Teaching Mental Computation Strategies
The development of mental math skills in the classroom should go beyond
drill and practice by providing exercises that are meaningful in a
mathematical sense.
While learning addition, subtraction, multiplication and division facts, for
instance, students learn about the properties of these operations to
facilitate mastery. They apply the commutative property of addition and
multiplication, for example, when they discover that 3 + 7 is the same as 7
+ 3 or that 3 x 7 = 7 x 3. Knowing this greatly reduces the number of facts
that need to be memorized. They use the distributive property when they
learn that 12 x 7 is the same as (10 + 2) x 7 = (7 x 10) + (2 x 7) which is
equal to 70 + 14 = 84.
Understanding our base-ten systems of numeration is
key to developing computational fluency. At all grades, beginning with
single-digit addition, the special place of the number 10 and its multiples are
stressed. In addition, students are encouraged to add to make 10 first and
then add beyond the ten. The addition of ten and multiples of ten is
emphasized, as well as multiplication by 10 and its multiples.
Connections between numbers and the relationship between the number of facts
should be used to facilitate learning. The more connections that are
established, and the greater the understanding, the easier it is to master
facts. In multiplication, for instance, students learn that they can get to 6 x 7
if they know 5 x 7 because 6 x 7 is one more group of 7.
In general, a strategy should be introduced in isolation from other
strategies. A variety of practice should then be provided until it is
mastered, and then it should be combined with other previously learned
strategies. Knowing the name of a strategy is not as important as knowing
how it works.
Are there any students in the class who already have a
strategy for doing the computation in their heads? If so, encourage them to
explain the strategy to the class with your help. If not, you could share the
strategy yourself.
Explaining the strategy should include anything that will help students see
its pattern, logic, and simplicity. That might be concrete materials,
diagrams, charts, or other visuals. The teacher should also “think aloud” to
model the mental processes used to apply the strategy and discuss
situations where it is most appropriate and efficient as well as those in
which it would not be appropriate at all.
In the initial activities involving a strategy, you should expect to
have students do the computation the way you modeled it. Later, however,
you may find that some students employ their own variation of the strategy.
If it is logical and efficient for them, so much the better. Your goal is to help
students broaden their repertoire of thinking strategies and become more
flexible thinkers; it is not to prescribe what they must use.
You may find that there are some students who have already mastered the
simple addition, subtraction, multiplication and division facts with
single-digit numbers. Once a student has mastered these facts, there is no
need to learn new strategies for them. In other words, it is not necessary to
re-teach a skill that has been learned in a different way.
On the other hand, most students can benefit from the more difficult
problems even if they know how to use the written algorithm to solve them.
The emphasis here is on mental computation and on understanding the
place-value logic involved in the algorithms. In other cases, as in
multiplication by 5 (multiply by 10 and divide by 2), the skills involved are
useful for numbers of all sizes.
In general, it is the frequency rather than the length of practice that fosters
retention. Thus daily, brief practices of 5-10 minutes are most likely to lead
to success. Once a strategy has been taught, it is important to reinforce it.
The reinforcement or practice exercises should be varied in type, and focus
as much on the discussion of how students obtained their answers as on
the answers themselves.
The selection of appropriate exercises for the reinforcement of each
strategy is critical. The numbers should be ones for which the strategy
being practiced most aptly applies and, in addition to lists of number
expressions, the practice items should often include applications in
contexts such as money, measurements and data displays. Exercises
should be presented with both visual and oral prompts and the oral
prompts that you give should expose students to a variety of linguistic
descriptions for the operations. For example, 5 + 4 could be described as:
• the sum of 5 and 4
• 4 added to 5
• 5 add 4
• 5 plus 4
• 4 more than 5
• 5 and 4 etc.
Many of the thinking strategies supported by research and outlined in the
curriculum advocate for a variety of learning modalities.
For example:
• Visual (images for the addition doubles; hands on a clock for the
“times-five” facts)
• Auditory (silly sayings and rhymes: “6 times 6 means dirty tricks;
6 x 6 is 36")
• Patterns in Number (the product of an even number multiplied by 5 ends
in 0 and the tens digit is half of the number being multiplied)
• Tactile (ten frames, base ten blocks)
• Helping Facts (8 x 9 = 72, so 7 x 9 is one less group of 9; 72 - 9 = 63)
Whatever differentiation you make it should be to facilitate the student’s
development in mental computation and this differentiation should be
documented and examined periodically to be sure it is still necessary.
