Archive for June, 2013

Faulty logic in the new Math Wars skirmish

Mathematicians love puzzling about self-referential statements such as “This sentence is false” (ask yourself if it is true or false and see what happens), and analyzing them has led to significant insights in the way mathematics is used to model real world phenomena. But I suspect the authors of a recent New York Times opinion piece did not see the ironical self-reference in their title The Faulty Logic of the ‘Math Wars’, published on June 16.

If ever there were an article that repeatedly utilized faulty logic, this was it. Evidently written from an advocacy viewpoint, the authors obviously got carried away, allowing their advocacy position to stretch and twist logic well beyond breaking point.

I don’t normally comment on math ed advocacy articles, since people tend to be so firmly entrenched in their position that no amount of evidence and reasoning will prompt them to reflect. But this particular article was so far off the mark, and the logic so abused, I could not resist picking up my teacher’s red pen and going through it paragraph by paragraph, annotating as I did.

(Actually I generally use a green pen, since red is known to have negative consequences on student motivation. In this case, the two authors are accomplished academics, well able to handle the back and forth of scholarly debate, where we attack one another’s ideas but not the people, so ink color is not an issue. Besides, for this medium I worked at a keyboard, using boldface to signify my comments to their normal-typeface article.)

So, original article in regular type, my commentary in bold. Here we go. (It’s long, as was the original article, much of which I have to quote in order to critique it.)

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There is a great progressive tradition in American thought that urges us not to look for the aims of education beyond education itself. Teaching and learning should not be conceived as merely instrumental affairs; the goal of education is rather to awaken individuals’ capacities for independent thought. Or, in the words of the great progressivist John Dewey, the goal of education “is to enable individuals to continue their education.”

This vision of the educational enterprise is a noble one. It doesn’t follow, however, that it is always clear how to make use of its insights. If we are to apply progressive ideals appropriately to a given discipline, we need to equip ourselves with a good understanding of what thinking in that discipline is like. This is often a surprisingly difficult task. For a vivid illustration of the challenges, we can turn to raging debates about K-12 mathematics education that get referred to as the “math wars” and that seem particularly pertinent now that most of the United States is making a transition to Common Core State Standards in mathematics.

At stake in the math wars is the value of a “reform” strategy for teaching math that, over the past 25 years, has taken American schools by storm. Today the emphasis of most math instruction is on — to use the new lingo — numerical reasoning.

No. Numerical reasoning is just one aspect of math instruction. Analytic reasoning, logical reasoning, relational reasoning, and conceptual understanding are just as important and equally stressed. The basic components of K-12 education were elaborated at length by a blue ribbon panel of experts assembled by the National Academies of Science in a 2001 National Academies Press volume titled Adding It Up: Helping Children Learn Mathematics

This is in contrast with a more traditional focus on understanding and mastery of the most efficient mathematical algorithms.

What is meant by “efficient” here? For many centuries, it was a crucial ability to be able to carry out numerical computations in the head or by paper-and-pencil. The “standard algorithms” were developed in India in the first centuries of the Current Era, and further honed by traders and engineers in the Iraq-Persia region, in order to make mental paper-and-pencil calculation most efficient. (The medium then was either a smooth patch of sand, a sandbox, a parchment, or some form of tablet.) 

Those standard algorithms sacrificed ease of understanding in favor of computational efficiency, and that made sense at the time. But in today’s world, we have cheap and readily accessible machines to do arithmetical calculations, so we can turn the educational focus on understanding the place-value system that lies beneath those algorithms, and develop the deep understanding of number and computation required in the modern world, and prepare the ground for learning algebra.

A mathematical algorithm is a procedure for performing a computation. At the heart of the discipline of mathematics is a set of the most efficient — and most elegant and powerful — algorithms for specific operations.

“At the heart of the discipline”! Totally untrue. This reduces mathematics to computational arithmetic. The standard arithmetical algorithms were developed by, and for, traders, to facilitate commercial activity. Those algorithms were never at the heart of mathematics, not even when they were developed. Anyone who says this, exhibits so little knowledge of what mathematics is, they should not purport to be sufficiently expert to pontificate on mathematics education.

