANTI-FEDERALISTS: Critics of the Constitution 1781-1788, Jackson Turner Main, p. ix]

The difference is that the Math Wars aren't over and the MC/HOLD warriors haven't won. Periodically, Wayne Bishop declares victory, tells us that this or that progressive idea, practice, program, etc., has gone "belly-up," and serves to make clear that he's whistling past yet another graveyard. Nonetheless, it's useful to remind oneself of the power of language and to warn people of good will not to let the bad guys get away with co-opting terminology so as to destroy all meaning.

Changing Order of Topics: An Example From Practice

Some day, my princess will come. And she'll be passionately interested in discussing questions of actual teaching practice in mathematics classrooms. Actually, of course, many such exist, but getting them together on this blog or in open, active public forums, having the time to participate actively, etc., isn't always a trivial task. For full-time teachers, time is always at a premium, and while there are many good resources on the Internet, tracking down what would really be productive for one's own work can be difficult.

From my perspective, one of the shortcomings of even the best conversations about practice, however, is the lack of access to specific classroom data, particularly video with supporting documentation (teacher journals, samples of student work, assessment instruments, observer notes, etc.) that would allow other practitioners to engage in deep analysis of what goes on in mathematics teaching based on common access to the same materials. As a former literature doctoral student, I am reminded of the countless frustrating conversations I had with undergraduate students of mine (and occasional fellow graduate students) about the absurdity of discussing literature without making much or even any direct reference to things in the text(s) under examination. Hearing someone offer up "personal reactions" to Dylan Thomas' "And Death Shall Have No Dominion," or Marlowe's DR. FAUSTUS (e.g., "I could really relate to that Mephostophilis character. He reminded me of my best friend in high school") just doesn't quite measure up to the level of discourse one hopes for in serious literary analysis. I kept wanting students to show me where IN THE TEXT they were getting their ideas and HOW what they said was going on actually worked. Generally, I was very disappointed even when I gave meaty selections from the texts that we'd dealt with in class on exams and asked students to offer some sort of analysis that stayed primarily "on the page."

To expect deep conversations about teaching, one must have similarly detailed examples of practice about which to converse. There's really no substitute for specifics and no specifics better than seeing what teachers do, with supporting documentation to flesh out and give depth to one's analysis. But as of this writing, there seems to be no readily-accessible, free web resource that could be used by groups of teachers, parents, researchers, and other interested stake-holders to discuss actual mathematics practice grounded in the sort of data needed for "close textual analysis." (Of course, perhaps I'm in error: if so, I hope to be flooded with information about just such resources and will, you can be sure, flock to them post haste).

While waiting for Godot (or, in fact, trying to figure out how to create just such a resource), I will resort to the less-satisfactory standby of anecdotal evidence, offered strictly in writing and through the admittedly-flawed lens of my own memory for something that's now about five or six years in the past. Any inaccuracies in the recounting are, I hope, due merely to incipient senility rather than purposeful misrepresentation of fact.

A Fortuitous Decision

My focus is the decision I made in the winter of the 2002-2003 academic year to change the order in which I taught two topics in an intermediate algebra course I'd been teaching with various materials for 2 1/2 years at a public charter high school in Ann Arbor. The goal of the course was to prepare high school sophomores (for the most part) to take the same course subsequently with college instructors for dual-enrollment credit towards both high school graduation and credit at the community college that sponsored and hosted our school. My course was intended to help bridge the gap between taking first year algebra over the period of one year at the typical pace of high school and taking a second algebra course at a college pace in one semester.

I had been using various books for the course, including the same text mandated for the college class, an indifferent text to my mind written by past and current members of the math department, designed for use both in regular classes and in self-paced courses. As a result, it was similar to programmed instruction texts from the 1960s to some extent. I had also used COMAP's Mathematics: Modeling Our World, a more progressive text that was heavily grounded in applications and mathematical modeling. While each book worked well with a subset of my students (and, frankly, with my own development as a teacher), neither was without flaws and neither reached a core group of students I had each semester who were low achievers and/or what might politely be called slackers when it came to school in general and mathematics in particular.

Although I eventually wound up using a third text (ALGEBRA 1: A PROCESS APPROACH, from the University of Hawaii Curriculum Research and Development Group) to reach that group in a shortened spring semester, when other students had gone on to the college course, I'm not sure it or any book I've seen would have been right for all my students or been fair to all these groups of kids I wanted to reach in ONE semester.

