EARLY CHILDHOOD ADVISORY GROUPMATH 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.

1. 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 teaching entails 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).

2. 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:

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

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 ideas that should be fostered during the early childhood years are 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

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..

1. Best practices. Knowledge of best practices includes understanding: a. 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.

b. 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

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

equal parts), which provides the conceptual basis or rationale for division, fractions, and measurement. Also critical for good teaching are connections 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

b. 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.

c. 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.

d. 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.

2. Children’s Mathematical Development. Knowledge of children’s mathematical development includes understanding how 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

may or may not be educational, what manipulatives best support a learning goal and why, the limitations and misuses of activities or manipulatives).

c. 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.

d. 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.

e. 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.

f. The importance of using ongoing assessment in planning and evaluating instruction, targeting student needs, and evaluating student progress.

2. Psychological development. Knowledge of best practices includes understanding:a. 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.

b. The importance of using developmental progressions in assessing developmental readiness (e.g., identifying whether developmental prerequisites for an instructional goal have been acquired), planning developmentally appropriate instruction, and deciding

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

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 concepts and skills.

the next instruction step or a remedial plan and how to use developmental progressions effectively.

c. 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.

d. 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 CardinalityI.A) Understands that subitizing—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 how subitizing 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, has children 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’s use 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

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

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

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

reading a story. (2) Next, at about 4 years of age, children discover the increasing-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 principle enables 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 principle to 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 to compare mentally two number neighbors and provides engaging 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) Understand that 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 I.E.1. Explicitly points out that we use numbers in several different ways—

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

(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).

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

(2) Reading numerals entails constructing a mental image that embodies distinguishing characteristics of a numeral—its parts (e.g., both 6 and 9 consists of a “stick” and a “loop”) and more importantly part-whole relations—how the parts fit together including their orientation (e.g., the loop of a 6 is at the bottom right of the stick; whereas the loop of a 9 is at the top left of the stick ).

I.E.2. For children who cannot read a numeral, explicitly helps them to identify the parts of a numeral by, for instance, color-coding them, encouraging children to do so, code, or prompting them to make the parts from clay. Explicitly helps children identify the part-whole relations of a numeral by, for instance, having them assemble the clay parts to form a numeral and explicitly discussing how to do so properly.

(3) Writing numerals requires learning a motor plan—a plan that translates the mental image into a series of motor actions that create the written numeral (e.g., for 6, start at the top right, draw a stick toward the bottom left, and make a loop to the bottom left).

I.E.3. For children who read, but cannot write, a numeral, use writing paper with directional clues and explicitly talk them through the motor plan and then encourage the child to specify this step-by-step plan for writing the numeral.

I.F) Understands the role of estimation (useful when exact answers are not possible or an approximate answer is sufficient) and why children resist estimating answers (fear of being wrong, obsession with the correct answer as reinforced by the guess-and-check).

I.F.1. With sets (number estimation) first and then operations (arithmetic estimation), helps children understand the value of estimation by explicitly discussing when estimation is useful and practicing estimation in contexts that imitate real situation (e.g., where there is a need for a quick approximate answer or an exact answer is not possible). I.F.2. With sets (number estimation) first and then operations (arithmetic estimation), exhibits best practices such as, *begins with the range of collections and arithmetic problems just beyond a child’s expertise to determine exactly and move to progressively larger collections.*encouraging children to give a range of plausible answers or the upper and lower limit of what would be reasonable, *providing feedback on the accuracy of estimates by using a range (e.g., a good educated guess would be ten to fifteen) * promoting the understanding and use of terms related to estimation (e.g., “about,” “near,” “closer to,” “between,” “a little less than”) by regularly modeling estimation efforts with such terms.

II. Operations and Algebraic ThinkingII.A) Understands what addition and subtraction concepts and skills children need to learn in early childhood. (1) Understand that the idea that “adding objects to a group increases quantity and taking objects away from a group decreases quantity” is the foundation for addition and subtraction. connect this idea to children’s experiences.

