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.

Warfield, Systems, G.S. Chandy, and Mathematics Teaching

Above is a picture of John N. Warfield, "the author of two U.S. patents on electronic equipment, and the inventor of Interpretive Structural Modeling, Interactive Management, and General Design Science." Professor Warfield's work has had a significant influence on the contributions of G.S. Chandy to the math-teach@mathforum.org list. In connection with my recent blog entries about "important open questions in mathematics pedagogy," ("Is Mathematics Teaching A Closed Book?"), Mr. Chandy has tried to offer an approach to discussing these and related points of contention on that most contentious of lists that is grounded in the work of Professor Warfield.

I do not profess to know Warfield's work, but I found what Mr. Chandy posted to be sufficiently heuristic to warrant posting what he wrote in its entirety in hopes of generating interest among readers of this blog, as well as those on various lists to which I contribute who many not have ventured into math-teach (probably with good reason).

G.S. Chandy's Post

The 'elements' below are generated from MPG's posting on Aug 22, 2008 8:43 PM at this very thread. (Many of the elements listed below have been generated by GSC - they should be properly articulated by real teachers of K-12 & other school math):

Possible Mission Statement:

M: "To put into place a highly effective way of teaching math (for teachers); a highly effective way of learning math (for students)"

First trigger question:
"What, in your opinion, are some THINGS TO DO to help accomplish Mission M?"
====
Some possible elements, generated in response to above trigger question - these elements largely derived from MPG's posting above-noted:
+++
1. To ensure that the ideas of those who actually teach K-12 are taken into the action planning for future teaching of K-12

2. To learn how to go beyond the lecture-driven, teacher-centered math classes

3. To ensure active student participation in the math class

4. To stimulate the student's genuine interest in math

5. To generate real interest from students

6. To ensure that students become actively engaged in mathematical thinking and math discourse

7. To provide (point to) ways demonstrating HOW students could become truly active in math

8. To ensure that 'grades' are not treated as jokes (by students or by teachers)

9. To use an effective grading system through which both students and teachers could benefit

10. To convince students that it is possible for each of them to become productive and creative in math

11. To convince students that it is inevitable that some would be more productive, others maybe less productive - but that all can be truly productive

12. To help each student 'fill in the gaps' in his/her understanding of math

13. To convince students of the real benefits that a sound understanding of math could bring to them

14. To convince students to put forth enough real effort to become productive and creative in math

15. To convince students of the value of math that goes beyond just 'book-learning' or 'exam-learning'

16. To ensure that students are fairly and effectively tested at various levels

17. To ensure that math-teaching should go beyond vapid 'multiple choice' questions

18. To enable and encourage students to take a real cract at truly challenging math questions (suited to their current level)

19. To ensure that all students always get needed further chances to improve their standings in math

20. To prevent students from attempting to cheat on math - demonstrate that it is really futile to try and cheat in math

21. .... (Etc).

+++
There will obviously be a great many such 'elements'. I suggest that generated such elements should be a continuing task, on which teachers and students could very beneficially spend a few minutes each day.

As we generate such elements, the most useful thing to do with them is to find out just how they may "CONTRIBUTE TO" each other (and to M).

This is done by asking questions like:

"Does, in your opinion,

Element X

CONTRIBUTE TO

Element Y?"
(The strength of the contribution relationship should initially be taken as small, i.e., we initially create the model predicated on the relationship "MAY contribute to". Later, as we become more certain of our understanding of the elements (and of the system in which those elements are active), we could strengthen the relationship to "SHOULD contribute to", and still later to "CONTRIBUTES TO".

Asking and answering those questions about these elements will give participants the opportunity to enhance their understanding of the whole system in which the 'teaching and learning of math' operate. They will also become clear in their minds about each separate element in itself, in its specific context.

(Almost) needless to state, each class would have its own listing, and the lists (and the models generated from them) would see considerable variation. Also, it is entirely possible that there would be considerable disagreement about the contributions or otherwise of various elements. Most of such disagreements can be worked out to the mutual satisfaction of all involved - and it is often possible to work out models representing a satisfactory consensus. Sometimes we cannot come to agreement - I suggest in such cases that the persons involved should create and follow their own models.

