The debate about multiplication and repeated addition is raging once again. Check out Michael's take. And read Burt in the comments.

I thought about writing something new, but those arguing for the wrongheaded notion that multiplication is repeated addition haven't offered anything new since I wrote the article reposted below (July of 2008). Enjoy!

It seems ironic that Devlin, who belongs to a group that "finally learned to sidestep that unanswerable 'What is it?' question," is now telling us what something is not--that multiplication is not repeated addition.

Not only is he confidently answering an apparently unanswerable question here, he's also dispensing advice based on that answer: "please stop telling your pupils that multiplication is repeated addition." This would seem to break—if not fundamentally, then at least technically—what Devlin refers to as a kind of 100+-year-old rule requiring mathematicians to never again think about or speak about the "what is it" that they're doing.

But it only seems that way.

However, even though I am not a professional mathematician, my saying that Devlin is right would only make me an accomplice to this unlawful behavior. Therefore, while I won't specifically say he's right, I will say—and have said—that he has the "right angle" on this issue.

Walking to Paris

From what I've seen, when people argue that it is okay to teach kids that multiplication is repeated addition, they generally try to do so in two ways (at least mathematically). They either argue that multiplication is defined as repeated addition, using the Peano axioms--or something similar—or they argue that multiplication and repeated addition both get the same results.

The second argument is undoubtedly true, but being so is not a justification for telling kids "multiplication is repeated addition"—even when you're talking about just the counting numbers (1, 2, 3 …). Yes, the multiplication function y = 3p (or 3 × p) and the repeated addition function y = p + p + p both have the same "functional value" when you plug in a number for p. (For example, 3 × 4 = 12, and 4 + 4 + 4 = 12.) But we can rewrite the repeated addition function above as y = p &minus (^-p) &minus (^-p). Shall we take this as a justification to tell kids that "multiplication basically reduces to the repeated subtraction of negatives"?

Even the simple multiplication function y = ax can be expanded to


If we plug in any counting number (1, 2, 3, …) for a and x into both equations, both will have the same "functional value." Does this mean that now we are justified in calling upon factorials, square roots, and squares to explain or define multiplication?

Of course not. Just because multiplication and repeated addition get the same results does not make them the same operation.

The first argument I mentioned—that multiplication is defined as repeated addition—is trickier, in part because it seems to depend somewhat on what text you're reading. I quoted from the text I have in front of me at the moment here. Ultimately, whether or not you agree or disagree with the idea that Peano—or anyone else—essentially defines multiplication as repeated addition (I would disagree), this definition is all but useless for solving problems.

For example, most sixth or seventh graders (and certainly some students in lower grades) would know that an equation like p + p + p + p + p = 85 can simply be rewritten as 5p = 85. Then, to solve the equation, they would know to divide (the inverse of multiplication) both sides by 5 to find p = 17. But it is impossible to solve that equation using only repeated addition (and its inverse, subtraction)—unless you guess or use a model or something like that. If all you know is repeated addition and subtraction, then you could not even get started by rewriting the expression p + p + p + p + p as 5p, because you simply wouldn't know to do that.

There are a lot of things in mathematics that you can "basically reduce down to" or "think about as" repeated addition. Theoretically, a plane trip somewhere is the exact same thing as a bunch of repeated steps to get you to the same destination. But if you're in the U.S., and that destination is Paris, they most certainly aren't the same. You can argue all day long that multiplication as repeated addition is "rigorously defined" or that the historical development of multiplication ran through repeated addition and then was extended to other number systems. Even if all that is true, understanding multiplication as repeated addition will leave you mathematically impotent in the modern world.

Multiplication is fundamentally different from addition (and of course repeated addition) because it helps us understand fundamentally different things. Things change. Get over it.

Mathematics Is Not a Matter of Opinion

From various comments on this issue I can see that a big stumbling block for many people is their assuming that (a) mathematics never ever changes, (b) that mathematics is nothing more than what is or has ever been written down, and (c) that mathematicians thinking about the subject today only have credibility by being entirely consistent with everything that has happened in mathematics for the past thousand years--no matter how unuseful it might be.

