Making Your Case
Jamie Walker, posted March 27, 2017 —
Due to a bit of bad luck, I had to spend a
little time in traffic court recently. While there, a gentleman was trying to
convince the judge that he was not guilty of a parking violation. It was very
clear by the terms he was throwing around that this gentleman watched a lot of Law and Order. “Your honor, it was not
clear beyond a reasonable doubt that I could not park on that side of the
street. No jury of my peers would convict me.” The judge did not buy his
argument, no matter how much legal jargon he used, and he was ordered to pay
the fine. While listening to this man, I was reminded of my students throwing
around math vocabulary, trying to explain what they had done, without fully
grasping the vocabulary terms they were using.
In my last post, I discussed developing a
classroom environment that encourages arguments and debates to deepen understanding.
But once you’ve established that environment, have you given your students the
tools they need to make their case? A firm understanding of the language of
mathematics gives students the ability to voice a stronger argument.
I suggest a few steps to help your students
build their vocabulary. Instead of pre-teaching vocabulary, introduce new terms during
a lesson when they are used. Give students the opportunity to practice using
this vocabulary and support them in finding the correct terminology. To demonstrate,
I refer to a lesson on similar polygons.
Rather than giving a list of vocabulary before
a lesson, introduce the term in context, which will allow students to develop
their own understanding and definition of the term. Starting a lesson by displaying
a slide with “Similar Polygons,” having the same shape but not necessarily the
same size, without putting the term in context, could lead to serious
misconceptions. According to this definition, a student might think that all
rectangles are similar to one another; they are all the same “shape,” right? Instead,
in this lesson, I pass around a 4 x 6 photo of my family and display two
enlargements on the board. One enlargement is correct, and one is distorted. Once
the class is engaged, I then explain that the correct enlargement is
geometrically similar to the photo. At that point, I introduce the definition,
then show more examples and model correct usage of the vocabulary.
When students show a basic understanding of the
new concept, let them try to make their case by explaining it with the correct
mathematical terms. Although students may be quick to use the new words, their
misconceptions will become clear when they try to justify their work. Personally,
this is my favorite part of the lesson.
These mistakes are wonderful opportunities for
growth if the student can learn from them. Instead of correcting the misuse of
a term, support the student in seeing his or her mistake. During our lesson on
similar polygons, students will often say that polygons are similar because the
sides are congruent. I know that my students understand congruence, and I also
know that they can clearly see that the sides are not of equal length. However,
when learning new terms, it is easy to mistakenly use words that have worked in
the past when relating two shapes. Rather than just correcting the student and
saying, “You mean the sides are proportional,” I will instead question them
about the sides and congruence, supporting the student until he or she sees the
My students have shown the most amazing insight
and understanding once they are given the proper tools to express themselves. It
is my hope that I can help my students see that their arguments are
strengthened, not by using big words but by using the correct words. Can’t we mathematicians
agree that the most beautiful answer is the most concise and precise argument?
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My students struggle with vocabulary when we started a lesson in Geometry. I started the lesson off wrong, by giving them a list of vocabulary words to study. When I started to teach the lesson and I asked several students questions about the vocabulary they could not remember. For example, we were working on different congruent angles. If there were a pair of angles that were congruent, I would ask the students, "why are they congruent?" The students would say "because they are equal." What I was looking for, was they are congruent because of alternate interior angles, alternate exterior angles, corresponding angles, etc. I redid the lesson the next day by giving the kids a picture of two parallel lines cut by a transversal and I ask them to describe a pair of selected angles. The students started thinking and then the hands went up. They shouted, "they are congruent." I said, "prove it?" The students started disscussing how to prove it. One set of students shouted, "because they are corresponding angles." We did this task for a few moments and the kids finally started to have a conversation with one another about how they could prove that the angles were congruent. It was awesome. When I stepped away from the debate and let them talk it out. The students started teaching each other.
I most certainly agree that the most beautiful answer is the most concise ad precise argument. I taught at a low income school and was forced to create a word wall for terminology. I felt I spent more time putting it together than the students did understanding or using the terms.
Most textbooks I have used have the vocabulary listed at the beginning of the chapter or introduction reinforcing that old school list and define type of comprehension.
I agree with your methodology of introducing vocabulary in context instead of listing it before or using a word wall. I think a word wall can be helpful if they forget the terms but I agree that using it correctly in context will help.