Of course, there may be people participating who are, so to speak, more interested in empty argumentation rather than productive resolution of disagreements and issues - just "trolls" - we have a few such in every set of participants. We need to learn how to handle these effectively: not easy at all to do, but it is possible to ensure that trolls do not disturb the flow too much. If we are all actively involved in generated useful elemetns, working on creating productive models of our perceptions about relationships of those elements, then the trolls will disturb working partipants much less. James Wysocki (and others) have suggested practical ways in which such trolls could be dealt with.
+++
Below my signature, I provide a list of some other useful 'trigger questions' about Mission M.

GSC
+++
Useful trigger questions (all are in relation to the Mission M identified above):

Creating the OPMS

Step 1: Identify a desirable Mission M:

MISSION: “______________________________________________”

(In case you have a problem, just put it into ‘Mission’ format. Various examples of ‘Missions’ are available in various examples illustrated).

Step 2: THINGS TO DO (to accomplish Mission):

Generated from “1st Fundamental Trigger Question”:

“What, in your opinion, are the THINGS TO DO to accomplish the above Mission?”

Step 3: Action Planning:

We shall construct an Interpretive Structural Model (ISM) with the responses to above 1st Trigger Question. This ISM (which is technically known as an ‘Intent Structure’) will develop into the ongoing Action Plan for the Mission.

The participants in the Mission should spend about 20-30 minutes each day to develop models relating to progress in their individual parts of this Mission (as seen in their individual One Page Management Systems that should start developing as the organization works towards the global Mission). As a whole, the organization should meet from time to time, as found convenient, to model progress towards the global Mission.



Step 4: Identifying DIFFICULTIES, BARRIERS and THREATS – and overcoming them

These are generated from the “2nd Fundamental Trigger Question”:

“What, in your opinion, are the BARRIERS, DIFFICULTIES & THREATS that may hinder or prevent accomplishment of the chosen Mission?”

Generally, responses to the above trigger question may be:

a) converted into appropriate THINGS TO DO, which would then be integrated into the ongoing Action Plan
b) inserted into a Field Representation – the OPMS software will then help users create Dimension Titles for the Field Representation and also to link up various elements in the Field with appropriate relationships. Various other actions are to be done with Field Representations, which are described in our Workbook.

c) These BARRIERS, DIFFICULTIES & THREATS are also most usefully inserted into a ‘Problematique’ (the governing relationship of which is “aggravates”). It takes a while to learn to develop and use a problematique effectively, but it will be well worth the effort, for the reasons noted in our ‘Basic Presentation’, a copy of which is provided for reference.

The above exercises would help us identify and implement, on a continuing basis, means to overcome BARRIERS, DIFFICULTIES and THREATS discovered during our progress towards our Mission.

Step 5: Identifying STRENGTHS (available/required)

Generated from “3rd Fundamental Trigger Question”:

“What, in your opinion, are the STRENGTHS (available/required) that could help accomplishment of the chosen Mission?”

Responses to the above Trigger Question, if sizable in number, may be inserted into a Field Representation.

The STRENGTHS identified as “Required, not available” would be translated into appropriate THINGS TO DO format and integrated into the ongoing Action Plan to enable users find ways to develop the required
STRENGTHS.

Step 6: Identifying WEAKNESSES – and overcoming them

Generated from “4th Fundamental Trigger Question”:

“What, in your opinion, are the WEAKNESSES that may hinder or prevent accomplishment of the chosen Mission?”

The means of handling WEAKNESSES are exactly the same as we have for handling BARRIERS, etc. That is, responses to the above trigger question would be:

a) converted into appropriate THINGS TO DO, which would then be integrated into the ongoing Action Plan
b) inserted into a Field Representation – the OPMS software will then help users create Dimension Titles for the Field Representation and also to link up various elements in the Field with appropriate relationships. Various other actions are to be done with Field Representations, which are described in our Workbook.

c) These WEAKNESSES are also most usefully inserted into a ‘Problematique’ (the governing relationship of which is “aggravates”). It takes a while to learn to use a problematique effectively, but it will be well worth the effort, for reasons cited in our ‘Basic Presentation’, a copy of which has been provided with our Workbook.

The above exercises help us identify and implement, on a continuing basis, means to overcome WEAKNESSES discovered during our progress towards our Mission.

For individuals as well as for organisations, correct identification of BARRIERS & WEAKNESSES (and defining ways to overcome them) is often a very long and painful task – particularly when organisational Barriers and Weaknesses are derived from an individual’s Weaknesses. It could take months – sometimes people NEVER learn to get over their Weaknesses even when they cause disaster. In fact, most (man-made) disasters are caused precisely because people have not learned how to handle their existing Weaknesses!

