Interesting article about encouraging the expansion of thought in math. Click on this link to read all about it.
My feelings about this article:
The article states, "If math were music, mastering the basic concepts would be like learning scales and leading students through discussions of open problems would be like playing songs".
If this means that students should memorize the basic operations (+, -, /, x facts) and be able to use this information in a variety of ways to solve problems, then I agree with you. I still think an effective math program needs a balance of traditional and contemporary methods to work. I do not support math programs that spend little or no time on the student's acquisition of automaticity of math facts and spend the majority of time on conjecture alone.
It's obvious that , for a student to arrive at an answer to the open question you posed (which I think is a very good example of an open problem)without undue frustration, that child should have a good "handle" on the basic facts.
I'm not saying to make the problem too easy or not a challenge at all. But we must be sure that our students have the tools (basic skills) to be able to even APPROACH the problem with some confidence. Yes, students can learn from their mistakes, but their patience and tolerance is limited (adults too-we're all human)and there needs to be some kind of closure and satisfaction. Children, in general, like a conclusion. They don't like the "unknown" to drag on and on. So, discovery and exploration of numbers is fine, but it shouldn't continue be encouraged beyond the attention span and should meet the "need to know" in a timely manner.
What do YOU think of this article? What do you think of my comment? I would love to hear your take on this!