Article Discussion #1

A Model for Understanding Understanding in Mathematics by Edward J. Davis

This was an article about the process of understanding mathematical ideas and content in the subject of math. The definition of understanding can vary greatly from person to person. In addition, the act of understanding is often seen as a process. It is very possible for someone to just partially understand something, instead of not understanding at all or completely understanding. The whole idea of understanding in mathematics is a very important topic for teachers to be concerned with. We must constantly try to assess our students' understanding of the material that is being presented to them. This article talked a lot about "moves", or the way that a teacher actually teaches mathematics. Extensive research has shown that "moves" contribute greatly to student understanding. Moreover, it has been found that teachers behave differently when teaching different mathematical content. The article discussed that teachers tend to instruct students differently when teaching mathematical concepts, like place value, than when teaching mathematical procedures, like long division. Additionally, the article touched on several areas of understanding in mathematics. Within each area, two distinct levels exist. Level 1 are the more basic elements of understanding, whereas level 2 are the higher-order elements of understanding. These areas include understanding mathematical concepts, understanding mathematical generalization, understanding mathematical procedures, and understanding number facts.

Gauging student understanding is a critical component of assessment in the classroom, and this article really dived into the logistics of understanding. I never realized how detailed and specific understanding really is. It was interesting to read about the various areas of understanding and the individual levels underneath each area. The author made sure to note that teachers should not be under the impression that students must achieve all aspects of Level 1 before moving onto Level 2. This is not realistic, as some students may be achieving parts of both levels simultaneously. In addition, the article discussed some key suggestions for creating "moves" while teaching mathematics. First, for younger students, it is important to begin with physical representations, move to picture representations, and finally, utilize symbolic representations of mathematics. Teachers should also emphasize to students the importance of continually utilizing physical representations and pictures when explaining their thinking in math. Correct answers are no longer sufficient enough, and students need to descriptively explain their thinking.

Discussion Questions:
- Why is understanding such an important element to take into consideration when teaching mathematics?

- Understanding is described as being a process. Why might this be? Does a student either not understand something or completely understand something, or is there room in between?

References:
Davis, E.J. (2006). A model for understanding understanding in mathematics. Mathematics Teaching in the Middle School. 12(4). 190-197.