Entry #2

"Relational Understanding and Instrumental Understanding" by Richard R. Skemp is an article discussing two different types of mathematics teaching. Skemp explains that relational understanding is what most of us would consider to be a true understanding. An understanding where the process of what to do, and the reason why it is to be done that way, are clearly understood. Skemp describes instrumental understanding as being a lesser understanding of the concept. He states that instrumental understanding is not knowing why you are doing something but just being able to apply a formula. The two might be thought of as follows, relational is understanding why you are doing what you are, and instrumental is being able to "plug and chug" when given a formula. Skemp presents three reasons why instrumental understanding might be advantageous and consequently some disadvantages to relational understanding. The first reason is that instrumental mathematics can be much simpler to understand than relational. He says that understanding some concepts relationally can be too complicated for pupils when it is initially presented to them. When a concept is simply too complicated to be taught relationally, instrumental instruction is beneficial. Another reason Skemp proposes that instrumental mathematics can be better is that it allows students to reach the correct answer quickly, rather than having to work their way through it. The final reason that instrumental can be beneficial is that it provides what we would call "instant gratification." Students get a page of right answers, feel successful and then are motivated to continue. Skemp states many times in his article that he is biased towards relationally understanding, he lists four reasons why relational mathematics is superior to instrumental mathematics. The first point he raises is that a relational understanding of a concept allows the pupil to apply the process to more abstract ideas and can thus succeed at a wider range of problems, whereas the student who has only an instrumental understanding would only receive frustration when he attempts a new type of problem. He then tells us that even though instrumental understanding is easier to learn than relational it is not easier to remember. Relational becomes superior in Skemp's opinion because there are no rules to be memorized just concepts to understand and then the rules can be derived. When talking about instrumental understanding Skemp noted that it could be beneficial because it provides instant gratification, on a similar note, Skemp tells us that even though it may take longer to do a problem with relational mathematics, learning it relationally acts as the goal itself and the need for outside motivation is reduced. His final reason for superiority is closely related to his previous argument about motivation. When the learning itself becomes the goal, the motivation is applied to learning more and more and even branching out of the limits of the classroom, leading to a more dedicated and interested pupil. When it comes to the actual application of these two types of mathematics, it is fairly common that mathematics will be taught from a strictly instrumental stance. Instructors will not even attempt to give an answer to the question of why something is performed a certain way. Relational understanding on the other hand usually encompass mostly concepts and understanding but it is likely that formulas will be introduced and used which come from instrumental mathematics. Skemp is strongly in favor of relational mathematics but is aware of the system and the difficulties that prevent relational instruction from occuring.

Entry #1

What is mathematics?

Mathematics is the study of patterns that occur in the world and the relationships of objects in our world. Mathematics is also the foundation of how we study science.

How do I learn mathematics best?

I learn mathematics best by having a brief lecture and working through examples in class. I think that I learn best this way because it gives me an opportunity to be introduced to the topic and then requires me to demonstrate what I have just learned. I believe that being required to demonstrate what I was recently taught helps to solidify what I need to know.

How will my students learn mathematics best?

I think that my students will learn math best by having an environment in which they participate heavily in the lecture. What I mean by this is working through the lecture and allowing them to help figure out what to do next. Even if the path that they ask to take is wrong, I think that the direction should be pursued in order to demonstrate why it was wrong. We learn best from our mistakes and if all my students see is me lecturing perfectly what they are to do, I don't think that they will learn as well what they are supposed to do.

What are some of the current practices in school mathematics classrooms that promote students' learning of mathematics?

In many of the mathematics classes I took in High School I was expected to demonstrate homework problems on the white board. That was part of our mathematics department's policy. This allowed me, and every student to not only gain confidence in our work but it also gave us an opportunity to teach each other instead of just having lectures. I think that that is a very effective part of a mathematics classroom.

What are some of the current practices in school mathematics classrooms that are detrimental to students' learning of mathematics?

This is probably something that many people will disagree with. I feel like mathematics in the schools currently places too much of an emphasis on the need to use a calculator. Granted I understand the value and importance of understand how to use a calculator, I feel that teaching those skills has trumped the necessity of teaching students basic arithmetic. A calculator is definitely useful when multiplying a 5 digit number by a 3 digit number but it does not save time to need a calculator to multiply 6 by 8.