Commentary on The Math Gene, by Keith Devlin (Basic Books. NY: 2000)

The ability to develop “patterns” in our minds is Keith Devlin’s theory of how people think “mathematically.” Pattern-making abilities, innate in our minds, permit us to reason abstractly. Some of us are better than others at it. He lists the mental abilities needed. Nine of the most obvious are: number sense; numerical ability; algorithmic ability; the ability to handle abstractions; a sense of cause and effect; the ability to construct and follow a causal chain of facts and events; logical reasoning ability; relational reasoning ability; and spatial reasoning ability.

These are listed and explained in Chapter 1, “A Mind for Mathematics.” This is not an easy book to summarize, but the dust cover tries. First, (Devlin) offers a new theory of how language developed in two stages and math evolved out of the same symbol-manipulating ability (as) true language. It goes on, “people who are adept at math have learned the knack of treating numbers somewhat like old friends. That permits us to reason mathematically.”

There is so much interesting and useful information in the book, that it sometimes seems to “get in the way” of the main idea; but Devlin has good reasons for writing this way. An “abstraction” is actually a shorthand way of thinking about more than one “thing” at a time. That is what he means by patterns.

My conclusion, probably what was intended, is that mathematics is difficult because thinking about more than simple, sensory objects requires having a name for each one, but also another name for what is “happening,” and still more words about the “context” and “relationships.”

Mathematical symbols do away with all that. What is left are “symbols,” “signs,” and “labels” that IF WE KNOW WHAT THEY REFER TO explain the “pattern.” The problem we have is that schools and society “expect” that we understand the totally abstract “language” of mathematics WITHOUT helping us learn it well enough. WE WILL ALWAYS HAVE PROBLEMS READING A LANGUAGE THAT WE DON’T UNDERSTAND. MATHEMATICS IS A LANGUAGE, BUT WE DON’T KNOW THE MEANINGS OF THE SYMBOLS AND THE SYNTAX (how the symbols are used when they are connected).

UNTIL SCHOOLS CHANGE THE WAY THEY PRESENT THE LANGUAGE OF MATHEMATICS, AND TEACHERS CHANGE THEIR EXPECTATIONS OF WHAT CHILDREN UNDERSTAND AT THE BEGINNING OF THEIR MATH CLASSES, THE MATH PROBLEM WILL NEVER BE RESOLVED!

I digress, only to indicate how complex are the ideas Devlin makes more clear, and why so many of us have problems learning and understanding math. A German philosopher of the late 19th-early 20th Century, Hans Vaihinger, wrote about mathematics as the perfect language of “as if.” Vaihinger indedpendently expanded 18th Century philosopher Jeremy Bentham’s perspective, in his book *The Philosophy of As If *(Routledge and Keegan Paul. London: 1924). That book explains how very much of language does NOT apply to specifics—that is, we talk about things *as if* the words are the reality. Other philosophers, semanticists and linguists have tried to explain this, including Korzybski, Chomsky, Whorf, Bickerton, etc.

Ludwig Wittgenstein, in his later work, suggested to philosophers that they “give it up.” He claimed to have reasoned to the ultimate purposes of the way we communicate—that conversations (and most communication) are “games” during which we make up the rules (and the meanings of the words we use and how we use them in phrases and sentences).

To go on with Devlin’s book. He realizes how difficult it will be to change the way we think about thinking about and teaching mathematics. That seems to be why he includes a history of communications development, math development, and theories of language. Even though he cannot prove scientifically that the theories of Pinker, Chomsky and others that a “language instinct, or gene” exists, he insists that what we do know is enough to think about it “as if” it does.

I agree. Mathematics IS not easy; if it were, more of us would be able to learn it more easily and there would be no “math wars.” Devlin proposes some realistic answers, as does Jo Boaler in another book reviewed in this blog. TREAT MATHEMATICS AS ANOTHER LANGUAGE; WHEN STUDENTS DON’T UNDERSTAND THE TERMS, TEACH THEM THE MEANINGS. DON’T ASSUME THEY WILL LEARN THEM FROM CASUAL DINNER TABLE CONVERSATION, GRAFFITI, “STREET LANGUAGE,” TV REALITY SHOWS OR THE DAILY NEWSPAPER.

To truly internalize Devlin’s thinking, you will need to read the book carefully at least once. I will address just one other major point in it. That is the idea that people think “on-line” when they are actively “doing” something or in a conversation that does not require many of the “skills” listed above. On –line thinking in Devlin’s view, is the kind of “proto-language” that D. Bickerton suggests was used when “the physical environment…generates—through input stimuli–the activation patterns necessary to start and maintain on-line thought.” (p. 244)

Off-line thinking is “Roughly speaking…the capacity to reason in an abstract “what if” fashion.” (p.172). Devlin asserts that “thinking off-line” is a “new capacity of the brain,” to handle grammar, syntax and abstract ideas. This differs from “on-line” thinking, and (may be the reason) why we can think abstractly. That happened, Devlin believes, accidentally but as a result of thinking “differently” by some humans, about 8,000 or so years ago. Devlin does not suggest that this development came quickly. He says, “…identifiable precursors of mathematical thinking were arguably present even in the dinosaurs.” (p. 173)

Of the nine attributes of a “mathematical mind” (listed above), Devlin says that human ancestors Homo Erectus lacked four of them: numerical ability, algorithmic ability, abstracting ability, logical reasoning ability. However, they had the beginnings of these, and, clearly Homo sapiens developed them. But just as learning to read requires being taught, so does learning to “think off-line.”

Perhaps a simple “off-line” example can help us understand the limitations of believing without basis really works. When president Kennedy said “We will reach the moon in 10 years,” he didn’t think “we” all would; just that it was possible for “someone” to do it. When Devlin says, “We” can learn math, it means that some of us can—hopefully, millions more than reached the moon.

My example is the effort, begun after WWII, to teach foreign languages in schools—so that we could communicate better. How did that work out? The effort to introduce Russian failed; very few students learned it well enough to use it. Chinese? Some schools still try, but there seems to be limited success. In Montgomery County, MD, Russian is no longer taught, the Italian embassy no longer subsidizes teaching of Italian, French has declined, and only Spanish (with a large number of poorly-literate Hispanics entering the schools) is increasing in enrollment.

Assume that mathematics is a “foreign” language.” How difficult is it? Children begin in pre-school to learn some of the concepts. By high school, there is some basic math “literacy” for most, but not all, students. Now, think of all the “variations” in “language” in the kinds of math taught in our schools—algebra, trigonometry, calculus, statistics—are a few, but there are many more to be learned on the way to a PhD in “math.” If our schools continue to consider mathematics as NOT A LANGUAGE, it might as well continue to treat them like art or physical education classes; the majority of students will NOT MASTER it.

Wishful thinking will NOT make mathematics easier. Continuing to do the same thing over and over even though it does not “work” well, is a symptom of a mental illness we don’t want to believe we have.

Let’s “get over” the self-delusion. IF we want more of our children to succeed in math, we know what we need to do. It begins with adequate planning, then financing, training, and finally doing it. The first step requires abstract, “off-line” thinking. The last four require “on-line” ACTION.

Personally, “thinking otherwise” about Devlin’s analysis tells me that, if they begin now, it will take several generations for public schools (and most private schools) to even approach an acceptable level of “mathematical literacy.” That’s why this blog suggests alternatives to parents—viable ones, so that those who want to make a choice for their children’s future can have that choice.

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