# Mathematics

The domain of Mathematics includes Preschool Benchmarks in: Sense of Numbers, Identification of Relationships in Objects, Concepts of Geometry, and Analysis of Data Information

## Introduction

Young children are natural mathematicians, fascinated by what is “bigger,” wanting “more” of their favorite things, and very concerned with whether the distribution of those things is “fair.” These kinds of observations of the world are mathematical at their core because they are about quantity and size. Preschool children’s experiences of the world are equally affected by ideas about spatial relationships and shape. They explore the concepts of geometry whether they are maneuvering through the living room, building a block tower, or choosing a puzzle piece. Such daily experiences are packed with mathematical concepts that fascinate and challenge young thinkers and can eventually prompt analytical thought, growing precision, and abstraction.

The major mathematical task of early childhood is to coordinate these natural interests and understandings with the beginnings of a useful knowledge of conventional math concepts and skills. Unfortunately, for many children, meaningful mathematical thinking is displaced too early by an emphasis on math “facts” (such as 2 + 2 = 4) and math “procedures” about what to do when. Too many young children learn how to say the counting words up to 20 without being able to successfully count out a set of five objects. While the procedures—such as the order of the count words—must be learned, it is crucial that they be meaningfully connected to things children understand and care about, such as “how many” children can fit at the play dough table or “how many” slices of apple they can have at lunch.

To effectively build on young children’s innate interests in quantity and space and move their thinking in conventional mathematical directions, the most important thing teachers can do is talk with them, helping them “see” the math in the world. When adults provide rich language to mathematical experiences, such as “thicker” or “longer” rather than simply “bigger,” children understand that there are many different types of attributes that can be compared and measured. When teachers ask, “How do you know the door looks like a rectangle?” they support children’s budding conception of geometric rules, such as a rectangle having four sides. When teachers count with one-to-one correspondence to find out “how many children are in the group today,” they demonstrate the use of whole numbers in a way that is very real to children and matters to them. These sorts of interactions, based on experiences that are a natural part of children’s everyday lives, are the best way to ensure the development of beginning mathematical understandings that inspire children to keep learning.

The mathematics standards in the IELDS are more detailed and developmentally informed than those in the previous version, reflecting our growing understanding of how children’s mathematical thinking develops during early childhood. We hope they will provide a useful guide to the kinds of mathematics experiences preschool children ought to have prior to their kindergarten year.

Math is a big part of every day in Ms. O’Brien’s Preschool for All classroom. She knows that understanding quantity for preschoolers goes far beyond reciting counting words, and she provides many opportunities for children to count in meaningful situations. Each day, the group counts how many children are present and how many are absent, how many steps from the door of their classroom to the playground, and how many plates and napkins are needed to set a table for snacks. They say the counting words in both English and Spanish. She provides many manipulatives that encourage children to use one-to-one correspondence as they sort, categorize, order, and build to create groups of objects and to connect numbers to quantities of objects. Shapes are everywhere in her preschool classroom, and Ms. O’Brien takes advantage of every opportunity she can to name the shapes for the children in both English and Spanish and to encourage them to explore, manipulate, and build with them. The children have also learned to love taking surveys. Ms. O’Brien has created clipboards with Yes/No graphs on them for children to interview each other about favorites. She loves to hear a child ask another, “¿Te gusta helado de chocolate?” (“Do you like chocolate ice cream?”) and see him note the answer under the Yes or No column. She makes sure to follow up on the results of his survey and have him present it to the class at large group time. Ms. O’Brien finds it easy to include math goals from the IELDS on her lesson plan for her play areas, daily routines, and group experiences because math is everywhere!

## Benchmarks

**Goal 6**

Demonstrate and apply a knowledge and sense of numbers, including numeration and operations.

**Learning Standard 6.A**

Demonstrate beginning understanding of numbers, number names, and numerals.

