Addition and Subtraction - Arithmetical Strategies
Conceptual Construct
Construct 0
Emergent Counting
Construct 1
Unitary
Perceptual Counting
Construct 2
Figurative Counting
Construct 3
Initial Number
Sequence
Composite
Construct 4
Intermediate Number
Sequence
Construct 5
Facile Number
Sequence
Brief Description
Cannot count visible items. The student either does
not know the number words or cannot coordinate the
number words with items or cannot use a cardinal
number to quantify the collection.
Can count perceived items but not those in concealed
collections. This may involve seeing, hearing or
feeling items.
Can count concealed items using a re-presentation,
but counting typically includes what adults might
regard as redundant activity. For example, when
presented with two screened collections, told how
many in each and asked to find the total, the student
will count from “one” instead of counting on.
The student counts-on rather than counting from “one”,
to solve addition or missing addend tasks. The
student uses a count-down-from strategy to solve
removed items tasks (e.g., 17 - 3 as “16, 15, 14; the
answer is 14!”).
The student counts-down-to to solve missing
subtrahend tasks (e.g., 17 - 14 as 16, 15, 14; the
answer is 14!”). The student can choose the more
efficient of count-down-from and count-down-to
strategies.
The student uses a range of non-count-by-one
strategies. For example, in additive and subtractive
situations, the student uses strategies such as
compensation, using a known result, adding to ten,
commutativity, subtraction as the inverse of addition,
and awareness of the “ten” in a teen number.
Development of Forward Number
Word Sequences (FNWS)
Level
Level 0
Brief Description
The student cannot produce the FNWS from "ten" to "one".
Emergent FNWS
Level 1
Initial FNWS
to “ten”
Level 2
Intermediate
FNWS to “ten”
Level 3
Facile with
FNWS to “ten”
Level 4
Facile with
FNWS to “thirty”
Level 5
Facile with
FNWS to “one
hundred”
The student can produce the FNWS from "ten" to "one". The student
cannot produce the number word just after a given number word in the
range "ten" to "one". Note: Students at Levels 1, 2, and 3 may be able to
produce FNWS beyond “ten”.
The student can produce the FNWS from "ten" to "one". The student can
produce the number word just after a given number word in the range "ten"
to "one", but drops back to generate a running count when doing so.
The student can produce the FNWS from "ten" to "one". The student can
produce the number word just after a given number word in the range "ten"
to "one", but drops back to generate a running count when doing so.
The student can produce the FNWS from “one” to “thirty”. The student can
produce the number word just after a given number word in the range “one”
to “thirty” without dropping back. Note: Students at this level may be able
to produce FNWS beyond “thirty”.
The student can produce FNWS in the range “one” to “one hundred”. The
student can produce the number word just after a given number word in the
range “one” to “one hundred” without dropping back. Note: Students at
this level may be able to produce FNWS beyond “one hundred”.
Development of Backwards Number
Word Sequences (BNWS)
Level
Level 0
Brief Description
The student cannot produce the FNWS from “ten” to “one”.
Emergent BNWS
Level 1
Initial BNWS
to “ten”
Level 2
Intermediate
FNWS to “ten”
Level 3
Facile with
FNWS to “ten”
Level 4
Facile with
FNWS to “thirty”
Level 5
Facile with
FNWS to “one
hundred”
The student can produce the BNWS from “ten” to “one”. The student
cannot produce the number word just before a given number word in the
range “ten” to “one”. Note: Students at Levels 1, 2, and 3 may be able to
produce BNWS beyond “ten”.
The student can produce the BNWS from "ten" to "one". The student can
produce the number word just before a given number word in the range
"ten" to "one", but drops back to generate a running count when doing so.
The student can produce the BNWS from "ten" to "one". The student can
produce the number word just before a given number word in the range
"ten" to "one", but drops back to generate a running count when doing so.
The student can produce the BNWS from “one” to “thirty”. The student can
produce the number word just before a given number word in the range
“one” to “thirty” without dropping back. Note: Students at this level may be
able to produce FNWS beyond “thirty”.
The student can produce BNWS in the range “one” to “one hundred”. The
student can produce the number word just before a given number word in
the range “one” to “one hundred” without dropping back. Note: Students at
this level may be able to produce FNWS beyond “one hundred”.