The development of mental math skills in the classroom should go beyond
drill and practice by providing exercises that are meaningful in a
mathematical sense.
While learning addition, subtraction, multiplication and division facts, for
instance, students learn about the properties of these operations to
facilitate mastery. They apply the commutative property of addition and
multiplication, for example, when they discover that 3 + 7 is the same as 7
+ 3 or that 3 x 7 = 7 x 3. Knowing this greatly reduces the number of facts
that need to be memorized. They use the distributive property when they
learn that 12 x 7 is the same as (10 + 2) x 7 = (7 x 10) + (2 x 7) which is
equal to 70 + 14 = 84.
Understanding our base-ten systems of numeration is
key to developing computational fluency. At all grades, beginning with
single-digit addition, the special place of the number 10 and its multiples are
stressed. In addition, students are encouraged to add to make 10 first and
then add beyond the ten. The addition of ten and multiples of ten is
emphasized, as well as multiplication by 10 and its multiples.
Connections between numbers and the relationship between the number of facts
should be used to facilitate learning. The more connections that are
established, and the greater the understanding, the easier it is to master
facts. In multiplication, for instance, students learn that they can get to 6 x 7
if they know 5 x 7 because 6 x 7 is one more group of 7.
In general, a strategy should be introduced in isolation from other
strategies. A variety of practice should then be provided until it is
mastered, and then it should be combined with other previously learned
strategies. Knowing the name of a strategy is not as important as knowing
how it works.
Are there any students in the class who already have a
strategy for doing the computation in their heads? If so, encourage them to
explain the strategy to the class with your help. If not, you could share the
strategy yourself.
Explaining the strategy should include anything that will help students see
its pattern, logic, and simplicity. That might be concrete materials,
diagrams, charts, or other visuals. The teacher should also “think aloud” to
model the mental processes used to apply the strategy and discuss
situations where it is most appropriate and efficient as well as those in
which it would not be appropriate at all.
In the initial activities involving a strategy, you should expect to
have students do the computation the way you modeled it. Later, however,
you may find that some students employ their own variation of the strategy.
If it is logical and efficient for them, so much the better. Your goal is to help
students broaden their repertoire of thinking strategies and become more
flexible thinkers; it is not to prescribe what they must use.
You may find that there are some students who have already mastered the
simple addition, subtraction, multiplication and division facts with
single-digit numbers. Once a student has mastered these facts, there is no
need to learn new strategies for them. In other words, it is not necessary to
re-teach a skill that has been learned in a different way.
On the other hand, most students can benefit from the more difficult
problems even if they know how to use the written algorithm to solve them.
The emphasis here is on mental computation and on understanding the
place-value logic involved in the algorithms. In other cases, as in
multiplication by 5 (multiply by 10 and divide by 2), the skills involved are
useful for numbers of all sizes.
In general, it is the frequency rather than the length of practice that fosters
retention. Thus daily, brief practices of 5-10 minutes are most likely to lead
to success. Once a strategy has been taught, it is important to reinforce it.
The reinforcement or practice exercises should be varied in type, and focus
as much on the discussion of how students obtained their answers as on
the answers themselves.
The selection of appropriate exercises for the reinforcement of each
strategy is critical. The numbers should be ones for which the strategy
being practiced most aptly applies and, in addition to lists of number
expressions, the practice items should often include applications in
contexts such as money, measurements and data displays. Exercises
should be presented with both visual and oral prompts and the oral
prompts that you give should expose students to a variety of linguistic
descriptions for the operations. For example, 5 + 4 could be described as:
• the sum of 5 and 4
• 4 added to 5
• 5 add 4
• 5 plus 4
• 4 more than 5
• 5 and 4 etc.
Many of the thinking strategies supported by research and outlined in the
curriculum advocate for a variety of learning modalities.
For example:
• Visual (images for the addition doubles; hands on a clock for the
“times-five” facts)
• Auditory (silly sayings and rhymes: “6 times 6 means dirty tricks;
6 x 6 is 36")
• Patterns in Number (the product of an even number multiplied by 5 ends
in 0 and the tens digit is half of the number being multiplied)
• Tactile (ten frames, base ten blocks)
• Helping Facts (8 x 9 = 72, so 7 x 9 is one less group of 9; 72 - 9 = 63)
Whatever differentiation you make it should be to facilitate the student’s
development in mental computation and this differentiation should be
documented and examined periodically to be sure it is still necessary.
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