- Mathematics6.A.ECa: Count with understanding and recognize “how many” in small sets up to 5.
- Mathematics6.A.ECb: Use subitizing (the rapid and accurate judgment of how many items there are without counting) to identify the number of objects in sets of 4 or less.
- Mathematics6.A.ECc: Understand and appropriately use informal or everyday terms that mean zero, such as “none” or “nothing”.
- Mathematics6.A.ECd: Connect numbers to quantities they represent using physical models and informal representations.
- Mathematics6.A.ECe: Differentiate numerals from letters and recognize some single‐digit written numerals.
- Mathematics6.A.ECf: Verbally recite numbers from 1 to 10.
- Mathematics6.A.ECg: Be able to say the number after another in the series up to 9 when given a “running start,” as in “What comes after one, two, three, four…?”.

## Example Performance Descriptors

EXPLORING | DEVELOPING | BUILDING |
---|---|---|

Recognize how many there are in a set of 1 or 2 without counting them (e.g., one car or two blue crayons). | Recognize how many there are in a set of 3 without counting them (e.g., three yellow beads). | Recognize how many there are in sets of 4 and 5 when presented in a nonlinear, organized fashion (like a die face). |

Point to or move objects around as though to organize without necessarily counting out loud. | Point to or move objects when counting out loud without effectively tracking items counted (may skip items or count items more than once). | Point to or move each object to make sure sure each is counted once and only once when counting in sets up to 5. |

Demonstrate an understanding of zero by making a comment such as “Now I don’t have any more” when finished with a snack of four crackers. | Demonstrate understanding of none by looking into an empty container and commenting that there is “nothing in there.” | Respond to a question about quantity, such as “How many red bears are left?” when none are left by saying: “None.” |

Confuse numerals and letters, saying number names occasionally when pointing to letters. | Say number names when pointing to numerals (but not letters), even if they don’t match. | Correctly identify the numerals 1, 2, and 3. |

Say some counting words when “counting.” | Recite counting words from 1-10, with 2-4 errors (e.g., skip numbers, mix up order) but also some number names in words in consecutive order (e.g., “one, two, five, four, six, seven, nine, ten”). | Recite counting words in order from 1-10 (with an occasional error). |

Fill in the next number when the teacher says, “one, two…” | Fill in the next number when the teacher says, “one, two, three…” | Fill in the next number when the teacher says, “three, four, five…” (not starting at “one”). |

**Learning Standard 6.B**

Add and subtract to create new numbers and begin to construct sets.

- Mathematics6.B.ECa: Recognize that numbers (or sets of objects) can be combined or separated to make another number.
- Mathematics6.B.ECb: Show understanding of how to count out and construct sets of objects of a given number up to 5.
- Mathematics6.B.ECc: Identify the new number created when small sets (up to 5) are combined or separated.
- Mathematics6.B.ECd: Informally solve simple mathematical problems presented in a meaningful context.
- Mathematics6.B.ECe: Fairly share a set of up to 10 items between two children.

## Example Performance Descriptors

EXPLORING | DEVELOPING | BUILDING |
---|---|---|

Combine items to create a new number (e.g., combine two blocks with a friend’s two blocks and say, “Now we have four.”) | Separate items from a set (e.g., with a set of three cups, takes one away and says, “Now we have two.”). | Recognize that combining sets always results in “more” and separating sets always results in “less.” |

Count out two objects correctly (e.g., count two crackers on plate at snack time). | Count out three and four objects correctly (e.g., count four blocks in a block tower). | Count out five objects correctly (e.g., count five children in a small group). |

Solve simple math problems (e.g., know that if one child is added to the group that makes one more). | Solve simple math problems (e.g., know that if one chair is taken away from the table that makes less). | Solve simple math problems (e.g., know that if one orange is taken away from a group of five, there are four oranges left). |

Divide a set of two to four objects between self and a friend evenly. | Divide a set of six to nine objects between self and a friend evenly. | Divide a set of 10 crackers between self and a friend evenly. |

**Learning Standard 6.C**

Begin to make reasonable estimates of numbers.

- Mathematics6.C.ECa: Estimate number of objects in a small set.

## Example Performance Descriptors

EXPLORING | DEVELOPING | BUILDING |
---|---|---|

Make reasonable estimates of small quantities of objects (e.g., guess “four” when asked how many peach slices are in the bowl). | Tell whether a set is more or less than 5. | Presented with a set of 7 or 8, estimate a number in the range of 5 to 12. |

**Learning Standard 6.D**

Compare quantities using appropriate vocabulary terms.