II.A.1. Provides preschoolers opportunities to (a) experiment with adding and subtracting 1 or a few items with small collections that they can subitize and discuss whether a collection got bigger or smaller and (b) solve simple non-verbal addition and subtraction problems involving adding the addition of one object and one object (1 & 1), 1 & 2, 1 & 3, 2 & 1, 2 & 2, 3 & 1, 2 objects take away 1 object (2-1), 3-1, 3-2, 4-1, 4-2, and 4-3 in the context of, for instance, the Hiding Game.

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

II.A.2. Encourages primary-grade children to use qualitative reasoning to predict the direction of an answer to word problems, including missing addend problems—that the sum (whole) of simple change add-to problems should be more than either number added (part) and that the difference (part) for change take-away problems should be smaller than the starting amount (whole) and later with a missing-addend problem that the missing part must less than the whole.

(2) Can describe, differentiate among, and illustrate with a word problem key meanings of addition and subtraction.

II.A.2. Initially encourages children to solve simple change-add-to with sums up to 5 and related change-take-away problems and then part-whole, equalize, and compare problems informally (by using fingers, objects, marks, or drawings to directly model the meaning of a problem). As children are ready, provide more challenging problems by increasing sums to 10 and then 18, introducing less familiar types of word problems (those with a part-whole, equalize, and comparison meaning), and more difficult types of word problems (missing addend problems).

(3) Understands how to link formal symbols of the plus sign and the minus sign to children’s existing, informal knowledge of addition and subtraction.

II.A.3. Helps children understand the formal representations of addition and subtraction by relating formal symbols of the plus sign and the minus sign initially to children’s informal view of these operations (change add-to or change take-away) and by encouraging them to translate change add-to or change take-away problems into written expressions or equations and vice versa. In time, relates other meanings of addition and subtraction to expressions and equations.

(4) Understand the phases of mental-addition/subtraction development and the central role of discovering patterns and relation and using reasoning strategies in achieving fluency with basic sums and differences.

II.A.4. Supports children’s initial use of counting strategies to solve word problems and expressions and the invention of more efficient counting strategies, such as counting-on. Then encourages children to search for patterns and relations, discuss/share discovered regularities, and devise and share/discuss reasoning strategies—strategies for logically solving unknown basic combinations by using a known relation and known combination (e.g., recognizing that near doubles such as 4+5 can be solved by using the known double 4+4=8 and add-1 combination 8+1=9). Only after children learned patterns, relations, and reasoning strategies encouraging fast responses.

(5) Understand the important role of composition an decomposition of numbers and other number for a name concept (e.g., 5 = 0+5, 1+4, 2+3, 3+2, 4+1, and 5+0)

II.A.5. Provides opportunities to decompose a number into parts or compose a number from parts in various ways. II.A.6. Encourages the use of making ten to solve relatively difficult unknown combinations with sums greater than 10 (e.g., 5+9 = 5+1+9 = 5+10 = 15

(6) Understand key principles of addition and subtraction in the early childhood curriculum, such as the difference between commutativity (changing the order in which numbers are added does not affect the sum) and associativity (changing the grouping of numbers such as 1+3+6 by adding 1 and 3 first and then adding 4 to 6 or adding 1 to 3+6 or 9 does not affect the outcome) and how addition and subtraction are related, including adding a number can be undone by subtracting the same number (inversion) and unknown subtraction problems can be solved by using related and known

II.A.6. With a nonverbal addition and subtraction task, provides opportunities for children to recognize that the sum of a number and adding nothing is still the original number (additive identity); the difference of a number and taking away nothing is still the original number (subtractive identity); the difference of a number and taking away an equal number leaves nothing (negation); the addition of a number and then taking away the same number leaves the original number unchanged (inversion).II.A.7. Helps children understand and learn the subtraction-as-addition reasoning strategy (e.g., 5 – 3 = ? can be viewed as ? + 3 = 5) by (a) providing examples of adding a number to an initial number and then reversing the process by subtracting the number to restore

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

addition combinations. the initial number (e.g., 5 + 3 = 8 and then 8 – 3 = 5; “empirical inversion”); (b) labeling combinations in an “addition-subtraction family” (e.g., 3 + 5, 5 + 3, 8 – 5 = 3 and 8 – 3 = 5) with part and whole labels (e.g., labeling the 3s as a “part,” labeling the 5s as a “part,” and labeling the 8s as a “whole”); and explicitly asking whether, for instance, 3 + 5 = 8 can help solve 8 – 5 = ? and why.

