Ready Rosie: Math Strand
When we think of math for the young child, we think of teaching shapes, colors and numbers. We think
of counting and simple arithmetic such as 1+1=2. Yet mathematics for the young child is so much more!
It is investigating, reasoning, thinking, communicating and solving problems. It is thinking of numbers in
flexible ways, such as 5 can be 1 and 4 but it can also be 3 and 2. Unfortunately, if you do an internet
search of “pre-k mathematics” you will find a large number of worksheets! And yet, math is all around
us! Young children are eager to learn and solve problems and experience numbers through investigating
problems. As parents and educators it is our job to foster that excitement and shape the young child’s
math experiences.
The National Council of Teachers of Mathematics (NCTM), which is widely supported in the mathematics
community, published Principles and Standards for School Mathematics (2000) with mathematical
standards for grades Pre-K to 12. Within these principles are five content standards: (1) number and
operations, (2) patterns, functions and algebra, (3) geometry, (4) measurement, and (5) data analysis
and probability. These 5 standards will be outlined and further explained below. A description of each
standard will be given, followed by a summary of the expectations for each. These expectations are for
Pre-k through 2nd grade and have been paraphrased when appropriate for ease of understanding.
Number and Operations
This standard is the core of instruction for the young child. Most children enter school knowing how to
count. While counting is an important part of number and operations, it is only a small piece of the big
picture. In her book, “The Young Child and Mathematics,” Juanita Copley outlines the development of
number:
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Counting – reciting the names of numbers in order without necessarily understanding the
quantities they represent. Research suggests that young children can also subitize, or recognize
small groups of objects without counting.
One-to-one correspondence - when counting objects, making the connection that each object
corresponds to one number word.
Keeping track while counting - to count correctly, the child needs to keep track of items already
counted. Many children “count in circles” repeatedly counting the same item over and over.
Quantity - understanding that the last number counted relates to the number of objects in a set
Part-part-whole relationships – understanding that all numbers are made of parts is central to
developing number sense. For example, knowing that 5 can be represented by 1 and 4, 2 and 3
and 0 and 5. This is central to composing (putting together) numbers and decomposing (taking
apart) numbers. Experience with composing and decomposing makes a smoother transition to
addition and subtraction. For this reason, children need many experiences in this area.
Using representations – a variety of representations are vital to developing number sense. For
example, to represent 10 one can use tally marks, fingers, a ten frame, dominoes, pictures,
numbers,etc.
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Change operations – the most common change operations in the Pre-K setting are “add to” and
“take away.” Solving such problems includes counting all objects in a set, adding on, and mental
arithmetic. Multiplication and division can also be introduced through simple activities such as
sharing goldfish (division) or joining sets (multiplication).
Comparison – when comparing 2 or more objects, comparison words such as “bigger than,”
“less than,” “heavier than,” etc. are very helpful to the young learner who may not have enough
experience with numbers to compare otherwise. This will also be helpful later when describing
the relationship between numbers such as 100 is 10 times bigger than 10.
Pre-K – 2nd grade Expectations in Number and Operations (NCTM, 2000)
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Count and understand “how many” in a set
Use multiple models to understand base-ten number system (ex: 10 fingers, 10- frame)
Develop an understanding of and the connections between:
o the relative position (ex: beginning, middle, end) and
o magnitude (how big) of whole numbers
o cardinal numbers (how many) and
o ordinal numbers (1st, 2nd, 3rd, etc.)
Use whole numbers in flexible ways including relating, composing and decomposing
numbers
Make a connection between the number and the quantity it represents
Understand and represent common fractions ( ½ , 1/3 and ¼ )
Understand adding and subtracting whole numbers and the relationship between them
Understand the effects of adding and subtracting whole numbers
Understand multiplication and division situations such as joining and sharing equally
Develop and use strategies for whole-number computations, with a focus on addition and
subtraction
Develop fluency with basic number combinations for adding and subtracting
Use a variety of methods to compute including: objects, mental, estimation, paper/pencil
and calculator
Patterns, Functions and Algebra
Patterns are a natural part of our everyday lives. There are patterns in fabrics, in nature, in tiled floors.
There are patterns to our day, in songs, and in the weather. Children see and understand many patterns
around them. By understanding how to find, repeat and extend, copy and create, and grow patterns,
the foundation is being laid to find patterns in numbers! Early experiences with patterns will later help
the child recognize patterns in counting, skip counting, multiplication, division and other areas of
mathematics.
Sorting falls under the patterns, functions and algebra standard. By sorting and understanding the
different attributes of objects (size, color, texture, etc), children are able to create more complex
patterns. For example, if I child could distinguish and sort shiny pennies from dull pennies, they could
recognize a pattern of “shiny, dull, shiny, dull.”