- Mathematics6.D.ECa: Compare two collections to see if they are equal or determine which is more, using a procedure of the child’s choice.
- Mathematics6.D.ECb: Describe comparisons with appropriate vocabulary, such as “more”, “less”, “greater than”, “fewer”, “equal to”, or “same as”.

## Example Performance Descriptors

EXPLORING | DEVELOPING | BUILDING |
---|---|---|

Match sets of things that go together, item to item (e.g., match one napkin to each of the place settings at the table). | Through words or gestures, identify which set has more. | Through words or gestures, identify whether sets have more, less, or an equal amount. |

Use the terms “more” or “same as” (e.g., acknowledge that one child has more pegs and another has the same number). | Use the terms “less”, “not as many”, or “fewer” (e.g., acknowledge that one child has less play dough than others do). | Use a variety of appropriate vocabulary to make comparisons of quantity (e.g., “more”, “less”, “greater than”, “fewer”, “equal to”, or “same as”). |

**Goal 7**

Explore measurement of objects and quantities.

**Learning Standard 7.A**

Measure objects and quantities using direct comparison methods and nonstandard units.

- Mathematics7.A.ECa: Compare, order, and describe objects according to a single attribute.
- Mathematics7.A.ECb: Use nonstandard units to measure attributes such as length and capacity.
- Mathematics7.A.ECc: Use vocabulary that describes and compares length, height, weight, capacity, and size.
- Mathematics7.A.ECd: Begin to construct a sense of time through participation in daily activities.

## Example Performance Descriptors

EXPLORING | DEVELOPING | BUILDING |
---|---|---|

Compare magnitudes of one object to another (e.g., line up two strings of beads to determine which is longer; stand next to peer to see who is taller). | Order multiple objects to compare magnitudes (i.e., arrange blocks from tallest to shortest). | Order multiple objects to compare magnitudes and describe comparisons (i.e., arrange blocks from tallest to shortest and describe). |

Use nonstandard means to measure items (e.g., using a piece of string or a long block as a measurement tool). | Use nonstandard units to measure items (e.g., use hands or small blocks to measure the length of a table). | Use nonstandard units to measure items and identify the quantity of units (e.g., may not be correct but attempt to count the number of hands or small blocks in the length of the table). |

Use appropriate vocabulary when making measurements, such as “small”, “big”. | Use appropriate vocabulary when making measurements, such as “small”, “big”, “short”, “tall”. | Use a wider appropriate vocabulary when making measurements, such as “small”, “big”, “short”, “tall”, “empty”, “full”, “heavy”, and “light”. |

Ask about the sequence of the daily schedule (e.g., “When will we have snack?” “When are my Mom and Dad coming?”). | Know the sequence of the daily schedule and guess the progression of the schedule throughout the day but not with accuracy (e.g., guess incorrectly that snack is after circle time yet knows that Mom or Dad will come after outside time). | Know the sequence of the daily schedule and begin to accurately gauge time by progression of the schedule throughout the day (e.g., know that naptime comes after lunch or that outside time comes after snack). |

**Learning Standard 7.B**

Begin to make estimates of measurements.

- Mathematics7.B.ECa: Practice estimating in everyday play and everyday measurement problems.

## Example Performance Descriptors

EXPLORING | DEVELOPING | BUILDING |
---|---|---|

Make predictions and estimations during play without much accuracy (e.g., estimate how many scoops of sand it will take to fill a small bucket at the sand table — “I think 100!”). | Make more accurate predictions and estimations during play without checking by counting (e.g., estimate how many pebbles will fill the balance scale cup, “I think 10” but without counting to check). | Make more accurate predictions and estimations during play and check them by counting (e.g., “I think it will take five scoops of sand to fill this cup – 1, 2, 3, 4, 5, 6 – oh, I was almost right!”). |

Estimate to solve a task without much accuracy (e.g., when setting table for snack, estimate how many napkins are needed. “I think 50.”). | Estimate to solve a task with more accuracy but without checking by counting (e.g., during block play, estimate how many blocks are needed to make the road being constructed reach the wall, “I think six” – but without counting to check). | Estimate to solve a task with more accuracy and check by counting (e.g., during block play, estimate how many blocks are needed to make the road being constructed reach the wall, then count to see how many it took). |

**Learning Standard 7.C**

Explore tools used for measurement.

