Making Sense of Number Sense
Lori Kruse
February 12, 2013
Cedar Lane Elementary
What is number sense?
How would you calculate?
3996 + 4246 =
3996 + 4246 =
• What would a mathematician or someone
with good number sense do?
• Look at number first and then decide on a
strategy.
• Because 3996 is close to a friendly number
4000, a more efficient strategy would be to
remove 4 from 4246 = 4242. Then add 4 to
3996 to get 4000.
• 4000+ 4242 is easy to calculate mentally
What is number sense?
• What if the numbers aren’t so friendly?
• Try 234 + 136
–Could use 235 + 135 (+/-1)
–Could use 240 + 130 (+/-6)
– If numbers can’t be made friendly and algorithm is
faster, it is probably a long series of numbers that is
better done on a calculator.
What is number sense?
• What about subtraction?
3400 – 189
– Could add 11 to each number 3411 – 200
Subtraction is really the difference
between any two numbers on a
number line
• Difference in age between a 28 year old and
a 7 year old
• In three years they will be 31 and 10 - the
difference between them remains the same.
• Slide the numbers back and forth on a
number line to reach a landmark number
that makes subtraction easier.
What is number sense?
• Many children get confused by 3400 –
189.
• They end up with 3389 because they
combine adding and subtracting.
• They lose sight of what they are doing
and don’t think about whether their
answer is reasonable.
Why is it important to have an
understanding of math?
Talk to your neighbor!
To be successful in today's world, we need a deep conceptual
understanding of mathematics. We are bombarded with
numbers, statistics, advertisements, and similar data every day...
We need good mental ability and good number sense in order
to evaluate advertising claims, estimate quantities, efficiently
calculate the numbers we deal with every day and judge
whether these calculations are reasonable, add up restaurant
checks and determine equal shares, interpret data and statistics.
We need a deep understanding of number and operation that
allows us to both estimate and make exact calculations
mentally. This understanding includes algorithms, but it places
emphasis on mental arithmetic and a repertoire of strategies.
Catherine T. Fosnot
Professor at City College of NY
So what is number sense?
• Looking at the number first and then
choosing a strategy.
• Knowing how numbers can be taken
apart and put together,
e.g. 7 can be 1+6, 2+5, 3+4, 0+7 and turnarounds
• Using that knowledge to solve problems
efficiently
• Having many strategies at your disposal.
What is an algorithm?
• A logical arithmetical or computational
procedure that if correctly applied
ensures the solution of a problem
• If number sense is not in place first, using
an algorithm is like walking off trail in the
woods.
• You might get lucky and reach your
destination, but more likely you will get
lost and have no idea how to get home.
What is an algorithm?
• It is important to build number
sense first.
• Then learn to apply the formula
or algorithm later.
How do we teach number sense?
• Tools:
• Arithmetic racks – used to teach
understanding of how numbers can be put
together and taken apart.
Arithmetic racks
• Allows children to build their
understanding of numbers using 5
e.g. 6 as 5 + 1, 8 as 5 + 3 or 4 as 5 - 1
• Supports learning doubles, near doubles,
e.g. 6 + 7 = 6 + 6 = 1
• Supports making tens,
e.g. 9 + 6 = 10 + 5
Games
• Allows children to build their
understanding of number sense
• Designed to support learning math facts
e.g. doubles, near doubles, + 1, - 1, sums
to 10
• Supports composing and decomposing
numbers
Games
Minilessons
• Strings of numbers in a series that:
• are designed to encourage students to
develop mental strategies to solve
problems
• are related so learning builds upon itself
• enhance students’ development of
numeracy
• are structured to expand the repertoire
of strategies for mental arithmetic
Minilessons
• encourages students to find Landmark or
friendly numbers.
