In a recent article in the Atlantic entitled The Math Revolution, Peg Tyre discusses the growth of extracurricular math programs. More and more students, Tyre reports, are able to advance to levels far beyond what their school math classes are taking them.

An article like this one presents an opportunity to critique the shortcomings of school math classes; unfortunately, Tyre, once again, misses the mark. Notice, for example, the subtle bias in this paragraph:

Broadly speaking, there have been two opposing camps. On one side are those who favor conceptual knowledge—understanding how math relates to the world—over rote memorization and what they call “drill and kill.” (Some well-respected math-instruction gurus say that memorizing anything in math is counterproductive and stifles the love of learning.) On the other side are those who say memorization of multiplication tables and the like is necessary for efficient computation. They say teaching students the rules and procedures that govern math forms the bedrock of good instruction and sophisticated mathematical thinking. They bristle at the phrase drill and kill and prefer to call it simply “practice.”Apparently there are no well-respected math gurus worthy of a second parenthetical who say that "memorization of multiplication tables and the like is necessary for efficient computation"--let alone for more advanced learning and for making math problems less tedious. For more on this issue, see Barry Garelick's excellent response to Tyre in Education News.

The extracurricular math programs Tyre focuses on include the Russian School, which she places squarely in the Constructivist camp:

The new outside-of-school math programs like the Russian School vary in their curricula and teaching methods, but they have key elements in common. Perhaps the most salient is the emphasis on teaching students to think about math conceptually and then use that conceptual knowledge as a tool to predict, explore, and explain the world around them. There is a dearth of rote learning and not much time spent applying a list of memorized formulas. Computational speed is not a virtue.

The pedagogical strategy at the heart of the classes is loosely referred to as “problem solving,” a pedestrian term that undersells just how different this approach to math can be. The problem-solving approach has long been a staple of math education in the countries of the former Soviet Union and at elite colleges such as MIT and Cal Tech. It works like this: Instructors present small clusters of students, usually grouped by ability, with a small number of open-ended, multifaceted situations that can be solved by using different approaches.It's all there, supposedly: "conceptual thinking," "problem solving," open-ended problems, multiple solutions, exploration, real-world application, and group activities. The only deviation from hard-core Constructivism is that the groups are "usually" homogenous in terms of ability.

Of course, there is plenty of conceptual thinking and problem solving in any good math curriculum, Russian Math included. But, here in America, the terms "conceptual thinking" and "problem solving," pedestrian though they may be, are by now so bastardized by Reform Math that anyone writing about them needs to be absolutely clear about what she is talking about. When it comes to non-American math (and pre-1970s American math), "conceptual thinking" and "problem solving" have very different meanings from those assumed by the mainstream American edworld. Tyre's article, failing to acknowledge this, will only further entrench current practices: particularly, the emphasis on group work; problems with high ratios of verbiage and "real-world" application to actual mathematical challenge; and requirements that students solve problems in multiple ways and explain their answers verbally and pictographically.

That is not what Russian Math is all about.

Tyre does cite one problem that she presents as representative of Russian Math and its ilk: a problem from the nascent math-and-science site Expii.com:

Imagine a rope that runs completely around the Earth’s equator, flat against the ground (assume the Earth is a perfect sphere, without any mountains or valleys). You cut the rope and tie in another piece of rope that is 710 inches long, or just under 60 feet. That increases the total length of the rope by a bit more than the length of a bus, or the height of a 5-story building. Now imagine that the rope is lifted at all points simultaneously, so that it floats above the Earth at the same height all along its length. What is the largest thing that could fit underneath the rope? The options given are bacteria, a ladybug, a dog, Einstein, a giraffe, or a space shuttle. The instructor then coaches all the students as they reason their way through. Unlike most math classes, where teachers struggle to impart knowledge to students—who must passively absorb it and then regurgitate it on a test—problem-solving classes demand that the pupils execute the cognitive bench press: investigating, conjecturing, predicting, analyzing, and finally verifying their own mathematical strategy. The point is not to accurately execute algorithms, although there is, of course, a right answer (Einstein, in the problem above). Truly thinking the problem through—creatively applying what you know about math and puzzling out possible solutions—is more important. Sitting in a regular ninth-grade algebra class versus observing a middle-school problem-solving class is like watching kids get lectured on the basics of musical notation versus hearing them sing an aria from Tosca.But this open-ended and verbiage-filled problem is not at all representative of the kinds of problems students encounter in Russian Math and the other high-level extracurricular math programs that Tyre is ostensibly writing about. Here's a link to some sample problems from the Russian School website. And here is one of the sample Russian Math problems given for 7-8th grade:

Solve: (x-4)(x-5)(x-6)(x-7) = 1680It is this kind of problem, not the Expii.com problem, that distinguishes Russian Math from contemporary U.S. math.

