Alternate Algorithms
By Michael Naylor

An algorithm is a step-by-step "recipe" telling what to do with number symbols. Rather than teach traditional algorithms in isolation, math educators and many of the newer curricula urge you to encourage students to invent their own algorithms and propose a variety of alternate algorithms.

Traditional algorithms have evolved over thousands of years and are now extremely efficient, but when it comes to student learning, this efficiency often comes at the price of loss of meaning. If efficiency is your goal at any cost, standard algorithms win. Students will initially be slower doing alternate algorithms than students who go straight to standard algorithms – it takes more time to develop number sense than it does to learn a few rules for sliding digits around.

Addition: from big to small
Humans naturally think about quantities from big to small. A child with no mathematics training will mentally add 23 and 34 by thinking "20 + 30 is 50, and 3 + 4 is 7, so the answer is 50 + 7 or 57."

The standard algorithm, however, goes from small to large. When the result of an addition is greater than 10, the last digit is recorded and you "carry the one," an algorithmic step that often has little meaning for students who have not developed strong number sense.

To develop good number sense when teaching addition algorithms, be sure to provide base-ten blocks, place-value mats and recording sheets.

Your students will want to add the larger pieces first – great! They may end up with 7 tens and 14 ones and then need to regroup after they've completed the addition, but that's okay. They're developing an algorithm that matches their thinking and makes sense.

Later as students become very comfortable with the algorithm, you can suggest doing the ones column first so there will be less crossing out.

Multiplication algorithms
If your students are just learning multiplication, these algorithms will expand their understanding of how digits interact when computing a product. If they've already learned the standard algorithm, these methods make excellent puzzles – require your students to explain (using base-ten) why the algorithms work.

Remind your students that one way to think about multiplication is with area; a rectangle measuring 4 units by 5 units contains 4 x 5 squares. The same idea helps form a basis for understanding multiplication algorithms.

Base-10 block models
Using base-10 blocks, model 12 x 13:

Emphasize that the four areas give four partial products. The result is the sum of these four products: 100 + 30 + 20 + 6 = 156.

Give your students several practice problems and ask them to model with pieces and then draw a picture showing the four areas.

Area grids
After working with the models, students can use a shorthand grid and write the tens and ones of each number to be multiplied on the sides of a four-part rectangle. Each portion of the grid should be filled in with a partial product and the four products are summed.

Lattice multiplication
An algorithm needs a method to organize the partial products so it's easy to add up the ones, tens and hundreds. Start with a grid as shown below. The two numbers to be multiplied are written across the top and down the side of the grid. Point out that there are four squares inside the grid – these are for the four partial products. The diagonal lines will help organize those products to make it easy to add.

Within each square, write the one- or two-digit product of the digits heading the row and column.

Once all four partial products are recorded, add the numbers along the diagonals or write the sums outside of the grid as shown in the final figure. The result wraps around the corner of the grid.

How does this work?
The bottom right diagonal contains the ones, the next diagonal the tens, then hundreds, then thousands. In the previous example, 3 x 4 is shorthand for 3 x 40 = 120. The 120 is recorded as a "2" in the tens diagonal and a "1" in the hundreds diagonal.

Sometimes, adding a diagonal results in a sum greater than 9. In the example below, one of the sums is 17. You can then decide how to handle it when writing your product – the 17 should be regrouped to make 7 tens and one additional hundred.

There are so many steps and rules and the digits must be written in exactly the right places, that students become very focused on not making a mistake and lose their focus on number sense. If you teach the standard algorithm, be sure to constantly make connections to students' previous models.

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.

April 2007, Vol.37, No.7