In 2023, Clements and Samra published a series of learning trajectories for many early math concepts working with students from birth to age 8. “Children follow natural developmental progressions in learning,” (Sarama & Clements, 2023) these trajectories corresponded with the number sense strands as well as other areas of mathematics. In taking a closer look at the trajectories directly related to number sense and how they have been divided into the number sense strands the skills listed in the trajectories match with the Western Australia Phases of Development, breaking the first three phases of development into concrete chunks. The challenge of the learning trajectories is that they do not give skill development guidance past age eight or grade three. While these are great tools for the early and primary years additional trajectories are needed for the intermediate and middle years.
Counting
“To count a set of objects, children must learn to count verbally – saying the list of number names in order, as well as learning the system that generates new number names. Counting includes 1) the ability to say number words in correspondence with objects (enumerate objects), 2) understanding that the last number word said when counting refers to how many items have been counted (cardinality), and 3) using counting strategies to solve problems.” (Clements & Sarama, 2023)
| Age | Range | Trajectory Level | Descriptor |
| 0 | 1 | Number Word Sayer | No verbal counting. Names some number words No sequence |
| 1 | 2 | Chanter | Chants number words May run number words together. May begin copying an adult placing objects one at a time |
| 2 | 2 | Reciter | Says distinct number words. May not say the words in the correct order. If more number names are known than objects complete the list If more objects than known number names repeats number names |
| 3 | 4 | Counting: Reciter (10) | Verbally counts to ten. Some correspondence to objects May skip or double count objects |
| 3 | 4 | Corresponder | Keeps one to one correspondence between objects and number words |
| 4 | 4 | Counter (Small Numbers) | Accurately counts objects to five. Knows that the last number said is the total in the collection |
| 4 | 4 | Counter (10) | Counts objects to ten. Understands cardinality. May read and write numerals 0-10. Developing ability to count to twenty verbally |
| 4 | 5 | Producer (Small Numbers) | Counts out objects to five. Knows they need to count when a set number of objects is needed. Produces a group of four objects |
| 5 | 6 | Counter And Producer (10+) | Counts out objects accurately to ten. May go beyond ten (possibly to 30) Explicit understanding of cardinality Tracks objects that have and have not been counted. Writes or draws to represent 1-10 (possibly to 30) Gives next number to 10 (possibly to 30) Separates tens and one’s parts of number words. Relates parts of a number word to quantity Recognizes errors in others counting. May correct errors in own counting if prompted |
| 5 | 6 | Counter Backwards Form 10 | Counts backward from 10 to 1 verbally. Counts backward to remove objects from a group |
| 6 | 7 | Counter From N (N+1, N-1) | Counts verbally and with objects from numbers other than one. Does not yet keep track of the number of counts. Immediately determines numbers immediately before or after |
| 6 | 7 | Skip Counting By 10s To 100 | Skips counts by ten to one hundred or beyond with understanding. Sees groups of ten within a quantity Counts groups of ten |
| 6 | 7 | Counter To 100 | Counts to one hundred Makes decade transitions starting at any number |
| 6 | 7 | Counter On Using Patterns | Keeps track of counting acts using numerical patters Uses patters to add from 1 to 3 |
| 6 | 7 | Skip Counter | Counts by fives and twos with understanding |
| 7 | 7 | Counting: Counter On Keeping Track | Counts forward or back from a given number keeping track of counting acts. Uses objects first. Begins to keep track of how many times they counted backwards |
| 7 | 8 | Counter Of Quantitative Units / Place Value | Uses base ten numeration and place value concepts. Includes counting in units, and multiples of tens, hundreds and ones. When counting groups of ten can decompose to ones if needed Uses the placement of a digit within a number to assign value. Counts unusual units such as wholes when shown combinations of parts and wholes |
| 7 | 8 | Counter Beyond 100 | Accurately counts beyond one hundred. Uses patterns of ones, tens and hundreds |
| 7 | 8 | Number Conserver | Consistently conserves number – recognizes that the number has been unchanged. Conserves number even when distracted or disrupted |
| 7 | 8 | Counter Forward and Back | Counts counting words in either direction. Recognizes that decade’s sequence mirrors single digit sequence. Is able to switch between sequencing and composing/ decomposing multidigit numbers easily |
(Clements & Sarama, 2023)
Subitizing
Subitizing is the ability to instantly recognize and name the number in a group without directly counting. Subitizing begins in infancy and is related to students ANS. Students begin subitizing small quantities and with practice begin to use groups and quickly combine them into one quantity.
