Valuing Mental Computation
What is Mental Computation?
Whereas mental arithmetic traditionally focuses on speed and accuracy, mental
computation has its emphasis on the processes used to achieve the answer. For example,
think of how you could mentally work out how many 45 cent stamps you could buy for $10.
As effective users of numbers we have developed our own way of solving number problems
'on the spot'. We have generally developed our mental computation strategies without
formal instruction.
Why teach mental computation?
Mental computation is now recognised as an important form of computation. The
importance can be summarised as follows:
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it is the most common form of computation used by adults.
("85% of calculations done by adults in their daily life." McIntosh and Northcote
1998)
it is used for estimation
it is needed to check calculator answers
students see it as an easy way of doing calculations
it is an excellent way of learning how numbers work
it is a creative and problem solving approach to number.
What to do in the classroom: Activity 1
Explain to your students (any age group) that you are going to ask a question you want them
to do in their heads. You do not want any answers for at least 45 seconds but after that
time, you will ask not for the answer but for the method they used to arrive at their answer.
The first question. Ask the students to think about “What do 56 and 25 add up to?” If you
want to write the numbers on the board please do this horizontally: 56 + 25. Vertical form
can encourage an algorithmic method.
Ask for a student to volunteer a method. Record this on the board with the student’s name.
Ask how many used that method and then “did any one do it differently?” Continue until
you have exhausted the methods. Lead a brief discussion on whether any one way was
better than another for this particular question. Try not to endorse one method over
another.
Common methods you might expect from your students
Once a student has mastered counting on by ones they are ready to move to more facile
strategies. Sharing the mental computation strategies in a whole class situation will facilitate
the gathering of methods as well as assisting students to give a clear argument for their
mathematical reasoning.
When you listened to the strategies your students chose to use for 56 + 25, you may have
seen the following:
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one of the numbers is kept as a basis and the tens and ones of the second number
added to the first
both numbers split into tens and ones, the tens and ones added separately and the
answers recombined
the use of some known fact such as “I know 50 is twice 25 so 56 + 25 is like three 25s
and 6”
bridging to the next ten eg “56 and 4 more is 60, and then I need another 21…”
Of course there may be other unusual strategies which may be prompted by the particular
numbers in the question. Compensation methods where students add to one number and
take away from the other, so that 6 + 8 may be thought of as the same as 7 + 7. When you
try a subtraction question another family of responses come in to play.
Jump or Split
The two main groups of strategies used for mental computation of addition and subtraction
questions are:
• the jump methods (Beishuizen, 1993) where students build on from one of the numbers
and
• the split methods where students will partition both numbers into tens and ones (Cobb
and Wheatley, 1988).
The use of the jump method or the split method is often a function of the instructional
sequence students have been through. Although weaker students have an observed
preference for the split method, Beishuizen and Anghileri argue that this is because students
initially find it difficult to increment by tens off the decade, such as 37, 47, 57….
Nevertheless, when encountering the more difficult 2-digit addition and subtraction
questions requiring regrouping, students using the split method are more likely to make
errors and to find it difficult to modify their strategies.
We need to encourage students to move beyond a count by one method but there is no
preference for teaching the students one facile method over another. Rather the value is in
students being comfortable to use which-ever method is more efficient for the particular
question.
What to do in the classroom: Activity 2
Repeat this process on another day and with another question.
You may like to try a subtraction.
Here are some questions you could choose from:
65 + 38, 52 – 24, 27 + 19, 74 – 15, 169 + 14, 149 + 149, 299 – 51
What to do in the classroom: Activity 3
If you feel students are still learning to express their strategies, try one or two more
questions of a similar nature to those in activities one and two.
If your students are really enjoying explaining methods, try extending the mental
computation to include items such as:
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a three number addition e.g. 13 + 35 + 17
a multiplication e.g. 199 X 4
a real life situation e.g. How many 45 cent stamps can you buy for $10.
Remember in all of these situations we are wanting to hear how the student worked the
answer out without the use on pencil and paper.
http://www.curriculumsupport.education.nsw.gov.au/secondary/mathematics/numeracy/mental/in
dex.htm
Some Common Methods: animations:
http://www.curriculumsupport.education.nsw.gov.au/vmco/examples.html
The empty number line
New teaching methods are being used as part of the Counting On project across the state.
For the first time ever, for many students, the invisible mathematics is starting to become
visible. Teaching in the Counting On project emphasises using problems and ways of
recording that allow students to share what they are thinking.
Think about adding together two numbers in your head, say 157 and 36. When this addition
is carried out mentally, we often start from the left rather than from the right. To get the
students to show their thinking, the teachers in the Counting On project have been using an
empty number line.
Each student can show how he or she thought about the problem by drawing and filling in an
empty number line.
A number line showing 30 and 6
One of the interesting things about mental calculations is that we do not all think the same
way. Some people start by breaking the 36 into 33 and 3. This turns the question into the
problem of adding 33 to 160.
A number line showing adding 3 and then 33
http://www.curriculumsupport.education.nsw.gov.au/secondary/mathematics/numeracy/countingo
n/index.htm