II.B) Understands the formal meaning of relational symbols and how these symbols are or can be interpreted by children. (1) Understands the difference between a formal, relational view of the equals sign (“same as” or “same number as”) and children’s informal, operational view of this key symbol (e.g., “adds up,” “makes,” “produces the answer”) and recognizes that ≠ indicates “not equal.”

II.B.1. Fosters a relational view of the equals and the not equals sign by (a) introducing these symbols in the context of equal and unequal collections (e.g., o

oo = 000; o

oo ≠ oooo), then

equal and unequal collections and numerals oo

o = 3; oo

o ≠ 4), a variety of everyday contexts (e.g., 7 days = a week, two 5¢ [nickels] = one 10¢ [dime]), and only then in the context of addition and subtraction; (b) refers to the equals sign interchangeably as “the same number as” or “same as” and “equals”; using atypical equations (e.g., ☐ = 5 + 3, 5 + 3 = 4 + ☐, 5 + 3 = 9 – ☐ as well as typical ones (5 + 3 = ☐).

(2) Can differentiate between the “greater than” and the “lesser than” symbols and be able to relate these relational symbols a meaningful analogy such as the “the mouth is open to the larger amount” (e.g., for 3 ☐ 5, the symbol that goes in the box is < because 3 is less than 5 and the mouth is opened toward the 5: 3 < 5)

II.B.2. Uses the “open-mouth” or other meaningful analogy to help children differentiate between and learn the meaning of the < and > greater symbols.

II.C) Understands what multiplication and division concepts and skills children need to learn in early childhood. (1) Can describe, differentiate among, and illustrate with a word problem key meanings of multiplication and division.

II.C.1. Provides opportunities to solve a variety of problems starting with groups-of multiplication problems (e.g., 4 groups of 2 items) and divvy-up division problems (e.g., If 8 items are shared fairly among 4 people, what size is each person’s share?”) and then area problems for both multiplication (e.g., “A garden plot is 7 feet long and 5 feet wide, what is the area of the garden plot?”) and division (e.g., The area of a garden plot is 35 square feet and its length is 7 feet long, what is the width of the plot?”) and measure-out division (e.g., If 8 items total are used to make shares of 2 items each, how many people can share the items fairly?” )

(2) Can show how manipulatives such as base ten blocks can be used to solve word problems that illustrate the various meanings of multiplication and division and describe how such problems and solutions can be translated into equations.

II.C.2. Encourages children to devise, share, and discuss informal solutions to a variety of multiplication and division problems, including the use of base ten blocks to solve area problems.

(3) Recognizes that symbolic multiplication should be related to what children already know (repeated addition or groups-of problems) and division (fair sharing involving divvying-up [problems).

II.C.3. Helps children understand the formal representations of multiplication and division by relating symbolic multiplication and division to what children already know (repeated addition or groups-of problems and fair sharing involving divvying-up problems, respectively) and by encouraging them to translate such problems into expressions or equations and vice versa.

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

(4) Understand the phases of mental-multiplication/division development and the central role of discovering patterns and relation and using reasoning strategies in achieving fluency with basic products and quotients.

II.C.4. Supports children’s initial use of counting strategies to solve word problems and expressions. Then encourages children to search for patterns and relations, discuss/share discovered regularities, and devise and share/discuss reasoning strategies—strategies for logically solving unknown basic combinations by using a known relation and known combination (e.g., recognizing that 2 times a number or versa has the same answer as a related addition double such as 2 x 7 or 7 x 2 = 7 + 7 = 14). Only after children learned patterns, relations, and reasoning strategies encouraging fluent (fast and accurate) responses.

(5) Understand the relations among addition, subtraction, multiplication, and division and how these relations can be used to check subtraction and division or use known multiplication facts or (factors of division) to deduce the quotient of unknown division facts.

