September 26, 2007

On communicating math standards

Florida's Board of Education has recently approved new state standards in math, and I think it's the most constructive long-term decision the board has made in years. My judgment isn't based on the fact that I'm a friend and colleague of one member of the group that drafted the new standards. I looked at the standards before the FBOE approved them, and from my lay perspective, I just breathed a sigh of relief. The emphasis is on a few "big ideas" for each grade up through eighth (with plenty of connections to other areas of math), and while the "big ideas" are in line with current grade-level expectations, it should provide focus.

Moreover, I think these standards have a much better chance at being adopted sans controversy than the National Council of Teachers of Mathematics efforts over the past 20 years. While I know NCTM didn't intend to do so, I suspect the somewhat jargon-laden language of number sense, measurement, etc., turned heuristic curriculum concepts into reified categories, at least in the public ear. To be blunt about it, no adult goes around thinking about things like number sense or measurement. If you're buying new windows for a house, you get an estimate of the number of windows, the size of each window, the cost, the total, etc. I can point to number sense and measurement in the bidding process for new windows, but in everyday life, math doesn't break down into the terms that NCTM has used. When combined with the idea that calculators are useful instructional tools (read by some as "kids don't need to learn multiplication and division"), and because the math-education terms don't align with the way academic mathemeticians think about their discipline, NCTM has been swimming upstream with the rhetoric, even if many of the concepts are sensible.

The difference with Florida's new standards is that they're more easily understood.  For those who think about math as conventional arithmetic and algorithmic operations, they will read phrases such as basic addition facts and related subtraction facts,  grouping by tens and ones (two of the big ideas for first grade), or quick recall of addition facts and related subtraction facts and fluency with multi-digit addition and subtraction (from one of the big ideas in second grade), and with luck they will be reassured that kids have to learn conventional operations. But there's plenty of strategies language as well, and hidden in there are terms such as mathematical reasoning, conjectures, axioms, proofs, etc. In other words, the standards document is flexible enough in language to satisfy everyone without raising red flags.

I suspect that math education people could be even more effective in communicating their instructional and curricular ideas. The following is a restructuring of what I've read plenty of times, though I don't think I've ever heard math educators organize their arguments in exactly the following way. So this is a completely amateurish PR effort rather than a statement about math curriculum itself. A good education in math covers the following six areas (and credit my math-major wife with convincing me that there were six rather than five: guess which was her insistent addition):

  1. Learning conventional interpretations of math objects and concepts. Students learn at least one standard way of understanding terms such as whole numbers, adding, subtracting, multiplying dividing, fractions, decimals, negative numbers, etc.
  2. Recognizing math objects and concepts in concrete applications and instantiations. Students learn ways of thinking about the world in mathematical terms, from measuring length to calculating speed, etc.
  3. Becoming fluent in efficient algorithms for solving mathematical problems. Students learn the procedures of algorithms, the fact that algorithms are not unique, and the ways that people make choices among different algorithms (such as efficiency, ease of operation, etc.).
  4. Recognizing reinterpretations of known math objects and concepts. When advancing in math, students must learn new perspectives on old terms. Multiplying is not just repeated addition but akin to stretching a rubber band or calculating area and volume. From everything I understand (from outside both math and math education), flexibility in reinterpreting objects and concepts is critical to learning new topics in math, even if the curriculum appears sequential.
  5. Making and testing conjectures. In a good math education, students are encouraged to speculate about mathematical objects and concepts and are required to test conjectures about math, to figure out how they can confirm or disprove what they think they know.
  6. Making deeper connections among different areas of math. Middle-school, high school, and college math courses are not just sequential in the sense of adding material but also make connections among different topics. A solid math education teaches students to recognize the multiple connections among different areas of math, and a great math education teaches students something of how math has and continues to evolve as a field in part by changing the relationships among different areas of math.

Most of NCTM's concepts focus on areas 1-3 and 5, but I've also seen many of my colleagues focus on the practical teaching skills of #4. One poster session at AERA this spring discussed the different ways in which geometry teachers used visuals, and the fluidity convinced me that great teachers have to master the reinterpretation of concepts, if for no other reason than multiple representations of the same concept is both a way to reach students with different prior understandings and also a way for other students to grasp some deeper math.

I'm convinced that #6 is not hard at all: it should flow easily from the rest as part of the occasional hey, look at how beautiful math is mini-lesson. I'm convinced that what students find much harder is learning how to make and test conjecturing. Because the legacy math curriculum places proofs in geometry (9th or 10th grade), plenty of students have little experience with conjectures and the like until geometry, at which point they hear that they are to craft proofs like the perfect classics they are then exposed to.  It is much like taking students to an illusionist's show and then asking them to perform sleight of hand tricks without any assistance... or crueler yet, showing students a tape of Sandy Koufax and then asking them to strike batters out. "Hey, you've just seen it done. You can do it."

Yes, I'll admit that these divisions are as artificial as the ones NCTM or anyone else has crafted. But I think these are reasonably consistent with what I have read and heard from colleagues in math educaiton, and I think that these are easily communicated (or more easily communicated than NCTM's).

Comments and kibitzing are most welcome!

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Posted in Education policy on September 26, 2007 12:53 AM |