As far as (a) is concerned, it seems popular to argue that every bit of mathematics that was ever thought of is simply an "extension" of some earlier mathematics. Obviously, this is complete rubbish. One would be hard-pressed, I think, to call Greek mathematics an "extension" of Babylonian mathematics unless one were using the word extension very, very loosely.

The assumptions in (b) and (c) are interrelated. For (b), I'm reminded of a great quote from Uri Leron:

According to the algebraic image of functions, an operation is acting on an object. The agent performing the operation takes an object and does something to it. For example, a child playing with a toy may move it, squeeze it, or color it. The object before the action is the input and the object after the action is the output. The operation is thus transforming the input into the output. The proposed origin of the algebraic image of functions is the child's experience of acting on objects in the physical world. . . . Inherent to this image is the experience that an operation changes its input—after all, that's why we engage it in the first place: you move something to change its place, squeeze it to change its shape, color it to change its look.

But this is not what happens in modern mathematics or in functional programming. In the modern formalism of functions, nothing really changes! The function is a "mapping between two fixed sets" or even, in its most extreme form, a set of ordered pairs. As is the universal trend in modern mathematics, an algebraic formalism has been adopted that completely suppresses the images of process, time, and change.

Now, certainly, the idea of a function didn't start out like this. It was almost certainly more informal, incorporating concepts like "process, time, and change." But eventually, as Leron notes, those concepts were stripped away from the pure mathematical notion of a function. So where are the axioms and theorems written down throughout the history of mathematics that TELL US as much? Where in the theorems and axioms of Dedekind or Peano or Landau can I directly find the notion that functions (or anything) really don't include the concept of "process, time, and change"?

Well, I'll tell you where. Nowhere.

But this idea that, for example, 5 &minus 2 isn't really considered to be some kind of process where 2 things get taken away from 5 is FUNDAMENTAL to our understanding of mathematics. Yet, some people would argue that because it isn't written down, it must not be true. And therefore it's okay to fill our kids' heads with wrong ideas.

Which brings us to (c). This assumption--that the ideas promoted by mathematicians today MUST be completely consistent with what you learned in Kindergarten or high school or college or else they're wrong--sounds a lot like the creationist argument that everything that can be explained about modern evolutionary theory we can find in The Origin of Species.

It's just not true. Both biology and mathematics--and a lot of other fields for that matter--are growing, learning, dynamic fields. I hate to use the ethotic argument here, but I find it interesting that people who aren't professional mathematicians--say, for example, a K-12 Web site developer or an actuarial analyst--would not first suppose that they didn't know everything about mathematics and start asking questions.

Unfortunately, it's not what happened.

Even something as simple as this (from the Park, Nunes paper I referenced earlier in the series):

"Amy's Mum is making 2 pots of tomato soup. She wants to put 3 tomatoes in each pot of soup. How many tomatoes does she need?

is a multiplication problem, whereas this (again, from the same paper):

Tom has three toy cars. Ann has three dolls. How many toys do they have together?

is a repeated addition problem. They are different ideas, fundamentally. The "processes" of finding a product and finding a repeated addition sum are the same for both problems, but the ideas involved--INCLUDING THE MATHEMATICAL IDEAS--are very, very different.

Mathematics is not the notation. Mathematics is the meaning behind the notation. And much of that meaning cannot be captured by the notation itself. No doubt, 3 + 3 = 2 × 3. But multiplication and repeated addition are STILL fundamentally different.

What we are doing now in elementary mathematics education is blurring (or even erasing) this distinction when we first introduce multiplication. And that's bad math, not good math.

Connections, Connections, Connections

Devlin supposes (I think correctly) that at least one reason why people have such a problem with this idea—that multiplication is not repeated addition—is because of their desire to make mathematics tangible:

Many people feel a need to make things concrete. But mathematics is abstract. That is where it gets its strength. Multiplication simply IS NOT a generalized addition, and exponentiation IS NOT a generalized multiplication. Just as you can't really say what the number 7 IS in concrete terms - it's a pure abstraction - so too you can't say what addition and multiplication and exponentiation ARE. They are BASIC, not derived. A significant part of mastering mathematics is coming to terms with that.

I blame it on the obsession with connections:

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

Just Accept the Challenge

But nowhere in Devlin's articles on this issue (the way I'm reading them) does he suggest that we teach multiplication without connections to repeated addition and without concrete representations. On the contrary,

(I do think that you need to present simple everyday examples of applications. Teaching a class of elementary school students about axiomatic integral domains is probably not a good idea! . . .

