167

I took three to four ratio—cause for every four pigs there's three cows, which is point

seven five. And so if you're given four pigs, four times point seven five is three, and so

you can keep increasing by point seven five.

While working on the value-prediction task, students used different strategies than they

used on the translation problems. Several students used a strategy that was similar to functional

reasoning but performed the operations *stepwise* instead of all at once. For example, one

student divided a group of objects into “test samples” and then counted several of these

samples. This corresponded to first dividing the original quantity and then multiplying the

result in contrast to multiplying by a fraction or ratio. In this example, the student was working

on the problem: “In New York there are three SUVs for every five cars. If there are 165 cars in

a parking lot, how many SUVs do you expect there to be?”:

Well because, it says for every five cars… I wanted to… take it out of that one sixty

five and once I was done, I was just going to multiply it back out. So... 165 divided by 5

is 33? Yeah. 33, and I just like left that number alone and then there's three for every

five, so I did a test sample kind of thing, it’s one thing of five cars, so if there's three for

every five I just multiplied the 3 by how many test samples there are—which is 33—

and I got 99.

Numerous students used *proportional reasoning* to make their prediction by constructing a

proportion from the written description and cross-multiplying:

Okay, there are three SUVs for every five cars, so you can put that into a proportion…

So if you set up a proportion and set them equal to each other, so you know like three

over five, is SUV—or, is SUVs per car, so you set it equal to x over one sixty five and

you can cross multiply and divide through everything and you can find that there's

ninety nine SUVs with the hundred sixty five cars.

**Student Performance**

Roughly half of the students gave correct answers on the translation and function-

construction items on the written assessment; roughly one third of the students gave an

incorrect answer involving a reversal error. This is in contrast to previous studies in which

students had more difficulty translating situations when one of the coefficients was not 1. On

the value-prediction item, students performed significantly better, with 70% of student giving

correct answers and only 11% supplying an answer that indicated a reversal error. Despite

students’ success with the value-prediction item, they had difficulty when asked to construct an

equation that represented the situation in this item.

While students were more successful predicting values than translating situations into

equations, there was no correlation between the strategy a student used to construct an answer

on the written assessment and their strategy on the same item during the interview. In addition,

there was little correlation between a student producing correct answers or making reversal

errors across multiple items. That is, there were few students who consistently produced correct

answers or consistently made reversal errors.

Students displayed a flexible conception of the equals sign. They avoided common

misconceptions, recognizing that (*a*+*b*)

2

is not equal to *a*

2

+*b*

2

and they did not describe *ac*

+*ab*=*a*(*b*+*c*) as “reverse distribution.” However, 20 out of 27 students asserted that the number

*Lamberg, T., & Wiest, L. R. (Eds.). (2007). Proceedings of the 29*

*th*

* annual meeting of the *

*North American Chapter of the International Group for the Psychology of Mathematics *

Education, Stateline (Lake Tahoe), NV: University of Nevada, Reno.