Step 7: Identifying OPPORTUNITIES available

Generated from “5th Fundamental Trigger Question”:

“What, in your opinion, are the OPPORTUNITIES available
that may help accomplishment of our chosen Mission? –
and what are the THINGS TO DO to avail the
OPPORTUNITIES discerned?”

Responses to above trigger question are inserted into a Field Representation showing OPPORTUNITIES available – and THINGS TO DO to avail the OPPORTUNITIES identified. The THINGS TO DO to avail the OPPORTUNITIES are also integrated into the Action Plan, to enable us to see how we may work towards availing of the OPPORTUNITIES that arise.



Step 8: Identifying EVENTS & MILESTONES

Generated from “6th Fundamental Trigger Question”:

“What, in your opinion, are the EVENTS/MILESTONES
that may occur during our progress towards our
chosen Mission?”

Responses to above trigger question are inserted into PERT/Gantt Charts (as is done in the conventional ‘Project Management’ software, to show the
status of Milestones during progress towards our Mission.

We observe here that the conventional Project Management software deals only with this single dimension of the OPMS (namely, the EVENTS Dimension). Obviously the name ‘Project Management’ software is a serious misnomer for such software – which may accurately be called ‘Event Management’ software.

The OPMS may be seen to fulfill the role of true Project Management, as it enables users to see all dimensions relating to a specified Mission. However, we prefer to regard the OPMS as an ‘aid to problem-solving and decision making’ (which includes ‘Project Management’, and much else besides).


Step 9, et seq: ‘System’ Dimensions of the OPMS

When sufficient numbers of elements have been generated in response to the Six Fundamental Trigger Questions as described above (and, further, those elements have been appropriately inserted into models AND linked up), the users would find it necessary and useful to start working on the ‘System Dimensions’ of the OPMS:

• PLANNING SYSTEM(S)
• INFORMATION SYSTEM(S)
• MARKETING SYSTEM
• PRODUCTION SYSTEM (for manufacturing organizations. For educational institutions, research institutions, etc., the title of this dimension may be modified appropriately).
• PROBLEM SOLVING SYSTEM AND LEARNING SYSTEM: The OPMS itself defines these two systems, and no work is required to be done by users in regard to these two closely coupled systems)
• MONITORING & EVALUATION SYSTEMS
• FINANCE CONTROL SYSTEM
• OTHER(s) (as required)

The OPMS would help users develop all above dimensions.

In general, the models in these ‘system’ dimensions of the OPMS are small, but powerful, ‘meta models’ derived from the larger models appearing above the System Tie Line. It is generally found that these models start developing effectively, in a very natural way after a certain ‘richness of connection’ has been established between the fundamental models above the System Tie Line.

The models within the OPMS are models of human perceptions relating to the chosen Mission. The purpose of creating such models is primarily to show a simple ‘action path’ to each person involved in accomplishment of the Mission. That is, these models show what each person and each group involved should do each day in pursuit of the Mission. The THINGS TO DO identified in the Action Planning structure as ‘FOCUS elements’ are the crucial activities at any point of time – these would be just a few (typically, 3 to 5 each day). These focus elements will naturally change from time to time as we progress towards accomplishment of the Mission.

More Problems of Remediation and Assessment

In response to one of my previous posts on the issue of meaningful open questions about mathematics pedagogy, an anonymous contributor wrote in part: "How can I teach a student at level N who hasn't mastered all the work of level N-1 (or maybe even N-2 for some things)? What can be done to help with the learning of the current content while remediating gaps in learning?"

This is a particularly nagging problem for most mathematics teachers in K-12 and beyond, though at the college level it is easier perhaps for instructors to be blithely condescending and dismissive of the problems of students who come to class ill-prepared for the level of coursework expected of them. College is, after all, a choice, and professors are not obligated to provide remediation, although there are generally non-credit classes offered that do just that, or at least purport to do so.

I wrote on math-teach@mathforum.org in response to the above question:

The first question is nearly ubiquitous in mathematics education and of course there is no simple answer to it (not that we won't likely here some in this venue from non-teachers).

I suspect that we'll never see a time where teachers don't have to ask that question for the simple reason that there will never be a "system" that eliminates differential intellectual development, readiness, or individual variability in motivation for learning any given subject. It seems inevitable that kids will not be equally ready in all sorts of ways for any given mathematical topic at a given age/grade level.

