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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?
Addition1
47+28 75
“7 + 8 = 15. Put down the 5 and carry the 1. 4 + 2 + 1 = 7”
Subtraction 7 13
83- 37 46
“I can’t do 3 – 7. So I borrow from the 8 and make it a 7. The 3 becomes 13. 13 – 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.
658 + 253 = 297 + 366 =
76 + 27 = 314 + 428 =
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
Some Examples of Invented Strategies for Addition with Two- Digit Numbers
Add on Tens, Then Add Ones
46 + 38
46 + 30 = 76 76 + 8 = 76 + 4 + 4
76 + 4 = 8080 + 4 = 84
Some Examples of Invented Strategies for Addition with Two- Digit Numbers
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 465 + 230 and 526 + 98 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.
636 - 397 = 221 - 183 =
502 - 256 = 892 - 486 =
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
Some Examples of Invented Strategies for Subtraction with Two- Digit Numbers
Some Examples of Invented Strategies for Subtraction with Two- Digit Numbers
Some Examples of Invented Strategies for Subtraction with Two- Digit Numbers
Some Examples of Invented Strategies for Subtraction with Two- Digit Numbers
Some Examples of Invented Strategies for Subtraction with Two- Digit Numbers
Another Look at the Subtraction Problems
636 - 397 = 221 - 183 =
502 - 256 = 892 - 486 = 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 enhancedThey are built on student understandingStudents 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)