Response to Mordy's post on Liping Ma's research study

Question: “Imagine that one of your students comes to class very excited. She tells you that she has figured out a theory that you never told the class. She explains that she has discovered that as the perimeter of a closed figure increases, the area also increases. She shows you this picture to prove what she is doing: [a 4cm by 4 cm square and a 4cm by 8cm rectangle, with respective perimeters and areas calculated correctly, and “in support” of her claim]. How would you respond to this student?”

Response

Mordy, this is an interesting question. Here are my initial thoughts. This is exactly how I reacted to this question and remember that I was coming from the perspective of a teacher candidate.

First, I immediately took out a piece of paper and drew the rectangles described in the question and concluded that her theorem was correct based on these two rectangles. Then I tried to look at other pairs of rectangles. Since my thought was that the theorem COULD have been correct I continued to find pairs of rectangles which "fit" to her theorem.

Then, I was not content to simply settle with the fact that a student could have potentially come up with a VALID theorem on her own. *Interesting that I was determined to try to DISPROVE the student rather than prove her theorem!

I continuously tried drawing countless rectangles to try and understand the properties and relationships between area and perimeter of a rectangle. I drew a single 1cm by 1cm square, then drew a 1cm x 2cm rectangle and further more, while calculating both the area and perimeter of each shape.

• Then I had an idea! What about a counter example! There are many rectangles I know that have the same area but different perimeters. Namely, a 2x8 rectangle (P=20 and A=16) and a 4x4 square (P=16 and A=16). There we go! I counter example to disprove her theorem! A-ha! Gotcha!

Now, isn't that CRUEL! I actually was happy to know that I had thought of a counterexample to crush her theorem to shreads. Thinking about this question even more, I realize that I have to come up with a way to "respond" to her claim. To be honest, I would simply tell her that it was good that she tried to come up with a theorem on her own, but then I would present her with the counter example. It would be important for me to know how she arrived at this claim (whether it was based off one example or many ideas). It would be a great learning moment to explain to the student why and how the properties of area and perimeter are related.

I am interested to know how other teachers responded to this question.

Mordy's Response

Thanks for taking the time to think about the problem!! Your thought process is interesting in the context of this study for a few reasons. First, your initial pedagogical reaction to the student was representative of almost all the teachers in the study - nearly all the participants said that they would praise the student for her independent thinking and inquiry. However, is this enough? Many of the teachers gave this praise but could not proceed to guide the student further.

Second, your solution of using a counterexample was the method that was used by all the teachers who solved the problem correctly. However, once the student's claim is shown to be false, differences in teacher understanding were revealed came through their attempts to follow up with an "unpacking" of the problem. You certainly refer to this type of unpacking when you speak about discussing how she arrived at this claim, and using her claim as a learning opportunity to explore how area and perimeter are related (e.g. if the perimeter increases, what can potentially happen to the area?).

Finally, the intensity (or is it cruelty? ;)) that you show by saying things like "crush her theorem to shreads" is indicative of the interest that you have in approaching a new mathematics problem! This was not the case with all of the teachers in the study; in fact, many of the U.S. teachers did not even attempt to approach the problem.

Have a look at my Wiki page entitled, Liping Ma's Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China & the United States, for a more detailed account of how the teachers approached this problem, as well as Ma's analysis of these results.

Tanya's Response

Mordy you wrote:

...According to Ma, these lecture-style lessons require the teachers to practise and train their mathematical communication skills. I have always felt that lecture-style teaching has its benefits (such as the suggested opportunity to organize and gather ideas), but I also feel that these benefits are not exclusively tied to lecture-style teaching, nor do I think that Ma is suggesting this. Specifically, acquiring well-developed communication skills in mathematics is possible through other teaching styles... under the heading Acculturation of Math teachers to the Mathematics Discipline – Thinking Mathematically

I am very interested in the idea that the ability to teach geometry MUST come with an excellent ability to communicate the meaning and reasoning behind the geometry. Since spatial reasoning is mostly understood either in the brain or in contrete materials, I believe that communication is key for this type of mathematics learning.

My question to you would be specifically in response to your comment saying, "...acquiring well-developed communication skills in mathematics is possible through other teaching styles" : What other teaching styles do you think specifically help in communicating and teaching geometry?

It is possible to develop diagrammatic ways to communicate (as well as verbal). This also takes lots of work, but can be learned.

In talking with Mordy last week, I wanted to connect this problem to some of the 'big ideas'. Here is one such connection: In the plane, if you cut the rectangle and past the pieces back together in a different way, the area is the same, but the perimeter typically changes.

This then starts an exploration of some connections between perimeter and area:

```  Can you make the perimeter very large, with a given area?
Can you make it small (a minimum)?
Under what conditions on the changes will you know for sure that the area increases?
How do those changes related to a smaller dimension?
What about surface area and volume?
```

Mordy's Response II

```My question to you would be specifically in response to your comment saying, "...acquiring
well-developed communication skills in mathematics is possible through other teaching
styles" : What other teaching styles do you think specifically help in communicating and
teaching geometry?
```

When I first read this section in Ma, I thought it strange that lecture-style teaching (which is the devil according to teachers' colleges here) could actually be a good thing! After some thought, I realized that it was the preparation that went into these lecture segments of a lesson that contributed to improved teacher communication skills. To be clear, lecture-style learning is not the only type of lesson used in China. However, Ma relates that its use for introducing new topics may have significantly helped teachers improve their communication skills. BUT I also felt that this sort of preparation can be accomplished via other teaching styles.

For example, I think that the use of technology can be a great way to teach geometry, but this teaching style still requires a lot of preparation on the part of the teacher. When using technology as a teaching tool, we WILL be asked questions by our students about both the technology itself and the material that it is trying to convey. Thinking about how to respond to students' questions, and even preparing handouts to guide students through the use of the software helps teachers to develop their communication skills. Even in the initial stages of planning a lesson, a natural question is: How can I best communicate or represent this topic?

Professor, I am thinking about your extension of surface area and volume... I will try to find or put together some sort of activity that addresses this question, and leaves the door open for more exploration. I think that the 3D software packages that I am looking into may be helpful in this regard.