II.C.5. Encourages children to connect and use the connections among multiplication, division, addition, and subtraction, including— *identifying multiplication-division families (3 x 5 = 15, 5 x 3 = 15, 15 ÷ 3 = 5, and 15 ÷ 3 = 5) and using such relations to solve missing number items (e.g., 15 ÷ 3 = ?, 3 x ? = 5, 5 = 15 ÷ ?); * view division as a missing factor problem (e.g., 15 ÷ 3 = ? can be thought of as ? x 3 = 15). * view division as repeated subtraction (e.g., 15 ÷ 3 = 15 – 3 – 3 – 3 – 3 – 3.

III. Numbers and Operations in Base TenIII.A) Understands, can identify, and can apply the fundamental concepts of grouping and place-value that underlie the Hindu-Arabic numeral system and operations with multi-digit numbers. (1) Understands the big idea that that progressively larger units can be composed from smaller units and that the position of a digit in a multi-digit number can represent these hierarchically larger units: (a) in our base ten system, each larger unit is composed of ten of the next smaller unit (i.e., is 10 times larger than the next smaller unit); (b) the position of a digit in a multi-digit number indicates the size of the unit (e.g., in 325, the digit 3 indicates 3 groups of 100, the digit 2 indicates two groups of ten, and the 5 indicates five singles; (c) the pattern “units, tens, and hundreds” repeats at each level (e.g. one thousands, ten thousands, hundred thousands); (d) each successively larger place is represented by power of tens that larger units in a base-ten system are created by grouping 10 or a power of 10 smaller units.

III.A.1. Using interlocking blocks, Egyptian hieroglyphics, or other models, creates opportunities for children to understand the initial steps toward a grouping and place-value concepts, including:

a) trading several smaller items for a larger one;b) grouping by fives or tens, including composing and decomposing

the numbers 11 to 19 using a group of ten and ones.c) recognizing “10 ones” can be traded for a “ten”d) understanding why 0 is needed as a place holder (e.g., to distinguish a mong 2, 20,

and 200). III.A.2. Helps children construct the understanding that other two-digit numbers can be thought of groups of ten and one; then three-digit numbers as groups of a hundred, groups of ten, and ones; and finally four-digit numbers as groups of one thousand, groups of a hundred, groups of ten, and ones.III.A.3. Provides opportunities to compare multidigit numerals meaningfully by initially modeling multi-digit numbers with base-ten blocks or other models such as Egyptian hieroglyphics and then with symbols >, =, and <.

(2) Understands the why and how of rounding whole numbers to tens and hundreds, including— (a) why the method of rounding (rounding up, rounding down,

III.A.2. Helps children discover that different situations require different approaches to rounding, even with 5, and how to round up, down, and up or down.

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

rounding up or down) depends on the tasks (e.g., round up to overestimate the amount so as to ensure a limit is not exceeded; round down to overestimate to ensure more than enough is available, round up or down for a relatively accurate estimate); (b) understands why there is not one correct way to round with 5 and why alternating between rounding 5 up and down is the most precise method of rounding. III.B) Understands the application of place value, properties of operations, and the relations between addition and subtraction to adding and subtracting multi-digit numbers up to 1000, including demonstrating and explaining renaming (carrying and borrowing) algorithms with base-ten blocks.

III.B.1. Create opportunities for children to learn the following mental-addition rules: decade after rule (adding 10 to a decade = the next decade in the counting sequence), its application to adding 10 to any two-digit number decade before rule (subtracting 10 from a decade = the decade before), its application to subtracting 10 from any two-digit number, and in time, the corresponding rules for adding or subtracting 100 and then 1000.III.B.2. Encourage children to devise, share, discuss, and evaluate informal strategies for adding and subtracting multi-digit numbers up to 1000 with base-ten blocks, Egyptian hieroglyphics, or other models/drawings and then to translate such informal models into written multi-digit procedures. III.B.3. Help children to use their understanding of informal written procedures to reinvent or learn the conventional renaming algorithms for adding and subtracting with multi-digit numbers up to 1000 and encourage them to justify the steps of such algorithms using base-ten blocks, Egyptian hieroglyphics, or other models/drawings

III.C) Understands the application of place value and properties of operations to multiply one-digit whole numbers and multiples of 10 up to 90 (e.g., 9 x 80), including demonstrating and explaining how a groups-of meaning of multiplication can be demonstrated with base-ten blocks.