Once you have established that there are two distinct (I don't say unconnected) useful operations on numbers, then it is surely self-evident that repeated addition is not multiplication, it is just addition - repeated!

But now, you have set the stage for that wonderful moment when you can tell kids, or even better maybe they can discover for themselves, this wonderful trick that multiplication gives you a super quick way to calculate a repeated addition sum.

What he's simply suggesting is that we introduce multiplication and addition as distinct operations and then come back to the connections between multiplication and repeated addition later. Perhaps he wasn't as clear as he could have been, but it seemed pretty straightforward to me.

And, given that, and the fact that he breathlessly disclaimed knowing much if anything about K-12 education, it would be silly to read his articles as though they were lesson plans and then turn round and declare either that he didn't give us enough help or that the whole thing is impossible.

My advice would be to just accept his challenge. I promise I'll do the same.

Correspondence

For those of you still holding on to the idea that while the math might be correct, exposing children to the idea that addition and multiplication are distinct operations might make their little heads explode, let me tease you with a neat little study that I referenced in an earlier post in this series. The authors are Jee-Hyun Park and Terezinha Nunes, and the title is "The development of the concept of multiplication." Here's the (chopped-up) abstract:

Two alternative hypotheses have been offered to explain the origin of the concept of multiplication in children's reasoning. The first suggests that the concept of multiplication is grounded on the understanding of repeated addition, and the second proposes that repeated addition is only a calculation procedure and that the understanding of multiplication has its roots in the schema of correspondence. . . . Pupils (mean age 6 years 7 months) from two primary schools in England, who had not been taught about multiplication in school, were pretested in additive and multiplicative reasoning problems. They were then randomly assigned to one of two treatment conditions: teaching of multiplication through repeated addition or teaching through correspondence. . . . At posttest, the correspondence group performed significantly better than the repeated addition group in multiplicative reasoning problems even after controlling for level of performance at pretest.

I should note here that, although it may read that way above, the ultimate aim of this study was not to compare the effectiveness of the correspondence and repeated addition treatments; it was to test two hypotheses about the "origin of the concept of multiplication in children's reasoning." Obviously, one of the hypotheses says that the origin is in repeated addition, and another says that it is in correspondence.

In other words, the "significantly better" performance of the correspondence group over the repeated addition group was taken by the researchers not as evidence of the superiority of the correspondence treatment, but as evidence of the fact that children begin to think about multiplication NOT as repeated addition but as a "one-to-many correspondence."

That's about it. It's been real.

Part I | Part II | Part III | Part IV | Part V | Part VI | Finale

Comments to follow soon.

Stewart:
Why is it so difficult to get change in the educational system in our country? That seems to be one of the most intractable systems, either because of the boards that are there or the unions or the—what is it about our education system that makes it so difficult to reform?

Gates:
Well, until recently there was no room for experimentation. And charter schools came in—although they're only a few percent of the schools—and they tried out new models. And a lot of those have worked. Not all of them. But that format showed us some very good ideas, and among those ideas is that you measure teachers, you give them more feedback. And--but people are afraid you'd put in a system that will fire the wrong person or have high overhead, and that's a legitimate fear. So actually having some districts where it works and then getting the 90% of the teachers who liked it, who thrived, who did improve to share that might allow us to switch—not have capricious things but really help people get better.

Stewart:
But don't public things like schools and medical care need to have the power to fail, need to fire the wrong person every now and again? It's never going to be perfect. Aren't people's expectations of what it's supposed to be so precious that you never get change in the positive direction?

Gates:
That's right. But you have to have a measure. And it's very tough to agree on a measure. You know, right now the health system rewards the person who just does more treatment, so it's quantity of output, not the kind of preventative care and measuring and saying, "Okay, you do that well." Or, "You teach this kid really well." We haven't been able to agree on that. And without that it's a problem.

Jon's comment—or question, rather—about the education system's lacking the power to fail struck me as being similar to what I said in this post:

Education seems unable to help but vacillate between its skepticism, which holds every idea (or none of them) to be right, and its particularism, which holds all of its own ideas to be right. This inability, in the end, makes it nearly impossible for education to decide before the fact that something can be wrong. And that is precisely what is wrong with education.

