Sunday, February 3, 2008
Discussing math concepts
I have always been fascinated by listening to young children explain concepts of regrouping and trying to grasp their understanding of our base ten system. Our readings, the video clips, and in-class discussion have all underscored the importance of the discussion of mathematical processes for me. I feel that way too much emphasis is put upon the omnipotent one and only correct answer, as opposed to the process itself. I feel that it is through our understanding of a student’s processes and conceptual knowledge that we can gain insight into misunderstanding and/or mis-connected ‘discrete’ concepts that underlie a student’s poor understanding of a big idea or concept. Hundreds of worksheet problems cannot tap into a conceptual misunderstanding or lack of understanding if the ‘right answers’ are all that are asked for by the assessment tool. Through conversation and dialogue, we negotiate our mathematical understanding and question the reasons why we need to find efficient problem-solving methods and tools. The process of sharing and comparing methods and ideas is the journey to a deeper, more integral understanding of a technique or concept. This personal realization leads me to question why we do not spend more time on this aspect of teaching elementary mathematics. By understanding exactly what our students’ conceptual understanding is based upon, we can then truly connect to their prior knowledge and build cognitive networks of mathematical meaning that make sense to them in a real-world context. Knowing how to calculate a correct answer using a traditional algorithm is not an accurate predictor of a student’s ability to apply a mathematical concept in their daily life. I wonder why we do not explore other number base systems and concepts at the elementary level in order to develop the ‘template’ of numerical base systems. It seems as though base systems are not explored until middle school and algebra-level coursework. Why do we wait so long to play with the concept of bases and pattern repetition?
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Your question about why we tend to wait to introduce kids to different bases is very interesting. I can't say that I know the answer but I suspect it has a lot to do with curriculum trends that emphasize learning the base ten system because that is what our culture uses. Babylonians used a base 60 system (evident in our time measures) so I'd imagine that they'd be more likely to focus on base 60 in their schools. But your question is bigger than that--why don't we show them other systems? I'd again guess that we fear that if we give time to other systems we'd be 'wasting' time on 'non-important' things. Notice my use of quotes! Your question relates to a specific field of study--curriculum theory. It tends to focus on issues of why we teach what we teach. Maybe you can do some research? I suspect though that it has to do with a sense of relative importance in our culture coupled with concerns over not having a lot of time. Interesting...
A major critique of our American elementary curriculum revolves around the crux of time versus topic, or more appropriately, breadth versus depth. I agree that culture determines curricular priorities, and unfortunately for us, it is politicians dictating educational policy priorities and an outdated system of property tax-based funding that dictates financial priorities for our public schools. Ignoring reality, and returning to a more universal perspective of 'WHY' we don't explore other base number systems at the K-8 level: I find myself lingering here in establishing a valid line of inquiry interview on place value with one of my second grade students. I agree that it appears that writing or reading three-digit numbers and explaining their place value is independent of showing the value of each number's place (or position), but I hesitate in asking a child who is new to our culture (and way of thinking) to test their emergent ideas on models that only represent our base ten system. I wonder if their culture has multiple lenses of place value so that demonstrating their reflective thinking with a model that I present will be an accurate and valid representation of their conceptual and procedural knowledge. Since even skillful use of a procedure will not help develop the related conceptual knowledge, how can I expect to attach meaning to a procedure that I am basing on on a conceptual model that is foreign to a child's mathematical and contextual culture? This is the struggle that confronts me in my design of of an effective interview tool. I am searching for a line of inquiry that taps into early number concept "webs" that will realistically communicate relational understanding of our base ten system in such a way that I will be able to "see" how this child's schema is assimilating and accommodating new information. It is only through this indepth "sight" that I can truly begin to to teach... in a constructionist way that can 'de-construct' current conceptual knowledge and allow for altering current schema to accommodate new understanding. This also happens to be the job description of one of our culture's lowest-paid and lowest-viewed professions.
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