| Age | Range | Trajectory Level | Descriptor |
| 0 | 1 | number senser | Inborn sense for number No explicit knowledge of number Distinguishes between groups on 1 and 2. May sense groups of two and three May show sensitivity to ratios of larger numbers. Pre-math skills |
| 1 | 2 | Very small number recognizer | Begins to connect small quantities to number words. Begins to understand cardinality. After first birthday typically learns the number words for one and two Begins to understand more and less. Begins to understand that all groups labeled with the same number word contain the same amount |
| 2 | 3 | Maker of small collections | Makes a small collection of 1-3. With the same number as another collection May match another collection. May not be verbal |
| 2 | 3 | Small collection namer | Names groups of one to3 with accuracy Begins to recognize and make groups of four. Can name small groups with out using a model and matching |
| 3 | 4 | Perceptual subitizer to four | Instantly names collections up to four when shown briefly (2-3 seconds) |
| 4 | 4 | Perceptual subitizer to five | Labels all groupings and arrangements to five. Able to identify the parts and name the wholes. Collections seen briefly (2-3 seconds) Extends known spatial and numeric structures |
| 4 | 4 | Conceptual subitizer to five | Verbally labels all arrangements to five when shown briefly (2-3 seconds) Identify whole quantity as a combination of smaller quantities |
| 4 | 6 | Conceptual subitizer to seven | Verbally label all arrangements to seven Shown briefly (2-3 seconds) |
| 5 | 6 | Conceptual subitizer to ten | Labels all arrangements from 1-10 when shown briefly (2-3 seconds) Recognizes most combinations to ten. Uses ten frames to help recognize larger quantities |
| 6 | 7 | Conceptual subitizer to twenty | Labels all structured arrangements to twenty shown briefly (2-3 seconds) Uses parts to know the whole. Knows common groupings and pairs to make 10 and 20 |
(Clements & Sarama, 2023)
Comparing numbers
Comparison involves students’ ability to order numbers, determine which is least and greatest and sequence numbers.
| Age | Range | Trajectory Level | Descriptor |
| 0 | 1 | Comparison sensor: foundations | Sense small changes in small collections. Sense large changes in larger collections. Make simple equivalence comparisons using senses |
| 1 | 3 | Early comparison corresponder | Recognizes small collections that have the same number. Intuitively matches or corresponds objects between each. May put objects, words or actions into one-to-one or many-to-one correspondence |
| 2 | 3 | Perceptual comparer | Compares collections that are very different (two or three times greater) Compares similar collections involving small numbers. Compares collections using number words one and two |
| 3 | 3 | First-second ordinal counter | Identifies the first and second objects in a sequence |
| 3 | 3 | Early comparer of similar items | Compares one to four items verbally or nonverbally by looking. Items in the collection must be the same. May compare the small collections using number words two and three |
| 3 | 4 | Early comparer of dissimilar items | Matches small equal collections of different items. Able to show they are the same number |
| 4 | 4 | Matching comparer | Compares groups of 1-6 by matching |
| 4 | 4 | Counting comparer (same size) | Compares accurately through counting when objects are the same way and groups are less than five. Size of objects and collection influence accuracy Less accurate with a large number of small objects Accurately counts two equal collections. Will say the group with larger objects has more |
| 5 | 5 | Spatial extent estimator – small/big | Estimates which set is more or less if differences are obvious. Names a small number less than five for sets that cover a small space and a big number (10-20) for sets that cover large spaces. Estimations of little / big may be influenced by the size of the objects |
| 5 | 5 | Counting comparer (5) | Compares with counting even when larger collections objects are smaller. Figures out how many more or less |
| 5 | 5 | Mental number line to five | Uses knowledge of counting number relationships to determine relative size and position with perceptual support |
| 5 | 5 | Serial orderer to five | Orders quantities and numerals to five Orders lengths marked into units |