III.C.1. Provides children with opportunities to devise, share, discuss, and evaluate informal strategies using base-ten blocks, Egyptian hieroglyphics, or other models/drawings to solve group-of multiplication problems involving one-digit whole numbers and multiples of 10 up to 90.III.C.2. Encourage children to apply their knowledge of decomposition and the associative property to devise the “add-0 shortcut” (e.g., 9 x 80 = 9 x [8 x 10] = [9 x 8] x 10 = 72 x 10 = 720 or more simply the product of 9 x 8 with a 0 in the ones place.

IV. Numbers and Operations—Fractions IV.A. Can explain two common meanings of fraction notation in terms of the conceptual basis for fractions (equal partitioning) using the informal analogy of fair sharing. (1) Can explain that a fraction may represent (among other things) division (e.g., 1/3 can be viewed as “one pizza shared fairly among three children).

IV.A.1. Encourages children to devise informal strategies to solve fair-sharing problems involving a whole or 2 to 8 wholes shared with 2 to 8 children and, in time, relate fraction notation to such division problems. For example, (e.g., 2/3: The 2 represents two same-sized wholes; the fraction bar can read as “shared fairly among”; and the 3 represents the number of people sharing the wholes).

(2) Can explain that a fraction may represent (among other things) a part of a whole (e.g., “If one pizza is shared fairly among three children, then each child gets one of three equal pieces—i.e., the

IV.A.2. After relating fraction notation to division, children have informally determined the size of each person’s share by using a picture or a manipulative, explicitly point out that the size of a share in a fractional fair sharing problem is a part of a whole divided into

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

size of each share can be represented as 1/3). equal-size parts (e.g., Each person’s share of 2 parts of a whole divided into 3 equal parts can be written 2/3).

IV.B. Can justify equivalent fractions in terms of the informal analogy of fair sharing. For instance, 1 whole shared fairly between 2 people is equivalent to (results in the same size share as) 2/4 (2 wholes of the same size shared fairly among 4 people), or 3/3 and 4/4 both result in a fair share of 1 (a whole).

IV.B.1. Explicitly points out that, with equivalent fractions or comparing fractions, the fractions must be from the same size whole or wholes of the same size.B.2. Encourages children to determine and justify fraction comparisons using a fair-sharing analogy.

IV.C. Can justify fraction comparisons in terms of the informal analogy of fair sharing. For example, that 1/2 is more than 1/3 of the same whole or sized whole because when the whole is shared between 2 children, each child gets a bigger share of the pizza than when the same-sized pizza is shared fairly among three.

IV.C.1. Encourages children to determine and justify fraction comparisons using a fair-sharing analogy (e.g., “The share when one pizza is shared fairly among four people is bigger than when one pizza is shared among five because you are sharing the pizza among fewer people,” or “One 3/5 is more than 4/9 because 3/5 is share that is more than half of the whole and 4/9 is a share that is less than half of whole”).

V. Measurement and DataV.A. Understands the general principles of measurement. V.A.1. Recognizes that that an object may have multiple measurable attributes (e.g., length, area, volume, capacity, weight, temperature, color intensity) and thus its measurement requires a choice of measurable attribute.

V.A.1. Encourages children to identify and describe the measurable attributes of objects and to consider which sort of “big” they need to measure (K.MD.A.1).

V.A.2. Understands that measurement involves a comparison on a selected attribute either through a direct comparison between objects or an indirect comparison between an object and a stand-in in form of a measurement unit.

V.A.2. Provides opportunities to— a. first directly compare two items that share a common measureable attribute to

determine which has more or less of the attribute (K.MD.A.2); b. then order three objects by length; c. compare the length of two objects that cannot be directly compared by comparing

each to the length of third object (“standard”).Note. b & c are CCSS Goal 1.MDA.1 and serve as a transition between direct to indirect measurement.