To make the similarity less philosophical (clearer, I hope), I've drawn two dartboards below. Take a look at these two dartboards. Using the dartboard on the right—a typical dartboard—we obviously do have the power to fail if our goal is to hit the bullseye. Using the dartboard on the left, we don't really have the power to fail—not because every throw will be considered a bullseye, but because we have not set out ahead of time what failure and success mean.


An important question we wrestle with in education, specifically with regard to instruction, is What kind of dartboard are we throwing at? Can we explain, before ever throwing a dart, what it means to hit the bullseye and how to get closer to it? If so, then we're throwing to the right; if not, then we're throwing to the left.

It seems right—er, correct—to say no, we can't really describe "bullseye" instruction before we deliver it (particularism) or at all (skepticism), because every student learns differently, there are multiple ways of delivering the same content, etc. For what seems like the same reasons, we can't really describe "bullseye" ice cream flavors or "bullseye" back massages. In other words, when it comes to instruction, the dartboard on the left seems to be the most appropriate.

Jon challenges this notion by asking, "But don't public things like schools and medical care need to have the power to fail, . . .? It's never going to be perfect. Aren't people's expectations of what it's supposed to be so precious that you never get change in the positive direction?"

For education—specifically, for instruction—shouldn't we be using the dart board on the right, not the one on the left? Shouldn't we have the courage to draw the bullseye somewhere, even if we know that we will sometimes unfairly exclude some good instruction and unfairly include some bad instruction? I would say yes.

Gates responds: "That's right. But you have to have a measure. And it's very tough to agree on a measure." Or, using the dart board analogy, we must have a way to decide exactly where to draw the circles, including the bullseye. The same sentiment was registered as a comment to an article I wrote here:

Difficult politically and otherwise to implement, because quality of instruction is non-quantifiable and seemingly unrelated to issues like teacher compensation.

I agree that it is difficult to agree on a measure and that quality of instruction is non-quantifiable, but I disagree that we should be looking for something so narrow as a measure (or even group of measures) or something necessarily quantifiable in the first place. What we should be looking for first are, as I stated in the post referenced above, clear, specific, acceptable principles of instructional quality.

Recently, I read an old article by Jerry Coyne and Richard Dawkins titled One Side Can Be Wrong, a piece that helpfully reminded me, more than four years after it was written, that something similar should be penned for education.

Why should that be so? Well, let's talk about that.

Dawkins and Coyne's article rather tidily does away with what had then become (and remains) the "teach the controversy" argument for intelligent design creationism (IDC)--the notion that IDC should be taught in science classrooms because it offers an alternative to the theory of evolution by natural selection as an explanation for the origins of different species on Earth. As the authors point out, their stance against "teach the controversy" seems counterintuitively closed-minded, but is demanded of them by the evidence--or, rather, lack of evidence:

So, why are we so sure that intelligent design is not a real scientific theory, worthy of "both sides" treatment? Isn't that just our personal opinion? It is an opinion shared by the vast majority of professional biologists, but of course science does not proceed by majority vote among scientists. Why isn't creationism (or its incarnation as intelligent design) just another scientific controversy . . .? Here's why.

If ID really were a scientific theory, positive evidence for it, gathered through research, would fill peer-reviewed scientific journals. This doesn't happen. It isn't that editors refuse to publish ID research. There simply isn't any ID research to publish. Its advocates bypass normal scientific due process by appealing directly to the non-scientific public and--with great shrewdness--to the government officials they elect.

Intelligent design creationists theorize that "certain features of the universe and of living things are best explained by an intelligent cause," yet they have not, at least as of this writing (2009), produced any positive evidence for this intelligent cause. It is this lack of evidence--not its character as an alternative explanation--which precludes IDC from acceptance in scientific circles and from "both sides" consideration. As Dawkins and Coyne note in the article linked above, alternative explanations based on actual evidence abound within evolutionary science and are thus far more worthy of debate than is IDC.

