Alternative Algorithms for Addition and Subtraction If we don’t teach them the standard way, how will they learn to compute?

Children’s first methods are admittedly inefficient. However, if they are free to do their own thinking, they invent increasingly efficient procedures just as our ancestors did. By trying to bypass the constructive process, we prevent them from making sense of arithmetic. Kamii & Livingston

What are the goals for students? Develop conceptual understanding Develop computational fluency

What is Computational Fluency? Fluency demands more of students than memorizing a single procedure does. Fluency rests on a well-build mathematical foundation that involves:  Efficiency implies that the student does not get bogged down in many steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily.  Accuracy depends on careful recording, knowledge of basic number combinations and other important number relationships, and verifying results.  Flexibility requires the knowledge of more than one approach to solving a particular kind of problem. Students need to be flexible to choose an appropriate strategy for a specific problem.

Stages for Adding and Subtracting Large Numbers Direct Modeling: The use of manipulatives or drawings along with counting to represent the meaning of the problem. Invented Strategies: Any strategy other than the traditional algorithm and does not involve direct modeling or counting by ones. These are also called personal or flexible strategies or alternative algorithms. U.S. Traditional Algorithms: The traditional algorithms for addition and subtraction require an understanding of regrouping, exchanging 10 in one place value position for 1 in the position to the left - or the reverse, exchanging 1 for 10 in the position to the right.

What do we mean by U.S. Traditional Algorithms? Addition “7 + 8 = 15. Put down the 5 and carry the = 7” Subtraction “I can’t do 3 – 7. So I borrow from the 8 and make it a 7. The 3 becomes – 7 = 6. 7 – 3 = 4.”

Time to do some computing! Solve the following problems. Here are the rules: You may NOT use a calculator You may NOT use the U.S. traditional algorithm Record your thinking and be prepared to share You may solve the problems in any order you choose. Try to solve at least two of them = = = =

Sharing Strategies Think about how you solved the equations and the strategies that others in the group shared.  Did you use the same strategy for each equation?  Are some strategies more efficient for certain problems than others?  How did you decide what to do to find a solution?  Did you think about the numbers or digits?

Some Examples of Invented Strategies for Addition with Two- Digit Numbers

Add on Tens, Then Add Ones = = = = 84

Some Examples of Invented Strategies for Addition with Two- Digit Numbers

Invented Strategies In contrast to the US traditional algorithm, invented strategies (alternative algorithms) are:  Number oriented rather than digit oriented Place value is enhanced, not obscured  Often are left handed rather than right handed  Flexible rather than rigid Try and Did you use the same strategy?

Teacher’s Role Traditional Algorithm Use manipulatives to model the steps Clearly explain and model the steps without manipulatives Provide lots of drill for students to practice the steps Monitor students and reteach as necessary Alternative Algorithms Provide manipulatives and guide student thinking Provide multiple opportunities for students to share strategies Help students complete their approximations Model ways of recording strategies Press students toward more efficient strategies

The reason that one problem can be solved in multiple ways is that… mathematics does NOT consist of isolated rules, but of CONNECTED IDEAS! (Liping Ma)

Time to do some more computing! Solve the following problems. Here are the rules: You may NOT use a calculator You may NOT use the U.S. traditional algorithm Record your thinking and be prepared to share You may solve the problems in any order you choose. Try to solve at least two of them = = = =

Sharing Strategies Think about how you solved the equations and the strategies that others in the group shared.  Did you use the same strategy for each equation?  Are some strategies more efficient for certain problems than others?  How did you decide what to do to find a solution?  Did you think about the numbers or digits?

Some Examples of Invented Strategies for Subtraction with Two- Digit Numbers

Another Look at the Subtraction Problems = = = = Now that we have discussed some alternative methods for solving subtraction equations, let’s return to the problems we solved earlier. Go back and try to solve one or more of the problems using some of the ways on the subtraction handout. Try using a strategy that is different from what you used earlier.

Summing Up Subtraction Subtraction can be thought of in different ways:  Finding the difference between two numbers  Finding how far apart two numbers are  Finding how much you have to “add on” to get from the smaller number to the larger number. Students need to understand a variety of methods for subtraction and be able to use them flexibly with different types of problems. To encourage this:  Write subtraction problems horizontally & vertically  Have students make an estimate first, solve problems in more than one way, and explain why their strategies work.

Benefits of Invented Strategies  Place value concepts are enhanced  They are built on student understanding  Students make fewer errors

Progression from Direct Modeling to Invented Strategies  Record students’ explanations on the board or on posters to be used as a model for others.  Ask students who have just solved a problem with models to see if they can do it in their heads.  Pose a problem and ask students to solve it mentally if they are able (may want to use hundred’s charts).  Ask children to make a written numeric record of what they did with the models.

Development of Invented Strategies  Use story problems frequently. Example: Presents and Parcels picture problems from Grade 2 Bridges  Multiple opportunities  Not every task must be a story problem. When students are engaged in figuring out a new strategy, bare problems are fine. Examples: Base- ten bank, work place games such as Handfuls of treasure and Scoop 100 from Grade 2 Bridges.

Suggestions for Using/Teaching Traditional Algorithms  Delay! Delay! Delay!  Spend most of your time on invented strategies. The understanding students gain from working with invented strategies will make it much easier for them to understand the traditional algorithm.  If you teach them, begin with models only, then models with the written record, and lastly the written numerals only.

Growing evidence suggests that once students have memorized & practiced procedures without understanding… …they have difficulty learning to bring meaning to their work. (Hiebert)