February 2013 Coaches Meeting

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.
Algorithms
Some algorithms taught
at home
No algorithms
17 students
19 students
20 students
12% correct
26 % correct
45% correct
Range of answers
9308-29
Range of answers
989-114
Range of answers
617-139
Research from Constance Kamii. Young Children Reinvent Arithmetic:
Implications of Piaget’s Theory, Columbia University, 2000




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.
Direct
Modeling
• Counts by ones
• Use of base-ten models
Invented
Strategies
• Supported by written recordings
• Mental methods
Traditional
Algorithms
• Usually requires guided
development






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 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?



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.
Add Tens, Add Ones and
Combine
46 + 38
40 + 30 is 70
6 and 8 is 14
70 and 14 is 84
Move Some to Make
Tens
46 + 38
Take 2 from the 46 and
put it with the 38 to make
a 40. Now you have 44
and 40 more is 84.
Take Tens from Tens,
Then Subtract Ones
73-46
70 minus 40 is 30
Take away 6 more is 24
Now add in the 3 ones
27
Take Away Tens, Then
Ones
73 – 46
73 minus 40 is 33
Then take way 6
3 makes 30 and 3 more is
27
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?