8. Explore alternative algorithms.

In math, an algorithm is a set of steps, used to solve a problem, that works for all numbers within a given set. Our algorithms are developed because they are often the most efficient way to solve the most difficult problems. But it's important to keep two things in mind regarding algorithms:

1. The traditional algorithms used in the U.S. are not the most efficient way to solve all problems. Take 399 + 299 for example. The traditional algorithm would have you rewrite these two numbers vertically, start with the ones, regroup into the tens, etc. In the amount of time it took one student to rewrite the problem, a fluent, flexible student could have this problem solved mentally. They might think about 399 and 400, and 299 as 300. 400 + 300 = 700. Then they subtract off the 2 extra that they added to get get a final answer of 698.

We don't want to push algorithms so much that we devalue students' flexible strategies. They need both. We can teach students algorithms, while continuing to strengthen their mental math skills and flexibility through regular number talks.

2. The traditional algorithms used in the U.S. do not necessarily help students make sense of the operation. Do your students know why they use a "place holder zero" in the traditional multiplication algorithm? Do they know why they "bring down the next number" in the long division algorithm? Our algorithms were created to make solving problems easier, but not necessarily to reveal what's actually happening and why.

If we want to teach conceptually for deep understanding, we then have at least 2 options: teach students to see and understand the math in the traditional algorithms OR teach alternative algorithms that incorporate more sense-making opportunities. Below are some strategies or algorithms you might try with your students.

Examples of Alternative Algorithms Your Students Might Develop

"My students might develop? What?!" Yes! When students are given opportunities to explore operations through practical problems and modeling, before they have been taught standard algorithms, they can develop their own invented strategies for computation. In younger grade levels, this could look like students who use "left handed strategies" of working with largest place values first. These strategies are not wrong; they're just different from our standard algorithm! To learn more about how you can encourage students as they develop their own invented strategies, check out Part 1 and Part 2 of Courtney Koestler's NCTM blog about this topic.

If you're on Twitter, follow Math Strategy Chat (#mathstratchat) with Pam Harris to learn even more strategies for computation.