STRATEGIES FOR WHOLE NUMBER COMPUTATION

February 2013 Coaches Meeting

CCSS

Fluently add and subtract within ____ using strategies and algorithms (3rd and 4th grade) based on place value, properties of operations, and/or relationship between addition and subtraction.

Solve 7 + 52 +186 without paper and pencil. Private think time

Share with your shoulder partner how you arrived at your answer.

Comparison of three 2nd grade classes

Algorithms Some algorithms taught at home

No algorithms

17 students 19 students 20 students

12% correct 26 % correct 45% correct

Range of answers9308-29

Range of answers989-114

Range of answers617-139

Research from Constance Kamii. Young Children Reinvent Arithmetic: Implications of Piaget’s Theory, Columbia University, 2000

Big Ideas

Flexible methods of computation involve taking apart and combining numbers in a variety of ways.

Invented strategies are flexible methods of computing that vary with the numbers and the situation.

Flexible methods for computation require a good understanding of the operations and the properties of operations.

The traditional algorithms are clever strategies for computing that have been developed over time. They work for all numbers but are often far from the most efficient or useful method of computing.

Three Types of Computational Strategies

• Counts by ones• Use of base-ten models

Direct Modeling

• Supported by written recordings

• Mental methods

Invented Strategies

• Usually requires guided development

Traditional Algorithm

s

Benefits of Student Invented Strategies

Students make fewer errors. Less re-teaching is required. Students develop number sense. Invented strategies are the basis for

mental computation and estimation. Flexible methods are often faster than

the traditional algorithms. Algorithms invention is itself a

significantly important process of “doing mathematics.”

Invented Strategies vs Traditional

Invented strategies are number oriented rather than digit oriented. 45 + 32 think of 40 + 30 rather than 4 +3

Invented strategies are left-handed rather than right handed.

Invented strategies are flexible rather than rigid. Try each mentally 465 + 230 and 526 + 98 Did you use the same method?

Algorithms

Students will not invent the traditional methods because they are right-handed methods and are simply not natural. This means that the teacher must introduce and explain each algorithm.

Students will sense that they are the “right way” or “real way” to compute and will abandon any flexible method they have been developing.

They will happen.

Invented Strategies for Addition

Add Tens, Add Ones and Combine

46 + 3840 + 30 is 706 and 8 is 14

70 and 14 is 84

Move Some to Make Tens

46 + 38Take 2 from the 46 and

put it with the 38 to make a 40. Now you have 44 and 40 more

is 84.

Invented Strategies for Subtraction

Take Tens from Tens, Then Subtract

Ones73-46

70 minus 40 is 30Take away 6 more is

24Now add in the 3 ones

27

Take Away Tens, Then Ones

73 – 4673 minus 40 is 33Then take way 6

3 makes 30 and 3 more is 27

Mental Strategies

Developing Fluency

Strategies

Strategies

Strategies

Strategies using Place Value

Yesterday there were 57 penguins sitting on the iceberg. Later 34 penguins joined them. How many penguins are now on the iceberg?

There were 72 penguins sitting on the iceberg. 49 penguins jumped into the icy water. How many penguins are still on the iceberg?