• Landmark = numbers which are friendlier or
easier to use. Usually a multiple of 10
• Using landmarks can help one add, subtract,
multiply or divide
e.g. 48 x 3 – difficult to do mentally,
but 50 x 3 – 6 is easy to do
Minilessons
An open number line encourages students to:
• think about landmarks on the number line
• take leaps mentally
• visualize landing points, rather than to count
by ones
Minilessons
• Splitting numbers – provides children
with visual images for discussion
Using ten and multiples of ten
• taking leaps of ten all at once and adjusting
e.g. 15 + 9 = 15 + 10 – 1
• moving to the next multiple of 10
e.g. 15 + 9 = 15 + 5 (to get 20 – the
next multiple of 10) + 4
• using compensation to make a problem with
10 in it
e.g. 15 + 9 = 14 + 10
Addition – keeping one number
whole and taking leaps of 10
15 + 10
15 + 9
15 + 19
28 + 19
(encourages use of leaps of 10)
(encourages leap of 10 - 1)
(encourages extension of previous strategy)
(challenges students to use an equivalent
expression: 28 + 20 – 1 or 30 + 17)
28 + 32 (challenges students further:
28 + 30 + 2 or 30 + 30)
39 + 21
(application of strategy)
Subtraction - keeping one number
whole and taking leaps of 10 back
36 – 10
36 – 20
36 – 24
43 – 30
43 – 39
57 – 21
(leap back 10)
(leap back 20)
(leap back 20 – 4)
(leap back 30)
(leap back 40 + 1)
(application of strategies)
Compensation & equivalency move to a landmark number
58 + 22
60 + 20
30 + 50
28 + 52
32 + 48
33 + 47 (supports thinking about equivalency taking from one to give to the other)
98 + 42 (application of strategies)
Regrouping
• When working with whole numbers, the
standard algorithm of lining the numbers up
vertically and regrouping is only one way to
subtract.
23
-8
• There is more than one way to solve the
problem!
Regrouping
Student could also think:
• I need 2 more to get to 10, then 10 more to
get to 20, then 3 more to get to 23.
2 + 10 + 3 = 15
• Student might know that 20 – 8 is 12, so they
need another 3.
20 – 8 = 12 + 3 = 15
• Student may break up the 8 to 3 + 5, take the
3 away from 23, and then take away the 5.
23 – 3 – 5 = 15
Regrouping
• It is important to relate subtraction to
addition by thinking about the grouping
that took place to build 23
• 23 ones can be 2 groups of ten and 3 ones
or 20 + 3
• 23 can also be 1 group of ten and 13 ones
or 10 + 13
• With 13 in the ones place, we can now
subtract the 8 ones.
Regrouping
• Research shows that young students struggle
to understand that a ten is simultaneously one
ten and ten ones.
• By keeping the emphasis on the place value,
composing and decomposing a ten, and
connecting to regrouping in addition, we try
to avoid having the standard algorithm
become just a procedure that students will
memorize and then forget.
Strategies
• learned in a developmentally
appropriate order:
–One-to-one tagging when counting
–Counting three times when adding
–counting 1 – 9 in sequence
–counting backwards
–using a 5 structure
–using doubles and near doubles
–using known facts
Strategies
–counting on
–making tens
–using compensation
–skip counting
–uses ten structure
–splitting numbers
–keeping one addend whole & moving to
a landmark number
–keeping one addend whole & taking
leaps of ten
Strategies
–taking leaps of ten back and adjusting
–varies adding on versus removing
–decomposing the subtrahend to get to
a landmark number
–regrouping
–using constant difference
–generalized use of a repertoire of
strategies for addition and subtraction
dependent on the numbers
Final Thoughts
• Number sense is developed through
conversation and exploration of ideas to
see what works best in each situation.
• Our goal is that students will look at the
numbers first and then decide what
strategy will work most efficiently.
• We want students to have a variety of
strategies in order to calculate efficiently.
Helping Your Child
• Play the games that first grade teachers send
home often
• Play mental math games
- I have 6 spoons. How many do I need to
make 10?
- There are 8 birds at the feeder and 3 fly away.
How many will be left?
• Don’t teach your child to do multiplication and
division until they have excellent number sense
• Celebrate your child’s learning where they are
today
Questions
Thank you for coming!