After sitting and pondering what I was going to write this week, I kept coming back to this quote. I guess we could say that by the standards of the GCSE’s the two approaches are not at different ends of a spectrum with regards to mathematical effectiveness. Both schools had relatively similar exam marks. However, we must keep in mind in our society, mathematical effectiveness is measured in terms of exam marks and therefore the different approaches are comparable in that regard.
However, I think Boaler sheds important light on the fact that the different approaches to instruction supports the development of different forms of knowledge. Amber Hill students were taught in a traditional manner and their knowledge was strictly procedural. Students had “learned the teachers’ methods and rules without really understanding them” (p.122). Whereas, the students at Phoenix Park who had been taught in a reform setting had developed knowledge that was internal and all encompassing, “students learned to adapt and change methods to fit the demands of different situations” (p.125). Students at Phoenix Park had “developed a usable form of mathematics in response to their project work,” unlike Amber Hill students who seemed lost and disconnected from real world mathematics. Students at Amber Hill “reported that school mathematics was unrelated to the mathematics problem they encountered in their jobs and lives, and when they want to use methods many of them reported they were unable to apply them” (p.123). If there were no cues for students to follow they were incapable of continuing with the problem or they could not interpret the demands of the question correctly. Boaler refers to the Cognition and Technology Group at Vanderbilt study as “they report that problem-oriented approaches to learning help students view mathematical concepts as useful tools that they can use in different situations. More traditional approaches to learning cause students to view concepts “as difficult ends to be tolerated rather than as exciting inventions (tools) that allow a variety of problems to be solved” (p.123). From this one must question what type of knowledge is most valuable? Students with procedural knowledge but cannot apply what they know within their real lives? Or, students who may not know every procedure but have inherent mathematical reasoning available to them to navigate their way through a problem? Both sets of students cultivated different identities as learners and users of mathematics because of the approaches to instruction that was integrated within each school. The way each set of students “connected and interacted with mathematics and formed mathematical relations was different because of the practices in which they engaged in school and the effect of those practices on the mathematical identities students developed” (p.135). Students at Amber Hill “did not regard themselves as mathematical problem solvers. [Meaning] that when they met new situations, they abandoned the mathematics they had learned in school and were forced to rely on their own invented methods” (p.130). Students had a general opinion that mathematics was fixed and was only relevant in their mathematics classroom. These students were set in the novice stage of learning, given little opportunity to gain an understanding of the mathematics they were learning. This is different from students at Phoenix Park who “had engaged as mathematical problem solvers, and they were willing to consider any mathematical situation and think about what was involved” (p.134). Students had “developed more effective forms of knowledge and more productive identities as mathematical learners” (p.134). Students saw no boundaries with regards to mathematics, giving them more access to their school-learned mathematics in their real lives. The students in our classrooms will be leading us into next generation; do we want students taught in traditional settings who are “received knowers-expecting to accept knowledge from authoritative sources without contributing to knowledge or making connections across areas”? (p.122). Or do we want students taught in progressive classrooms who are empowered by mathematics, who identify themselves as mathematical problems solvers, and who have “developed a predisposition to think about and use mathematics in new and different situations”? (p.126).
Just some thoughts!References:
Boaler, J. (2002). Experiencing School Mathematics. New York: Routledge.
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