Methodists, Particularists, and Skeptics

Yet some may argue that while it may be true that IDC is unscientific, it does not follow from that observation alone that it is wrong. And, indeed, Dawkins and Coyne make no such claim explicitly in the article. Instead (again, one may argue), the authors simply hold up IDC to certain criteria of philosophical empiricism--that knowledge is derived from sense experience in the natural world--and describe how the theory fares (not well).

Chisholm categorized empiricism of this variety as a form of what he termed "methodism"--one of three possible solutions to the problem of distinguishing what is true from what is not:

(A) What do we know? What is the extent of our knowledge? (B) How are we to decide whether we know? What are the criteria of our knowledge?

If you happen to know the answers to the first of these pairs of questions, you may have some hope of being able to answer the second. Thus, if you happen to know which are the good apples and which are the bad ones, then maybe you could explain to some other person how he could go about deciding whether or not he has a good apple or a bad one. But if you don't know the answer to the first of these pairs of questions--if you don't know what things you know or how far your knowledge extends--it is difficult to see how you could possibly figure out an answer to the second.

On the other hand, if, somehow, you already know the answers to the second of these pairs of questions, then you may have some hope of being able to answer the first. Thus, if you happen to have a good set of directions for telling whether apples are good or bad, then maybe you can go about finding a good one--assuming, of course, that there are some good apples to be found. But if you don't know the answer to the second of these pairs of questions--if you don't know how to go about deciding whether or not you know, if you don't know what the criteria of knowing are--it is difficult to see how you could possibly figure out an answer to the first.

Particularists and particularist philosophies (described in the second paragraph above) decide first which are the good and bad apples--or what is true and what is not, or what we know and what we don't--and then shop around for a sorting system that reliably turns out results consistent with those decisions. Empiricist, or "methodist," philosophies, in contrast, (described in the final paragraph above) find their answers to the first question (which are the good apples?) by first answering the second question (how are we to decide whether we have a good or bad apple?).

Thus, Dawkins and Coyne, as loyal empiricists, reject IDC as a bad apple--not, as the argument might go, because they believe it actually is a rotten apple (the authors subscribe to a philosophy which does not permit them to discern that directly) but because the method they have decided upon to sort the apples (quantity or quality of evidence, naturalism, the scientific method, etc.) leads them almost inevitably to this conclusion.

Our third choice, according to Chisholm, is skepticism. The skeptic adroitly recognizes that in order to determine whether or not we possess in each case a good or bad apple we require a method to justify our choice and that in order to select a reliable method we need to know the difference, ab initio, between good and bad apples, and she therefore concludes that there is no way to decide.

The Truth Is Out There

It is an admixture of the skeptic's and particularist's philosophies which most closely resembles the weak orthodoxy of American K-8 education--a system (if one could be so generous as to describe it as such) often characterized, certainly not thoroughly but perhaps most aptly, by its ability to not distinguish between good and bad apples (refresh page if video below does not start with roundtable):


One can see evidence for this strange orthodoxy not only in the way the "system" administers itself, but in more abstract ways as well. I highlighted in this post, for example, part of a "skeptico-particularist" argument that is, in one form or another, very popular among professional educators as a defense against the evils of generalization and standardization:

It is simply not possible to prove that an approach to teaching and learning will be effective before the fact.

Education as a scientific discipline is a young field with an active community focused on R&D--research on learning coupled with the development of new and better curriculum materials. In truth, however, much of the work is better described as D&R--informed and thoughtful development followed by careful analysis of results. It is in the nature of the enterprise that we cannot discover what works before we create the what.

Similarly, James and Dewey—two of educational psychology’s founding philosophers—though not self-identified skeptics or “particularists” in any strict or relevant sense, were not exactly warm to a “methodist” approach to discerning truth. Author and Aggie John J. McDermott said it this way:

James has a name for . . . methodological anality. He calls it "vicious intellectualism" by which we define A as that which not only is what it is but cannot be other. Proceeding this way, answers abound and clarity holds sway. Missing is surprise, novelty, the wider relational fabric, often riven with rich meanings found on the edge, behind, around, under, over the designated, prearranged conceptual placeholders. Percepts are what count, and the attendant ambiguity in all matters important, presage more and deeper meaning not less. Following John Dewey, method is subsequent and consequent to experience, to inquiry. Method can help fund and warrant experience, but it does not grasp our doings and undergoings in their natural habitat. For that, we must begin with and experimentally trust our affections--dare I say it, trust our feelings. They may cause trouble, but they never lie.

