MATHEMATICS CONTINUUM
1. AWARENESS STAGE
Rote counts to ten or more but with little understanding of the fact that each is one more than
the preceding number
Sorts objects, not by common characteristics but by perceptual and positional relationships --
making designs or pictures with names
2. EMERGENT STAGE
In this stage of mathematical development, children rely on concrete objects to explore,
communicate, and learn about mathematics. Children begin to sort, recognize and create patterns,
count, and solve problems using objects. They describe their thinking using pictures, numbers, and
words.
Sorts by one attribute or more e.g., big red ones, little red ones, big blue
Relies heavily on perceptual clues -- what something looks like, for example two strings of the
same length may be perceived as different in length depending on whether they are stretched
out or coiled in some way.
Rote counts but just beginning to understand the concept of numberness -- what makes 5 or
what makes 6 for example.
3. EARLY STAGE
In this stage of mathematical development, children apply numbers to represent groups of
objects. Measurement is used to compare and estimate quantities, while basic geometric attributes
become more familiar. Their ability to see the relationship between addition and subtraction
becomes more apparent as they work with objects. They use patterns, models, and stories to
explain mathematical thinking. They construct strategies for single digit addition and subtraction
using objects to model the action represented in story problems.
Uses concrete objects to represent cardinal numbers.
Can order a group of objects by one attribute, perhaps by color or by shape or by size
Can solve simple story problems (n+n= __) by modeling the quantities involved. May begin to use
counting strategies (e.g., counting up from the first quantity)
Begins to write and solve simple subtraction and addition problems
4. DEVELOPING STAGE
In this stage of mathematical development, children transition from objects to symbolic
representation using numerals, symbols, diagrams, charts, and models. Students recognize, create,
and illustrate problems by joining and separating groups. They construct strategies for additon and
subtraction using counting up or down strategies and begin to derive answers using facts they know.
For example, 6+5=11 because 5+5=10. They are developing an understanding of the number fact
familes through 9s and are beginning to develop place value concepts using objects.
Can sort a collection of objects by several different attributes e.g., first by color, then by shape
Can name and order the fact familes through 9
Uses objects and symbols to represent the fact families from 10 - 19
Can write addition and subtraction story problems
Uses objects to develop place value concepts
Begins to tell how they solve various types of story problems
Understands that multiplication is successive addition and division is successive subtraction
using objects and drawings
5. TRANSITIONAL STAGE
Children at this stage are using counting and deriving strategies to solve various types of story
problems and can communicate their cognitive process - how they know - both verbally and by
using numerals and symbols. They are extending their understanding of place value using objects
connected to symbols .
Can create a matrix by organizing a set of objects by two attributes simultaneously e.g., rows
might be color designations and columns might be shape designations
Can select a solution strategy for many types of addition and subtraction story problems
Can solve some problems by deriving the answer from known facts
Begins to solve 2-place addition and subtraction problems mentally.
Begins to solve multi-step problems choosing from a variety of strategies
Can name and list all the fact families through 19
Begins to add and subtract 2-place and 3-place quantities using symbols and numerals
6. INDEPENDENT STAGE
In this stage students use symbolic language to represent mathematical thinking. They use
diagrams, arrays, and models to clarify and communicate. Students recognize, create, and illustrate
problems using all four operations: addition, subtraction, multiplication, and division. They develop
skills with multi-digit addition and subtraction. They use rounding, estimation, and place value to
make appropriate predictions, solve problems, and recognize the reasonablenss of their answers.
They can write 7-digit numbers correctly with ease.
Uses deriving strategies for all types of addition and subtraction story problems when solving
mentally
Understands place value computation through tens and hundreds both mentally and symbolically
