First of all I want to thank you for work in bringing us together through the Math Coach Chronicle. I am looking forward to watching it grow. I have shared it with my colleagues and am encouraging them to participate in that growth.

This year I have made a commitment to myself to improve the way I teach math. I have been reading a lot about ‘conceptual’ teaching and it makes sense to me. I am beginning to see the value in having my students understand the ideas behind the math I’m teaching, rather than just memorizing rules like I did in school.

The problem is… I don’t know what happens in the period between finding inspiration in a great activity or strategy and launching it in my classroom, but the end result usually falls sadly short of my expectations, and my well-rehearsed lesson, complete with ‘conceptual questions’, is met with blank stares. Invariably, after a few attempts to adjust my offering, I find myself going back to what I have always done… showing them how. My students show visible signs of relief and get to the task of applying my instructions, and everything returns to ‘normal’.

I am beginning to lose the courage of my conviction about this approach to teaching math and am left wondering if I am doing more harm than good. Are there others out there having similar experiences? Help!

Good Intentions in Pittsburgh

Dear Good Intentions,

Congratulations on your commitment to enrich your students’ mathematical experience… hats off to you!

Let me response to the end of your letter first. You are absolutely not alone in what you are describing here. You haven’t said what grade level you are teaching but I will guess that it is upper elementary (3-6).

The shift to a ‘conceptual’ approach is nothing more than teaching math for meaning. That sounds easy enough but it is a significant stretch from what we have traditionally done and the transition isn’t as simple as deciding to do it. I have seen that ‘thousand-yard stare’ coming back from students who are being asked to think past the symbols to the underlying idea. And why wouldn’t they? From learning to count, to memorizing number sequences (7+5=12) to following prescriptive procedures (steps in an algorithm), the majority of students have focused on following and applying instructions to arrive at the ‘answer’, commonly perceived as the whole point of the exercise. Suddenly they are being asked to think, and they aren’t sure how to do that. But they can and, with persistence, they will!

A student’s apparent confusion when asked to think conceptually is likely the result of a variety of factors; here are a few:

a) ‘You can’t get there from here’. Foundational understandings may be missing from a previous grade level. The only cure for that is to reach back and pick them up. That’s easier said than done of course, but very little is gained from trying to forge ahead without them. Following is a typical exchange I’ve had on numerous occasions on this topic.

T: “My students just aren’t getting it and I’m falling behind with the curriculum.”

M: “They are likely missing some foundational ideas from a previous year. You may need to identify those and reach back to pick them up.”

T: “How can I do that? I’ll never cover my curriculum in time!”

M: “Are you covering it now?”

(This last question is usually followed with a bit of reflective silence.)

b) Shift in purpose. Students may need some time to adjust to a shift in the purpose of the math, from ‘getting the right answer’ to ‘understanding the idea behind the numbers’ and how it applies to real situations. Many students experience some frustration when direct procedural instruction is even partially withdrawn and they are asked to apply intuition to come to a conclusion.

To help students bridge this gap, try putting the pencils and paper away and engaging them in an investigation where they use physical materials to explore the idea and draw conclusions. Following is an example of such an investigation, from “I Get It!” Math, 4th level:

You may choose to ask students to record their results and compare their findings. You will be pleasantly surprised at how much math talk can happen as a natural extension of an engaging investigation!
Hint: Always complete the investigation yourself before giving it to your students. Get the materials out and move them around. You might be amazed at what you discover in the process!

c) What you say makes all the difference. Make a conscious effort to use natural, make-sense language to initially replace formal terms and rote rules. Students will be much more likely to engage in conversation involving language that is familiar to them. That is not to say that correct mathematical terminology is not important, but rather that this effort will help to break down existing barriers to mathematical discussion within your classroom. Once understanding is in place, it is easy enough to reintroduce those terms as a way to represent and communicate the conclusions they have drawn.

Most of all, I encourage you to continue along your path. At times it may be difficult to believe, but if you persist I can promise you that you will make that difference! Persist, insist but never desist!

Maggie

Whatever you have on your mind, chances are you are not alone!
We are looking forward to hearing from you. Email Dear Maggie letters to contributions@igetitmath.com

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