Development of
Numeral Identification
Level
Level 0
Brief Description
The student cannot produce the FNWS from “ten” to “one”.
Emergent
Level 1
Numerals
to “10”
Level 2
Numerals
to “20”
Level 3
Numerals to
“100”
Level 4
Facile with
FNWS to “thirty”
Level 5
Facile with
FNWS to “one
hundred”
The student can produce the BNWS from “ten” to “one”. The student
cannot produce the number word just before a given number word in the
range “ten” to “one”. Note: Students at Levels 1, 2, and 3 may be able to
produce BNWS beyond “ten”.
The student can produce the BNWS from "ten" to "one". The student can
produce the number word just before a given number word in the range
"ten" to "one", but drops back to generate a running count when doing so.
The student can produce the BNWS from "ten" to "one". The student can
produce the number word just before a given number word in the range
"ten" to "one", but drops back to generate a running count when doing so.
The student can produce the BNWS from “one” to “thirty”. The student can
produce the number word just before a given number word in the range
“one” to “thirty” without dropping back. Note: Students at this level may be
able to produce FNWS beyond “thirty”.
The student can produce BNWS in the range “one” to “one hundred”. The
student can produce the number word just before a given number word in
the range “one” to “one hundred” without dropping back. Note: Students at
this level may be able to produce FNWS beyond “one hundred”.
Development of Backwards Number
Word Sequences (BNWS)
Level
Level 0
Emergent
Level 1
Facile
Structures to 5
Level 2
Intermediate
Structures to 10
Level 3
Facile
Structures to 10
Level 4
Intermediate
Structures to 20
Level 5
Facile
Structures to 20
Brief Description
The student can subitize only small quantities (ip to 3) and relies on
counting to quantify larger groups. The student builds finger patterns by
raising fingers sequentially.
The student can subitize regular spatial patterns to 6 and irregular spatial
patterns to 5. The student can create finger patterns in the range of 1 to 5
by raising fingers simultaneously. The student is able to combine and
partition numbers in the range of 1 to 5 without counting.
The student can recognize spatial patterns and create simultaneous finger
patterns in the range of 6 to 10 using both five-wise (5-plus) and pair-wisw
(doubles) structures. The student may be unable to give other
combinations and partitions in this range.
The student is able to use the sub-base of 5 and other structures to
combine and partition numbers in the range of 1 to 10 without counting.
The student is able to combine 10 and a number in the range of 1 to 10,
without counting. The student is able to give partitions involving 10 for
numbers in the range of 11 to 20. The student may be unable to give other
combinations and partitions in this range.
The student is able to use a base of 10, the sub-base of 5, and other
structures to combine and partition numbers in the range of 1 to 20 without
counting.
Classroom Instructional Framework for Early Number (CIFEN)
Number Words
Numerals
Addition and
Subtraction
Structuring
Numbers
Place Value
Multiplication and
Division
Chapter 3
Chapter 3
Chapters 4 & 6
Chapters 5 & 7
Chapters 18 & 9
Chapter 10
Forward/Backward
Number Word
Sequences to 10
Emergent Counting
Early Spatial Patterns
and Finger Patterns
Perceptual Counting
Facile Structures to 5
Numerals to 5
Forward/Backward
Number Word
Sequences to 30
Numerals to 10
Forward/Backward
Number Word
Sequences to 100
Numerals to 20
Figurative Counting
Intermediate
Structures to 10
Numerals to 100
Initial Number
Sequence
(Counting-on and
Counting-back)
Facile Structures to 10
Forward/Backward
Number Word
Sequences for
Composite Units
Numerals to 1000
and beyond
Facile Number
Sequence
Tens or Ones
Perceptual Counting
in Multiples
Tens and Ones with
Materials (Jump &/or
Split)
Figurative Composite
counting
Tens and Ones
with/out Materials
(Jump &/or Split)
Repeated Abstract
Composite Grouping
Mental computation
involving addition
and subtraction with
2- and 3- digit
numbers
Multiplication and
Division as
Operations
Intermediate
Structures to 20
Facile Structures to 20
Number Word
Sequences to 1000
and Beyond
Initial Grouping