Esther said: I read this posting with interest and wondered what would be a similar debate in numeracy instruction? Skills vs. Context? Skills vs. Conceptual knowledge? I learned recently that in the UK the term "numeracy" may be being phased out in favor of "functional mathemematics." Is teaching numeracy very different from teaching literacy? read this posting with interest and wondered what would be a similar debate in numeracy instruction? Skills vs. Context? Skills vs. Conceptual knowledge? I learned recently that in the UK the term "numeracy" may be being phased out in favor of "functional mathemematics." Is teaching numeracy very different from teaching literacy? What would be similar research findings in the field of numeracy instruction?
And then I said: I realized there are some questions embedded in the above commentary which are older than the "nature vs. nurture" debate: it just seems to go in cycles, and get's re-discovered. Does (can?) a label, a title make a difference? If I cll it skills or knowledge, where's the line and who decides? I can add 2 + 2 as a skill but once I've done it, is it a skill or knowledge?
SO... I'd like to note that I will be addressing this somewhat in a presentation this summer at the Adult's Learning Math conference in Ireland. But, my approach is simply to hold a mirror to what students think and do. It's not my level of competence as a mathematician or my aggregate experience as an instructor that I believe to be the critical factors. I believe it's what students bring to the classroom in the form of strategies of thinking, observing, participating, mathemeizing and understanding their role in the classroom. What I principally need to do is foster an environment where not only can they learn "skills" and "knowledges" but also they can learn to participate, interact and verify how to examine questions for validity. Rather than my only teaching the time tested and true methodologies for solving certain types of problems, I need to also teach how to identify what questions to listen to and how to help students develop and generalize what they already know in forms which might be unfamiliar to me as a person and as an instructor.
Rather than talking about it in excruciating detail and risk numbing your sensibilities, let me simply state an example.
We were addressing the type of problem called mixture problems: an example (from a 1950-something text) is - a grocer wishes to obtain 40 pounds of coffee worth 44 cents a pound by mixing 48 cent coffee and 32 cent coffee. How many pounds of each should he use? I have yet to see a textbook or a chain of thinking by an instructor which doesn't define "x" as the amount of 48 cent coffee (or 32 cent) and "40-x" as the amount of 32 cent ( or 48 cent) coffee. This is done because the total amount of coffee in the final mixture is 40 pounds. As we discussed various strategies for approaching this problem and worked through several concepts, a student asked "Isn't the important thing that the amount of 48 cent and 32 cent coffee add to 40 pounds?" to which I answered yes, and the student's follow-up was "well, can we just use "20 + x" and "20 - x" - "that adds to 40"? Well, can we? The point of the story is that this student had an idea and in my view it was critical to pursue it because it was a student's question, not mine, and letting all students see that questions are worthy of examination is, in my view in the classroom, more critical than the skills or knowledge I might teach. Quite honestly, I had never considered the students perspective on the problem, but the student was correct and it engendered a cascade of questions in this and subsequent classes. Did I miss an opportunity to teach some more skills and knowlege by not sticking to the curriculum outline and the lesson plan? Yes, but it was more an opportunity to demonstrate not only the power of questioning but the importance of refining questions in order to answer inquiries that might lead to interesting and powerful strategies ... later, mark