- Mathematics7.C.ECa: With teacher assistance, explore use of measuring tools that use standard units to measure objects and quantities that are meaningful to the child.
- Mathematics7.C.ECb: Know that different attributes, such as length, weight, and time, are measured using different kinds of units, such as feet, pounds, and seconds.

## Example Performance Descriptors

EXPLORING | DEVELOPING | BUILDING |
---|---|---|

Incorporate teacher-introduced standard measuring tools into play without attention to quantity. | With teacher assistance, use standard measuring tools without expressing interest in quantity (e.g., teacher suggests they see how many rulers high the shelf is; child helps with measuring). | Ask teacher to help with using standard measuring tools and figuring out quantities (e.g., use a measuring tape and ask how long the two blocks are). |

Learn the vocabulary words “thermometer” and “clock”. | With teacher assistance, explore measuring hot and cold with thermometers. | With teacher assistance, learn that clocks measure time. |

With teacher assistance, use a balance scale to compare weights of objects in the classroom. | With teacher assistance, use a scale that provides numerical weight to compare weights of objects in the classroom. | With teacher assistance, use a variety of similar tools for measurement
of weight (e.g., use both balance scales and scales that provide a numerical weight to explore objects in the classroom). |

**Goal 8**

Identify and describe common attributes, patterns, and relationships in objects.

**Learning Standard 8.A**

Explore objects and patterns.

- Mathematics8.A.ECa: Sort, order, compare, and describe objects according to characteristics or attribute(s).
- Mathematics8.A.ECb: Recognize, duplicate, extend, and create simple patterns in various formats.

## Example Performance Descriptors

EXPLORING | DEVELOPING | BUILDING |
---|---|---|

Match similar objects when an attribute is named (e.g., “Which rocks are smooth like this one?” “Can you find another ball that’s this big?”). | Compare and describe various objects, identifying one of their attributes (e.g., describe different rocks by referring to their size, shape, or weight). | Compare and describe various objects, identifying at least two of their attributes (e.g., describe different rocks by referring to their size and shape or texture and weight). |

Match similar objects (e.g., putting all the toy cars together or lining up plates on a table). | Sort objects by a single attribute (e.g., ordering fire trucks from shortest to longest or ordering rocks from smooth to rough). | Sort objects according to two different characteristics and describe a sorting strategy (e.g., sort crayons by color and size, “Here are the big red ones and there are the little blue ones”, or sort blocks by shape and color, “These are all yellow triangles and these are the green rectangles”). |

Attempt to create a simple A-B repeating pattern using early childhood materials but without maintaining the repeating pattern (e.g., makes colored marks on the white board beginning with black, green, black, then adds red, green, black, blue, black). | Successfully create a simple A-B repeating pattern using classroom objects (e.g., build a tower of alternating blue and red cubes). | Create a simple A-B-C or A-B-B repeating pattern using classroom objects (e.g., lines up people figures with small, medium, large as the repeating pattern; strings beads on a necklace with one yellow, two orange in a repeating pattern). |

Replicate a simple pattern in music following the beat by clapping or tapping foot lightly. | Replicate patterns in music by playing finger games such as “Open, Shut Them.” | Replicate patterns in music by singing repetitive songs such as “B-I-N-G-O.” |

**Learning Standard 8.B**

Describe and document patterns using symbols.

- Mathematics8.B.ECa: With adult assistance, represent a simple repeating pattern by verbally describing it or by modeling it with objects or actions.

## Example Performance Descriptors

EXPLORING | DEVELOPING | BUILDING |
---|---|---|

With adult assistance, describe a pattern in words (e.g., “tall, short, tall, short, tall, short” or “red, blue, yellow, red, blue, yellow, red, blue, yellow”). | When presented with a visual “red-blue, red-blue, red-blue” repeating pattern and told “do a clap for red and a tap for blue,” produce clap-tap, clap-tap, clap-tap with adult assistance. | When presented with a visual “circle-square, circle-square, circle-square” repeating pattern and told “do a green bear for circles and a yellow bear for squares,” produce green bear-yellow bear, green bear-yellow bear, green bear-yellow bear pattern with adult assistance. |

**Goal 9**

Explore concepts of geometry and spatial relations.

