Early Childhood Advisory Group

Math Standards

Effective early childhood teachers must have strong knowledge of (1) the mathematics they teach, (2) children’s mathematical development, and (c) best practices for ensuring that mathematics instruction meshes with and fosters children’s mathematical development.

Foundational mathematical knowledge. Professional development of foundational mathematical knowledge should go beyond what has traditionally been called mathematics content knowledge and focus on “mathematics knowledge for teaching.” Mathematics knowledge for teachingentails more than learning formal mathematical concepts and procedures, it also involves understanding how to translate this formal knowledge into everyday terms or analogies children can understand and learn meaningfully (e.g., relating formal concepts such as division, fractions, and measurement to everyday situations familiar to children such as fair sharing).
Pedagogical knowledge.Professional development of pedagogical knowledge must go beyond general teaching strategies and include best practices for promoting specific and key mathematical concepts and procedures. Understanding such best practices is intimately tied to knowledge of how children’s mathematical knowledge develops.

In effect, knowledge of children’s mathematical development is a cornerstone of professional development because it is necessary for mathematics knowledge of teaching (the mathematics that must be taught) and understanding best practices (knowledge of for effective instruction)
Foundational Mathematical Knowledge
(“mathematics knowledge for teaching”)
Effective professional development must foster the mathematical proficiency of pre-service early childhood teachers. Strong mathematical knowledge for teaching includes (1) mathematical proficiency and (2) a sound grounding in children’s development of mathematical proficiency:

Mathematical Proficiency.Mathematical proficiency involves both conceptual and procedural content knowledge, process capabilities, and affective capacities. Specifically, it encompasses:

Conceptual content. Conceptual understandingof the mathematical content that is taught during the early childhood years (preschool to grade 3) and just beyond (grades 4 to 8). Understanding such content includes an ability to explain and apply mathematical concepts and procedures. As making connections is the basis for conceptual development, the big ideasthat should be fostered during the early childhood yearsare particularly important. Big ideas are general concepts that underlie and connect concepts and procedures within and across domains. An example of a relevant big idea is equal partitioning (dividing a quantity into equal parts), which provides the conceptual basis or rationale for division, fractions, and measurement.Also critical for good teaching areconnections to everyday mathematical applications or real-world analogies. Such knowledge is essential for translating formal mathematical content into instruction that children can understand and learn meaningfully. For example, the formal concept of equal partitioning can be related to the experience familiar to all children of sharing something fairly
Procedural content. Fluency with the mathematical procedures taught during the early childhood years and just beyond. Fluency requires that procedural knowledge be linked to conceptual understanding so each step in a procedure can be explained or a procedure can be readily adapted to solve a novel problem.
Process capabilities. Facility with the processes of mathematical inquiry (mathematical problem solving, reasoning, communicating, and justifying)—particularly those processes that are appropriate for fostering at the early childhood level.
Affective capacities. A productive disposition includes the positive beliefs about mathematics (e.g., nearly everyone is capable of understanding elementary-level) and the confidence to tackle challenging problems and teach mathematics.

Children’s Mathematical Development. Knowledge of children’s mathematical development includes understandinghow children develop mathematical proficiency from birth to age 8 and what conditions foster or impede this development. This involves understanding how informal mathematical knowledge based on everyday experiences develops and provides a basis for understanding and learning formal (school-taught and largely symbolic) mathematics during the early childhood years and beyond. It also involves understanding the developmental progressions of key early childhood conceptsand skills.

/ Pedagogical Knowledge
(“knowledge for effective instruction”)
Effective professional development must foster pre-service early childhood teachers’ understanding of (1) best practices and (2) how best practices are tied to psychology of mathematical development..

Best practices. Knowledge of best practices includes understanding:

The importance of using a variety of teaching techniques (including regular instruction that specifically targets mathematics, integrated instruction, and unstructured and structured play)and how to systematically and intentionally engage children with developmentally appropriate and worthwhile mathematical activities, materials and ideas; take advantage of spontaneous learning moments; structuring the classroom environment to elicit self-directed mathematical engagement; how games in particular can serve as the basis for intentional, spontaneous, or self-directed learning.
The importance of using instructional activities and materials or manipulatives thoughtfully and how to do so for key concept and skills (e.g., why fun activities may or may not be educational, what manipulatives best support a learning goal and why, the limitations and misuses of activities or manipulatives).
The importance of focusing on the meaningful learning of both skills and concepts (not memorization of facts, definitions, and procedures by rote) and thus developing concepts and skills together (e.g., helping children learn the rationale for a procedure and its steps as well as the steps of a procedure) and how to do so with key concepts and skills.
The importance of engaging children in the processes of mathematical inquiry (problem solving, reasoning, conjecturing and communicating/justifying or “talking math”) and how to do so effectively.
The importance of fostering a positive disposition and how to do so effectively (e.g., encouraging children to do as much for themselves as possible), including how to prevent or remedy math anxiety.
The importance of using ongoing assessment in planning and evaluating instruction, targeting student needs, and evaluating student progress.

