Common Core math: deep or dull?

According to a New York Times article by Motoko Rich, parents and students are finding Common Core math not only confusing but tedious and slow.

To promote “conceptual” learning, many Core-aligned textbooks and workbooks require steps that may be laborious for students who already get it. A second-grade math worksheet, pictured in the article, includes the question: “There are 6 cars in the parking lot. What is the total number of wheels in the parking lot?” To answer the question, the student drew six circles with four dots within each. (Actually, this doesn’t seem new; it reminds me of “New Math” and “constructivist” math.)

One nine-year-old, apparently weary of this kind of problem, stated that she grew tired of “having to draw all those tiny little dots.”

Students with good understanding may be put through steps that seem redundant to them. If they skip those steps, they may be penalized.

“To make a student feel like they’re not good at math because they can’t explain something that to them seems incredibly obvious clearly isn’t good for the student,” said W. Stephen Wilson, a math professor at Johns Hopkins University.

One reason for emphasizing “conceptual” learning is that employers apparently are demanding critical thinking. Several questions remain to be answered, though: (a) whether Common Core math–in its current forms–really is promoting conceptual learning; (b) if so, whether it also promotes math proficiency; (c) whether the current approach is benefiting students at the upper and lower ends–and those in between, for that matter–or holding them back; and (d) whether this is the kind of “critical thinking” that will serve students well in college, the workplace, and elsewhere.

I will comment briefly on the first question; I welcome others’ insights.

Tedium and depth are not the same. One can go through a long explanation of a problem without gaining any understanding; one can solve a problem quickly and come to understand a great deal.

In sixth grade, in the Netherlands, I learned mental arithmetic: I learned to add, subtract, multiply, and divide double-digit numbers in my head, using all kinds of tricks that the teacher taught. Those tricks enhanced my understanding of what I was doing. I enjoyed the swiftness and ingenuity of it; I would have detested it, probably, if I had to write it all out, step by step, and illustrate the steps with circles and dots.

Detailing and explaining your steps is a worthwhile exercise. But part of the elegance of math has to do with its mental leaps. Sometimes, when you do steps in your head, or when you figure out which steps in a proof are assumed, you not only understand the problem at hand, but also see its extensions and corollaries. Sometimes this understanding is abstract, not visual or even verbal.

There seems to be an unquestioned assumption that one comes to understand math primarily through applying it to real-life situations; hence the Common Core emphasis on word problems. While word problems and practical problems can lead to insights, so can abstract reasoning, and so can models that bridge the abstract and the concrete, like the multiplication table.

Yes, the multiplication table–horrors, the multiplication table!–abounds with concepts. If you look at it carefully (while committing it to memory), you will see patterns in it. You can then figure out why those patterns are there (why, for instance, any natural number whose digits add up to a multiple of 3, is itself a multiple of 3). (Something similar can be said for Pascal’s triangle: one can learn a lot from studying the patterns.)

In other words, conceptual learning can happen in the mind and away from “real-life situations”; it need not always be spelled out at great length on paper or illustrated in terms of cars and wheels. Nor should students be penalized for finding shortcuts to solutions. Nor should memorizing be written off as “rote.” Yes, it’s good to understand those memorized things, but the memorization itself can help with this.

In ELA see a similar tendency toward laboriousness (that likewise long predates the Common Core). Students are required to “show their thinking” in ways that may not benefit the thinking itself. For example, they may be told to explain, at great length, how a supporting quotation or detail actually supports their point–even when it’s obvious. Students with economy of language (and, alas, clarity of thought) may lose points if they don’t follow instructions. Instead of being at liberty decide whether an explanation is needed, they receive a message along the lines of “Explain, and explain again, and then explain that you have explained what you set out to explain.”

Critical thinking is important–and one should think critically about how it is conveyed and taught.

Can math scaffolding hinder learning?

When I was fourteen, we spent a year in Moscow. I attended a Soviet school that “specialized” in French–that is, it taught French from the early grades. The other subjects (math, literature, history, technical drawing, geography, physics, chemistry, and biology) were in Russian. No one expected me to participate in class, but I insisted on being added to the class list and asked teachers to treat me like a regular student. I was eventually doing the work in all of my subjects except for chemistry and biology, where I lacked the necessary background knowledge and was usually a bit lost. (I barely got by in physics, but I did learn something.)

My favorite classes were math and French. Here is a picture of the math textbook. It took us through algebra, beginning calculus, and some trigonometry. Its 220 pages contained more substance than many a hefty textbook I’ve seen since. When I returned to the U.S., I was ready for calculus but had to take a year of precalculus first, along with my classmates. (It didn’t hurt, as I got to do more trigonometry.)

Recently I have been wondering how this textbook manages to convey so much in such short space, and how I learned so much without finding it particularly difficult. To answer this question well, I would have to work my way through the textbook again, this time with pedagogy in mind. That’s a project for another time. In the meantime, I’ll toss out a few hypotheses.

Well, one obvious reason we were able to learn so much is that there was a standard curriculum through the grades. All students came to this course with similar knowledge and practice. Some were better at math than others, but it wasn’t because they had better preparation. (Of course this isn’t entirely true, as some students had additional resources at home and elsewhere.)

It could also be that the curriculum included fewer topics than math courses in the U.S. do; thus there was more time to learn them thoroughly.

But what strikes me about this little textbook is that it plunges right in. The first chapter talks about inductive proofs. The second goes into combinatorics. There are no pictures except for graphs of functions (and a few circles and rectangles). There are word problems, but they are relatively few. There are no needless “scaffolds.”

Scaffolds in instruction are temporary supports intended to bring students to the point of self-sufficiency. All good instruction uses them to some degree.  But certain kinds of “scaffolds” can actually  become barriers, complicating the student’s entry into the subject matter. In mathematics, excessive reliance on “visuals,” “manipulatives,” and “real-life” applications can stand in the way of the math itself.

This textbook, by contrast, “scaffolds” the instruction in one way only: it builds from simpler problems to more complex ones. It  lacks the “scaffolding” that plagues many a math textbook that I have seen: those colored graphics, tips and strategies, needless word problems, and so on. It has a few word problems, but there are reasons for them to be word problems. The vast majority of the problems use mathematical notation. Thus, students become fluent in it and learn to think in it.

I was recently looking at AMSCO’s Geometry–better than many in terms of presentation. Very little clutter. But even AMSCO has word problems like this: “Amy said that if the radius of a circular cylinder were doubled and the height decreased by one-half, the volume of the cylinder would remain unchanged. Do you agree with Amy? Explain why or why not.” There is no reason to bring Amy into this; Amy’s presence does nothing for the problem. Also, turning this into a matter of opinion (“do you agree or disagree”) confuses the matter. Instead, the student should be asked whether the statement is correct or incorrect.

In bending over backwards to make math “accessible,” we may actually make it inaccessible. What do you think?