An example of a difficulty none of the books seemed to address adequately was teaching absolute value inequalities and their graphs, as well as the broader question of helping students see the connections between linear and quadratic equations and their respective graphs. But merely solving absolute value inequalities was always a sticking point for all but my best students. For reasons I had not yet determined, this topic always came after we had covered quadradic equations and their graphs (though not immediately thereafter). And even for students who did reasonably well with the pure algebraic manipulation and solving required in the units on quadratics, understanding how changes in the coefficients impacted their graphs and vice versa proved more difficult. So I had two stubborn but seemingly unrelated problems to deal with in these units.

For reasons that have become lost in the mists of time (I wish I could say that I had some sort of brilliant insight following much mediation or had been led to an epiphany through conversations with colleagues, but if either was the case, I have no recollection to report), I decided to try teaching the absolute value inequality unit before we did quadratics. It may be that I just wanted to get it out of the way and move on, or that I wanted to weed out a few students who were annoying me. But most likely it was nothing quite so insidious or inspired.

To what I recall as my great surprise and greater pleasure, the students on the whole seemed to do much better on both units, including how they approached solving the absolute value inequalities. But the biggest payoff, based both on the sorts of things they said in class and how they performed on assessments was in their understanding of the graphs and their connections with their symbolic representations. In particular, ideas about horizontal shifts, which heretofore always seemed extremely elusive for the majority of my students, now seemed to make sense to a lot more of them. And when we revisited these and related issues in a subsequent unit on quadratic equations (and even later on quadratic inequalities), they were to a much greater degree than in previous semesters I'd taught able to grapple successfully with what was going on with the graphs as we fiddled with the parameters.

To what do I attribute the apparent changes? Naturally, I wish I could say it was a vast improvement in the quality of my instruction, and perhaps that was true in one particular way. Because as I taught the transformations of the graphs of the absolute value inequalities, it hit me for perhaps the first time (or at least for the first time so overwhelmingly clearly) that there were a host of parallels between these changes and those associated with quadratics. In fact, to my mind, it seemed that as we covered the topic in the unit on absolute value inequalities, it was somehow easier to communicate what was happening with the graphs as we changed the values of the coefficients.

What occurred to me, though I only had scattered informal feedback from some students to confirm this (and it was in retrospect, when we treated the quadratics unit later), is that absolute value graphs are extremely simple because they consist only of two straight lines. Students are comfortable with straight lines compared with curves, and of course a linear relationship is easier to plot and to draw the graph for with complete confidence compared with graphing any curve, which always seems imprecise by comparison when drawn by hand or even with tools to help aid in precision on graph paper. Students quite quickly "get" the symmetry in the absolute value graphs, and of course the fact that there are two predictable (and opposite-signed) slopes operating on the two "legs" of the graph makes hand-graphing a snap.

While it may seem like a minor point, contrast that with graphing parabolas. Although the symmetry is there, of course (in fact, ALL the same properties are there EXCEPT for the straightness), it's harder to graph individual points except for lattice points with complete confidence (it can be argued that the same is true with absolute value graphs, too, but consider how the issue of slope comes into play and it is likely that one will see how relatively easy these are compared with any graphs involving curves). Moreover, and this may be the crucial point, it's easier to picture mentally and to anticipate what's happening on a point-by-point basis with the absolute value graphs, which in their simplest forms (with "leading coefficient" of one) are, after all, pieces of y = x and y = -x, possible shifted vertically and/or horizontally.

And it's particularly those horizontal shifts that are difficult for students to grasp. The idea that adding a positive quantity to the independent variable "inside" the function results in a left shift and adding a negative quantity produces a right shift is counter-intuitive at first blush. Without having to deal with the complication of squaring, many students seemed more able to just think about the impact of this one change on the resulting y-values. Later, when the same issue arose with quadratics, most students seemed able to transfer what they'd already learned from their previous experiences with the absolute value graphs.

Research "Inside" Education?

Is the above "proof" that what I observed is definitely the case and that all teachers should follow suit? Of course not. It's simply food for thought and, I hope, someone's formal research. But this is one example of where classroom practice could inform research and that the subsequent research could later influence broad-scale practice. If someone decides to set up and conduct a sound comparative experiment, it is conceivable that they could do more deeply the kind of analysis that Deborah Ball and Francesca Forzani call research "inside" education rather than about it ("What Makes Education Research 'Educational'?", looking at the interplay between content, teacher, and student on this and related issues. Thus, I suggest that while what I have presented here is not a substitute for either formal research or the kind of deeper and broader data-base I mention in the beginning of this piece as a source for practitioners and others looking to improve classroom practice, what I've offered is a meaningful contribution to research and practice.