It is easy for students to get fixated on the "how" of mathematics. As much as we try to emphasize thinking and discussion and as often as we praise clear explanations and celebrate insights, there is still a lot of pressure from parents and siblings and tutors and society at large to focus on getting the answer. And let's be honest, for many students (especially the middle schoolers I see each day), any schoolwork is time taken away from the truly important stuff. So just tell them what they need to do so they can be done and move on.
In trying to get them to value both the process of exploring and the connections that come from it, I realized that they might not actually see the difference between rote procedures and conceptual understanding the way I do. After all, from a student's point of view, if she knows how to do a problem, then she understands it. So this year I put a poster up in my room that says Little Steps / Big Ideas. When we started using fraction operations at the beginning of the year, and we re-examined the equivalence of multiplying by 1/3 and dividing by 3, or why we can multiply just the numerator in a problem like 3 · 4/5, we were able to talk about the Big Ideas behind the Little Steps and the difference between the two.
Area formulas are a perfect opportunity to highlight the difference between the memorized rules and the geometric principles behind them. Once students appreciate the difference, then they can see the value of assessment items that look for evidence of understanding the Big Ideas. Someone once told me the students know what teachers value by what they test and grade. Is that Big Idea going to be on the test? If you give tests, it should be.
I like to remind them of the long history of development in mathematics; that the procedures we have today weren't always obvious to even full-time mathematicians, and took a long time to evolve. Anytime your students' explorations culminate in a procedure or a "shortcut" that makes calculations quicker and easier, you can bring into relief the algorithm that makes the mathematics handy, and the big ideas that justify it and connect it to the rest of the field of mathematics.
In trying to get them to value both the process of exploring and the connections that come from it, I realized that they might not actually see the difference between rote procedures and conceptual understanding the way I do. After all, from a student's point of view, if she knows how to do a problem, then she understands it. So this year I put a poster up in my room that says Little Steps / Big Ideas. When we started using fraction operations at the beginning of the year, and we re-examined the equivalence of multiplying by 1/3 and dividing by 3, or why we can multiply just the numerator in a problem like 3 · 4/5, we were able to talk about the Big Ideas behind the Little Steps and the difference between the two.
Area formulas are a perfect opportunity to highlight the difference between the memorized rules and the geometric principles behind them. Once students appreciate the difference, then they can see the value of assessment items that look for evidence of understanding the Big Ideas. Someone once told me the students know what teachers value by what they test and grade. Is that Big Idea going to be on the test? If you give tests, it should be.
I like to remind them of the long history of development in mathematics; that the procedures we have today weren't always obvious to even full-time mathematicians, and took a long time to evolve. Anytime your students' explorations culminate in a procedure or a "shortcut" that makes calculations quicker and easier, you can bring into relief the algorithm that makes the mathematics handy, and the big ideas that justify it and connect it to the rest of the field of mathematics.
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