The most efficient algorithm for addition, for instance, involves stacking numbers to be added with their place values aligned, successively adding single digits beginning with the ones place column, and “carrying” any extra place values leftward.

Actually, these are only the most (computationally) efficient algorithms if the computation is done using paper-and-pencil. For mental calculation, left-right algorithms are far more efficient. But this is a red herring, since the focus in education should be learning and understanding, and there are algorithms that are far more efficient in achieving those goals.

What is striking about reform math is that the standard algorithms are either de-emphasized to students or withheld from them entirely.

De-emphasized, yes, for the reason I alluded to above.  The need for a strong focus on those particular algorithms evaporated with the dawn of the computer age. No good teacher would withhold mention or discussion of the standard algorithms, not least because they have huge historical significance. That last remark of the authors is simply not true (though I dare say you could find the occasional teacher who acted is such a way).

In one widely used and very representative math program — TERC Investigations — second grade students are repeatedly given specific addition problems and asked to explore a variety of procedures for arriving at a solution. The standard algorithm is absent from the procedures they are offered.

Note that this is what is done in the second grade, when students are just starting out on their mathematical learning.

Students in this program don’t encounter the standard algorithm until fourth grade, and even then they are not asked to regard it as a privileged method.

So much for the authors’ earlier assertion about the standard algorithms being withheld! As to those algorithms not being treated as privileged methods, that is as it should be in today’s world. They were (rightly) privileged for many centuries when calculation had to be done by hand. But those days are gone.

The battle over math education is often conceived as a referendum on progressive ideals, with those on the reform side as the clear winners. This is reflected, for instance, in the terms that reformists employ in defending their preferred programs. The staunchest supporters of reform math are math teachers and faculty at schools of education.

Just stop and think about this for a moment. Where would you expect to find people who know most about mathematics education? Dare I say, math teachers and faculty at schools of education? By way of analogy, consider this statement. “The staunchest supporters of the need for cleanliness and the use of improved medical procedures are the doctors and nurses who work in hospitals.” You don’t say! If you get sick, you consult a medical professional – someone who has spent years studying the subject and has demonstrated their knowledge and ability. Why not proceed in the same fashion when it comes to education?

While some of these individuals maintain that the standard algorithms are simply too hard for many students, most take the following, more plausible tack. They insist that the point of math classes should be to get children to reason independently, and in their own styles, about numbers and numerical concepts. The standard algorithms should be avoided because, reformists claim, mastering them is a merely mechanical exercise that threatens individual growth.

This is such a blatant misrepresentation, I am suspicious of the authors’ motives in writing this. The standard approach in current beginning mathematics education is to begin by providing opportunities for children to reason independently (a hugely valuable ability in today’s world!), and then introduce algorithms, starting with algorithms designed for educational efficiency, and then moving on to algorithms optimized for hand-calculation efficiency. (It is arguable that it would make sense in today’s world to spend some time also looking at the algorithms used by computers, since they are not the standard paper-and-pencil algorithms, and comparison of different algorithms can help students gain deep understanding of number and computation. That could perhaps come later in the educational journey.)

The authors end the paragraph by repeating once more their false claim that “the standard algorithms should be avoided”. The “threatening growth” comment would have substance if mechanical mastery of the standard algorithms were the students’ only exposure to computational methods. But they are not.

The idea is that competence with algorithms can be substituted for by the use of calculators, and reformists often call for training students in the use of calculators as early as first or second grade.

No they do not. I do not know a single teacher who advocates calculator use in the second grade. I can’t say with certainty that you won’t find a self-proclaimed “reformist” who has made such a call, but it definitely is not “often”.

Reform math has some serious detractors. It comes under fierce attack from college teachers of mathematics, for instance, who argue that it fails to prepare students for studies in STEM (science, technology, engineering and math) fields.

You can find (a few) college professors who say evolution is false, but they are not in the mainstream. College professors enjoy enormous freedom in what and how they teach, so you can find all kinds of examples. But as someone whose career is almost exclusively spent in academia, who travels extensively and meets other academics across the US and around the world, I have yet to meet anyone who argues strongly that reform math fails to prepare students for studies in STEM. What I do hear a lot is complaints about a lot (not all) of K-12 education failing to prepare students adequately. My sense is the problem is quality of teaching as much if not more than curriculum, though the two are not necessarily independent.