As long as we try to mass educate under the current model, with bizarre expectations that we can legislate kids to level N or intimidate schools, parents, teachers, or kids to some pull rabbits out of hats in an attempt to pretend that all kids are at level N at the appointed age/grade level, we'll miss the boat. A saner approach is to do the best we can to get kids ready for school and then teach them where they actually are, using differentiated instruction and methods to brings them as far as we can given where they start. That requires more flexible ideas about content and sequence, about instruction and tools, about the nature of classrooms, and about meaningful, useful assessment than we can reasonably expect to see given conservative/ reactionary opposition to sensible approaches to public school in particular, and education in general.

Richard Strausz replied:

Michael, I agree with what you say. Some critics use such real-world observations as more ammunition for their crusade against public education.

However, in talking with math teachers in Catholic and Jewish high schools, I hear similar situations in their classrooms.

And I commented:

I'm not surprised in the least. Didn't mean to suggest there was something unique going on in secular schools. I suspect we'd hear similar issues and concerns from private, non-sectarian classrooms.

Of course, the usual suspects in mathematics education, right-wing ideologues who pretend that all was once well in math teaching back in some imagined day, and/or that it will all be well again if only we had "real standards," (who decides and why they are qualified to do so always turns out to be those same ideologues and their like-minded brethren), "real accountability" (to whom is never quite clear, but it generally means to people who are at best marginally involved in actual teaching), "real textbooks" (and of course, nothing BUT textbooks, generally of the skill-based, routinized kind best exemplified by "teacher-proof," learning-proof Saxon Mathematics products), "real methods of instruction" (direct instruction with the teacher firmly in the middle, doing most of the talking and most of the mathematics), then all would be well.

Wayne Bishop Chimes In

In that light, the following predictable nonsense was offered up by the reliable anti-reformer, Wayne Bishop:

You are right about nonpublic schools but you seem to have misinterpreted the real problem with your identification of this problem being "more ammunition for their crusade against public education". It's ammunition for "their" crusade against colleges of education, the problem so well identified by Reid Lyon. The aversion against standards-based education (prior to the collegiate level where, at least in mathematics, standards tend to be used and accepted) is a direct consequence of the teaching of schools/colleges of education. For inexplicable reasons, almost religious in appearance, the industry prefers to place students by age-level even when it is obvious that students are being placed into situations where they are doomed to fail if honest performance standards are maintained.

It's always fascinating to hear Dr. Bishop cite, directly or otherwise, the execrable Reid Lyon, a former Bush education "expert" who was in part responsible to the multi-million dollar boondoggle known as READING FIRST, a project that appears to have been both rife with corruption and utterly ineffective, neither of which comes as a shock to Lyon's critics or fair-minded observers of how the current administration has lowered to every opportunity to help those most in need of its assistance. It was of course Mr. Lyon who said "You, know, if there was any piece of legislation that I could pass, it would be to blow up colleges of education" (McCracken, Nancy. "Surviving Shock and Awe: NCLB vs. Colleges of Education." English Education, January 2004, 104-118.) Scratch a right-wing ideologue these days, you may just turn up an anti-democratic terrorist with his hands in the public trough up to his shoulders.

My Proposal

In any case, here is my response to Dr. Bishop's latest screed:

For those of us who actually teach K-12, the issue that is far more onerous is not what book was used, what teaching methods were most prominent (well, that's a concern if one is using eclectic instructional approaches with students who've been taught in totally lecture-driven, teacher-centered classrooms in which student passivity is so ingrained that ANY requirement that they become actively engaged in mathematical thinking and mathematical discourse is useless until the teacher shows students HOW to become active in mathematics and convinces them that it is necessary and productive to do so), or even to some extent whether the previous class covers every topic that s/he would have. Rather, it is the passing of students from previous classes with grades of D and, generally, C. Frankly, I'd say that in my experience, unless we're talking about an honors course of some sort or an extraordinary school, anyone coming into Algebra II with a grade less than a solid B in Algebra I is going to have more mathematical holes in his/her head than it takes to fill the Albert Hall or are found in the typical argument from members of Mathematically Correct and NYC-HOLD about any aspect of mathematics education. And that's a lot of holes, let me tell you.

