In my high school days, I was good at math… well… I got good marks in math. Mathematics came really easy to me because I had a good memory. (Age is starting to have something to say about that.) Teachers would show me how to “do it” and I would remember how to “do it”. I didn’t have to study for a test until I reached my OAC year. (OAC? – I am old.) I never thought about teaching math until late in the university application process because it was just something that came so natural to me.
My first math teaching assignment however, was a different story. The first part of most lessons went really well – cruising like a locomotive on the track. Then came the “train wreck” – questions. “What do you mean you don’t understand – just do it like I did.” “But sir, how did you do that?” My answer, as I pointed to my chalk writing – “Like that”. I was finished once I taught it like I knew it. My only move was talking louder and slower, repeating what I had already said, tracing over what I had already written. Good grief!
I was procedurally fluent in mathematics but what I spent many late nights doing in my first few years of teaching was learning conceptual understanding in mathematics. I needed to be able to show another way or show students the “why” of the math. What’s becoming more apparent to me these days is that a more intentional, concrete approach is needed in order to teach mathematics.
This school year I have assumed a different special assignment position than I had been doing the last 2 years. My assignment this year is that of the Secondary Numeracy Lead. This probably fits better with my previous 18 years in teaching mathematics but I’m also looking forward to applying the last 2 years of technology “coaching” experience to this assignment.
Intentional teaching is two-fold (at least). Intentional in that, we need to be able to anticipate where students will have difficulty so we can properly address those issues. Intentional teaching also means getting to know who our learners are and how they learn best. Without knowing this, we will just be spinning our wheels in trying to help them. Using concrete teaching methods helps students to conceptualize their understanding. Some students will have great difficulty simply remembering what circumference is. If we describe and show the circumference of a circle as being the crust on a pizza, that may help some students to visualize what circumference is. And, in the process, you’re not harming those students like me who don’t need these aids but it’s good for me too.
Here’s an example where just knowing how, may actually confuse a student.
Solve for x.
One method involves multiplying the numerator and the denominator of by 8. This results in x = 64.
This same teacher may also show students a method where they equate the cross products.
Recently, I watched a student draw diagonal lines on the proportion like they were going to equate the cross products but then multiply the numerator and denominator of by 6 because the denominator, 8 in the first ratio, times 6, equals the numerator, 48 in the second ratio. So, x = 36. (numerator, 6 in first ratio, times 6)
It’s pretty obvious that this student was mixing up the two methods. Helping this student by simply untangling their solution doesn’t help them for tomorrow or for next week’s test. They need to know the “why” of each of these methods so they don’t mix them up.