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.)

* * * * * * * * *

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.**

Great post. I read the op-ed in the NYTimes the day it came out. I had just finished Jo Boaler’s book “What’s Math Got to Do with it?” and I saw this as just another example of muddy thinking. As a practicing (non-academic) mathematician for decades, I know that most of the algorithms I learned in school, including such arcana as manually computing square roots and inverting matrices by hand, are entirely obsolete. I learned maths in spite of the algorithm heavy school system, but I was lucky.

What we need today is people — not just those in STEM areas — who know what mathematics is really about.

THANK YOU so much for this. I was so angry about the article I could barely sleep last night. You have done a fantastic job articulating all of the things the authors got wrong.

I think you fail to give the last paragraph the credit it deserves. As the conclusion of the article, it must deliver on what was promised in the title. And so it’s reasoning goes:

“Now that we have established our definition of mathematics (math = traditional algorithms), we call in the authority of Plato himself to convince you that this kind of math is an exalted form of intellectual exercise. So don’t believe those nasty reformers who try to convince you the standard algorithms are some sort of mechanical exercise. Instead, believe us when we say that the standard algorithms demand a distinctive (and exalted — don’t forget Plato!) kind of thought.”

Keith, I’m disappointed by your response. You are a leader in the math world and can play a crucial role in helping people understand both sides of this discussion. There needn’t be math wars and we should work to prevent them as it only hurts our ability to improve math education. The original article was too strongly worded, but reflects the reality of how reform math has been implemented in some (many?) places in the classroom. To dismiss these concerns entirely is a mistake.

Comments like “Actually, what the preceding reflection indicates is that the authors haven’t any real understanding of mathematics or mathematics learning” don’t help to forge mutual understanding and further improvements in math education. With one of the authors being a respected mathematician, who clearly has a real understanding of mathematics, such comments are out-of-place and can only be seen as personal, rather than professional, attacks.

Thomas, I think you have misread my remark, which is, as I say, a statement of what the authors’ words

. Elsewhere in the commentary I note (more than once) my puzzlement that a successful research mathematician should co-write such an article, and I even speculate as to the reason. You may be able to “only see” my comment as personal, but I doubt that is a common reading. It’s simply two people expressing different views — the very essence of scholarly debate.indicateDear Keith,

If you’re going to cast a critical eye on each phrase in the original article, I think you have to be much more careful about what you write. Otherwise, one can make the same type of criticisms of what you wrote, and that doesn’t help improve math education.

Clearly, at least one of the authors has a real understanding of mathematics. So the statement “X implies the authors haven’t any real understanding of mathematics or mathematics learning” is argumentative and personal as you are questioning whether the authors have a real understanding of mathematics. You aren’t questioning the statement they have made. You are making a judgment about their knowledge. I did read your puzzlement about the second author’s comments elsewhere, but that doesn’t make your claim here any less personal. If you want to question assertion X, one should address X. One doesn’t need to bring the authors into this (by the way, I should add that I never heard of these authors until the original article was published).

These debates would be more productive if there weren’t these personal attacks (I know you didn’t start this discussion, but you do control your responses). Even in your response to me, you question my comment by insinuating that my interpretation must be uncommon. If it is uncommon, I must live in a strange part of the world as I asked three other colleagues how they would read your statement and they read it the same way I did. Even if it were uncommon, I would think professionalism requires us to not express ourselves in ways that could be construed as insulting.

Scholarly debates are healthy, but they shouldn’t involve my questioning your understanding of mathematics, nor you questioning mine.

-Thomas

Thomas, I repeat, I did not suggest the author did not know what mathematics is, I said that is what the article indicated. Moreover, I did so in counterpoint to the final paragraph which itself provided the authors’ summary of what they believed they had indicated. Both their paragraph and my counterpoint were about what their argument indicated. They based their conclusion on what they had previously written, and so did I. I simply adopted their argument structure.

Someone can be respected as a mathematician and still advocate positions about mathematics education that appear to reflect little or no understanding of mathematics or its teaching/learning.

