I'd like to pretend that that disaster has disappeared, but it still hangs over math education like the Sword of Damocles:
California’s proposed math curriculum framework has ignited a ferocious debate, touching off a revival of the 1990s math wars and attracting national media attention. Early drafts of the new framework faced a firestorm of criticism, with opponents charging that the guidelines sacrificed accelerated learning for high achievers in a misconceived attempt to promote equity.
The new framework, first released for public comment in 2021, called for all students to take the same math courses through 10th grade, a “detracking” policy that would effectively end the option of 8th graders taking algebra. A petition signed by nearly 6,000 STEM leaders argued that the framework “will have a significant adverse effect on gifted and advanced learners.” Rejecting the framework’s notions of social justice, an open letter with over 1,200 signatories, organized by the Independent Institute, accused the framework of “politicizing K–12 math in a potentially disastrous way” by trying “to build a mathless Brave New World on a foundation of unsound ideology.”
About once every eight years, the state of California convenes a group of math educators to revisit the framework that recommends how math will be taught in the public schools. The current proposal calls for a more conceptual approach toward math instruction, deemphasizing memorization and stressing problem solving and collaboration. After several delays, the framework is undergoing additional edits by the state department of education and is scheduled for consideration by the state board of education for approval sometime in 2023.
It's a lengthy article, heavy on details. Subsections are entitled Historical Context, Addition and Multiplication Facts, Standard Algorithms, Research Cited By The Framework, Research Omitted By The Framework, and Bumpy Road Ahead. Great information.
I want to comment here on just one point, the memorization of addition and multiplication facts. I've long believed this, but now there's research to back up what has always been obvious to me:
Fluency in mathematics usually refers to students’ ability to perform calculations quickly and accurately. The Common Core mathematics standards call for students to know addition and multiplication facts “from memory,” and the California math standards expect the same. The task of knowing basic facts in subtraction and division is made easier by those operations being the inverse, respectively, of addition and multiplication. If one knows that 5 + 6 = 11, then it logically follows that 11 – 6 = 5; and if 8 × 9 = 72, then surely 72 ÷ 9 = 8.
Cognitive psychologists have long pointed out the value of automaticity with number facts—the ability to retrieve facts immediately from long-term memory without even thinking about them. Working memory is limited; long-term memory is vast. In that way, math facts are to math as phonics is to reading. If these facts are learned and stored in long-term memory, they can be retrieved effortlessly when the student is tackling more-complex cognitive tasks. In a recent interview, Sal Khan, founder of Khan Academy, observed, “I visited a school in the Bronx a few months ago, and they were working on exponent properties like: two cubed, to the seventh power. So, you multiply the exponents, and it would be two to [the] 21st power. But the kids would get out the calculator to find out three times seven.” Even though they knew how to solve the exponent exercise itself, “the fluency gap was adding to the cognitive load, taking more time, and making things much more complex.”
So what's the problem?
California’s proposed framework mentions the words “memorize” and “memorization” 27 times, but all in a negative or downplaying way. For example, the framework states: “In the past, fluency has sometimes been equated with speed, which may account for the common, but counterproductive, use of timed tests for practicing facts. . . . Fluency is more than the memorization of facts or procedures, and more than understanding and having the ability to use one procedure for a given situation.” (All framework quotations here are from the most recent public version, a draft presented for the second field review, a 60-day public-comment period in 2022.)
The bottom line for cognitive load is, if you're spending all your brainpower on simple arithmetic calculations, you're not going to have any brainpower left over for using the results of those calculations. Why not just let a calculator do those arithmetic calculations, then? Because if you don't understand that 3*4=12, then you won't understand why 4*3=12, or why 1/4 of 12 is 3, or why 1/3 of 12 is 4. The addition and multiplication facts have to be memorized, or anything after that will be so much tougher.
1 comment:
People always look at the detracking movement as holding back higher achievers, but is also a disaster for kids on the other end of the scale. It ends up putting kids with learning difficulties in classrooms with curricula several years beyond their abilities. How are they supposed to progress? Math, reading, and pretty much everything else, builds on itself. Simply ignoring this, and believing that all our children above average, condemns slower learners to be no learners.
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