V.A.3. Understands that measurement essentially entails equally partitioning a measureable attribute, which cannot be counted, into equal size parts (units) that can then be counted.

V.A.3. Prompts children to use informal “units” such as and 1-inch square tiles for area) and informal methods— a. initially by using multiple informal unit of length, such as “pencil lengths” to

completely cover a length (e.g., pencils laid end-to-end to divide up without gaps and without overlap lengths; and then areas, into equal size pieces or “units” and specify both the number and name of the “units” (MD.A.2);

b. later by using a single informal unit of length and applying it iteratively and completely (without gaps or overlap) to a length;

c. next by using an informal unit of area, such as 1-inch square tiles, to cover an area without gaps or overlap;

d. finally, using a single informal unit of area and applying it iteratively and completely to an area.

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

V.B. Explicitly understands the whys and how of measurement commonly used in everyday life including the how to derive formulas for area and perimeter. V.B.1. Understands the procedure for measuring length with standard tools such as a ruler and relate it to equal partitioning (e.g., the ruler serves to divide a length in equal parts or units and so the end of the ruler that indicates 0 is placed at one of the length to ensure there is not a gap).

V.B.1. Provide opportunities to learn how to measure length in a meaningful and purposeful manner by— a. justifying how to measure a length in terms of equal partitioning;

b. selecting and using appropriate standard tools (e.g., using a ruler to measure items in inches when 1 foot or less and yardsticks when more than 1 foot; 2.MD.A.1);

c. recognizing measurement equivalents and understanding that the using of a larger unit results in fewer units, such as 2 feet = 24 inches (2.MD.A.2).

V.B.2. Understand how the big idea of equal partitioning applies to time measurement.

V.B.2. Provide opportunities to learn how to tell and to record time in a meaningful and purposeful manner by— a. helping children recognize that a day is divided up in 24 equal parts called “hours,”

that hours are divided up into 60 equal parts called “minutes,” and minutes are divided up into 60 equal parts called “seconds.”

b. helping children learn how to tell and write time in hours and half-hours using digital and then analog clocks by relating time keeping to real events and to each other (1.MD.B.3) and then doing so to the nearest minute (3.MD. A.1);

c. providing problems that entail the addition and subtraction time intervals in minutes and helps children represent problems using, e.g., a number line (3.MD. A.1).

V.B.3. Recognizes how money concepts can be related to other aspects of mathematics including skip counting, multiplication, renaming units, and the relational meaning of the equals sign.

V.B.3. Provide opportunities to learn money concepts in a meaningful manner by— a. relating the tally of pennies, nickels, and dimes to counting by ones, fives, and tens,

respectively, or multiplying by 1, 5, and 10, respectively; b. providing opportunities to solve money-equivalent problems/equations and use $ and

¢ symbols appropriately (e.g., “A dime and nickel are how many cents?”; 2.MD.C.8).

V.B.4. Specifies how manipulatives can be used to help children understand the geometric measurement concepts of area, volume, and perimeter.• (3.MD.C.7a) and 3.MD.C.6)

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V.B.4. Help children construct explicit and exact concepts of area, volume, and perimeter and by—

a. recognizing area and perimeter as an attribute of plane figures (3.MD.C.5a) and volume is an attribute of solid objects;

b. relating these ideas to concrete models, such as noting 1-inch square tiles have sides of 1 inch each and the area of the square tile is 1 square inch, and thus, a rectangle that can be covered without gaps or overlap by, for instance, 6 square tiles has an area of 6 square inches (3.MD.C.5a); (3.MD.C.6);

c. measuring area by counting unit squares using various units, such as, square inches, square feet, square cm, square meters, and improvised units (3.MD.C.6).

V.B.5. Demonstrates how the formulas for area, volume, and perimeter can be derived using manipulatives or graph paper.