**Learning Standard 9.A**

Recognize, name, and match common shapes.

- Mathematics9.A.ECa: Recognize and name common two- and three-dimensional shapes and describe some of their attributes (e.g., number of sides, straight or curved lines).
- Mathematics9.A.ECb: Sort collections of two‐ and three‐dimensional shapes by type (e.g., triangles, rectangles, circles, cubes, spheres, pyramids).
- Mathematics9.A.ECc: Identify and name some of the faces (flat sides) of common three‐dimensional shapes using two-dimensional shape names.
- Mathematics9.A.ECd: Combine two-dimensional shapes to create new shapes.
- Mathematics9.A.ECe: Think about/imagine how altering the spatial orientation of a shape will change how it looks (e.g., turning it upside down).

## Example Performance Descriptors

EXPLORING | DEVELOPING | BUILDING |
---|---|---|

Identify the shape of various twodimensional items in the early childhood environment (e.g., state that the clock is shaped like a circle or that the table top is a rectangle). | Identify the shape of various twodimensional items in the classroom and describe their attributes (e.g., state that a square block has four sides and a triangle block has three sides). | Identify the shape of various twoand three-dimensional items in the early childhood environment and describe their attributes (e.g., “I used all these ‘rolling blocks’ (cylinders) to hold up my bridge.”). |

Match triangles to triangles, squares to squares, circles to circles, and rectangles to rectangles. | Match triangles to triangles, squares to squares, circles to circles, and rectangles to rectangles even when size (or proportion) differs among examples. | Match cubes, spheres, and pyramids, even when size differs among examples. |

Match the face (flat side) of one common three-dimensional shape to another (e.g., match the face of one cube to another or one cylinder to another). | Describe the face (flat side) of one common three-dimensional shape (cube or cylinder) using two-dimensional shape names (square or circle). | Describe the faces (flat sides) of more than one common threedimensional shape, such as cubes and cylinders, using twodimensional shape names, such as squares and circles. |

Use one common two-dimensional shape to create simple representations of things in the real world (e.g., line up several rectangle blocks to make a “road”). | Use more than one common two-dimensional shape to create representations of things in the real world (e.g., place small square blocks on the “road” to be the “cars”). | Use common two-dimensional shapes to create more complex representations of things in the real world (e.g., place triangles around a circle to make a “flower”). |

Rotate and flip shapes, such as blocks and puzzle pieces, to make them “fit.” | Rotate and flip a shape to create something different (e.g., place the rectangle on its short or long side). | Discuss with teacher how rotating and flipping a shape will create something different (e.g., Teacher: “What do you think will happen if you turn the triangle upside down? Let’s try it.” Child: “It stands up by itself!”). |

**Learning Standard 9.B**

Demonstrate an understanding of location and ordinal position, using appropriate vocabulary.

- Mathematics9.B.ECa: Show understanding of location and ordinal position.
- Mathematics9.B.ECb: Use appropriate vocabulary for identifying location and ordinal position.

## Example Performance Descriptors

EXPLORING | DEVELOPING | BUILDING |
---|---|---|

Respond appropriately to request to place an object somewhere in space in relation to other objects (e.g., put doll in front of pillow; place shoes under table). | Respond to questions about location of an object (e.g., respond correctly to questions such as “Which colored block is on top?”). | Respond to questions about ordinal position of an object (e.g., respond correctly to questions such as “Who is first in line?” or “Which car came in third?”). |

Attempt to use vocabulary for location during play activities, not always correctly (e.g., when asked, say the doll is under the pillow when she is in front). | Use appropriate vocabulary for location during play activities (e.g., in conversations, use terms such as “near” and “far”, “over”and “under”). | Use appropriate vocabulary for ordinal position during play activities (e.g., in conversations, use terms such as “first” and “last”, “second” and “third”). |

**Goal 10**

Begin to make predictions and collect data information.

**Learning Standard 10.A**

Generate questions and processes for answering them.