The surest evidence, however, for the antagonism between Chisholm's "methodism" and American education can be found through experience and observation. A small helping only of each of these is enough, I think, to convince most rational people that at nearly every turn, education steers itself craftily away from the advisement of all but the vaguest and easiest criteria: How shall we teach? What shall we teach? Who shall we reward? punish? What shall we value and devalue? Education will provide answers to these questions or it won't, but it never has a way to decide, a methodology, a set of criteria it refers to. Its nascent science works at the fringes, in obscurity.

To come, finally, full circle, education seems unable to help but vacillate between its skepticism, which holds every idea (or none of them) to be right, and its particularism, which holds all of its own ideas to be right. This inability, in the end, makes it nearly impossible for education to decide before the fact that something can be wrong. And that is precisely what is wrong with education.

References:
Chisholm, R.M. (1982). The problem of the criterion. In L. Pojman (Ed.), The theory of knowledge, second edition (pp. 26-35). Belmont, CA: Wadsworth.

McDermott, J. (2003). Hast Any Philosophy in Thee, Shepherd? Educational Psychologist, 38 (3), 133-136 DOI: 10.1207/S15326985EP3803_2

Photo Credits:

http://www.flickr.com/photos/craftygoat/ / CC BY-NC 2.0

http://www.flickr.com/photos/mat-/ / CC BY-NC-ND 2.0

A little praise for yours truly.


And, by extension, that praise goes out to any and all editors (who are often teachers also) in math education who work tirelessly--and with little credit for it--to get things right and to make things clear for students.

I've met many math ed. editors with whom I have disagreed, often strongly. But I've never met one who doesn't care about education, about kids, about math. So, I'm extremely grateful to Keith Devlin for making mention of their work.

They deserve this praise.

To quote myself from last time:

To each writing and editing task, I bring four important resources: (1) my own knowledge about the lesson that I am writing, (2) a list of related lessons that have likely already been presented, (3) a list of related lessons that will likely be presented in the future, and, most importantly, (4) a memory not only for the content of each of the lessons mentioned above, but also a memory of all the different ways those lessons have been presented.

And then, from the same source:

Even if intelligent and/or experienced discussants were to restrict themselves to a specific lesson at a specific grade level and bring only those four resources mentioned above into a debate, one could write ahem, a lot, about--and education could gain a lot from--the ensuing conversation.

But, really, what could we talk about that actually matters if we couldn't discuss research or results or various ways of delivering instruction, etc.?

Well, I'd like to offer just one example here (with a few hypothetical details) that might demonstrate to readers what such a conversation would look like--and why such a conversation would be important.

But before I do that, I think I should do something to try to convince skeptical readers that such an exercise is not an alien way of thinking, not a new-fangled concoction from some no-name blogger looking to air yet another opinion about education. To do that, I will turn briefly to the law as an analogy.

The Law as Analogy

When a case is brought before a court, many, if not all, of the facts of the case under consideration are related to events in the past. That is, there is generally a temporal distance between the facts considered by a court and the events to which those facts relate. What's more, there is an informational distance between the actual events and the facts considered. Certain aspects of the events are submitted for consideration while others are not, and certain aspects of the events are deemed irrelevant while others are relevant. Courts may also find other areas of law not mentioned in a dispute both relevant and applicable.

These distances (among others) between the facts (and contexts) before a court and the events (and contexts) to which they relate serve collectively to separate the thinking of actors in legal proceedings and other people, because the distances describe different inputs to that thinking.

Given these distances and differences in "thinking," one would be right to wonder why everyday folks grant any legitimacy at all to the courts of law that function to settle their disputes. The answer, of course, in part, is that both citizens and courts operate under a set of general guidelines, or principles. Citizens agree that these principles apply to them, and courts work to interpret and apply them to settle disputes.