The obvious solution is to stop making grades the joke they are and instead go to minimum exit/entrance exams with multiple sorts of assessment employed to determine whether students are ready to move on. I suggest both exit and entrance exams so that there is a double check in place: one on leaving level N-1 and one on beginning level N. And also so that those who marginally fail to gain exit at level N-1 can get another shot at the beginning of the following year. Should they have gotten themselves up to speed by then, they should not be denied a chance to prove themselves ready to proceed.

Naturally, I'm not calling for a nationally-normed testing instrument here, especially if it's strictly some vapid multiple-choice test that values only one narrow set of skills. But within each state's standards, if there is a real attempt at meaningful formative assessment that leads directly to re-teaching areas of weakness before allowing students to take a crack at more challenging mathematics that builds directly on previous knowledge, then such a system would potentially reduce the number of students who are simply pushed further down the rabbit hole with nothing to hold onto.

For such a system to work, however, it can't be linked to a bunch of politicized punishments intended to help ideologues and politicians use teachers, schools, and kids as footballs for making propaganda.

Minimally, that would mean that people like Wayne Bishop would need to be keep as far as possible from influencing the assessment process. Instead, knowledgeable assessment experts who actually understand and stick to psychometric principles must have major input into each state and district's development of assessment tools. Close attention must be paid to some of the issues raised in two papers by SUNY @ Stony Brook mathematician Alan Tucker about cut scores, the theory of performance standards, and related issues. A balanced approach to assessment, regardless of the religious objections of the nay-sayers, must be used, and any structure that has been proven to promote wide-scale cheating due to absurd political pressure on educators, parents, and students, must be closely examined and, if found incapable of improvement, abandoned.

Is authentic, meaningful assessment expensive? You bet. But then, finding out what's really going on in a complex world is always dearer than looking for surface data that tells you what you already believed to be the case beforehand. And crafting real solutions rather than destroying public education so that the rich can get richer is always unpopular. . . among the powerful and their lackeys and mouthpieces.

Keith Devlin Continues on Multiplication

The multiplication issue continues to live on in various quarters, and now Keith Devlin, who started the current round of debates, discussions, and arguments, has posted a third column on the subject, "Multiplication and Those Pesky English Spellings." I'm still chewing over what he says there, but I can report with pleasure that this blog, along with three others, is mentioned favorably in his new column:

Of the blogs I looked at, which had threads devoted to my "repeated addition" columns, the following all had some good, thoughtful comments by their owners and by some of their contributors - by no means all agreeing with me - though the discussion in "Let's Play Math" soon descended to uninformed and repetitive name calling, and the owner eventually closed the thread, which unfortunately soon reappeared elsewhere.

* http://letsplaymath.wordpress.com
* http://www.textsavvyblog.net
* http://rationalmathed.blogspot.com
* http://homeschoolmath.blogspot.com/

Those blogs each have interesting and useful things to say on other math ed topics as well. Most of the others I saw seem to be little more than sounding boards for people who are so convinced that the overwhelming mass of evidence must all be wrong - since it runs counter to their beliefs - they don't even bother to read it. Those (other) blogs are not information exchanges from which you can learn anything, but platforms for people to espouse their own particular, unsubstantiated and often wildly wrong beliefs. Mathematicians who care about our subject and who like to think that the students who pass through our classrooms emerge with a good understanding of the mathematics we taught them, should be advised that they venture into any other mathematics education blog at their own risk.

Hard not to feel pleasure at getting that kind of comment from someone of Prof. Devlin's stature, and I like the other blogs he mentions as well, so to be put in their company is doubly flattering.
******************************************************************************************

Meanwhile, as I try to digest Devlin's latest column, I thought I'd let readers of this blog know that I recently made a very fortuitous discovery at a used book sale while visiting Fenton, MI last Saturday: about 50 issues of THE MATHEMATICS TEACHER from 1965 to 1973. There are so many remarkable things in these magazines that are worth considering in light of the last 20 years or so that I may well have material for several months' worth of blog posts just from what is in them.

As a teaser, the following is from an Educational Testing Service (Cooperative Test Division) advertisement in the February 1967 issue:

Cooperative Mathematics Tests

This series reflects the ferment and change in the mathematics curricula. At the same time, content has been carefully selected to ensure appropriateness of the tests for most students.

Ability to apply understanding of mathematical ideas to new situations and to reason with insight is emphasized. Factual recall and computation are minimized.

No, I didn't make that up. No, I'm not mistyping the date. That is 1967, not 1997 or 2007. It would be fascinating to see these testing instruments, and if anyone knows of them, please let me hear from you in the comments section or directly by e-mail