What makes this particular piece dangerous is that how it misrepresents both mathematics and the views of an amorphous, easily attacked shadow group labeled (by the authors, not any member of the group that I know of) “reformists.” The very choice of that word is a political one, implying by the ending “-ists” that the folks being spoken about are somehow the ones who have an agenda, an implied “-ism” they are pushing that is outside the mainstream of, well, the proper view of anything and everything under discussion by the authors. That’s a cheap rhetorical trick, and they deserve to be called on it, as well as on their distortions of quite a few matters. Keith does a yeoman job of the latter and I commend him for doing it.

SInce the viewpoints and comments he’s rebutting are clearly those of the two authors, however, he is not making up straw folk to insult, but rather addressing directly the published statements of TWO people, no more and no less (regardless of whether others agree with some or all of those statements). That may make any criticism of the article sound “personal” but I think that’s a far cry from making up a group (“reformists”), making exaggerated and/or false claims about them, and hiding behind the fact that no specific persons are named. That, too, is a rhetorical trick, not too far from holding up lists of “Communists” (another popular group of “-ists” to find lurking in the woodwork) in the government (or, most recently, in the Progressive Caucus).

There are very definite reasons that there has been a Math War for the past 20+ years in this country. Having studied carefully how it evolved, I have a clear sense of why it has existed, still exists, and is likely to continue, even as the distractions of the Common Core wax and, it is to be hoped, wane. Reading the majority of the comments on the NYT article, most people will be struck by how strongly a vocal group of people in this country agrees with the mischaracterizations offered by the authors. It’s wonderful that Keith chose to speak out about the illogic of this advocacy piece. If the authors’ hair was mussed, I can only suggest that they asked for it.

As an addendum to Michael’s comment, I should note that the NYT did not accept my counter-article for publication, though I dare say they get sent a lot of submissions, and in all likelihood no one ever got round to looking at it. Thus the points I make will not be seen by the majority of people who read the original piece, who may be left with the impression that the article’s representation of actuality is correct.

I am sorry that you felt the need to respond to the authors in the article. You had already expressed your sense well in the last paragraph of the segment named Secret Sauce, Devlin’s Angle, Jan 2009. As you suggested, let’s not be distracted from creating a better situation for children to learn a proper sense of mathematics.

My first comment–responding to a couple of queries from friends–is a clarification: the comment above from James King is from another James King, not me, a mathematician at the University of Washington. (James King is a very common name.) Our state has an extensive history with the mathematician co-author of the NYT article, who was hired as an influential outside consultant to vet math standards and textbooks at the height of our local math wars. At that time I did some writing with a different perspective.

Personally, what I am sorry about is that this article appeared in the New York Times with a tone that appears to be a gratuitous attempt to re-ignite the very destructive math wars. And this comes at a time when I had hoped the math community and general public had moved on to more constructive work around the Common Core math standards. Given the existence and prominence of the NYT article, a rebuttal was certainly called for, though one can regret the necessity for it.

I feel Keith’s response is appropriate and insightful. I am very grateful that he took the trouble to critique in such detail the infuriating NYT article.

This is a really good post. I tend to agree with your commentary, although I am unfamiliar with US Math[s] Ed policy, the reform math[s] and indeed “The Math[s] war”, but this has really piqued my interest, do you have any suggestions for further reading on the subjects?

Oh, lord, not the math wars again! If there’s anything that math has taught me it’s the humility to admit that I don’t know a lot, despite having achieved a master’s degree in math and having taught math for some 30 years. From my experiences teaching, I can say that students will use the calculator as a crutch instead of a tool, 99.9% of the time. I taught linear algebra last term (using Strang!) and found students using a calculator to add, say, 1/2 and 2/3. What they lose is the concept of LCM and GCD and just the confidence to know that you can work the algorithms and maybe even find some nifty shortcuts like 97*99 = 98^2 – 1 = (100-2)^2 – 1 = 10000 – 400 + 4 – 1 = 9603…I think. What I’ve done when I teach remedial courses is use a text book that assumes the students have a calculator available, and then forbid them to use it. That way you get interesting numbers (which many texts shy away from) and plenty of practice with the algorithms. Another thing I do is teach the concept of significant digits, which many math texts ignore entirely. So if you assume that you have 3 digit approximations and you multiply, then you’re only after the the 3 most significant digits and a left to right algorithm will save you some time.

