Mathematical Strategies
Teaching K-2 Mathematical Strategies
Emergent
Students at emergent counting stage:
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may be able to say some number words and may relate those words to the
process of counting
are not able to:
- count forwards or backwards to/from 10
- count collections of up to ten objects
- consistently identify or name the numerals one to ten or beyond
- correctly use counting to state the number of items in a collection and identify or
name the numeral in order to label that collection.
TEN, DOE(2014)
Emergent
Students who are emergent need to focus on:
● counting collections
● identifying numerals
● labelling collections
Students who are emergent do not participate in addition or
subtraction activities until they have one-to-one object
identification
TEN, DOE(2014)
Emergent
Teaching focus - Developing Strategies
● Students working within the emergent counting stage always
rely on a single strategy when dealing with number activities.
● Numeral cards are not always presented in sequence.
TEN, DOE(2014)
Perceptual
Students at perceptuaL counting stage:
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are able to count collections of objects that they can see, hear or feel
rely on concrete representations of numbers
consistently apply the one-to-one principle of matching one number word to each
object
rely on count by one strategies and always begin at one.
TEN, DOE(2014)
Perceptual
Students at perceptual counting stage are working towards:
● adding two collections of items
● counting without relying on concrete representations of numbers
● visually recognising standard patterns for a collection of up to ten
items without counting them
● consistently saying the forward and backward number word
sequences correctly.
TEN, DOE(2014)
Perceptual
Teaching focus: Language Development
● Students at the perceptual counting stage need to be taught
the explicit mathematical language to enable them to describe
the activity and the strategy.
● The naming of number words in the teens can be a problem,
as can reversals for teen numbers.
TEN, DOE(2014)
Perceptual
Teaching focus: Numeral Identification
● Activities for developing numeral identification can be modified
for perceptual students by focusing on a smaller range of
numerals targeted.
● Explicit teaching the teen numbers is commonly needed.
TEN, DOE(2014)
Perceptual
Teaching focus: Re-representation of Concealed items
We are wanting to train students to remember what they have seen in concealed
tasks.
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Students at this level will usually re-represent what they have seen using fingers
or counters and count from one to find their answer. We are not expecting them
to count on.
TEN, DOE(2014)
Perceptual
Teaching focus: Strategy Development
Students start to learn number facts such as:
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Friends of five
Friends of ten
These are learnt for eventual instant recall when completing addition and subtraction
problems.
TEN, DOE(2014)
Perceptual - Now it’s your turn!
Blocks in a Bowl
You need:
● A partner
● A bowl
● 10 blocks
Figurative
Students at figurative counting stage:
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are able to count collections which are totally or partially concealed
do not need to see, feel or hear the items in a collection to be able to count the
collection
rely on the simple strategy of counting by ones, starting from one to find a total
have an understanding of numbers as entities however when adding numbers to
find a total, they will still start from one
TEN, DOE(2014)
Figurative
Students at figurative counting stage are working towards:
● using counting on to solve addition tasks
● using counting down to solve subtraction tasks
● developing base ten knowledge
● forming equal groups and finding their total.
TEN, DOE(2014)
Figurative
Teaching focus: Strategy development
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The main focus for teaching should be on developing the counting on procedure.
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Be aware that students who demonstrate the ability to use counting on may
revert to the strategy of counting from one when faced with a difficult task.
TEN, DOE(2014)
Figurative
Teaching focus: Number identification
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Whilst using the arithmetical strategies associated with the figurative stage,
students may show different levels of knowledge of numerals.
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The identification of numerals does not develop uniformly.
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Students may be able to identify some of the numerals beyond twenty before
they can identify all the numerals up to twenty.
TEN, DOE(2014)
Counting on and back
Students working within the counting on stage are able to use their knowledge of both
the forward and backward sequences of number words to solve addition and
subtraction questions.
Strategies typically used include:
•counting up from
•counting up to
•counting down from
•counting down to.
TEN, DOE(2014)
Counting on and back
Counting on is a count-by-one mental strategy that
involves starting with the highest number in an
addition equation and then counting on.
5 + 28 =
Start at 28 and add 5.
TEN, DOE(2014)
Counting on and back
Counting back is a count-by-one strategy of learning subtraction
by counting backwards from the highest number in the problem to
get to the answer.
15 - 6 =
Starting at 15 and counting backwards by ones to 9
TEN, DOE(2014)
Counting on and back - counting up
Counting up is a count-by-one strategy of learning
subtraction by counting up from the smaller number
to the bigger number.
23 - 18 =
Start counting up from 18 and stop at 23 = 5
TEN, DOE(2014)
Counting on and back
Teaching focus: Strategy development
As students develop a wider range of arithmetical strategies, teachers need to model
and explain the appropriate use of these procedures in problem solving.
Students need to become competent in selecting and using the most effective
strategy.
TEN, DOE(2014)
Now it’s your turn!
Card Flip
You need:
● A pack of cards
● A whiteboard
Facile or Flexible
The flexible or facile counting stage is characterised by using number properties
combined with number facts.
Flexible strategies make use of the properties of numbers and do not employ counting
by ones. For example, jump strategy is a flexible arithmetical strategy.
Students employ a range of non count-by-one strategies to solve problems.
TEN, DOE(2014)
Mathematical strategies
Moving to non count-by-one strategies
● Doubles & near doubles
● Bridging to 10
● Jump strategies
Doubles & near doubles
Doubles are the same number added together. 3+3=6 is the
same as saying double 3 is 6.
Near doubles is when you use the doubles fact and you then
adjust it by adding or subtracting.
See: 6 + 7
Think: Double 6 + 1 or double 7 - 1
See: 40 + 42 Think: Double 40 + 2
Bridging to ten
Bridging to Ten for addition is where we count on to the next
ten and then count on what is left.
17 + 6 =
Now it’s your turn!
Use Bridging to Ten to solve...
14 + 8 =
Bridging to ten
Bridging to Ten for subtraction is where we count down to the
next ten and then count down what is left.
23 - 8 =
Now it’s your turn!
Use Bridging to Ten to solve…
32 - 7 =
Jump Strategy
Look at 57 + 22.
1. Split the smaller number into tens and ones.
2. First we jump by tens.
3. Then we jump by ones.
*You can also use a hundreds chart...
Now it’s your turn!
Use the jump strategy to solve...
97 + 36 =
Jump strategy- subtraction
Look at 99 - 42.
1. Split the smaller number into tens and ones.
2. First we jump back by tens.
3. Then we jump back by ones.
Now it’s your turn!
Use the jump strategy to solve...
86 - 27 =
Thank you for coming!
Further strategies will be discussed at 2-6 Mathematical
Strategies Parent Information session
Wednesday 3rd August
9:30-10:30