V.B.5. Help children to understand the basis of geometric measurement formulas by providing children an opportunity to re-invent the formula for—

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

a. area of a rectangle by using square tiles or graph paper to determine area, discovering the shortcut that the total number of square tiles (area) is simply the length of the top row x the length of a side row (length x width), and summarizing this shortcut algebraically as the formula A = l x w (3.MD.C.7a);

b. volume of a cuboid by using square blocks by using cubes to determine volume, discovering the shortcut that the total number of cubes (volume) is simply the area of the top layer (l x w) x the number of layers (height of cuboid or h), and summarizing this shortcut algebraically as the formula V = l x w x h;

c. perimeter of a rectangle by measuring each side, discovering the shortcut that the total distance around is twice the length plus twice the width, and summarizing this shortcut algebraically as the formula P = 2l + 2w or 2(l + w).

V.B.6. Be able to solve a variety of measurement problems, including those that involve estimation, recognizing that multiplication can have an area meaning (i.e., the factors can represent the length and width in linear units) of a rectangle and the product can represent the area in square units), or the application of formulas

V.B.6. Provide opportunities for children to solve problems involving measurement, measurement estimation, relating measurement to operations, and the application of formulas, including—

a. estimating lengths using units of inches, feet, centimeters, and meters (2.MD.A.3);

b. measuring to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit (2.MD.A.4);

c. relating length to addition and subtraction within 100 by solving word problems that involve operating on lengths of the same units, e.g., by using drawings (such as drawings of rulers), equations with a symbol for the unknown number to represent the problem (2.MD.B.5), or a number line diagram (2.MD.B.6);

d. measuring and estimating liquid volumes and masses of objects using standard units such as cups, liters, ounces, pounds, grams, and kilograms (3.MD.A.2);

e. adding, subtracting, multiplying, or dividing to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem (3.MD.A.2);

f. solving for the area of rectangles with whole-number sides lengths in the context of solving real world and mathematical problems (3.MD.A.7b);

g. using tiling squares or base-ten blocks and an area model to represent the distributive property (e.g., to show that 12 x 3 = [10 x 3] + [2 x 3];

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

3.MD.A.7c);h. solving problems that require finding the areas of rectilinear figures

by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems (3.MD.A.7d);

i. solve real world and mathematical problems involving perimeters of polygons (3.MD.D.8).

V.C. Understands the role of data, data analysis, and data representations (e.g., graphs, tables) in solving problems, raising or addressing (e.g., scientific, social, economic, or political) issues or questions, and informing or others and the importance of involve participants in actually collecting their own data (as opposed to using canned data) and analyzing data.

V.C.1. Promotes understanding of the role of data (e.g., solve problems, raise or address issues or questions, inform or persuade others) data analyses (e.g., find patterns or relations, and data representation (visually make clearer patterns/relations, make predictions, facilitate communicating information) by actually embedding data instruction in addressing real problems, issues, or questions proposed by the children themselves or of interest to children.V.C.2. Provides opportunities to learn specific data collection, analysis, and representation skills, including—a. classifying objects by categories, tallying the number of objects (up to 10) in each

category, and sorting the categories by count (K.MD.B.3);b. organizing, representing, and interpreting data with up to three categories, including

asking and answering questions about the total number of data points, how many in each category, and how many more or less are in one category than in another (1.MD.C.4);

c. drawing a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories and using the data to solve simple put-together, take-apart, and compare problems (2.MD.D.10);

d. generating measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object and representing the measurements by making a line plot, where the horizontal scale is marked off in whole-number units (2.MD.D.9);

e. drawing a scaled picture graph and a scaled bar graph to represent a data set with several categories in order to solve one- and two-step “how many more/less” problems using (3.MD.B.3);

f. generating measurement data by measuring lengths using rulers marked with halves and fourths of an inch and representing the data with a line plot, where the horizontal scale is marked off in fourths (3.MD.B.4).

VI. GeometryVI.A. Understands the van Hiele developmental levels of geometric thinking and achieves at least Level 2 themselves:

VI.A.1. Provide children with opportunities to recognize, name, and compare common two- and three-dimensional shapes in order to progress to van Hiele’s level 0, including—.