So please, let’s not dredge up the false either/ors again. You need to work all the various types of reasoning, from mechanical algorithmic reasoning to logics to analytics to heuristics and back around again.

I think I love you. KEEP WRITING!!!!

I lost you at left to right progression is more efficient for mental calculation- maybe, if you are estimating and not trying to get an exact number. The break apart method is one method they use in common core aligned textbooks. This is where you break up 212 and 314 into 200+10+2 and 300+10+4 and then you start adding or subtracting in the hundreds and move your way to the right to the ones place. This is for children who are 8 or 9 and do not understand nor have they learned the standard algorithm. By the time they get to the standard algorithm, which is last, they are programmed to start on the right and some always struggle with it. You basically have to break a bad habit.

Do you honestly believe this method is quicker or more efficient for children in school? Are they to do this break apart method or another nonstandard algorithm in later grades to the side when figuring long division? Or would they draw blocks to represent the numbers and mark them for – or count them for +? If all algorithms are of equal quality and the standard algorithm is obsolete, then I would assume you are implying that this is an appropriate method for students to use through out their math studies year after year. It would take all day if each time they had to + or- a multi digit number they substitute a “strategy” for an algorithm.

No. I’m obviously not a mathematician, I am a parent who has seen what this actually represents outside the academic ivory tower. It sucks and nobody likes it. Everyone’s at Kumon or hiring private tutors- those who can afford it, the others are stuck with it. I can predict the achievement gap is going to skyrocket. Thanks!

Hi Erin,

My point, and that of the folks who developed the alternative approaches, is that computational speed and efficiency are no longer important. Our cheap machines and cloud services do computation WAY faster and MUCH more efficiently, and accurately, than the human brain.

In today’s world, understanding number and having the capacity for algebraic thinking are the crucial abilities. It must be decades since I ever had any need to use any of the standard algorithms! Yet I use mathematics every day in many aspects of my life.

Paying a company like Kumon to train a child to be a poor substitute for a $10 calculator seems to me to benefit no one apart from Kumon’s investors.

The world changed in the middle of the twentieth century, and what was an important skill before then rapidly became obsolete — with a major side-effect of that change being the rapid growth in the importance of number comprehension and algebraic thinking.

“Reform math” was not an arbitrary decision about what to teach made on a whim by mathematics educators. The world changed! For ever. The shift in educational priorities were simply a response to a major change in the world.

The linear thought process of the standard algorithm and the mental development process that occurs with mental mastery of math facts, etc. can’t be accomplished on a computer. It is a physical development of the brain and promotes independent intelligence.

I liken it to learning a musical instrument and how it has an affect on mental development in children. Should we no longer have children struggle through learning notes, scales and chords? I mean they could turn on the radio or iPod and cut and blend together recordings to put together songs, but without an independent ability they can’t CREATE music of their own original thought process down to the simple chord or in math computation.

Erin, I totally agree about the importance of mastering algorithms in learning mathematics, but (as I say in my article) there is nothing special about the standard algorithms in that regard, and it makes educational sense to use algorithms that bring out the place-value structure of the number system. Thanks for the comment.

Learning to play an instrument is a poor analogy to learning maths. To play an instrument, you need to train muscles, and this takes time and a lot of practice. A better analogy is learning how to write music. While many composers also play, it is fundamentally a different activity. You could practice scales all day, and not know how to construct a decent melody or write a counterpoint. These activities also take practice, but it is not the same thing. Playing scales is like syntax; learning to compose is semantics.

As someone who was trained as a (classical) composer and also does maths and programming, I can say that inside my head, the same parts of the brain are involved in composition and programming (and in higher maths) and they are not involved in learning to play.

What computers have done for music is to make it possible to compose music without going through the medium of learning an instrument first. What computers have done for mathematics is to make it possible to get at the real meat of mathematics without having to deal with the minutiae of doing sums and quotients.