Back to Education

What I have in the past proposed (and will do so again here) are principles for mathematics instruction which can be easily accepted as applicable to daily classroom teaching and interpretable at some temporal and informational distance from that teaching. Here they are again (ignore the Distributor/Producer and Consumer labels):


Teachers would agree, I imagine, that the content of their teaching should be presented accurately (precision), such that it is understandable to their students (clarity). And I think there would be little disagreement that the elements of their instruction should be arranged so that students can see the relatedness of ideas (packaging, or cohesion) while also moving in steps from known to unknown (order). In other words, teachers would have little trouble considering that these principles apply to the instruction they present. [The descriptions I included for the principles above are just my own interpretations. There are certainly others.]

It is also possible, in my view, for a diverse and intelligent group of people--at some temporal and informational distance from the activities of a classroom--to usefully interpret these principles as they apply to classroom instruction without requiring the full range of inputs that are available to a classroom teacher's thinking.

This brings me to the example I mentioned above of the kind of conversation that can take place regarding mathematics instruction--a conversation that can be principles-based and, while "disconnected" (temporally and informationally) from the classroom, can be practical and useful.

Area of a Parallelogram -- The Facts

A very common way of teaching students the reasoning behind the formula for the area of a non-rectangular parallelogram is to demonstrate how to take apart such a parallelogram and turn it into a rectangle. Some states' elementary and middle-school mathematics standards are fairly explicit about this "way" of teaching. Here are two examples (emphases mine):

Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by cutting and pasting a right triangle on the parallelogram). --California, 1997, Grade 5 (PDF)

Derive and apply formulas for areas of parallelograms, triangles, and trapezoids from the area of a rectangle. --Florida, 2007, Grade 5

The NCTM's 2000 publication Principles and Standards for School Mathematics (PSSM) also endorses this method as consistent with stimulating students' understanding of, and investigation into, area (emphasis mine):

One particularly accessible and rich domain for such investigation is areas of parallelograms, triangles, and trapezoids. Students can develop formulas for these shapes using what they have previously learned about how to find the area of a rectangle, along with an understanding that decomposing a shape and rearranging its component parts without overlapping does not affect the area of the shape. --PSSM, 2000, p. 244

And, from the third edition of this well-known book by Van de Walle, we have the following:

Parallelograms that are not rectangles can be transformed into rectangles having the same area. The new rectangle has the same height and two sides the same as the original parallelogram. Students should explore these relationships on grid paper, on geoboards, or by cutting paper models, and should be quite convinced that the areas are the same and that such reassembly can always be done with any parallelogram. As a result, the area of a parallelogram is base times height, just as for rectangles. (p. 334)

(I should note that, of the four snippets I present above, only Van de Walle's even makes an attempt at a consistent distinction between non-rectangular parallelograms ["parallelograms that are not rectangles"] and rectangles. But then even in the Van de Walle snippet, this distinction breaks down in the last sentence. Already not a good sign if we have precision in mind.)

Consistent with the standards and the specific observations and suggestions in the widely referenced publications listed above, textbook lessons typically introduce students to the idea that the area of a non-rectangular parallelogram can be described by the same formula as that used to describe the area of a rectangle with the same base and height, using one or more illustrations like the one below, accompanied by written instruction to help students understand the illustration:


Generally, the purpose of this written instruction is to clarify for students that (a) a part of the non-rectangular parallelogram was simply removed and then re-attached somewhere else on the figure--a transposition that does not change the area of the figure, and (b) the resulting figure is a rectangle whose height and base are the same as that of the non-rectangular parallelogram.

The Analysis

It's worth taking a moment--before proceeding to an analysis of the above instruction--to refresh one's perspective with the words of the late Isaac Asimov; these from an article titled The Relativity of Wrong.

The basic trouble, you see, is that people think that "right" and "wrong" are absolute; that everything that isn't perfectly and completely right is totally and equally wrong. However, I don't think that's so. It seems to me that right and wrong are fuzzy concepts. . . .

Nowadays, of course, we are taught that the flat-earth theory is wrong; that it is all wrong, terribly wrong, absolutely. But it isn't. The curvature of the earth is nearly 0 per mile [0.000126 per mile], so that although the flat-earth theory is wrong, it happens to be nearly right. That's why the theory lasted so long. . . .

[However], although the flat-earth theory is only slightly wrong and is a credit to its inventors, all things considered, it is wrong enough to be discarded in favor of the spherical-earth theory.

And yet is the earth a sphere?

No, it is not a sphere; not in the strict mathematical sense. . . . Actual measurements of the curvature of the earth were carried out in the eighteenth century and Newton was proved correct.