These professors maintain that college-level work requires ready and effortless competence with the standard algorithms and that the student who needs to ponder fractions — or is dependent on a calculator — is simply not prepared for college math.

The first part of this statement is totally false. (Unless the phrase “these professors” refers to a couple of professors the authors happen to know.) Familiarity with the standard algorithms plays no role in college STEM. To say mastery of those particular algorithms is crucial to STEM is like saying using a Mac better prepares you for STEM than using a PC. Having a good sense of, and facility with, number, including fractions, is absolutely vital, and the algorithms currently taught in K-12 were developed to maximize that outcome. 

Having been teaching university level mathematics around the world for 45 years, I know that in the days when the standard algorithms were the main focus, the results in terms of college-preparedness were terrible. Except for a few students (that few very likely including the article’s authors), the classical teaching methods simply did not work. If they had worked, you would not find so many adults who say they cannot do math! That failure of the old method is what led to the introduction of alternative approaches using algorithms optimized for learning.

They express outrage and bafflement that so much American math education policy is set by people with no special knowledge of the discipline.

Really? I mean, really? Other than a few outliers, I have not heard a deluge of outrage. Reform math is a result of an extensive collaboration between math teachers, mathematics education faculty, and mathematicians, including that blue-ribbon committee of the National Academy of Sciences I mentioned earlier, which published that huge volume on the basic of mathematics education almost fifteen years ago. Hardly “people with no special knowledge of the discipline.”

Even if we accept the validity of their position, it is possible to hear it in an anti-progressivist register. Math professors may sound as though they are simply advancing a claim about how, for college math, students need a mechanical skill that, while important for advanced calculations, has nothing to do with thinking for oneself.

I doubt they would sound that way. In any case, I am not sure what point the authors are trying to make here.

It is easy to see why the mantle of progressivism is often taken to belong to advocates of reform math. But it doesn’t follow that this take on the math wars is correct. We could make a powerful case for putting the progressivist shoe on the other foot if we could show that reformists are wrong to deny that algorithm-based calculation involves an important kind of thinking.

What seems to speak for denying this? To begin with, it is true that algorithm-based math is not creative reasoning. Yet the same is true of many disciplines that have good claims to be taught in our schools. Children need to master bodies of fact, and not merely reason independently, in, for instance, biology and history. Does it follow that in offering these subjects schools are stunting their students’ growth and preventing them from thinking for themselves? There are admittedly reform movements in education that call for de-emphasizing the factual content of subjects like biology and history and instead stressing special kinds of reasoning. But it’s not clear that these trends are defensible. They only seem laudable if we assume that facts don’t contribute to a person’s grasp of the logical space in which reason operates.

I still don’t know for sure what the authors are trying to say here. To my knowledge, no teacher has ever said facts are not important. I can only assume that authors are erecting a huge straw man, but it’s so ludicrous it does not deserve more than this brief dismissal.

The American philosopher Wilfrid Sellars was challenging this assumption when he spoke of “material inferences.” Sellars was interested in inferences that we can only recognize as valid if we possess certain bits of factual knowledge. Consider, for instance, the following stretch of reasoning: “It is raining; if I go outside, I’ll get wet.” It seems reasonable to say not only that this is a valid inference but also that its validity is apparent only to those of us who know that rain gets a person wet. If we make room for such material inferences, we will be inclined to reject the view that individuals can reason well without any substantial knowledge of, say, the natural world and human affairs. We will also be inclined to regard the specifically factual content of subjects such as biology and history as integral to a progressive education.

More of the same. The horse is long dead. It was never born, for heavens sake. Stop flogging it.

These remarks might seem to underestimate the strength of the reformist argument against “preparatory” or traditional math. The reformist’s case rests on an understanding of the capacities valued by mathematicians as merely mechanical skills that require no true thought.

The second sentence here is the exact opposite of actuality. The reformist case rests on knowing that mathematicians value mechanical skills that are based on sound understanding and can be utilized in a creative, thoughtful, reflective way.

[I am going to skip the authors’ next few paragraphs as theoretical cognitive philosophy. You can the entire article in its original posting.]