And one of the biggest reasons why U.S. math is underserving American kids (of all levels of talent) is because it has too many group-centered discovery problems of the open-ended, verbiage-intensive variety, and not enough problems of the sort that are truly representative of Russian math.

## 8 comments:

Most students can memorize the number facts and algorithms involved in "drill and kill" activities without a huge amount of effort. This leaves plenty of time for the much harder work of understanding when to apply the algorithm, how to do it, and why it works (when that is appropriate. I for one do not care why long division works). There are indeed some students who have great difficulty in remembering number facts and algorithms; this is a difficulty that resembles the difficulty some other students have in remembering the correct spellings of words. It's a learning disability, and we need to be supportive of such students because it will take them longer, and they will forget more easily. And there are other students who simply resist memorization for a variety of reasons (sometimes because they can get away with it). But we can't build a curriculum approach around these minorities of students (learning disabled and resistant). We have to make sure that all of the students who are capable of memorizing these very useful facts and techniques, do so.

It wont help to have more problems like (x-4)(x-5)(x-6)(x-7)=1680 posed. These types of

Problems used to be available via math club or extra credit for the interested. The interested had to struggle with them long enough to come up with insight and number sense that wasnt developed in k to 5 as they memorized their facts and took timed test after timed test. Now they are spending k5 helping 'those who struggle', rearranging kidney beans for the 20 millionth time for group members who will need another few years of practice with concrete objects to grasp the part to whole concept.These students are being shorted of several years worth of curriculum, as their test scores show. Its a shame, as the problem is easy to solve mentally for anyone who has been thru Singapore Math and understands the concepts.

Challenging problems AND proper foundations in basics are key. Yes, problems like the one posted above won't do any good if the basics are not in place, but Katharine is saying kids need both. Peg Tyre seems to think that the basics come as a "byproduct" of challenging problems.

E.g.,

"The new outside-of-school math programs like the Russian School vary in their curricula and teaching methods, but they have key elements in common. Perhaps the most salient is the emphasis on teaching students to think about math conceptually and then use that conceptual knowledge as a tool to predict, explore, and explain the world around them. There is a dearth of rote learning and not much time spent applying a list of memorized formulas. Computational speed is not a virtue."and

" “Cram schools,” featuring a mechanistic, test-prep approach to learning math, have become common in some immigrant communities, and plenty of tutors of affluent children use this approach as well, but it is the opposite of what’s taught in this new type of accelerated-learning program. To keep pace with their classmates, students quickly learn their math facts and formulas, but that is more a by-product than the point."I absolutely agree. I've known quite a few students who had to go back and do Kumon while working through the Art of Problem Solving's "Intro to Algebra" book (I believe AoPS is mentioned in the Ms Tyer's article as well.) Those kids did understand the concepts presented in the book well, but they weren't able to come to the right answers consistently on their own due to their lack of foundational skills (=arithmetic.) It is really hard to make the 6th and 7th graders do Kumon worksheets - they think it's unbelievably boring. It's much easier when a student is younger (up to 4th grade.)

It doesnt read that way to me for the enrichment, but it does for the 'cram school'. Some students do acquire the concepts and from coming to know the concepts, acquire their facts as a byproduct. For ex, in first grade students may group manipulatives to show all the two group combos that add up to four. In the process of doing so, the students internalize the facts and have no need to spend time memorizing fact families. The major point was learning the part-whole relationships, the byproduct was computational skills, whether the child was at the 'count up' or 'count on ' or some other stage.Some children will not be able to remember, and will need more time to get their facts learned, and to understand part/whole relationships. When they get to enrichment such as the Russian School problem posed, they have all the underlying concepts plus number sense needed , and they are reminded to approach the problem conceptually, not begin computation with the numbers they see or mechanically draw a factor tree. But maybe my English is bad and she means they will figure out a formula to use in the future. But my math club experience says no, they are after use of number theory. What formula or rote procedure does one get out of this problem?