One more point — children seem to do a pretty good job of making their own music. They can make up songs, and really get involved. Some children are fortunate in that this interest is developed and nurtured. For most of the rest, it withers. Children are natural scientists (and mathematicians) in a true sense. What endless drill on the ‘basics’ does is to kill this natural interest. What children and most adults get of mathematics is like trying to learn music by only playing scales and never hearing Mozart or Bach.

George, Thanks for the comment. Apropos of which, let me mention that I’ve been working with colleagues to try to create the mathematical equivalent of musical instruments, the first of which is due to be released soon. (We are finishing up final tweaks following user testing.) See

http://innertubegames.net/about.html and

http://www.americanscientist.org/issues/pub/the-music-of-math-games

for details. (Of course, this uses the math-music analogy as a design methodology for building digital interactives. We are not making any deep claims about learning.)

It is my understanding that the ‘traditional’ algorithms were developed when all paper was hand made and very expensive.

They were designed to save paper. I imagine if paper cost 2 dollars a sheet nowadays, most people would favour those algorithms which used least paper.

Hi Steven. The algorithms were developed long before paper was invented. Sandboxes were one medium for “scratch work”. Parchment was used for more permanent writing, and that was indeed expensive, but parchments could be washed off and re-used. I doubt the cost of the medium was a major factor, since the obvious way to handle that would be to do most of the work on a sandbox and then write up the key steps for the record. Given the efficiency of the algorithms for human calculation, I’ll stick by my statement that efficiency was the driver. (But I am not a historian, so cannot claim any authority here.)

Thanks for writing your response Keith. I was incensed by the NYT article as it is so inaccurate – as you point out well. I don’t know any reformers who don’t teach procedures, they just teach them with meaning, so that students can understand what they are doing. It is great that you took the time to respond.

I appreciate your response, but also like the mild rebuttal that James King shared (that you wrote!). I think that the false reform / preparatory dichotomy dissolves when three things happen: Progress is made when we expect teachers to learn their craft very well, treat teachers as professionals, and provide teachers with the resources to learn their profession deeply. As a high school math teacher at a very successful independent school, our shift from more traditional to more progressive pedagogy came from sharing, collaborating, and thinking about what was best for the students: not from fighting the so-called traditionalists. I remember a time about a dozen years ago when a previous department chair was more concerned about “fighting the battle for the progressives” in our department. Even if he was right, he was wrong: he dismissed colleagues, used others as “agents of change,” and alienated a department. Our current chair dropped the politics, listened to everybody, and helped us develop a collaborative culture where we all learned what everybody could contribute. We soon realized that the labels of “reform” and “traditionalist” started to mean very little. Does this mean we all agree? No. Do we all think as deeply about teaching math as we could, no, but we have all become more thoughtful. We committed to respect our colleagues and learn from each other. The “reform” evolved from that shift. I think that experience convinced me that “becoming the change you want to see” is truly the best way to “win the war.”

Hi William. I agree. I’m fairly familiar with the much-lauded Finnish education system (and am on the Advisory Board of the Finnish CICERO Research Learning Network), and their approach is build on exactly what you say in your second sentence. It absolutely works. I generally stay out of the MW debate precisely because there is too much polarized posturing. But the June 16 NYT article had so many falsehoods, distributed to a large audience through a major newspaper, I felt I had to use what pulpit I have to try to provide a counter.

And it’s appreciated. I’m wondering what compelled the two authors of the piece to speak out so strongly, especially with little evidence that they had researched the groups they are criticizing. No offense to the authors, but who are they? I have done a good bit of ed research, and I don’t think I know their work. How did they earn the opportunity to amplify their thoughts to nytimes.com? What gives their voice this elevated status?

Thank you, Keith, for being willing to do such a thorough debunking of the original piece. I was one of hundreds able to comment briefly on it, but you have done us all a service by taking so much of your time to do this.

I am a math educator with 15 years experience. Currently, I teach a Math Methods and Curriculum course for students in a elementary school credentialing program. The course briefly discusses math education history, the math wars and proposes concrete-representational-abstract methods or activities as part of a balanced approach to teaching mathematics. It also advocates using educational psych ideas to promote collaborative learning in groups and teaching math through problem solving.

