*If 49+83=132 is true, which of the following is true?*

*49=83+132*

*49+132=83*

*132–49=83*

*83–132=49*

Only 61% of 13-year-olds chose the right answer, which again is considerably lower than the percentage of students who can actually compute the result.

Another example is a multiple-choice problem in which students were asked to estimate The choices were 1, 2, 19, and 21. Fifty-five percent of the 13-year-olds chose either 19 or 21 as the correct response.^{63} Even modest levels of reasoning should have prevented these errors. Simply observing that and are numbers less than one and that the sum of two numbers less than one is less than two would have made it apparent that 19 and 21 were unreasonable answers. This level of performance is especially striking because this kind of reasoning does not require procedural fluency *plus* additional proficiency. In many ways it is less demanding than the computational task and requires only that basic understanding and reasoning be connected. It is clear that for many students that connection is not being made.

A second kind of item that measures adaptive reasoning is one that asks students to justify and explain their solutions. One such item (Box 4–5) required that students use subtraction and division to justify claims about the population growth in two towns. Only 1% of eighth graders in 1996 provided a satisfactory response for both claims, and only another 21% provided a partially correct response. The results were only slightly better at grade 12. In this item, Darlene’s claim is stated somewhat cryptically, and students may not have understood that they needed to think about population growth not additively—as in the case of Brian’s claim—but multiplicatively so as to conclude that Town A actually had the larger rate of growth. But given the low levels of performance on the item, we conclude that Darlene’s enigmatic claim was not the only source of difficulty. Students apparently have trouble justifying their answers even in relatively simple cases.

Research related to productive disposition has not examined many aspects of the strand as we have defined it. Such research has focused on attitudes