We have had great experiences with Russian teachers. My child had an amazing math teacher in the US and now in Israel, where we are currently working, math education is dominated by Russian immigrant teachers we continue to be impressed. However, our experience has been not the “creative, cooperative, out of the box thinking” but rather really solid work with some thinking problems on the side. I sent Katherine a page from my child’s 7th grade book which showed one “creative” type problem on the same page with 22 “regular” problems that looked more like:

2/3(3x-15)-3/4(4+8x)-3/5(5+10x) =-2(5x+8)

Creative isn’t treated as a substitute for basic skills rather it’s complementary after those skills have been mastered.

Peg Tyre confuses two levels of understanding and thinks that what mathematicians talk about is what K-8 educators talk about. When my son was in fifth grade (with Everyday Math), a new family came to the school and the father had a master's degree in applied math from Stevens. We had a meeting at the school about EM and he thought that it sounded pretty good. He offered to start an after-school problem solving club to build on their efforts. He soon realized that his ideas didn't line up with those of EM. He had to work on basic skills and understandings. It's not just about memorizing the times table or mastery of traditional algorithms, there is a whole world of bottom-up understanding that exists between EM and STEM preparation, let alone the problems of AOPS and AMC. K-6 is officially a non-STEM zone. The solution, even for "promising" (?) kids, is not to offer them after-school math clubs. Who would pick these kids? It won't be the math-trained experts, but the K-6 educational pedagogues who have no clue.

Daytime K-6 school math classes should provide a proper curriculum for all kids to prepare them for the STEM level math track splits that happen in 7th and 8th grade. This preparation is not the domain of after school programs, especially ones where the kids have to be selected as being promising. They will be the ones getting basic skills practice at home. My son never did after-school math leagues or clubs. He was too busy with music. However, I prepared him for the math tracking split in 7th grade and his proper textbook regular classes after that did the rest so that he got a 5 in AP Calculus BC. He got to the top AIME level, but never prepared for it. Unfortunately, I think it hurt him when he applied to college because some schools ask for your AMC scores as if that defines post AP/SAT II level math ability.

AMC is just a competition. It covers a limited set of material to an anal level where preparation and speed matter. Some like it. Being a star football player does not define a top athlete or the only path into sports. My son liked to play with GeoGebra and study string theory and not the AMC idiosyncrasies of logarithms. While I support schools that provide math leagues and AMC development, that is an optional role for after-school clubs. It does not define the only path to success in math as a career.

I've never liked math competitions. My approach has always been slow and methodical. AMC is not so much about problem solving new and different problems, but content and skill preparation of their limited domain of material. You prepare by studying all of the past test questions. If you don't, you will fail. You don't prepare by developing some sort of general-like problem solving skills. Like the SAT, you have to practice and "grok" the test.

Peg Tyre can go ahead and push for after-school math clubs and trying to identify promising students, but it's no grand or even part solution. It will mostly find kids who have already gotten skill help at home, and it ignores the huge "understanding" definition take-over by K-6 educational pedagogues. They are not pals with math content experts or even high school math teachers. The high school AMC prep club advisors are not the ones pushing EM and TERC. They are the ones using proper math textbooks and nightly homework sets in class as the basic requirement to be prepared for the after-school work. Bottom-up.

I tutored an 11 year old a while back for the ISEE (jr high independent school entrance exam). It quickly became apparent that she was very weak on her basic math facts. Since I only had 9 hours with her to prep her, there was no way I could do much in the way of arithmetic drills, and walked her through math concepts instead. I gave her some worksheets and phone apps to drill with, but it was up to her and her parents to get the drill work done. I don't think she ever took the time to do it.

I'm having a similar problem with both out 13 and 15 year olds in a completely different subject: Latin. They both can learn the grammar, but if they don't drill the vocabulary (and they don't), they can't get very far. At least in language classes, they seem to understand that drilling the boring stuff is necessary, and both their teachers said they should spend 5-10 minutes every night just doing vocabulary drills. I can see how much this would benefit them both, but neither ends up putting in the time.

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