This approach has been described in detail by Constance Kamii, a leading student of Piaget's, in a series of three books about how children in first, second, and third grade, respectively, can "reinvent arithmetic.
Ultimately, of course, it matters whether students come up with the right answer, but if they're led to think that's all that matters, they're unlikely to understand what's going on.
Thus, says Kamii
, "if a child says that 8+5=12, a better reaction would be to refrain from correcting him and . . . ask the child, 'How did you get 12?' Children often correct themselves as they try to explain their reasoning to someone else." Because that "someone else" can be a peer, it often makes sense for children to explain their reasoning to one another.
On a smaller and more informal scale, the constructivist theorist Constance Kamii
has tested a few elementary classrooms in which children worked all problems on their own, without being given any algorithms.
Consistent with the other studies, she
discovered that two constructivist second-grade classes did about as well as two conventional classes on a standardized achievement test but performed better on measures of thinking. A subsequent comparison of third graders also found that the "Constructivist Group
used a variety of procedures, got more correct answers, and made more reasonable errors when they got incorrect answers.
The Comparison Group
by and large had only one way of approaching each problem - the conventional algorithm - and tended to get incorrect answers that revealed poor number sense."
One last point, which is not so incidental: a teacher working with Kamii
commented that after she
adopted the nontraditional approach to instruction, her
classes "displayed a love of math that I had not seen during my first decade of teaching." While there are no hard data to confirm this impression (as there are with Whole Language), it certainly matches what the Purdue researchers witnessed in their experimental classrooms.
, 1994, pp.
(e.g., 1994, pp.
, 1985b, p. 3.
For examples of fortuitous events that can provide the opportunity for children in first grade, second grade, and third grade to think about numerical concepts, see Kamii
, 1985b, pp.
123-35; 1989, pp.
91-97; and 1994, pp.
Like some other constructivists, Kamii
also swears by the use of certain games -- such as those involving dice or play money -- for teaching purposes.
, 1985b, pp.
constructivist premises have led Kamii
to offer only a partial endorsement for the NCTM standards.
argues that, despite their emphasis on deeper understanding of mathematical truths, the standards still reflect an empirical view that those truths have a reality entirely independent of the knower.
Further, while collaboration among students is recommended, Kamii
believes the standards fail to reflect a constructivist appreciation for the necessity of understanding through resolving conflict among disparate ideas (see Kamii
, 1989, pp.
29. Not every math educator agrees that primary-grade children shouldn't be given algorithms at all, but Kamii
makes a strong case for this position.
31. "Research has shown, however, that most children think that the 1 in 16 means one, until third or fourth grade" (Kamii, 1989, p. 15).
"Even in fourth and fifth grades, only half the students interviewed demonstrated good understanding of the individual digits in two-digit numerals" (Ross, 1989, p. 50).
32. Vygotsky, 1978, p. 84.
33. Linda Joseph's account appears in Kamii
, 1989, p. 156.
, 1989, pp.
48. Kamii, 1994, p. 205.
49. Linda Joseph, quoted in Kamii
, 1989, p. 155.