Interview Report: Proportional Reasoning
Name: Emma Shrank
Age: 11 years old
Grade: 6th
Subject: Pre-Algebra
Teacher: Molly O’Neil
Pertinent Information: Child’s mother is a UCSC professor
In the
interview, Emma works on the Mr. Tall and Mr. Short proportional reasoning problem.
Emma refers a lot to the paper clips because the paper clips gives her a better
representation of the problem. Emma arranges the paper clips in a way that
measures Mr. Short’s height by positioning one end of the paperclip after
another end until enough paper clips cover the height of Mr. Short. The paper
clips are the hands-on tool Emma uses to communicate her thoughts along with
verbal communication. It helps her quantify what the information given in the
proportional reasoning problem.
During
the interview, I notice that Emma understands the problem and knows how to do
it. To be certain of her knowledge and to test her knowledge about the
question, I take the paper clips away so that she can give me a verbal
explanation of her reasoning. Without the use of paper clips or any objects to
help her think, I am getting the purest form of what Emma knows. The
explanation she gives is based on the paper clips because Emma needs something
solid to base her theories and opinions and in this case, paper clips are the
objects that solidify her thinking.
The
explanation consists of Emma adding two feet (biggies) to Mr. Tall’s height and
her adding three smallies to match the addition of two biggies. As you can see,
she uses an additive relation to think proportionally. Although this method
works, it only works well with cases that deal with small numbers and not big
numbers. In an attempt to squeeze more thought out of her, I try to get her to
provide some form of formula for figuring out Mr. Tall’s height given Mr. Short’s
height in a general case. Although it is not in the video, she explains to me
that she keeps adding to determine Mr. Tall’s height. Procedurally she shows
proportional reasoning, but when asked for a general formula, she conceptually
does not know two biggies equal three smallies. Emma sees the relationship of
for every two biggies added to Mr. Tall, then there are three smallies added as
well. The only thing she is missing is that she does not have a strong
connection to see both numbers as a fraction or percent. Building the
connection between ratios and fractions or percent will be one of the next
steps to do with students like Emma learning proportional reasoning.
My
reaction is that she is very dependent on objects and is a student that learns
from doing rather than seeing and listening. Emma is a very capable student who
knows how to do the proportional reasoning problem. What surprise me is that
Emma notice that for every two biggies there are three smallies, but when asked
for the general case, she could not produce the same result. I learn that Emma
has some background about proportions because she is able to do the problem
flawlessly without error. As a result from seeing Emma’s dependence on paper
clips, it definitely shows me that most students may rely on concrete objects
to get an understanding of proportional reasoning. This can also be applied to
other topics in math at the high school and middle school level. A follow up
for this lesson is for students to create their own word problems after doing
several proportional reasoning problems. I will have students work in groups
and come up with one word problem where they show various methods on how to
solve it such as paper clips or other objects that help unitize.
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