Psychological development. Knowledge of best practices includes understanding:

The importance of building on what children already know, so that instruction is meaningfully, and how to accomplish this goal—in particular how to relate or connect formal terms and procedures to children’s informal knowledge.
The importance of using developmental progressions in assessing developmental readiness (e.g., identifyingwhether developmental prerequisites for an instructional goal have been acquired), planning developmentally appropriate instruction, and deciding the next instruction step or a remedial plan and how to use developmental progressions effectively.
The limitations of children’s informal knowledge (e.g., misconceptions can it cause) and how developmental inappropriate instruction can cause misconceptions or other learning difficulties and how to address common learning pitfalls.
The progression in children’s thinking from concrete (relatively specific and context-bound) to abstract (relatively general and context free), including the need to help children “mathematize”situations (going beyond appearances to consider underlying commonalities or patterns).

I. Counting and Cardinality
I.A) Understandsthatsubitizing—immediately and reliably recognizing the total number of items in small collections of items and labeling the total with an appropriate number word—is the basis for a learning trajectory of verbal-based number, counting, and arithmetic concepts and skills. Can specify why, for example, learning to subitize goes hand in hand with constructing exact concepts of small number numbers (see A.1 and A.2) and howsubitizing can help children understand one-for-one object counting(see B.5) and that addition makes a collection larger and subtraction makes it smaller (basic concepts of addition and subtraction). / I.A.1 Labels or, better yet, haschildren label a wide variety of examples of a number with an appropriate number word in the following developmentally appropriate order: 1 and 2, next 3 and 4, and then 5 and 6. For instance, while reading a story, prompts children to name all examples of “three” in a picture or asks how many items are depicted in pictures of different things.
I.A.2. Explicitly points out or has children point out non-examples of a number to facilitate the development of exact concepts of small numbers in, for instance, everyday situations (That’s two crackers, not three crackers), while reading stories, or by playing dice games or games such Number—Not the Number.
I.B) Understands the requirements/components/principles of meaningful object counting: / I.B.1. Uses examples and violations of the principles to underscore the requirements of accurate object counting.For example, while playing Tell Me When I’m Wrong, the teacher models using the correct counting sequence and incorrect sequences (skipping or repeating a number word, inserting a number word out of order) or correct and incorrect object counting (e.g., skipping an item, saying two number words while to pointing to an item, coming back to item and giving it a second number-word label).
(1) the same verbal non-repeating counting sequence used must be used every time (stable order principle); / I.B.2. Frequently uses and frequently encouraging children’suse of the counting sequence to facilitate the memorization of the first 10 (and then 12) number words in the counting sequence in the correct order.
I. B.3. Promotes the recognition of counting patterns,the use of rule-governed counting, and awareness of the exceptions to counting rules. For instance, the teens are formed by adding “teen” (four + teen = fourteen; but fif- instead of five is added to teen to make fifteen).
(2) each item in a collection can be labeled with one and only one number word (one-for-one principle); / I.B.4. Can identify violations of the one-to-one principle and provide appropriate scaffolding. For example, if a child fails to keep track of which items have already been counted and recounts previously one or more previously counted items, encourage the child to (with pictured collections) cross out an item as it is counted or (with real objects) to clearly separate the counted items from those that need to be counted.
(3) the last number word used indicates the total or cardinal value of a collection (cardinality principle); / I.B.5. Models one-for-one object counting with small collections that a child can subitize to help him/her see that counting is another way of determining the total (cardinal value) of a collection and understands why the last number word has special significance and is either emphasized or repeated (the last number denotes the total, as well as identify/name the last item).
(4) the order in which the items of a collection are counted does not matter (i.e., does not affect the collection’s cardinal value) as long as the one-to-one principle is observed (order-irrelevance principle); / I.B.6. Asks children to predict what the total will be if a different one-for-one count is used (e.g., “How many will there be if I start with the last item I just counted and count the other way?”).
(5) different types of items (e.g., a block, ball, and top) can be viewed as “things” and counted as members of a collection of “things” (abstraction principle). / I.B.7. After children are successful in applying the first three counting principles with sets of like items (homogeneous collections), provides experience with small and then larger sets of different items (heterogeneous collections).
I.C. Can identify key, more advanced verbal and object counting skills on the learning trajectory for counting and cardinality and how these skills are logically and developmentally related.
(1) counting out a collection of a specified number of items (from a larger collection) is a more challenging both conceptually and skill-wise than counting a collection one-to-one; / I.C.1. Introduces counting out a collection (e.g., “take two crayons” or “give me three pencils”) with small collections a child can subitize and with larger collections only after the child reliably uses the cardinality principle of one-for-one counting.
