Math Manipulatives That Require No Manipulation

July 22, 2010

A lot of schools send reading lists home with students for the summer, which is fantastic! But does your school also provide them resources for maintaining or sharpening their math skills over the summer? If not, and you can still get in touch with students, I highly recommend Utah State University’s National Library of Virtual Manipulatives (NLVM).

All NLVM activities are free, and are organized by the five major strands (Numbers & Operations, Algebra, Geometry, Measurement, Data Analysis and Probability) within each of four grade level ranges (PreK-2, 3-5, 6-8, and 9-12). And there’s a good balance between activities that target traditional math skills and those that involve critical thinking. 

Best of all, when it comes to student engagement, there’s no need to resort to manipulation with these manipulatives. NLVM is so fun that my own children (10 and 8) choose it over other popular (and less educational) kid websites (NLVM has an electronic version of the logic game Mastermind that my kids are especially hooked on).

And whether or not you’re able to connect students with NLVM this summer, I encourage you to play around with it as you prepare for the fall, since it’s a great supplemental resource for multiple purposes: remediation, reinforcement, and enrichment (perfect for early finishers if you have computers in your room). You can even incorporate it into your direct instruction if you have an LCD projector.


“Math Teachers at Play” Blog Carnival

June 21, 2010

Math Teachers at Play is a blog carnival where each month a different blogger introduces and provides links to math-related posts from other bloggers. Last month’s carnival was hosted at math hombre and this month’s is at Ramblings of a Math Mom. Check them out and you’ll notice a couple of my posts, but more important (since you already know about my blog), you’ll discover other blogs with interesting and helpful ideas related to math instruction.

Must-Read for Math Teachers

June 11, 2010

Liping Ma’s Knowing and Teaching Elementary Mathematics (note: this link takes you to the 2nd edition at the publisher’s website; you can still purchase the 1st edition from other sources including Amazon) is a fantastic book for anyone who teaches math—elementary or secondary.

Briefly, Ma contrasts how math is taught in China with how it’s typically taught in the U.S. by asking teachers from both countries how they teach four topics: subtraction with regrouping, multi-digit multiplication, division by fractions, and the relationship between area and perimeter. Read this book and, at a minimum, you’re going to teach these four topics differently—and more effectively!—than you ever have. (Light bulbs especially went on for me when I read the subtraction and fraction chapters!). And to the extent you can generalize to your practice the ways in which Chinese teachers’ approaches are more effective than those of U.S. teachers, you’re going to be a better math teacher overall.

So if you’re looking to enhance your math instruction for next year, put Knowing and Teaching Elementary Mathematics at the top of your summer reading list.

Cancel Out “Cancel Out”

May 26, 2010

It’s important when demonstrating mathematical procedures that you stress the mathematics behind those procedures (i.e., the “why” behind the “what” or “how”), since failing to do so can cause or reinforce students’ misconceptions. Take, for example, multiplying fractions, as in the case of 3/5 x 5/8. Commonly—and correctly—teachers note that “the fives cancel out,” leaving 3/8 as the product.

The problem is that unless students understand why the fives cancel out, they see it as some mechanical (if not magical) step in the process. A common mistake, among others, I’ve seen students make as a result has been to also cancel out when adding or subtracting fractions—such that the sum of 3/5 and 5/8 would then equal the product of 3/5 and 5/8. To prevent this and other related errors, explain the mathematical basis for cancelling—i.e., you’re not really cancelling common factors, but rather dividing them to yield a quotient of 1 (represented as 1/1). Then show students (or better yet, have them discover themselves) that while this works when multiplying fractions, it doesn’t work when adding or subtracting them.

Is it ever ok to mutter “cancel out the 5s” as you work through a problem or just execute the cancellation without a word like your teachers may have done? Sure, once you’ve prevented or alleviated any misconceptions related to this. But until then, cancel out “cancel out,” and instead talk to students about the math behind it.