The earth has an equatorial bulge, in other words. It is flattened at the poles. It is an "oblate spheroid" rather than a sphere. . . .

Even the oblate-spheroidal notion of the earth is wrong, strictly speaking. In 1958, when the satellite Vanguard I was put into orbit about the earth, it was able to measure the local gravitational pull of the earth--and therefore its shape--with unprecedented precision. It turned out that the equatorial bulge south of the equator was slightly bulgier than the bulge north of the equator . . .

There seemed no other way of describing this than by saying the earth was pear-shaped, and at once many people decided that the earth was nothing like a sphere but was shaped like a Bartlett pear dangling in space. Actually, the pearlike deviation from oblate-spheroid perfect was a matter of yards rather than miles, and the adjustment of curvature was in the millionths of an inch per mile.

What I will want to do first in the next few paragraphs is argue that the instruction referenced above is wrong. And what I mean by wrong, at least at the very beginning of my argument, is not that everything about the instruction is "all wrong, terribly wrong, absolutely," but that it is "wrong enough to be discarded" in favor of better instruction.

Let's start with a very simple idea: rectangles and squares are special kinds of parallelograms. Deriving or developing a formula to describe the area of certain non-rectangular parallelograms based on what is taught about the area of rectangles--as is often done in this instruction--is a bit like arguing that the numbers below are all integers because you can count up from them or count down from them and reach the integer 9.

3, 1, 11, 45
It turns out in both cases that what is stated is true--non-rectangular parallelograms and rectangles share the same area formula, and the numbers above, along with the number 9, are all integers. And it so happens that in both of these cases, the reasons are difficult to argue with as well--transforming a non-rectangular parallelogram into a rectangle without gaining or losing area is certainly a convincing demonstration, and so long as one construes "counting" as being restricted to integers, the reason given above for the "integer-ness" of 3, 1, 11, and 45 is satisfactory, albeit sloppy.

But, still, those reasons are wrong. The "integer-ness" of integers has nothing to do with the number 9, and the reason the area of a parallelogram can be described by the formula A = bh or A = lw has nothing to do with rectangles and everything to do with the fact that parallel lines are at all corresponding points the same distance apart:


This idea--this more precise idea--awaits actual fleshing out in lessons, where the other principles (clarity, order, and cohesion), along with precision again, must be considered. Note that the new arrangement above is suggestive of reworked orders and packagings of content. If, for example, we are to keep this topic (parallelogram area) where it typically falls in a mathematics curriculum, then we would need to move up discussions about parallel lines from where they typically fall (order change), and such discussions could no longer be considered in isolation (cohesion change)--two extremely small examples of the enormous ground mathematicians, teachers, education specialists, etc., could cover in the kinds of discussions I recommend in this article.

Conclusion

I have been short on posts in 2009, but I have had one main idea--the method of education. What I tried to argue, beginning with the Garfunkel's Syndrome series and ending, for the most part, with this post (though these ideas go back a long way), was that part of the answer for just some of the troubles of education--specifically mathematics education--could be found in a methodology for approaching its questions.

And I'd like to think I'm a little bit closer now to figuring out what such a methodology could look like.

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

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

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

And that's pretty much it.

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

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

Bollocks

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

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

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

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

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

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

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

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

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

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


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

The Basics

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


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

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


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


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

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

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

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

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


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


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

Potential Problems

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

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

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

Lisa's Method

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


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

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

Scientific Thinking

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

Let's take that up next time.

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

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


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

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

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


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

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

Here are some other planets we have discovered.

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

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

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


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

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

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

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

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

45-Degree Rotations

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

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

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


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


Okay, so that was the easy part.

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


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

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

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


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

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

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

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


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

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


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


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

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

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

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


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


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


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


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


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



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

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

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

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

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

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

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

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

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

The Setup

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


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


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

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

The Question

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

The "Answer"

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

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

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

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

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

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

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

The Consequence

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

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

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

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

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

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

Stay tuned.

Garfunkel's Syndrome, Part III

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

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

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

Soup Sequitur

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

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

Instructions

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

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

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

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

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

Stir in the sausage.

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

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

"And a Spaceship Lands."

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

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

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

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

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


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