Although I am usually able to win skeptics (i.e. “I learned math perfectly find this way, why can’t our kids?”) over in the first few weeks, this article made getting through a class more difficult. Students were more willing to listen to the NYT than to the profs at the college they were paying.

Thanks for taking the weight of my shoulders. Because of this blog, whenever they start up again on that ridiculous article, I have a great link to send them. :)

How disappointing the New York Times has been lately. First there was the June 9 Hacker/Dreifus piece, then the Crary/Wilson mess that you’ve so thoroughly debunked, and now the NYT won’t print your response? For what it’s worth, I will reblog as well. In a just few weeks, I’ll be working with practicing K-12 teachers who spend long hours, year after year, to provide the best mathematics education they can. Most of them don’t pay much attention to diatribes against never-born horses (I laughed out loud at that paragraph — spot on!), and good for them, but also good for you for taking the time you did.

“My sense is the problem is quality of teaching as much if not more than curriculum, though the two are not necessarily independent.”

Your “sense” is that you just don’t really know.

The old joke is that traditional math is fundamentally flawed, but reform math just needs more teacher preparation. I think you should apply your same logic to those posters whose team you have publically joined.

Some of us are not part of any team, but see exactly what’s wrong with the system because our kids are currently suffering through it. We don’t have a “sense”. We have reality. My goal is not to defend Crary and Wilson as if this is some sort of battle of sides. Debunking C/W isn’t all you have to do.

The reality is that traditional math in K-6 has been gone for a long time. Our schools have used Everyday Math for 10 years, and for many years before that, they used MathLand, a curriculum so bad that nobody claims responsibility for it anymore. Where are the results?

Everyday Math cycles through material so fast that they must think kids are nervous chipmunks. EM tells teachers specifically to keep moving and to “trust the spiral”. It doesn’t work. Trusting the spiral doesn’t require more teacher training to get it to work right. It’s a curriculum copout. As they pump kids along, the range of ability widens, and part of that is due to the fact that some of us parents teach math facts at home. Teachers can’t possibly differentiate for that wide range of ability, so they are glad to have a curriculum that tells them to just keep moving along. Kids will learn when they are ready, and if they don’t, it must be an IQ thing.

It’s not as if teachers don’t value skills. It’s just that they don’t want to do it themselves. They send home notes to parents telling them to do the job and they talk about having sessions for parents to help them “understand” the math curriculum and prepare them to help their kids at home. That’s quite a change in educational pedagogy, guaranteed to widen the achievement gap. Everyday Math talks about having students select their own algorithm to master, but nothing ensures that even that happens. Are there studies that show a legion of skilled “lattice technique” users?

When my son was in fifth grade, we had a teacher/parent meeting about Everyday Math and everyone talked about how important “balance” was. Of course, nobody defined what that meant or who had the responsibility to get the job done. My son’s teacher, however, did accept the responsibility because she saw bright students who still did not know the times table. She didn’t “trust the spiral”, worked on the neglected skills portion of the balance, but did not get to 35 percent of the material. Good for her, but there was no systemic or curriculum-based discussion at the school.

So now we have CCSS and the PARCC test’s highest PLD level (“distinguished”) only means that kids will be successful in a college algebra course. They specifically avoid mentioning STEM preparation in K-6. The curriculum is statistics driven, not individual driven. There is no STEM path in K-6, so it’s all over by 7th grade for many students. I got to calculus in high school with no help from my parents. My son got there only with my direct help at home with “math facts”. When I was young, schools would never dream of sending home notes telling parents to work on math facts, but I have had many. It would be easy to survey parents about what math help they provide at home or with tutors. Nobody does that and I’m sure that my son will be used as a poster boy for Everyday Math.

What is 6*7? What is 2/3 divided by 4/7? What is 1/x^2 divided by (x-1)^3/2x? Whose responsibility is it to ensure these skills? I don’t care if you use hand puppets, video games, or deep thinking. Does the job get done or do you just pump kids along and then blame the student, parents, or society?

