## Thursday, January 14, 2010

### Two Types of Understanding

Students can learn mathematics in one of two ways, namely relational understanding and instrumental understanding. Instrumental understanding simply teaches the students how to find the correct answer, no more no less. Relational understanding also teaches the student how to find the correct answer, but it goes beyond that and teaches the student why that is the way to find the correct answer. Thus we see they are both effective in getting the student to pass the class, or maybe just a particular assignment, but in the long run relational understanding is more beneficial to the student's learning. The positive sides to instrumental understanding is that the results are seen quickly, and the rules are usually easier to understand than are the concepts behind those rules. Because of these benefits, students are able to find answers more quickly. However, with relational understanding students are able to adapt to new tasks more easily. They can see a new subject build off a previous one. Though relational concepts are not easier to understand, they are easier to remember because the student learns the big picture, and is then able to apply that bigger idea to smaller scenarios. It is easy to see issues with instrumental understanding; the students forget more easily, they do not know how to evolve what formulas they learned, etc. However there are several disadvantages to relational understanding. It takes a long time to achieve relational understanding, sometimes the level of difficulty exceeds the students' abilities, and often it is hard for teachers to teach relational understanding when so many of their colleagues are not doing so. However, it is still a good procedure to help students understand the fundamental ideas behind the method for solving a problem. Instrumental understanding is embedded within relational understanding, we simply need to broaden what we teach so that students can truly understand what they learn.

Haley

1. Wow, you do a wonderful job of capturing so many of Skemp's arguments about the pros and cons of the two types of understanding! I can tell you really understand this part of the paper.

The first part of the paper where you define the two types of understanding was confusing to me. I always think of these two terms as describing what someone knows about mathematics. The way that these terms are used in the beginning, however, seems to indicate that they are ways of teaching mathematics. Which way do you think Skemp uses the terms?

2. You did a great job including the advantages and disadvantages of both methods! I especially like how you said relational understanding is "harder to understand but easier to remember." Well said!
I think that for somethings maybe it is ok to teach instrumental mathematics,maybe just to give some variety. I guess I just think that leaning too much either way would be bad. What do you think?

3. First of all, I really like how you went into great detail about the advantages and disadvantages of instrumental and relational understanding. I think your most interesting point was when you mentioned the difficulty for teachers to make that jump from the current teaching techniques, which are more focused on instrumental learning, to a new innovative way of teaching focused on relational understanding. I liked how this statement sort of implied that the newer teachers, and us who are studying to become teachers, need to step up to the plate and take on this responsibility in order to improve mathematical learning for our students.

The only statement of yours that gave me some pause was that you mentioned that the level of difficulty of relational understanding may exceed the student's ability. I think this may be a bit of a misconception. Although Skemp explains how developing relational understanding is at first more difficult, I believe that this is partly due to the fact that we have grown accustomed to an instrumental way of thinking when it comes to mathematics. I think that if students began mathematics viewing it from a relational understanding perspective they would be able to catch on relatively easily because they wouldn't know any other way. This relates to the idea that you can't teach an old dog new tricks, because change is hard. However, if there is nothing to change maybe it would be easier then we might think.

4. By reading your summary of Skemp's article i felt like you completely hit every important part of his explantions when it comes to how we understand as students. The only difficulty I had while reading your blog was the length. I felt like a summary should be more precise and to the point but I still enjoyed reading it overall.