Background
Information
The student I interviewed is currently
a 9th grader who is taking geometry. The student is from my focus
class, so from my observation, I know that he has a good background in
mathematics. His math teacher is Ms. Mendez, which is my cooperating teacher (CT).
The child is 14 years old. In the interview, I was supposed to chain up the
paper clips when I do the demonstration of using the big paper clips to measure
Mr. Short. However, I decided to draw a hat on top of Mr. Short to make sure
that 4 big paper clips will be the size of him without being chained because I
did not want the student to waste time to chain the paper clips. The theory and
activity remained the same to a certain extent.
Proportional
Reasoning
Proportional
reasoning involved comparison of two or more extensive measures. For example,
when Lamon (1993) talked about the associated sets, he talked about using pizza
to teach the students about proportion reasoning. The pizza question asked,
“The student is shown a picture of 7 girls with 3 pizzas and 3 boys with 1
pizza. Who gets more pizza, the girls or the boys?” (p. 5). These kinds of
questions can help students think critically while figuring out an answer
because it requires understanding of proportional reasoning. Lamon (1993)
claimed, “Proportional reasoning was said to occur when a student could
demonstrate understanding of the equivalence of appropriate scalar ratios and
the invariance of the function ratio between to measures spaces, whether or not
the student could represent these relationship symbolically.” When the pizza
question was being asked to the girl, the girl provided a logistic explanation.
However, this would not make sense if the student uses math symbols to
symbolize her words. But the key point in proportional reasoning is that the
students have to use clear explanation to show their understanding of the
question instead of answering the question using procedural method.
A student is using proportional
reasoning when the student can provide accurate calculation to the problem.
Furthermore, the student should also be able to explain what he or she is doing
in the problem. By just computing the answer, the student shows that he has
strong procedural skills, such as computing, seeing patterns, and etc. In addition,
if the student is able to explain his or her answer with good reasoning and
logic, then it shows that the student is having some sort of conceptual
understanding in proportional reasoning. In proportional reasoning, conceptual
understanding is far more important than procedural fluency because they have
to be able to understand the concept in order to have good reasoning.
Through the interview, I was not
able to see that the student has much proportional reasoning because he simply
did not use anything related to ratios. However, the student was at the
pre-Proportional Reasoning Stage. Lowery (1974) claimed, “A student is at the
pre-Proportional Reasoning Stage of development if his or her explanation
refers to estimates, guesses, appearance, or extraneous factors without using
the data of if he or she uses the data in an illogical way” (p.18). The student
told me that the answer is “eight” because he simply added 2 small paper clips
to find Mr. Tall’s height when Mr. Short is 6 small paper clips. The student is
having his own guesses and reasoning to figure out the answer, but it shows no proficient level of proportional reasoning ability. Furthermore,
this is an illogical way to use the data. However, the student does tend to use
the most obvious solution to answer the question. Hence, the student used the
additive property instead of using proportional reasoning to solve the problem.
Next Step
As the interview assignment
mentioned, students usually do not master proportional reasoning until they are
16-18. My student was 14 years old, so it makes sense that he might be a little
confused with problems like that. If I was teaching the student’s class, I
believed that I would need to introduce the topic proportional reasoning starting
from the beginning because it seemed like the student did not have much
experience in proportional reasoning back in middle school. I would not use
textbooks to teach the students proportional reasoning because it’s simply too
procedural. Cramer et al. (1993) claimed, “Proportional reasoning abilities are
more involved than textbooks would suggest” (p. 169). Textbook would not be
able to teach students proportional reasoning ability, instead I would start
off by providing hands-on activities to the class to get the students engaged, so they will be willing to learn proportional reasoning.
My activity can simply consist of
coins of dimes and quarters. Since most students use money in a daily basis, they
would find this activity relates to them, so they are more likely to be engaged
in this activity. They would also like to play with money, since everyone likes
money. Using coins, the students will see the ratio 2:5, which means every 2
quarters is the same as 5 dimes in terms of values. By giving out this activity and providing proper scaffolds to the students, the students will
have an early foundation of proportional reasoning.
Reactions
I actually interviewed two students
from the same classes at a different time, and I learned that they made the
same mistakes. Both of my students gave me the answer 8 because they thought
you are supposed to add 2 instead of using proportional reasoning. Surprisingly,
they were also some of my top students in the class, so I thought they would be
able to answer my question. This makes me wonder did they learn proportional
reasoning back in middle school. Was the teacher in their middle school
unqualified to teach proportional reasoning? Was this the reason why students
did not master the topic proportional reasoning? Hence, I believe that it is necessary
to reteach proportional reasoning in high school, so that students can become
proficient in proportional reasoning.
Works Cited
Cramer, K.,
Post, T., Currier, S. (1993). Learning and teaching ratio and proportion:
Research implications. In D. Owens (Ed.) Research ideas for the classroom:
Middle grades mathematics. NY, NY: Macmillan Publishing Company.
Lamon (1993)
Ratio and Proportion: Connecting content and children’s thinking. Journal for
Research in Mathematics Education, Vol. 24(1), 41-61)
Lowery, L.
(1974). Proportional Reasoning. In Learning About Learning Series. Berkeley, CA
Univ. Of California, pages 17-20 and 37-38
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