# 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

Example Performance Descriptors can be found on the List of goals, standards, and benchmarks page.

**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.

**Mathematics****6.A.ECa:**Count with understanding and recognize “how many” in small sets up to 5.**Mathematics****6.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.**Mathematics****6.A.ECc:**Understand and appropriately use informal or everyday terms that mean zero, such as “none” or “nothing”.**Mathematics****6.A.ECd:**Connect numbers to quantities they represent using physical models and informal representations.**Mathematics****6.A.ECe:**Differentiate numerals from letters and recognize some single‐digit written numerals.**Mathematics****6.A.ECf:**Verbally recite numbers from 1 to 10.**Mathematics****6.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…?”.

**Learning Standard 6.B**

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

**Mathematics****6.B.ECa:**Recognize that numbers (or sets of objects) can be combined or separated to make another number.**Mathematics****6.B.ECb:**Show understanding of how to count out and construct sets of objects of a given number up to 5.**Mathematics****6.B.ECc:**Identify the new number created when small sets (up to 5) are combined or separated.**Mathematics****6.B.ECd:**Informally solve simple mathematical problems presented in a meaningful context.**Mathematics****6.B.ECe:**Fairly share a set of up to 10 items between two children.

**Learning Standard 6.C**

Begin to make reasonable estimates of numbers.

**Mathematics****6.C.ECa:**Estimate number of objects in a small set.

**Learning Standard 6.D**

Compare quantities using appropriate vocabulary terms.

**Mathematics****6.D.ECa:**Compare two collections to see if they are equal or determine which is more, using a procedure of the child’s choice.**Mathematics****6.D.ECb:**Describe comparisons with appropriate vocabulary, such as “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.

**Mathematics****7.A.ECa:**Compare, order, and describe objects according to a single attribute.**Mathematics****7.A.ECb:**Use nonstandard units to measure attributes such as length and capacity.**Mathematics****7.A.ECc:**Use vocabulary that describes and compares length, height, weight, capacity, and size.**Mathematics****7.A.ECd:**Begin to construct a sense of time through participation in daily activities.

**Learning Standard 7.B**

Begin to make estimates of measurements.

**Mathematics**Practice estimating in everyday play and everyday measurement problems.

7.B.ECa:

**Learning Standard 7.C**

Explore tools used for measurement.

**Mathematics****7.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.**Mathematics****7.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.

**Goal 8**

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

**Learning Standard 8.A**

Explore objects and patterns.

**Mathematics****8.A.ECa:**Sort, order, compare, and describe objects according to characteristics or attribute(s).**Mathematics****8.A.ECb:**Recognize, duplicate, extend, and create simple patterns in various formats.

**Learning Standard 8.B**

Describe and document patterns using symbols.

**Mathematics****8.B.ECa:**With adult assistance, represent a simple repeating pattern by verbally describing it or by modeling it with objects or actions.

**Goal 9**

Explore concepts of geometry and spatial relations.

**Learning Standard 9.A**

Recognize, name, and match common shapes.

**Mathematics****9.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).**Mathematics****9.A.ECb:**Sort collections of two‐ and three‐dimensional shapes by type (e.g., triangles, rectangles, circles, cubes, spheres, pyramids).**Mathematics****9.A.ECc:**Identify and name some of the faces (flat sides) of common three‐dimensional shapes using two-dimensional shape names.**Mathematics****9.A.ECd:**Combine two-dimensional shapes to create new shapes.**Mathematics****9.A.ECe:**Think about/imagine how altering the spatial orientation of a shape will change how it looks (e.g., turning it upside down).

**Learning Standard 9.B**

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

**Mathematics**Show understanding of location and ordinal position.

9.B.ECa:**Mathematics**Use appropriate vocabulary for identifying location and ordinal position.

9.B.ECb:

**Goal 10**

Begin to make predictions and collect data information.

**Learning Standard 10.A**

Generate questions and processes for answering them.

**Mathematics**With teacher assistance, come up with meaningful questions that can be answered through gathering information.

10.A.ECa:**Mathematics**Gather data about themselves and their surroundings to answer meaningful questions

10.A.ECb:

**Learning Standard 10.B**

Organize and describe data and information.

**Mathematics****10.B.ECa**: Organize, represent, and analyze information using concrete objects, pictures, and graphs, with teacher support.**Mathematics****10.B.ECb**: Make predictions about the outcome prior to collecting information, with teacher support and multiple experiences over time.

**Learning Standard 10.C**

Determine, describe, and apply the probabilities of events.

**Mathematics**Describe likelihood of events with appropriate vocabulary, such as “possible”, “impossible”, “always”, and “never”.

10.C.ECa:

## 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.