It is important to teach [the standard algorithms] because, as we already noted, they are also the most elegant and powerful methods for specific operations. This means that they are our best representations of connections among mathematical concepts. Math instruction that does not teach both that these algorithms work and why they do is denying students insight into the very discipline it is supposed to be about.

The first sentence is okay if you delete the qualifier “the most elegant”. The second sentence displays total ignorance of mathematics. (A very odd thing, since the second listed author is an accomplished research mathematician.) The third sentence needs a bit of analysis.

The standard algorithms are a very good historical hack, improved over many generations, that enabled people to do complex arithmetic calculations with paper-and-pencil (or its early equivalent). As long as students learn at least one algorithm for each basic arithmetical operation that gives them an understanding of number and the number system, they will gain insight into number and arithmetic. But number and arithmetic are not what the discipline [mathematics] “is supposed to be about.” Again, the authors’ words indicate that have not a clue what mathematics is about. (Once more puzzling, given the second author’s credentials.) As it happens, there are algorithms that are better suited than the standard ones for gaining insight into number and arithmetic, and those are the ones currently used in “reform mathematics education.” Teaching other methods, including the standard algorithms, can increase that important insight, but the justification for including the classical algorithms is largely historical.

(Reformists sometimes try to claim as their own the idea that good math instruction shows students why, and not just that, algorithms work. This is an excellent pedagogical precept, but it is not the invention of fans of reform math. Although every decade has its bad textbooks, anyone who takes the time to look at a range of math books from the 1960s, 70s or 80s will see that it is a myth that traditional math programs routinely overlooked the importance of thoughtful pedagogy and taught by rote.)

Nonsense. No one makes such a claim. There have always been good teachers providing good education. There have always been, unfortunately, poor teachers providing poor education. This parenthetical paragraph is another straw man.

As long as algorithm use is understood as a merely mechanical affair, it seems obvious that reformists are the true progressivists. But if we reject this understanding, and reflect on the centrality to mathematical thought of the standard algorithms, things look very different. Now it seems clear that champions of reform math are wrong to invoke progressive ideals on behalf of de-emphasizing these algorithms. By the same token, it seems clear that champions of preparatory math have good claims to be faithful to those ideals.

I cannot imagine a paragraph that is more the exact opposite of actuality.

There is a moral here for progressive education that reaches beyond the case of math. Even if we sympathize with progressivists in wanting schools to foster independence of mind, we shouldn’t assume that it is obvious how best to do this. Original thought ranges over many different domains, and it imposes divergent demands as it does so. Just as there is good reason to believe that in biology and history such thought requires significant factual knowledge, there is good reason to believe that in mathematics it requires understanding of and facility with the standard algorithms.

The authors were doing fine until they wrote the second part of this last sentence. The standard algorithms offer no privileged insight into arithmetic, let alone the far broader discipline of mathematics. Their value today is primarily historical. For the period after our ancestors developed symbolic writing up to the invention of the modern computer, the standard algorithms were of great value. They no longer offer anything of unique pedagogic value other than variety. Other algorithms are better suited to learning.

Indeed there is also good reason to believe that when we examine further areas of discourse we will come across yet further complexities. The upshot is that it would be naïve to assume that we can somehow promote original thinking in specific areas simply by calling for subject-related creative reasoning. If we are to be good progressivists, we cannot be shy about calling for rigorous discipline and training.

An apple pie paragraph that everyone will agree with.

The preceding reflections do more than just speak for re-evaluating the progressive credentials of traditional, algorithm-involving math. They also position us to make sense of the idea, which is as old as Plato, that mathematics is an exalted form of intellectual exercise. However perplexing this idea appears against the backdrop of the sort of mechanical picture favored by reformists, it seems entirely plausible once we recognize that mathematics demands a distinctive kind of thought.

Actually, what the preceding reflection indicates is that the authors haven’t any real understanding of mathematics or mathematics learning. (Second author puzzlement again.) They certainly give the impression they have no idea what reform mathematics is about, since the reformists’ position is as far removed from a “mechanical picture” as can be imagined.


I’m Dr. Keith Devlin, an emeritus mathematician at Stanford University, an author, and was for many years “the Math Guy” on NPR’s Weekend Edition. Off duty, I’m an avid cyclist. (The header photo is me halfway up Mt. Baldy in Southern California.)

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