Thank you for the excellent rebuttal. It just blew my mind when I read the claim that algorithms are “at the heart of the [mathematical] discipline”. What I encounter in my experience with students is that many of them find it difficult to connect mathematical methods (even simple arithmetic operations) with real world problems. When the problem requires multiplying a daily rate by 365 to calculate a yearly rate, and then dividing it by the number of people to get a per-capita rate, I couldn’t care less if they can do it on paper (in fact I encourage the use of a spread sheet because it makes it much easier to keep track of the steps and recheck for mistakes). What I want is that they understand what numbers need to be multiplied/added/divided. In my experience, many students find that challenging.

In my teaching, I have developed a number of practice problems designed to foster quantitative competence (numeracy) as well as critical thinking and systems thinking. See some examples at http://www.slideshare.net/amenning/quantitative-problems-food-security. Any comments are welcome.

Michael Goldberg: I like the lattice algorithm a lot. It seems more intuitive (but that may be subjective) and more transparent than the “standard” algorithm. In particular, it naturally illustrates distribution and commutativity. A great counter-example to the false “standard algorithm” claim.

In reply to Erin Tuttle and Steve H:

One big point to get clear on: the debate about algorithms is NOT about the role “basic facts” play in computational efficiency or deep conceptual understanding. I can’t imagine that either Keith Devln or Michael Goldberg would not want all students (or mathematicians) to have relationships such as 6*7 immediately available without reconstructing them through repeated addition. As for expressions such as 2/3 divided by 4/7 and 1/x^2 divided by (x-1)^3/2x, I hope both of them and both of you would want students to understand how to simplify them efficiently without error. The “debate” about algorithms, when you boil it down, is really about something else: methods for computing the results of arithmetic operations involving multiple digits. As Keith Devlin points out, the “standard algorithms” were designed for rapid handwritten computation. Another extremely efficient alternative to them in other cultures (with a traditional standing that goes back much farther than the mid-20th-century textbooks most “math traditionalists” use as a reference) has been the abacus, the use of which requires at least as much genuine conceptual understanding as the “standard algorithms.” Never mind that many of the adults who claim that 50s-style U.S. math instruction was “good enough for them” and “if they could do it, so should today’s kids” can’t, or won’t, do the math when it comes to environmental change or economics.

Steve, our district dropped Everyday Math for many of the reasons you mention. When our district adopted it, teachers who had enjoyed teaching 1st- and 2nd-graders to love math using truly progressive (and very far from traditional) teacher-developed approaches such as Math Their Way for more than a decade suddenly found themselves struggling to keep kids who were learning to dread math in touch with a bewildering maze of specialized workbook presentations of a curriculum that felt to students and teachers alike miles wide and barely inches deep. When it was clear we needed a different program, we quickly rejected Mathland for a multitude of reasons, choosing Investigations not for its progressive-sounding name but because it offered intellectually rigorous, developmentally-appropriate activities that provided conceptual discovery, reflection, and practice to strengthen skills in applying concepts. Investigations isn’t perfect. And our district is now trying to figure out how Investigations will fit into meeting the challenges of the Common Core standards, not to mention the tools being developed to evaluate student progress (a whole ‘nuther HUGE topic). One of my beefs (beeves?) with many of the commentors to the NYT column is that they tend to lump everything that isn’t their idea of traditional math program, from the New Math of the late 50s to Common Core, together as if it was a monolithic juggernaut bent on promulgating some subversive progressive educational (read: political/cultural) agenda. That’s like lumping everything from free-form post-bop jazz through atonal symphonic pieces through advertising jingles to rap music together, calling it “modern” and criticizing it because it’s not music from the canon of European classical composers.

It’s notable that the references by Crary/Wilson to Dewey, Sellars, Wittgenstein, and Plato didn’t get much reaction pro or con from academicians or practitioners in the field of philosophy as opposed to math/education — but I strongly suspect that Wittgenstein at the very least would drop his jaw in disbelief at how his extremely unconventional approach to the intellectual basis for mathematical thought had been mis- or re-interpreted and reduced to a couple of one-liners to defend the use of the “traditional algorithms” against what Crary/Wilson want us to believe is a simplistic progressive interpretation of them as “merely mechanical.” And I feel even more strongly that neither Crary nor Wilson would want any part of a math curriculum as envisioned by Wittgenstein.

