Sunday, January 27, 2013

Yet another false dichotomy: facts, facts, facts, vs. higher-level thinking:

I was astounded to read in this week's Education Week an article on a study that points to the benefits of memorizing math facts:

Students who performed well on the math section of the PSAT showed more activity in brain areas linked to memory of math facts. Those with lower math PSAT scores had less brain activity in those areas and more in areas associated with processing number quantities.

The findings suggest that the high-achieving students knew the answers by memory, while lower-performing students were calculating even low-level problems.
Less astounding, but no less compelling, is an article in the winter issue of the City Journal by E.D. Hirsch, Jr, on the benefits of factual knowledge and vocabulary size (thanks to one of my readers for alerting me to this article). Vocabulary size, writes Hirsch, predicts intellectual aptitude and future earnings:
Vocabulary size is a convenient proxy for a whole range of educational attainments and abilities—not just skill in reading, writing, listening, and speaking but also general knowledge of science, history, and the arts.
Vocabulary doesn’t just help children do well on verbal exams. Studies have solidly established the correlation between vocabulary and real-world ability.
There’s no better index to accumulated knowledge and general competence than the size of a person’s vocabulary. Simply put: knowing more words makes you smarter. And between 1962 and the present, a big segment of the American population began knowing fewer words, getting less smart, and becoming demonstrably less able to earn a high income.
Why is vocabulary so important? As Hirsch explains:
The space where we solve our problems is called “working memory.” For everyone, even geniuses, it’s a small space that can hold only a few items in suspension for only a few seconds. If one doesn’t make the right connections within that space, one has to start over again. Hence, one method for coping and problem solving is to reduce the number of items that one has to make sense of at any moment. The psychologist George A. Miller called that process “chunking.”

...As long as you hold in your long-term memory a lot of associations with that name [a particular vocabulary item], you don’t need to dredge them up and try to cram them into your working memory. The name serves as a brief proxy for whatever aspects will turn out to be needed to cope with your problem. The more readily available such proxies are for you, the better you will be at dealing with various problems. Extend this example to whole spheres of knowledge and experience, and you’ll realize that a large vocabulary is a powerful coping device that enhances one’s general cognitive ability.
But our nation's vocabularyis nowhere near what it once was. While scores on vocabulary tests rose steadily for 50 years up until 1967, ever since then, thanks to the thorough penetration in our schools of a so-called "progressive" educational philosophy that proclaims "the unimportance of factual knowledge and book learning," we've had a “historically unprecedented" decline in verbal SAT scores. And, since the 1980s, an persistent nadir in those scores.

Some have tried to blame this on the widening pool of SAT test-takers, which include more and more students from disadvantaged backgrounds. But the same decline has occured on the Iowa Test of Educational Development, a test, Hirsch  notes, that is "given to all Iowa high school students, who were 98 percent white and mostly middle-class in attitude."

Correlated with the anti-book learning philosophy of so-called "progressivism," and with the decline in our nation's verbal scores, is the "dumbing-down of American schoolbooks." Cornell sociologist Donald Hayes, Hirsch writes, has found that:
publishers, under the influence of progressive educational theories, had begun to use simplified language and smaller vocabularies. Hayes demonstrated that the dilution of knowledge and vocabulary, rather than poverty, explained most of the test-score drop.
The remedy involves a thorough reversal of current trends. As Hirsch notes:
The fastest way to gain a large vocabulary through schooling is to follow a systematic curriculum that presents new words in familiar contexts, thereby enabling the student to make correct meaning-guesses unconsciously.
Vocabulary tests, one of the few remaining relics from the past, simply don't cut it:
Spending large amounts of school time on individual word study is an inefficient and insufficient route to a bigger vocabulary. There are just too many words to be learned by 12th grade—between 25,000 and 60,000.
A large vocabulary results not from memorizing word lists but from acquiring knowledge about the social and natural worlds.
Again, we need a complete reversal of current trends. We need to move away from theme-based materials and assignments and generic, de-contextualized "skills," "processes," and "higher-level thinking." We must return to systematic, fact-rich, content-based instruction.

Returning to the first article, I suspect that some of what E.D. Hirsch says about vocabulary applies to math facts as well: not just that knowing them fosters higher test scores and broader intelligence, but also that they are best learned not as lists of "math facts," but in the course of doing actual, systematically structured math. Which, like systematic, fact-rich, content-based instruction, would be another complete reversal of current trends.


Anonymous said...

Arithmetic facts and basic procedures are like the alphabet: they have to be memorized (by whatever means works for the learner) before higher-level activities such as reading (or using math to solve problems) can occur. For some, this memorizing seems burdensome and goes slowly either in the realm of literacy or numeracy. But skipping it has great costs.

Anonymous said...

Knowing math facts and procedures is certainly important. There are exceptions. I know one student who simply could not memorize math facts as isolated facts (he did understand and apply procedures). Neither could he easily memorize spelling and dates. Had he been in a system that measured his math ability by how well he could memorize, he would have failed the system. However, he went on to do 3 semesters of college level calculus while still in high school, getting A's, and majored in Engineering. Skipping memorization of isolated math facts did not have great cost to him, making that the main goal in early years would have possibly. Of course, he did know some facts, and he was quite competent at calculating other facts from the facts he knew, but he did not have them memorized. He did not use a calculator until high school. On the other hand, I know another student very good at memorizing math facts, but not so good at mathematical thinking and did not go on in math. My impression from this is that mathematical thinking and memorization of isolated facts are different skills. The latter facilitates speed and accuracy for the former is all. It is not necessary for high level activities for at least some students. I took calculus in high school, got a 5 on AP calculus test, but I still could for the life of me remember what 8 + 5 is without doing a quick computation of making 8 ten leaving 3 or using my fingers. And this was back before calculators. I do use a calculator now, but I am not bad at math concepts or higher level problem solving.