Message from NCSM President, Paul Gray




April and May, at least in the parts of the United States and Canada between the Rocky and Appalachian Mountains, are severe thunderstorm season. These storms don’t just happen. The entire reason we have weather in any part of the globe is fairly straightforward: because the Earth is round and the angle of sunlight is different at different latitudes, the equator is warm and the poles are cold. Our amazing atmosphere doesn’t like that. At all. So, it uses fluid dynamics to even out that temperature difference. Warm air moves toward the poles and cold air moves toward the equator. Because the Earth turns, that creates some spin in the air. Alas, we have weather.


Nature prefers balance. Come to find out, so do mathematics educators and leaders.


We often couch debates about education and instructional strategies in binary terms of “either/or.” Conceptual understanding versus procedural fluency. Use a calculator or don’t. Direct instruction versus inquiry-based instruction.


Here’s the unpleasant reality. In any of these dichotomies, we actually need both ends in order to successfully teach and learn meaningful mathematics.


Let’s talk for a minute about teaching concepts and skills. Students need strong understandings of both. That’s not just me talking; the National Research Council identified conceptual understanding and procedural fluency as two of the strands of mathematical proficiency in their 2001 report, Adding It Up. One of the most effective ways to teach a concept is inquiry-based instruction where students explore the mathematics a little, notice and wonder about what they are doing, and then observe some patterns and trends.


For example, students often struggle with the question, “what is area?” Conceptually speaking, the area of a figure is the amount of flat space it covers. To help students understand that, you can arrange color tiles in rectangles and have them count the tiles. If you ask students to record the dimensions of the rectangle next to the area and they have a familiarity with some multiplication facts, they’ll notice that the number of tiles is equal to the product of the length and width of the rectangle. Every time. Hence, we can use a formula, A = lw, to calculate the area of a rectangle every time. That’s the conceptual understanding of area and the area formula.


Now, here’s where the skills come in. Using that formula to determine the area of a variety of rectangles is important skills practice. Students need to have that skills practice. But not every student needs the same number of practice problems to sharpen the skills. But don’t skip the skills practice or you’ll shortchange your students.


When I’m learning a new concept, I need to explore it. I need some time to kick it around, maybe consider some examples and non-examples. I’ll formulate some ideas and check resources like online sources or experts I trust and see how valid those ideas are. I’ll take their feedback and use it to refine the ideas. That’s an effective way for me to learn a new concept. Dollars to donuts, it works for you and for your students as well. There is a legitimate place for inquiry-based learning, but it can’t be the only instructional strategy that a teacher uses.


When I’m learning a new skill, say how to use a chainsaw, I don’t want to explore that. I want someone who knows what they’re doing to show me. Then guide me. Then watch me while I do it and provide corrective feedback as needed. There is a legitimate place for guided practice, or the much-maligned and often over-emphasized “I do-We do-You do” model. But it can’t be the only instructional strategy that a teacher uses.


Effective teachers know when to use what approach. Effective leaders help teachers build that practical knowledge. It’s all about balance. Like Mother Nature uses balance to even out temperature differences, teachers use balance to even out students’ mathematical understandings.


Y’all be careful. We’ll touch base again in June!



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