Perhaps the most troubling implications of viewpoints such as the Crary/Wilson column is what may not be articulated explicitly, but hinted at by the stress on terms such as “traditional” and “progressive” which are code words for a much larger debate within American society. This debate is really about submission to personal authority. The stereotype of the mid-20th century teacher is the battleax who drills the students on math facts and algorithms whether or not it’s easy or makes sense or is fun with suitable punitive measures as needed to keep thing moving because “it’s good for them, whether they know it or not.” Actually understanding the methods and goals of “progressive” math approaches isn’t important — all you need to know comes from buying into the symbolic image: progressive math = adults wanting to please kids (who “know” little) instead of kids wanting to please adults (who “know” best). It’s spookily reminiscent of the analysis Swiss psychologist Alice Miller made of destructive family authority dynamics in Books such as “For Your Own Good.” Fear begets, and wants to beget, fear.

“… terms such as “traditional” and “progressive” which are code words for a much larger debate within American society. This debate is really about submission to personal authority.”

This thinking is why I support school choice. Do you? I’m part of a big world out there that is not going to hop on any bus, progressive or traditional; democratic or republican. I know what needs to get done in math education to keep all STEM doors open. With the PARCC test for CCSS, many have shown their true colors. It’s not progressive or traditional. It’s low expectations for all.

I’m a big fan of El Sistema, which many claim to be very progressive. But is it? Many won’t “get it” if they are stuck with a traditional versus progressive mindset. Something else is going on. “Music for the poor should not be poor music”. Likewise, academics for the poor should not be poor academics, no matter how much anyone talks about understanding and critical thinking.

I was surprised to learn that the “old way” of adding (or multiplying) n-digit numbers is an algorithm. We didn’t worry about algorithms in 3rd grade.

But still, isn’t the use of those methods a good way to instill the “powers of 10″ idea? I saw a YouTube video of a student demonstrating a “new way” of adding 3 several-digit numbers (during which she said “we weren’t allowed to stack them”) by drawing and combining rectangles. After 4 or 5 minutes, she got the wrong result. I don’t see this as a great leap forward.

I will certainly agree that the standard methods are not the most efficient or elegant – just ask Dr Benjamin of Harvey Mudd. But they are simple, they work, and they’re easy to teach.

For those of us who have left them far behind, and rely on the ubiquitous calculator, they’re relics. The problem is that the calculator gives us a false sense of accuracy (when precision is something different).

Those methods are in fact the original algorithms! The name algorithm is derived from al-Khwarizmi, the 9th century mathematician in Baghdad whose book on those methods introduced them to the Muslim Empire, from where they spread to Europe in the 13th Century.

The fact that they do not work well educationally is made abundantly clear by the fact that, though they were used as the standard educational procedure for arithmetic for hundreds of years, the majority of people, even today, are not proficient in arithmetic and exhibit little real understanding of the place-ten number system.

They may be “easy to teach”, as you note, but the overwhelming evidence is that they are difficult to learn, and indeed, most students

! (They certainly did not when I was in elementary school, and I am now in my mid-sixties! I remember being one of the few students in my class who “got them”, and only after a long, hard struggle.)do not learn themKeith says that “The fact that they do not work well educationally is made abundantly clear by the fact that, though they were used as the standard educational procedure for arithmetic for hundreds of years, the majority of people, even today, are not proficient in arithmetic and exhibit little real understanding of the place-ten number system.”

The Chinese seem to have no problem with them. Nor the Japanese or Koreans. So is it a genetic fault of the Americans, or is it bad teaching?

I hope you do better proofs in math.

I have traveled to Japan and Korea many times and colleagues there tell a different story. I have no direct information about China, but you are down to one example.

At this stage, I think the familiar flags have been raised up the flagpoles, and further exchanges will add little, if anything. I made my initial commentary on what I saw as a badly flawed article. When comments started to come in, I decided to let them run until someone went beyond the bounds of civilized discussion and started to make cheap, facetious or sarcastic comments, as I feared would happen. We are now at that point, so I am closing comments on this forum.