(2) specifying the number after a given without counting from “one” is more difficult than doing so by counting from “one” (using a “running start”); / I.C.2. Provides practice specifying number-after relations first with a running start (e.g., “After one, two, three, comes what when we count?”) and—when a child can do this, then without a running start e.g., “After three, comes what when we count?”).
(3) counting on from a number more difficult than counting from 1; / I.C.3. Provides practice counting on after a child can specify number-after relations (e.g., “Start with three and keep counting”).
(4) counting backwards (from up to 10) is more difficult than counting on. / I.C.4. Use a number list, a microwave time, or a count down to an event (e.g., “The race will begin in 10, 9, 8…1 second, go”) to provide practice counting backward after a child knows the counting sequence well enough to count on.
I.D) Understands how children’s ability to make verbal-based magnitude comparisons develops, including the mathematical ideas this entails.
(1) Children first learn relational terms as “more” or “fewer.” / I.D.1. First ensures that children understand relational terms as “more” or “fewer” by using collections that are obviously different (collections involving one to three items or with any two collections in which the larger is more twice as large as the smaller) in the context of everyday situations, playing math games, or teaching other content, such as reading a story.
(2) Next, at about 4 years of age, children discover theincreasing-magnitude principle(the later a number word comes in the counting sequence, the larger the collection it represents). This probably occurs first as a result of subitizing small numbers and then generalizes to the rest of the counting sequence. The increasing-magnitude principleenables children to compare number words that are widely spaced apart in the counting sequence. / I.D.2. Next, helps children construct increasing-magnitude principle with small collections they can subitize and by specifying which collection is more. Provides practice comparing relatively small numbers (e.g., “Who is older a child that is 3 years old or 2 years old?”) or numbers well separated in the counting sequence (“Who is older a child that is 9 years old or 2 years old?). If a child can read numerals, uses a cardinality chart to underscore the increasing-magnitude principle.
(3) About 5 or 6 years of age, children use one-for-one counting and the increasing-magnitude principleto determine the larger of two (non-subitizable) collections that are not obviously different. / I.D.3. Encourages children to use counting to determine the larger of two collections first with small collections they can subitize and then with larger collection. For example, if Jacob has a score of five (represented by five blocks) and Derye has six (represented by six blocks), the children can each count their collection of blocks to see who counted the furthest. If necessary, provides scaffolding by counting Jacob blocks and then counting Derye’s blocks and emphasize that Jacob’s count has been surpassed: “Jacob has five, and Derye has ‘one, two, three, four, f-i-v-e, SIX. If further, explicit scaffolding is needed, the teacher adds: “Six is more than five, because six comes after five when we count.”
(4) Between 5 and 7 years of age, children can fluently determine which number comes after another in the counting sequence and thus mentally compare even two number words that are counting neighbors and do so efficiently. / I.D.4. After ensuring that a child knows the increasing-magnitude principle and is fluent with number-after relations (e.g., knows that “when we count, after seven comes … eight), encourages the child tocompare mentally two number neighbors and providesengaging practice.Engaging practice might involve a math game, such as Dominoes-Number After (like dominoes except that the number after is matched to an end number) or Car Race(in which a player draws a card and must decide which of two number neighbors is larger in order to move his/her racecar the most spaces on a racetrack).
(5) Understands that the importance of the successor principle (each succeeding number word in the counting sequence is exactly one more than its predecessor) as a key connection among counting, magnitude representations, and addition. / I.D.5. In building a cardinality chart or other model of the counting sequence, explicitly points out that that one must be added to make the next number or, better yet, asks children, for example, “If you already have 4 chips, how many more chips do you need in order to have 5 chips?”
(6) Understandthat identifying “first” and “last” is the first important step toward understanding the ordinal numbers. / I.D.6. Uses and encourages children to use “first” and “last” appropriately as needed in everyday situations.
I.E) Understands why written numbers (numerals) are valuable tools (e.g., can serve as a memory aid; make written calculation with large numbers easier or even possible) and how to promote the meaningful learning of the numeral reading and writing to 10.
(1) Numbers and thus their written representations (e.g., numerals)have four meanings/roles: indicate the total of a collection (cardinal meaning); specify position, order, or relative size (ordinal meaning); represent a measurement (measurement meaning); or serve as a name (nominal meaning; e.g., Bus 4). / I.E.1. Explicitly points out that we use numbers in several different ways—to tell us: how many (the total of a collection), where (order), how much (a measure), or what (a name).The relation between the cardinal and the ordinal or measurement meanings of numerals can be underscored by using a cardinality chart and later a number list