Lamon
(1993) mentioned that there are four types of semantic problem in analyzing the
problem typically by a portion. The four semantic types are well-chunked
measures, part-part-whole, associated sets, and stretchers and shrinkers. These
four semantic problems can help students develop meaningful logic in ratio and proportion.
It also helps students rethink the key idea of proportion. Furthermore, the
semantic problems can be really applicable to real-life. For example, Lamon
used the eggs example, which is really applicable to real-life and relevant to
the students since most of the students eat eggs for breakfast assuming that
they are part of the American cultural.
According
to Lamon (1993), proportion reasoning involved comparison of two or more
extensive measures. For example, when Lamon talked about the associated sets,
he talked about using pizza to teach the students 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. 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
mathematical symbols.
Students
often have troubles using algebraic expression to translate phrases into
mathematical symbols. However, when the students are allowed to use
proportional reasoning, the students are able to justify and explain their
answer through reasoning. This can address some difficulties of the students
that have in mathematics because students only have to explain their reasoning
for the problem. It does require conceptual understanding, but the students
would not have to write long proofs or detailed math solutions for the abstract
problems.
Pictures
and word problems are the best way to teach proportional problems. For example,
pizza is one of the best ways to teach proportional because of the feature of a
pizza. A pizza is usually in a circle shaped that is cut into equal slices. In
this case, students can use fractions in explaining the slices of pizzas.
Fractions are really similar to proportion, and this can help the students get engaged
using proportional reasoning for the word problems. Also, students can identify
patterns through looking at word problems. For example, if three oranges cost
as much as four apples, then how much oranges would 8 apples equal to? The
students know that if three oranges is equal to 4 four apples, then there is
going to be more than 3 oranges if there are 8 apples. The good part is that
students can make sense in their mind using proportional reasonings.
I think word problems is the best way to practice proportional reasoning because
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2. a lot of thinking is required to understand what the problem is asking
3. pictures are drawn to help recreate the situation and better see what is happening
4. execution. picking the right method and being able to show why it works with numerical expressions, etc.
Word problems work on a variety of skills necessary for students to move on in math. Proportional reasoning is especially important because it is linking multiple ideas such as rate, time, ratio, and units.
Stephen, I like that you are emphasizing the real life connection and how this is valuable to students understanding. Using objects that students can manipulate is a great first step in developing proportional reasoning. I too would start here and build students confidence. Allowing them to make the connection is much more powerful than you doing it for them. I find that many students have an initial anxiety when it comes to word problems. Too often there is a disconnect between reading and comprehension in mathematics maybe because students do not get enough of it or maybe because students get too much of it in other subject areas. I really like to approach mathematics in a way that is similar to a science experiment. As the teacher what would you do to promote understanding and how will you guide students who are struggling?
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