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Nail Down the Misconception:
by Su Liang, California State University San Bernardino 
Having taught Intermediate Algebra in University of Connecticut for three semesters, I found that a quite number of students kept making a common mistake:
Even in a Calculus class, this mistake is not uncommon in students’ work. Whenever they have an algebraic expression containing like terms in the numerator and denominator, they simply cancel them out. I had been thinking about how to correct the misconception effectively. In a pedagogy class, we were assigned a project to find a common misconception and to design some effective activities so that the misconception can be corrected by letting students make sense out of it. I shared this problem with two other mathematics teachers in my group. We decided to use this problem as our project. Luckily, when I had a chance to teach Intermediate Algebra in the fall 2008, I modified our design and applied it to my teaching. The result is significant for my class. In the beginning of the semester, about 1/3 of the students had the misconception. In the end of the semester, no students repeated this type of mistake. I would like to share our approach for this problem.
To tackle a problem, we have to know the underlying reason. Why are students tempted to make this type of mistake?
In the beginning of the semester, I let the students form their own groups to do group projects together through the whole semester. Each group had three or four students. In total we had 7 groups, 25 students. I held a group discussion and raised this question: Is
Given that bâ? c and neither a nor b is 0? I asked the groups to discuss this question and to reach an answer with full explanation. Each group would have a representative present their ideas. After about a 10minute discussion, I asked the two groups who gave the ’yesâ? answers to give their reasons. They reasoned that because ’a’ is in the numerator and also in the denominator, by division, it can be cancelled out. Then I asked the groups who said ’noâ? to explain their answer. They gave the reason that ’a’ can’t be cancelled because the operation sign between ’a’ and ’c’ and ’a’ and ’b’ are addition, not multiplication. Students could not wait to hear my judgment. I told them that I would not tell them who is correct just yet. By giving them the following work sheet, I let the groups work on it and report their result when they finished.
Please use the given values for a, b, c to
calculate , ,




































































































After 15 minutes, all the groups finished their work. I asked the groups one by one to report what they found. They all reported that
I didn’t stop there. Instead, I raised another question: Can we prove our conclusion algebraically? Again, I let the groups work on their own. I gave a hint to the groups who did not have a clue what to do (Hint: using cross multiply). Ten minutes later, all groups explained their ideas of proof. Following their explanations, I wrote down the proof step by step on the board. First, we suppose that
I addressed a summary to my students: ’As you can see, according to the data from the work sheet of table, the algebraic proof, and the operation rule, they all give us the same conclusion:
I used a red chalk to
write this in a big size of letters on the board. Furthermore, I asked the
groups to write a report reasoning why
Finally I gave them two questions and let them write the answers:
1. Is (2x + 1)/(2x +2) = 1/2 ?
2. Is 2x/3x = 2/3 ?
I checked their answers one by one. Every student in the class replied correctly. Through the end of this semester, students did not repeat the type of mistake
It is worth mentioning that although my work focused on a specific example, the method  having students work out numerical examples, then discussing the algebra, then looking at the differences between the situations where what the student wants to do is CORRECT versus where it is not, and finally practicing with some more sophisticated examples  is a promising one for other common student errors as well.
I would to thank my classmates Carla Ryall and Christopher Dailey who worked with me for the group
project ’ finding a common misconception and designing some effective
activities so that the misconception can be corrected by letting students make
sense out of it in our pedagogy class in spring, 2008.
Su Liang
(liangs@csusb.edu)
graduated
from China University of Politics Science and Law with a B.A. in Business Law
and from University of Connecticut with a M.S. in mathematics. She is completing
her PhD. in Mathematics Education at University of Connecticut in summer 2010
and will be an assistant professor in California State University at San
Bernardino in fall 2010. Her research interest is K12 mathematics teacher
preparation.
The Innovative Teaching Exchange is edited by Bonnie Gold.