- Mathematics10.A.ECa: With teacher assistance, come up with meaningful questions that can be answered through gathering information.
- Mathematics10.A.ECb: Gather data about themselves and their surroundings to answer meaningful questions

## Example Performance Descriptors

EXPLORING | DEVELOPING | BUILDING |
---|---|---|

With teacher assistance, identify a “yes” or “no” question to ask a peer and report verbally to teacher. | With teacher assistance, identify a “yes” or “no” question to ask multiple peers, recording on a “yes” or “no” chart or clipboard. | With teacher assistance, formulate questions of personal interest (make a list of things to find out about, such as favorite cookies or how children get to school each day) and conduct surveys on charts or clipboards. |

Notice a change in the environment and comment (e.g., “We need more paintbrushes at the easel.”). | Discuss one aspect of their environment and then collect relevant information with teacher assistance as needed (e.g., discuss whether trees have buds yet and go outside to check). | Discuss more than one aspect of their environment and then collect relevant information with teacher assistance as needed (e.g., discuss what kinds of insects live on the school playground and then go outside to investigate). |

**Learning Standard 10.B**

Organize and describe data and information.

- Mathematics10.B.ECa: Organize, represent, and analyze information using concrete objects, pictures, and graphs, with teacher support.
- Mathematics10.B.ECb: Make predictions about the outcome prior to collecting information, with teacher support and multiple experiences over time.

## Example Performance Descriptors

EXPLORING | DEVELOPING | BUILDING |
---|---|---|

Organize materials with teacher support to prepare for graphing (e.g., sort leaves by color, sort fruit by type). | Participate in creating a data display using concrete objects or pictures with teacher support (e.g., organize children’s favorite fruit in rows to demonstrate whether more children prefer apples or oranges). | Compare numerical information derived from graphs to find answers to questions with teacher support as needed (e.g., use information depicted on a chart or graph to describe which classroom games are most popular). |

With teacher support, begin to predict the outcome of an activity (e.g., predict there are more boys than girls at the snack table). | With teacher support, provide a reasonable prediction or guess for the outcome of an activity (e.g., predict that the class collected more yellow than red leaves on the nature walk before sorting and counting them). | With teacher support, predict with more accuracy the outcome of a counting or comparison activity (e.g., predict how many more chairs, when three are already there, are needed for the small group table so that six children can all have a seat). |

**Learning Standard 10.C**

Determine, describe, and apply the probabilities of events.

- Mathematics10.C.ECa: Describe likelihood of events with appropriate vocabulary, such as “possible”, “impossible”, “always”, and “never”.

## Example Performance Descriptors

## Notes

The Mathematics goals, standards, and benchmarks align with the following sections of the Kindergarten Mathematics Common Core:

- Goal 6: Counting and Cardinality, 1-7, Operations and Algebraic Thinking, 1-6.
- Standard 6.A: Counting and Cardinality, 1-7.
- Benchmark 6.A.ECa: Counting and Cardinality, 4-5.
- Benchmark 6.A.ECb: Counting and Cardinality, 4-5.
- Benchmark 6.A.ECc: Counting and Cardinality, 3.
- Benchmark 6.A.ECd: Counting and Cardinality, 3-4.
- Benchmark 6.A.ECe: Counting and Cardinality, 3 & 7.
- Benchmark 6.A.ECf: Counting and Cardinality, 1.
- Benchmark 6.B.ECa: Operations and Algebraic Thinking, 1-5.
- Benchmark 6.B.ECb: Operations and Algebraic Thinking, 1-2.
- Benchmark 6.B.ECc: Operations and Algebraic Thinking, 1-4.
- Benchmark 6.B.ECd: Operations and Algebraic Thinking, 4-5.
- Standard 6.D: Counting and Cardinality, 6.
- Goal 7: Measurement and Data, 1-3.
- Benchmark 7.A.ECa: Measurement and Data, 2.
- Benchmark 7.A.ECc: Measurement and Data, 1.
- Goal 8: Measurement and Data, 1-3.
- Benchmark 8.A.ECa: Measurement and Data, 1.
- Goal 9: Geometry, 1-6.
- Benchmark 9.A.ECa: Geometry, 1-2.
- Benchmark 9.A.ECb: Geometry, 3-6.
- Benchmark 9.A.ECc: Geometry, 2-5.
- Benchmark 9.A.ECd: Geometry, 5-6.
- Benchmark 9.A.ECe: Geometry, 4-6.
- Standard 9.B: Geometry, 1.