An Important Point About Teaching Decimals

April 17, 2010

There’s really no such number as “point three seven five.” Yet that’s how a lot of students say .375, and a big reason for this is that a lot of teachers say it that way too—me included until I realized this perpetuated students’ difficulties with decimals.

So from that point on, I stopped saying “point” and instead referred to decimal numbers correctly—“three-hundred seventy-five thousandths,” for example—and insisted students do so too. And not only did their grasp of decimal place value improve as a result, but so did their computational skills. Even better, they became much more proficient at something that trips kids up as much as anything: converting between decimals, fractions, and percents!

Look for more on decimal-fraction-percent conversions in a future post. For now, though, make it a point to stop saying point when referring to decimals (including mixed numbers—i.e., go with “and” rather than “point;” example: read 15.03 as “fifteen and three hundredths”).

Estimation Before Computation

March 29, 2010

You don’t need to be a baseball fan to know that the distance from third base to first base is a lot less than three miles. Yet that’s what several students came up with (16,200 feet to be exact) in a class I recently visited. And though it may seem like their answer came way out of left field, these students were actually just a calculator button away from hitting a home run on a pretty challenging problem. In other words, they knew they were dealing with a right triangle whose legs were both 90 feet long (the distance between home plate and first base, and home plate and third base), and they applied Pythagorean Theorem accordingly. They just stopped a step short—failing to take the square root of 16,200—and were thus off by some 16,073 feet.

What bothered me most about this wasn’t that students skipped a step, but that they accepted 16,200 feet as the correct answer. Until, that is, I said to them, “So what you’re telling me is that it’s 90 feet from third base to home plate, and 16,200 feet from third base to first base. No wonder some of those guys take steroids.” To which a couple students replied, “That can’t be right.” A minute or two of troubleshooting later—with no help from me—students had correctly changed their answer to 127 feet.

This experience illustrates why we should teach students to treat every problem as an estimation problem. Model this before you work through a problem with students by asking, “What would you expect the answer to be?,” and then recording their estimates. Do this until students start doing it themselves as they solve problems on their own. And if they’re resistant or forgetful, you might want to make it a required first step in the problem solving process. Whatever it takes to get students in the habit of estimating before computing so that they have a reliable ballpark answer to compare their actual answer to for reasonableness. And so that they can then recognize when they’ve committed an error—before it’s too late to correct it.

(With standardized tests coming up, this is a particularly good time to stress this idea to students. You can be sure, for example, that 16,200 feet would have been a “trap” choice had the baseball field problem appeared on a multiple choice test.)

Management or Math? Both!

March 18, 2010

Be sure to read the recent New York Times Magazine article, Building a Better Teacher, if you haven’t already. The article features two approaches toward improving instruction: Uncommon Schools founder Doug Lemov’s taxonomy of 49 effective teaching practices (Teach Like a Champion) and University of Michigan School of Education dean Deborah Loewenberg Ball’s Mathematical Knowledge for Teaching.

Briefly, Lemov focuses more on classroom management, while Ball focuses more on content. As for which camp I’m in, I can honestly say both. On the one hand, like Lemov’s taxonomy, my work as a teacher and coach has always reflected meticulous cause-effect analysis and intentionality when it comes to teaching decisions that transcend content. But like Ball, I also focus on developing teachers’ knowledge of math and how kids think about it.

What struck me most in the article was the extent to which Lemov has identified key teaching behaviors. And though I’ve yet to see his list in its entirety, it sounds promising, since examples cited in the article coincide with strategies that have worked for me and teachers I’ve coached—e.g., “don’t do two things at once” (I’m always cautioning teachers about multi-tasking); and “Cold Call,” where teachers choose who to call on from the entire class rather than calling on hand-raisers (see my earlier post, Assessing Through Asking).

And then there’s #46 on Lemov’s list, which includes injecting joy in the classroom by giving students nicknames. I did this too, and it was great for classroom culture. Be forewarned, though, that some parents may object, as was the case when a mother took exception to me calling her daughter “Blizzard.” (Hey, what do you expect when you name your kid “Winter”?)

Again, read the article!