When we started designing concept-based curriculum, the folks in the humanities jumped on board pretty quickly. Our science teachers, especially earth science and biology, were game right away too. The procedure-heavy disciplines, though, were tougher.
Many foreign language teachers were used to assigning vocab lists and conjugation charts for students to memorize. Many math teachers would teach the same GRR (gradual release of responsibility) lesson every day: teacher models a procedure, students and teacher practice together, students practice on their own.
Needless to say, no one was hitting it out of the park with these methods. Our kids weren’t acing their math exams or speaking fluent Spanish. The teachers knew it wasn’t working. But, while most of our humanities teachers could remember learning and practicing their disciplines in conceptual ways — writing essays on Macbeth’s vision of power and justice, debating the visions of equality that underpin capitalism and communism — many of our math and language folks could not. When they learned math, it was naked procedure after naked procedure. When they learned language, it was memorize, memorize, memorize. At least middle and high school levels.
But here are a few things we know to be true:
- Most procedures are rooted in powerful and important conceptual understandings.
- When you teach procedures as a series of steps to follow, ignoring the underlying concepts, kids will be limited in their ability to use the procedure flexibly and accurately.
I recently came across a blog about math written by a Wisconsin elementary teacher. This woman is amazing, and she chronicles many of her ideas, projects, and lessons here on her blogsite.
One of the posts that caught my eye highlighted the problem of teaching procedures without building students’ conceptual understanding AND demonstrated ways to improve student understanding through conceptual inquiry. The goal for students was to be able to round numbers to the nearest 10 or 100.
Traditionally, many teachers present rounding as a series of rules and steps. Round 13 to the nearest ten? First, identify the number in the 10s place, then look at the number to its right. Is this number 5 or higher? If so, round the number in the 10s place UP. Is this number 4 or lower? Keep the number in the 10s place as is.
Most of us do this type of math without even thinking about it. We know that 13 is closer to 10 than 20, that 198 is closer to 200 than 100, etc. The concepts at play — our deep understanding of place value, our strong number sense, our knowledge of how rounding helps us estimate in our everyday lives — are not visible to us when we perform this procedure ourselves. So we often teach the procedure without attention to those underlying concepts.
But Jen, the author of my new favorite math blog, has some great ideas for building the concept of rounding for young students. And as her own classroom data shows, nearly ALL kids became proficient in the procedure of rounding when she started with the concept!
What did she do? Instead of starting with rounding, she started with place value. The kids counted by 10s, counted by 100s, made number lines that showed 10s and 100s and discussed what “nearest 10” or “nearest 100” meant. If she asked them to round 13 to the nearest 10, the kids took the time to create a number line that counted by 10s. Then they plotted 13 on the line and counted to see which 10 it was closest to. When she asked them to round 162 to the nearest 100, they made a number line that counted by 100s and did the same.
image credit: beyondtraditionalmath.wordpress.com
Eventually, kids realized that it was tough to round 5 to the nearest 10, or 50 to the nearest 100. Only then did Jen teach kids that the math community has made a rule that we round 5s up — not because they’re actually closer to the next highest 10, 100, etc, but because we need a consistent way to do it.
Did this take more time? Absolutely! Does it seem like you could teach rounding faster if you taught kids the shortcut that doesn’t require them to draw a freaking number line every time they round a number? Sure. But will kids actually master the concept and procedure of rounding that way? Our experience tells us that some students will get it, but many — maybe most — will not retain this procedure or be able to use it flexibly (what about rounding to the nearest 1000, 10,000?) to solve problems.
So, take some time as you plan how you’ll teach your next procedure. What is actually going on behind the steps you want students to follow? How can you make those concepts visible and central to the lesson you teach?