A r c h i v e d  I n f o r m a t i o n

What Makes Counting Easy, Subtraction Hard?

The project's principle investigator, Stellan Ohlsson, has centered his exploration on the HS system, a sort of "electronic learner" that he designed in order to "teach" it the concepts of counting and arithmetic. HS would then perform calculations based on its understanding of these lessons, and Ohlsson could observe the system's errors, self-correcting procedures, and resulting efficiency and accuracy. HS thus simulated the cognitive learning and rule-building processes of actual learners, providing what Ohlsson calls, "a model that starts with knowledge of the concepts and principles and learns how to do arithmetic tasks correctly."

Ohlsson's first sessions with HS were on counting, which he says is "the simplest of all arithmetic procedures--one that children learn quite well, unlike the arithmetic in school." He found a direct connection between the skill of counting and the principles of counting, a perfect overlay of a conceptual template upon a procedural activity. "Learning to count," Ohlsson says, "has all the nice properties you would want school instruction in arithmetic to have. For instance, children can, if you change the counting task on them in some unusual way, readily adapt to this. They don't fall apart. So, you ask a child to count the pens here on my desk, and he does so and tells you there are nine. You can then say, count them again and make sure the red one is number three--and the child will be able to adapt his counting to that constraint."

Such flexibility apparently comes straight out of the direct, inflexible connection between the principle and the skill of counting. "The source of children's success with counting," Ohlsson says, "is that the idea constrains the behavior." In fact, the idea virtually is the behavior. "The concept of one-to-one mapping that underlies counting is reflected in the performance," Ohlsson continues. "Every step in counting is dictated by the principles, so if you do any step but the right one, you will be violating the one-to-one mapping." To learn counting, then, is to learn the principle, even though a child enumerating the pens on Ohlsson's desk might not be able to say why he does it the way he does.

Subtraction is a different story. In order to investigate the connection between the principles and the procedures in one topic of subtraction, Ohlsson taught HS how to subtract using two different procedures, or algorithms, and he taught each algorithm both conceptually and mechanically. The question was, which algorithm, taught by which method, could HS learn and perform better?

The algorithms taught to HS were regrouping and augmenting. Either can be used to solve what is called a non-canonical subtraction problem; that is, one like 42 minus 19, in which a digit in the subtrahend (19) is larger than the corresponding digit in the minuend (42). Regrouping, familiar to Americans, involves "borrowing" from the position immediately to the left of the too-small number in the minuend. In the example here, this would regroup the components of the minuend, 42, from 40 + 2 to 30 + 12. The 9 in 19 could then be subtracted from 12 instead of from 2. Augmenting, taught in Europe, simply adds equally to the minuend and the subtrahend. Thus, 42 and 19 might each be increased by 5 (or by any number from 1 through 8 that would rephrase the problem as a canonical one--such as 47 minus 24--without changing its outcome).

A comparison of the effectiveness of the four lessons not only moved toward settling an old dispute in early math education but also suggested a hypothesis for why arithmetic is so difficult for children to learn. Close observation of the steps, errors, corrections, and rule refinements that HS went through revealed that augmenting, taught mechanically, was most easily learned; regrouping, taught conceptually, was hardest to master. This ran counter to traditional wisdom in the field, which had held that regrouping was cognitively simpler to learn. The significant difference, Ohlsson found, lay neither in the concepts nor in the algorithm itself, but in the "attention allocation" of the electronic learner HS. "What we observed while running the model is that there are more complicated attention allocations in regrouping; in other words, the eyes move about over the display in a much more complicated pattern--and, of course, knowing the mathematical principles of subtraction isn't going to help with attention allocation."

Regrouping and augmenting are not the only two algorithms that one can use to perform mathematically correct subtraction. And there are many algorithms for many other kinds of arithmetic problems. Once a learner confronts a situation in which the behavior is not entirely constrained by principles, as it is in counting, then, says Ohlsson, "choosing the best algorithm is guided by expediency considerations." Thus, the conclusion he has drawn from running HS is that "even if you understand the underlying concepts and principles, there is still the problem of deciding what to do. You can't derive the most efficient algorithm from the mathematics, because the mathematics doesn't talk about that. We now think that is why it is so hard to learn subtraction, or other math, in a conceptual fashion. We think that is why so many instructional interventions fail, because they go in and try to teach the concept of place value, for example, and then find that the students still don't perform the calculations any better."

Because Ohlsson is concerned with efficiency in instruction, he questions whether time and effort involved in teaching arithmetic conceptually are well spent; yet he also concedes that "when you teach mechanically, that tends to rob the child of his own authority over the correct procedures." This matter of "authority," considered in the light of other work at NRCSL, may be what fills the gap between concepts and procedures in arithmetic.
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