Alternative Algorithms for Addition and Subtraction

1
Alternative Algorithms for Addition and
Subtraction

• If we dont teach them the standard way, how will
  they learn to compute?

2

• Childrens 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

3
What are the goals for students?

• Develop conceptual understanding
• Develop computational fluency

4
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.

5
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.

6
What do we mean by U.S. Traditional Algorithms?

• Addition
• 1
• 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 cant do 3 7. So I borrow from
• the 8 and make it a 7. The 3
• becomes 13. 13 7 6.
• 7 3 4.

7
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

8
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?

9
Some Examples of Invented Strategies for Addition
with Two- Digit Numbers
10
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 80
• 80 4 84

11
Some Examples of Invented Strategies for Addition
with Two- Digit Numbers
12
Some Examples of Invented Strategies for Addition
with Two- Digit Numbers
13
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?

14
Teachers 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

15
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)

16
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

17
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?

18
Some Examples of Invented Strategies for
Subtraction with Two- Digit Numbers
19
Some Examples of Invented Strategies for
Subtraction with Two- Digit Numbers
20
Some Examples of Invented Strategies for
Subtraction with Two- Digit Numbers
21
Some Examples of Invented Strategies for
Subtraction with Two- Digit Numbers
22
Some Examples of Invented Strategies for
Subtraction with Two- Digit Numbers
23
Some Examples of Invented Strategies for
Subtraction with Two- Digit Numbers
24
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, lets
  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.

25
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.

26
Benefits of Invented Strategies

• Place value concepts are enhanced
• They are built on student understanding
• Students make fewer errors

27
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
  hundreds charts).
• Ask children to make a written numeric record of
  what they did with the models.

28
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.

29
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.

30

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