Portfolio 2: Thoughts on Base Ten
February 7, 2008 by leynafaye
When I learned about the base ten system in my math classes here, I was completely amazed. It had never occurred to me just how arbitrary a numerical symbol could be. For the first time, I realized that math was about relationships and “negative space,” so to speak. Most people think of math as being absolute–of having a definite, uncompromising answer. In reality, this is not entirely true: the relationship is definite, the “number” is not.
Surprisingly, this ties in with Van de Walle’s discussion on the two types of mathematical knowledge. Most of my (and I’m assuming many others as well) experience with math has been purely in the form of what he calls procedural knowledge. Procedural knowledge can be understood in terms of rote learning: carrying out a specific set of rules or procedures to achieve a given task. This is why many instructors rely on nothing more than worksheets and repetition. They are trying to drill the procedure in through rote repetition, which is effective in this type of situation. Unfortunately, this provides no opportunities for the child to really understand the concepts of mathematics so that they can begin to build and modify schema.
Conceptual knowledge, on the other hand, is an understanding of the relationships between numbers, symbols, or ideas. This is a much more difficult type of understanding to come by, as it undergoes an endless journey of adaptation, modification, and application. Conceptual knowledge is never completely the same. A rectangle can represent a whole, a half, a third, a quarter, or any imaginable value depending on its context. Likewise, numbers exist within a specific context: a fairly universal context, but still a context. Numbers exist within a base ten framework. This means that all rebundling, regrouping, and breaking apart occurs with the assumption that the “whole” is a unit of ten. One square into ten rods. One rod into ten blocks.
With this in mind it no longer becomes surprising that I only understood the absolutes of mathematics. Many teachers spend so much time on procedure that there is no time for conceptualization. However, in order for our students to become truly fluent in the world’s “universal language,” it is imperative that they learn to manipulate the relationships between the absolutes and recognize and make use of the negative space that exists. The issue of negative space could lead into some fascinating discussions and lessons for upper-el students!
At this point, I am completely on-board and excited about the richness that conceptualized understanding could bring to mathematics. However, I am unsure as to how to weigh this against procedural knowledge, which must still be covered. I am nervous about teaching math for precisely this reason. I want my children to understand and really love math in a way that I never could; unfortunately, there are certain standardized checkpoints that I must meet in order for them to be successful in future classrooms. With such limited time and resources, how can I be sure that my children have opportunities to build conceptualizations, frameworks, and schema? Also, what is the best means of instruction? Didactic, inquiry-based, or idea-based? Why does this model have value over the other two?
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You raise so many issues I’m not sure where to begin. I love your ideas–we’ll start there! One that I find interesting is you talk of math as so contextualized and the example of the rectangle is beautiful, but you also call math a ‘universal language.’ There is a movement in math called ethnomathematics that tries to show how “universal language” is not so universal. Granted. lots of people do math activities in their lives all over the world but their point is to show how contextualized it is within certain communities, meaning that each group “thinks and talks” math differently. The movement is not to suggest that some groups can’t do Western math (what you and I likely think of when we think of math), but that when we assume the world thinks of math like we do, we are behaving like typically close-minded, imperialistic Westerners! Did you know math theory and education could be so fun and so political? The classes offered to undergrads (sadly) only touch on the big theories of reforming math education.
But to your points about teaching in “real” classrooms. Yes, there will be constraints such as tests and textbooks and parents. I want you to be able to make the best decisions you can given the constraints you end up facing. There are so many variables in each school site that it is impossible to account for them all. In general I say strive for understanding. The technical/procedural details can work themselves out usually (considering how much time will likely be spent on that at home, in other classrooms, etc.) Check out nctm.org’s pages for comments related to what you ask about.