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New Perspectives On the Student-Professor Problem
Unformatted Document Text:  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.

Authors: Weinberg, Aaron.
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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.


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