unprompted. Further, he did not expect that the question had to be answered by the teacher. Rather, he was confident that he had the tools, ideas, and concepts that would help him navigate his way to the answer. We also see that Zach rigorously assessed the reasonableness of his answers and that he used his knowledge of translating among the various representations to help him solve the problem. I conclude with this charming vignette as an illustration of the potential support our curriculum appears to offer to students beginning their learning of rational number.

Students then go on to learn algorithms that allow them to calculate a number like 83.3 percent from 5/6 efficiently. But the foundation in mathematical reasoning that students like Zach possess allow them to use those algorithms with understanding to solve problems when an algorithm has been forgotten and to double check their answers using multiple methods. The confidence created when a student’s mathematical reasoning is secure bodes well for future mathematics learning.

NOTES

1.  

Armstrong and Bezuk, 1995; Ball, 1990; Post et al., 1991.

2.  

Moss, 2000.

3.  

Carpenter et al., 1980.

4.  

Ball, 1993; Hiebert and Behr, 1988; Kieren, 1993.

5.  

Lamon, 1999.

6.  

Hiebert and Wearne, 1986; Wearne and Hiebert, 1988.

7.  

Hiebert and Behr, 1988.

8.  

Kerslake, 1986.

9.  

Heibert, 1992.

10.  

National Research Council, 2001.

11.  

Carpenter et al., 1993.

12.  

Lamon, 1999.

13.  

Lesh et al., 1988.

14.  

Baroody, 1999.

15.  

Lamon, 1999.

16.  

National Council of Teachers of Mathematics, 1989, 2000.

17.  

Markovits and Sowder, 1991, 1994; Sowder, 1995.

18.  

Cramer et al., 1989.

19.  

Sowder, 1995.

20.  

Sowder, 1992.

21.  

Hiebert and Behr, 1988.

22.  

Behr et al., 1983, 1984, 1992, 1993; Kieren, 1994, 1995; Ohlsson, 1988.

23.  

Hiebert and Behr, 1988.

24.  

Kerslake, 1986.

25.  

Behr et al., 1984; Case, 1998; Hiebert and Behr, 1988; Lamon, 1995; Mack, 1990, 1993, 1995; Resnick and Singer, 1993.



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