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

(Level 0—visual level). Uses appearances to recognize and name common two- and three-dimensional shapes;(Level 1—analysis level). Identifies and describes the attributes (e.g., number of sides, straight or curved sides)of common two- and three-dimensional shapes;(Level 2—informal-reasoning level). Identifies critical or defining attributes of two-dimensional shapes (the characteristics shared by all examples of a concept) and can use this knowledge to deduce the identity of a shape and understand the relations among shapes (e.g., a square is a special kind of rectangle because it has all the attributes of rectangle (e.g., 4 sides, parallel opposite sides, sides at a right angle) plus the additional critical attribute that all sides are equal.

a) identifying objects in their environment in terms of various two- and three-dimensional shapes (e.g., state that the clock is shaped like a circle or that the table top is a rectangle; K.G.1);

b) identifying some of the faces (flat sides) of common three-dimensional shapes using two-dimensional shape names;

c) identifying whether a shape is two-dimensional (“flat”) or three-dimensional (“solid”; K.G.3);

d) correctly naming shapes regardless of orientation (e.g., a square is a square whether on its side or turned to stand on a vertice; K.G.2)

VI.A.2. Provide children with opportunities to recognize, name, and compare the attributes of common two-dimensional shapes in order to progress to van Hiele’s level 1, including— (a) analyzing and comparing 2-D and 3-D shapes of different orientation and sizes and using informal terms to describe attributes, similarities, and differences (K.G.4); (b) identifying the attributes of various two- and three-dimensional shapes in their environment (2.G.1); (c) drawing a shape given its attributes (2.G.1).VI.A.3. Provide children with opportunities to identify the critical attributes of common two-dimensional shapes, use such attributes to reason about the relations among shapes in order to progress to Van Hiele Level 2, including—

a) using a variety of examples and non-examples to identify the critical or defining attributes of a 2-D shape;

b) distinguishing between defining and non-defining attributes (1.G.1)c) using concept maps, Venn diagrams, or other representations to

specify the relations among categories of 2-D shapes (CCSS Goal 3.G.1).

VI.B. Understand how the big ideas of composition and decomposition and equal partitioning apply to geometry and the developmental trajectory children follow in becoming competent composers and decomposers. compose and de-compose shapes; describe and explain the resulting new shapes using appropriate geometric concepts

VI.B. Provide children with opportunities to— (a) create models of real-world shapes by using available materials (e.g., clay, rods) as the component parts (K.G.5) (b) explore composing composite 2-D and 3-D increasing complex shapes from component shapes or decomposing a composite shape into component shapes (K.G.6; 1.G.2);

EARLY CHILDHOOD ADVISORY GROUPMATH STANDARDS

(b) partition a shape into equal size parts and label each part with a fraction (2.G.3 & 3.G.2).

VI.C. Understands basic geometric concepts, such as angle, parallel, and perpendicular and can describe these ideas in terms of an informal analogy (e.g., an angle is the “amount of turn”).

VI.C. Helps children understands basic geometric concepts, such as angle, parallel, and perpendicular by first relating them to an informal analogy and then linking a formal definition to a familiar informal analogy.

VI.D. Understands and can summarize and illustrate the cognitive developmental progression from visual to descriptive to analytic to abstract characterizations of shapes; use this progression to understand children’s thinking

VI.D. Encourage children to think about spatial relations by— (a) describing the relative location of objects by using such relation terms as “above,” “below,” “beside” or “next to” (K.G.1); (b) creating and using simple maps; (c) imagining/predicting how altering the spatial orientation of a shape will change its appearance (e.g., turning it upside down) and then provide them with manipulatives to examine various orientations of shapes such as rotating and flipping shapes, such as blocks and puzzle pieces, to make them “fit.”

VI.E. Understand the importance of precision in describing and reasoning about spatial locations and relationships, including

a) descriptive power of prepositions (and their imprecise mapping among languages and dialects)

b) mathematically precise tools, such as measurements, grids and the coordinate plane

VI.F. Understand that spatial relationships can be manipulated mentally and that point of view affects both experiences and representations of spatial relationships

VI.G. Describe the connections (relationships) between geometric properties and arithmetic and algebraic properties, and adapt a problem in one domain to be solved in the other domain