| 5 | 5 | Counting comparer (10) | Compares with counting even when larger collections objects are smaller. Up to ten |
| 5 | 5 | Ordinal counter | Identifies and uses ordinal numbers from first to tenth |
| 5 | 5 | Mental number line to ten | Uses internal images and knowledge of number relationships to determine relative size and position |
| 6 | 6 | Serial order to 6+ | Orders quantities or numerals to six and beyond Orders lengths marked into units |
| 6 | 6 | Spatial extent estimator | Extends sets and number categories to include small numbers which are subitized not estimated, middle sized numbers (10-20) and larger numbers. The arrangement affects the difficulty |
| 7 | 7 | Place value comparer | Compares numbers using place value understanding |
| 7 | 7 | Mental number line to one hundred | Uses knowledge of number relationships and mental images Including how ones can be imbedded in tens to determine relative size and position |
| 7 | 7 | Scanning with intuitive quantification estimator | Scans a group of objects and relates the results toa mental number line to perform useful numerosity estimations |
| 7 | 8 | Mental number line to one thousand | Uses mental images and knowledge of number relationships, including place value to determine relative size and positions |
| 8 | 8 | Benchmarks estimator | Counts a portion of the to-be-estimated collection and uses that as a benchmark from which an estimate is made. May work intuitively or using repeated addition or multiplication |
| 8 | 8 | Composition estimator | Decomposes or partitions the to be estimated set into convenient subset sizes then recomposes the numerosity. Initially done with regular arrangements using repeated addition and multiplication Develops into working with irregular arrangements and children use multiplication skills consistently to recompose. |
(Sarama & Clements, 2023)
Fractions
“The development of conceptual and procedural knowledge of fractions is critical for children. An understanding of fractions will include clear concepts of whole numbers versus fractions, including the idea that a single unit fraction of 1/2 is larger than 1/4, despite the denominator in one-fourth being larger.” (Clements & Sarama, 2023)
| Age | Range | Trajectory Level | Descriptor |
| 0 | 3 | Early proportional thinker | Has an intuition about proportions |
| 3 | 3 | Shape equipartitioner | Can equipartion a whole shape such as a circle or rectangle |
| 4 | 5 | Half recognizer | Recognizes halves at least in continuous area representations. In the context of fair shares Recognition of the need for ½ when sharing an odd number of objects Institutively and visually combines regions that are part of a whole. Showing initial foundations for addition |
| 4 | 5 | Unit fraction recognizer | Recognizer in simple discreet (countable) and maybe continuous (area) representations for ½, 1/3, and ¼ and understands intuitively that they are formed by dividing a whole into equal parts. Names these shares |
| 5 | 6 | Fraction recognizer | Recognizes simple (small number denominator) fractions in familiar continuous and discrete contexts |
| 6 | 7 | Fraction maker from units | Creates a fraction representation with equal parts and the correct number of repetitions of a unit or non unit fraction (as long as they are not greater than the whole-fraction my be only a part-of-a-whole) Compare simple common fractions using physical models |
| 7 | 8 | Fraction repeater | Creates fraction repetitions of unit and non-unit fractions, including results greater then a whole. Moves beyond fraction as a part-of-a-whole to a fraction with the same denominator can be added or subtracted using the units of the same unit fraction. Compares simple common fractions using models such as the number line – understands that two factions are equal when they represent the same portion of a whole or have the same length on the number line |
| 8 | 8 | Fraction arithmetic +/- | Adds and subtracts simple common fractions using physical models |
| 8 | 8 | Fraction arithmetic x/ ÷ | Multiplies simple common fractions using rectangle arrays model |
| 8 | 8 | Fraction and integer sequencer | Represents simple ratios as percentages, fractions and decimals. Orders integers, positive fractions and decimals |
(Sarama & Clements, 2023)