How much of a class should be conceptual and how much should be procedural?

December 20, 2011 at 11:24 pm 3 comments

This question came up at a professional development session I led last Saturday. I tried to answer it by sharing my daily class routine and giving some examples of how I’ve taught for conceptual understanding. But as I left the session, I found myself wanting to consider the question in greater detail.

Ratio is a hot topic in pedagogy these days. Teachers think about the ratio of student talk to teacher talk. They think about the ratio of lower-order thinking questions to higher-order thinking questions. Ratio is important because it ensures differentiated instruction.

But as I tried to figure out an appropriate ratio of teaching for conceptual to teaching for procedural understanding, I realized that it couldn’t be done. Teaching for conceptual understanding is like being a vegetarian: Vegetarians don’t eat a high veggie to meat ratio; they stop eating meat. Not only that, most vegetarians stop eating meat for a reason, and that reason impacts many of their choices, not just what they choose to eat.

The same commitment applies to teaching for conceptual understanding. If you believe that math is something that can and should be understood by all students then you give up teaching rote procedure. Conceptual understanding becomes the foundation of everything you do and all the choices you make. Sure, just like vegetarians might have a little fish or egg now and then, the conceptual teacher might have to do a little “drill and kill.” But the basic belief about math making sense is always present.

This doesn’t mean that a teacher who is new to teaching for conceptual understanding can’t shift approach gradually. It means that teaching for conceptual understanding requires more than a five minute addition to the class routine. It requires a shift in mindset. To get yourself in that mindset, let me share a few things that I often see teachers who are committed to conceptual understanding do:

1. Introduce every topic in a way that relies on student sense-making.

“See how this looks like an array? That means it’s multiplication!”

2. Correct every misconception by reminding the students “why.”           

“You forgot the zero. Remember how this 3 is really 3 tens? When you multiply it, it’s still going to be tens.”

3. Encourage students to ask questions and make connections.

It’s almost the New Year. Lots of people make resolutions about their diet. Maybe this year it’s time to go on a different kind of diet.

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Replicable Practice Oh, the Places You’ll Go!

3 Comments Add your own

  • 1. Peter Price  |  January 28, 2012 at 10:00 am

    I couldn’t agree more!
    I read a lot of math teacher’s blogs, and enjoy reading them all. But I rarely agree so completely with a post as I do with this one. Seriously, I thought it reads the way I speak when teaching preservice teachers to teach math.
    Your final three markers for teaching for conceptual understanding are brilliant!
    Thank you!

    Reply
  • 2. Wil  |  February 1, 2012 at 11:23 am

    Perhaps my biggest concern about “shifting” is the palatability to those who either don’t think it’s necessary, to whom it hasn’t occurred, or who don’t like change. I’m guessing this is going to be admin folks, traditionally minded teachers, and especially legislatures fearful of deviating from the status quo.

    How do we move that direction in the classroom, in the standardized testing, and ultimately in the foundation of the culture?

    A post I recently wrote of similar content: http://mathfour.com/research/performance-understanding

    Thanks for a concept-supporting post!

    Reply
  • 3. Whitecorp  |  February 7, 2012 at 1:55 am

    I am never a sucker for procedural teaching, simply because it cultivates the complacency of rote learning (where students eventually do not ask “why”). But still at times specific maths formulas (especially the more complex mores) have to be communicated to them kids without confusing them with the intense background knowledge from which these formulas are derived.

    Thanks for the nice article btw. Peace.

    Reply

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