Saturday, 28 April 2012 17:45

Awesome Algorithm Alternatives (Part 2: The “Advanced Deal ’em Out” Method of Division) (Teaching Tip #68)

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Last week I shared that my colleagues and I have spent the past few years learning about the Cognitively Guided Instruction (CGI) approach to the teaching and learning of mathematics. A huge emphasis of this philosophy is the need for students to understand math concepts on a deep level and use strategies that make sense to them. Encouraging students to use a wide variety of strategies to solve problems is a practice that stands in stark contrast to the traditional way that most of us were taught. When I was a student, I learned a series of algorithms that I was expected to follow, step-by-step, whenever I needed to add, subtract, multiply, or divide large numbers.

The advantage of these algorithms, of course, lies in the fact that we can carry out calculations quickly. If we follow the steps precisely and avoid making mistakes with our number facts, we will consistently generate correct answers. The disadvantage of using algorithms is that genuine understanding of math concepts is less likely to occur. When using algorithms, we are merely applying a series of memorized procedures, whether we understand their rationale or not. I remember, for example, being taught as a kid to “carry the one” when multiplying a one-digit number times a two-digit number, but I never understood why. I just did it because I was told it was the way to get right answers.

A child using algorithms can quickly solve problem after problem, but one tiny computational or procedural mistake will lead to an incorrect answer, and the student applying that procedure may never notice this error.

Since my school adopted the CGI approach, I have been exploring with my students a wide variety of alternatives to these traditional algorithms. We are always on the lookout for strategies that are reliable, efficient, and rely on our number sense, not the application of memorized procedures that we may or may not “get.”

“Advanced Deal ’em Out”

In this tip I highlight the “Advanced Deal ’em Out” Method of Division. In December of each school year, my third graders engage in a divison unit that features several strategies that help them understand the concept of division. In this unit the numbers we use remain small, and the kids are never asked to solve a problem with more than a two-digit dividend. The strategy that most students prefer is one I like to call “deal ’em out.” When I introduce this strategy, I pretend I am the dealer in a card game, and my job is to distribute a given number of cards to each player. As I explain this strategy, I simulate a card game with a few students, and I deal out one card at a time to the kids until everyone has the right number of cards. Kids understand this strategy easily and enjoy the analogy between card games and division.

Later in the year, after units on fractions, decimals, geometry, and multiplication with larger numbers, our focus shifts to division with larger numbers. Now, the kids need to solve problems with a three-digit dividend and a single-digit divisor. Traditionally, this is when many students learn the steps of “long division.” After much discussion with a few colleagues, I decided a few years back not to teach this strategy because it suffers from the same drawbacks that all algorithms suffer from. Specifically, the algorithm relies on the application of a memorized series of steps, does nothing to develop number sense, and reinforces the notion in students’ minds that math is something they have to “get.” In other words, math is about understanding other people’s ideas, rather than constructing our own ideas and using strategies that make sense to us.

As I demonstrate “Advanced Deal ’em Out,” let’s use the example 135 divided by 5. It would take a very long time to draw five circles and deal out one dot at a time until all 135 dots have been distributed. When my students first start using this strategy in December, they do deal out one dot at a time because the numbers are smaller. In addition, I think it’s OK for students to spend a little extra time on a strategy when they are first learning it so they gain comfort with it and develop confidence. In April students generally will not want to deal want to deal out one dot at a time, and this week’s strategy provides them with an effective alternative.

With “Advanced Deal ’em Out,” the object is to deal out the largest possible quantities. When we do this, it is helpful to choose “friendly” numbers, generally those with a “0” or “5” in the ones place. In this problem some students like to deal out 20 to each circle as their first move, while others may prefer to deal out 25 or 10. After the first number is dealt to each circle, the goal is to determine how many dots remain and then decide how many dots to deal out with the second move. If a student deals out 20 the first time around to each of the five circles, there would be 35 dots remaining. Students who know their basic facts well will deal out 7 dots to each circle and be done with the problem, while those who don’t may deal out 2 or 5 to each circle and then take one or two more moves before completing the problem. The final step, of course, is to see how many total dots are in each circle, and that number is the quotient.

I love this strategy because it relies on and helps strengthen students’ number sense. In addition, the strategy is very flexible and forgiving. With long division one wrong procedural or computational move, and that’s all she wrote. With Advanced Deal ’em Out students can pick numbers in their comfort zone and proceed in a maner that makes sense to them. Highly skilled students may be able to deal out all the dots in 2-3 moves, while those with emerging skills may need to adopt a more cautious approach using smaller, safer numbers. Either way, our efforts to develop number sense are being helped, not thwarted.

New Teaching Tips appear every Sunday of the school year.