17.  

Alibali, 1999; Lemaire and Siegler, 1995; Siegler and Jenkins, 1989.

18.  

Researchers have shown clear disconnections between students’ “street mathematics” and school mathematics, implying that skills learned without understanding are learned as isolated bits of knowledge. See, for example, Nunes, 1992a, 1992b; Saxe, 1990. It should be emphasized that, as discussed above, conceptual understanding requires that knowledge be connected. See Bransford, Brown, and Cocking, 1999; Hiebert and Carpenter, 1992.

19.  

Saxe, 1990.

20.  

Carpenter, Franke, Jacobs, Fennema, and Empson, 1998.

21.  

See Schoenfeld, 1992; and Mayer and Wittrock, 1996, for reviews.

22.  

Wiest, 2000.

23.  

Such methods are discussed by Schoenfeld, 1988.

24.  

Mayer and Hegarty, 1996.

25.  

Hagarty, Mayer, and Monk, 1995.

26.  

Bransford, Brown, and Cocking, 1999, pp. 19–38. See also Krutetskii, 1968/1976, ch. 13.

27.  

For each of the five levels in the stack of blocks, there are two options: red or green. Similarly, for each of the five toppings on the hamburger, there are two options: include the topping or exclude it. The connection might be made explicit as follows: Let each level in the stack of blocks denote a particular topping (e.g., 1, catsup; 2, onions; 3, pickles; 4, lettuce; 5, tomato) and let the color signify whether the topping is to be included (e.g., green, include; red, exclude). Such a scheme establishes a correspondence between the 2×2×2×2×2=32 stacks of blocks and the 32 kinds of hamburgers.

28.  

Pólya, 1945, defined such problems as follows: “In general, a problem is called a ‘routine problem’ if it can be solved either by substituting special data into a formerly solved general problem, or by following step by step, without any trace of originality, some well-worn conspicuous example” (p. 171).

29.  

Siegler and Jenkins, 1989.

30.  

English, 1997a, p. 4.

31.  

English, 1997a, p. 4. See English, 1997b, for an extended discussion of these ideas.

32.  

For example, Inhelder and Piaget, 1958; Sternberg and Rifkin, 1979.

33.  

Alexander, White, and Daugherty, 1997, p. 122.

34.  

Davis and Maher, 1997, p. 94.

35.  

Davis and Maher, 1997, pp. 99–100.

36.  

Davis and Maher, 1997, pp. 101–102.

37.  

Alexander, White, and Daugherty, 1997, propose these three conditions for reasoning in young children. There is reason to believe that the conditions apply more generally.

38.  

Carpenter and Levi, 1999; Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, and Human, 1997; Schifter, 1999; Yaffee, 1999.

39.  

Maher and Martino, 1996.

40.  

There is a precedent for this term: “Students come to think of themselves as capable of engaging in independent thinking and of exercising control over their learning process [contributing] to what can best be called the disposition to higher order



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