A passage from Derek Stolp’s Mathematics Miseducation: The Case Against A Tired Tradition (pages 70-73)
This proposed set of adjustments to our current program of study will inevitably invite the question, “What about the basics? Aren’t they important?” Well, indeed they are. But let’s see if we can agree about what we mean by the “basics.” The usual understanding is that these are computational skills such as addition and multiplication facts. The assumption is that one cannot go on to higher levels of mathematics without having first learned one’s facts. The work of Stanislas Dehaene and his colleagues, however, strongly suggests that learning these particular sets of facts do not strengthen one’s mathematical skills because learning them involves storing them in one’s verbal memory. Experiments on bilingual adults demonstrated that when making use of times tables, for example, the speed at which the computations were made was far greater in their original language, suggesting that they were making use of verbal rather than mathematical skills. (Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999) To convince yourself of the plausibility of this, try to perform some computations without saying, for example, nine times eight is seventy-two, if not aloud, then certainly “silently.” It is likely that, even if you are proficient and quick at these, your mind, if not your ears, must “hear” the sentence. Dehaene argues that since the human mind is not well designed for the task of memorizing discrete number facts, it compensates by using verbal memory. And what is “wrong” with the design?
If our brain fails to retain arithmetic facts, that is because the organization of human memory, unlike that of a computer, is associative: It weaves multiple links among disparate data. Associative links permit the reconstruction of memories on the basis of fragmented information… It is a strength again when it permits us to take advantage of analogies and allows us to apply knowledge acquired under other circumstances to a novel situation. Associative memory is a weakness, however, in domains such as the multiplication table where the various pieces of knowledge must be kept from interfering with each other at all costs. (Dehaene, 1997, pp. 127-128.)
When attempting to quickly recall from memory the product 8 x 9, it is not helpful to make the association with 8 x 8. (Ironically, if the child cannot recall the product of 8 x 9 but knows that the product 8 x 8 is 64 and realizes that he merely needs to add another 8 to 64, he is showing a deeper understanding of mathematics than another child who simply recalls the product 8 x 9. But the teacher who demands immediate recall regards this strategy as a weakness!) While it is handy to know number facts when, for example, we divide 8650 by 9, we must wonder to what extent that task improves one’s mathematical sense. When the child mindlessly performs the division algorithm (divide into, multiply, subtract, bring down, divide into, and so on), she is certainly performing numerical operations but far less efficiently than could a hand-held calculator, and she is gaining no real insight into numerical relationships. On the other hand, if she knows that the answer has to be larger than 865 (because 8650 divided by 10 is 865), and it must be less than 1000 (because 9000 divided by 9 is 1000), she is demonstrating excellent number sense.
This brings us to the question, “Should we focus our time and energy upon mechanical tasks that are more efficiently performed by machines or upon the complex tasks for which our associative minds are better suited?” Those of us who learned mathematics before the advent of calculators may have come to identify mathematical thinking with the ability to perform numerical calculations and symbolic manipulations, but we have, in recent years, begun to realize that there are some higher orders of thinking that have received short shrift. The complete solution of a multi-step problem, for example, requires that we analyze the problem, sort through the available information, apply the appropriate computations (numeric or symbolic), and write an argument (a proof) in defense of the solution. It has been vigorously argued that our kids should not be allowed to use calculators to do their thinking for them, but the kind of thinking done by calculators is simply recall of isolated facts and reproduction of standard procedures, not the kind of complex thinking of which humans are capable.
So what are the “basics”? If we mean by that term those skills that are elementary, then we should certainly rehearse simple techniques that will create a basis for understanding and hence a platform for further study. (Memorizing and recalling number facts does not create a basis for understanding.) But we also mean by the “basics” those skills that are essential for one to become a resourceful, mathematically literate citizen, in which case we need to move beyond the elementary; we need to have our students perform complex tasks as well as the simple mechanical ones. These might include the following: “Can you read a problem and distinguish between what’s important and what’s incidental?” “Can you search through your collection of intellectual tools and find ones appropriate to solving a problem?” “Can you discern similarities between this problem and another that we have seen before?” “Can you generalize from a set of specific instances?” “Can you create a mathematical model that allows you to understand a set of relationships and even allows you to predict?” “Can you communicate your reasoning, orally and in writing, so that others can follow it?” These skills are also basic, in that they are essential, and too often they have been ignored in traditional approaches to this discipline.
One cornerstone goal of any academic discipline should be to help students acquire a language and a set of concepts through which they can make sense of the world in which they live. Traditional instruction has lost sight of this mission: its excessive emphasis upon elementary computational techniques (and those trivial, though difficult problems that support those techniques) has kept our students from experiencing the true power and reach of this discipline. Too many have left with the sense that mathematics is little more than a set of rules which are often unwieldy, sometimes redundant and often inapplicable, and that mathematics depends less upon creative, thoughtful reflection than upon the ability to manipulate symbols (and quickly).
The calculator doesn’t substitute for the student’s important work; instead it increases opportunities for students to shift their focus from the tedious to the significant, from the computation to the problem, from the symbol to its meaning. It reminds us that mathematics is a lens through which we can see and begin to make sense of the complex world in which we live. It allows us to shift our gaze from that lens to the world.
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