Function is a relationship between two numbers. For example, if 1 person has 2 feet, 2 people have 4
feet, 3 people have 6 feet, etc. Another example of function is to observe change. Measuring the
growth of a plant over time and recording it would produce a function.
Algebra is generally thought of as an upper level math topic, with balancing equations and solving for
the unknown variable. For the young child, this can be as simple as understanding equality. For example,
2 blocks stacked end to end (up and down) are the same as 2 blocks stacked side by side. They can also
experience unknown variables with simple games at the dinner table. “I see 5 peas on your plate, when
I close my eyes, I want you to eat some.” When parent opens their eyes they can say, “Now I see 3 peas
on your plate. I think you ate 2 peas.” In later years, this simple problem would produce the equation
“5-x=3” where x=2.
Pre-K – 2nd grade Expectations in Patterns, Functions and Algebra (NCTM, 2000)
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Sort, classify and order objects by size, number and other properties
Recognize, describe and extend patterns
Analyze how repeating patterns are made (includes linear pattern such as pink, yellow, pink,
yellow)
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Analyze how growing patterns are made (ex:
)
Make generalizations about number properties (such as commutative property of 3+1=4 so
1+3=4)
Use concrete (objects), pictorial (pictures), and verbal representations to develop an
understanding of invented and conventional symbolic notions (ex: the concept of equality;
3+3=6 and 1+5=6 . In later grades, the child will see that 3+3=1+5)
Model addition and subtraction of whole numbers, using objects, pictures and symbols
Describe qualitative change (ex: “ I am taller than him”)
Describe quantitative change (ex:” I am 2 inches taller than my brother”)
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Geometry Standard
Most people think of geometry and the young child as naming shapes. While this is a small component,
the Geometry Standard is composed of two main ideas: (1) geometry - shape, size, position, direction
and movement and (2) spatial sense – an awareness of the space around us. Children need
opportunities to “handle, manipulate, draw and represent shapes in a variety of ways” (Juanita Copley,
The Young Child and Mathematics, 2000).
Pre-K – 2nd grade Expectations in Geometry (NCTM, 2000)
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For two- and three-dimensional shapes, the child should
o Recognize
o Name
o Build
o Draw
o Compare
o Sort
Describe attributes and parts of two- and three-dimensional shapes (two-dimensional example:
a triangle has 3 sides; three-dimensional example: the faces or sides of a cube are squares)
Investigate and predict what happens when you put together and take apart two- and threedimensional shapes
Describe, name and understand relative position in space and apply those ideas (ex: on, off,
over, under, in, out, into, out of, beside, between…)
Describe, name and understand direction and distance and apply those ideas (ex: near, far, up,
down, ….)
Find and name locations with simple words such as “near to” and “far from” using maps, etc.
Recognize and apply flips, slides and turns
Recognize and create shapes with symmetry
Measurement Standard
Measurement is a natural connection between geometry and number/operations. For example, finding
how many strawberries fit in a container combines counting, geometry (container is a rectangular
prism), and measurement (volume). However, research suggests that measurement is a difficult
concept for children and requires more time to teach than what is generally given (National Research
Council, 2009, Mathematics Learning in Early Childhood: Paths towards Excellence and Equity).
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Comparing and Ordering – early on, children compare objects by using words such as “heavier
than,” “shorter than,” etc. Once they learn to compare two objects, they are able to order 3 or
more objects. Such as ordering books on bookcase from shortest to longest.
Length – the most common measurement in the pre-k classroom is linear. Both standard (ex:
simple ruler) and non-standard units (ex: laying sticky notes end to end to measure the table)
are encouraged
Area – a difficult concept for young children as it requires 2-dimensions (length and width).
Experiences with placing square tiles to cover a surface help reinforce the idea of area. Looking
at a tile floor at home or school is another easy way for a child to “see” area.
Volume – adding a 3rd dimension (height) also makes volume difficult to understand. Children
enjoy experimenting with water and sand to find out the capacity (liquid measure) or volume of
a container. Cooking also lends itself nicely to measuring volume.
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Weight – children primarily use comparison terms such as heavier and lighter. Scales can also
be introduced.
Time – not much emphasis is put on learning time measurement as it is too hard for the pre-k
child. Instead, words such as “soon, tomorrow, early, late, fast, slow” help the child learn about
what time is.
Pre-K – 2nd grade Expectations in Measurement (NCTM, 2000)
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Recognize the attributes of length, volume, weight, area and time
Compare and order objects based on the above attributes (ex: order family members from
tallest to shortest)
Understand how to measure with standard and non-standard units
Choose the right tool for what is being measured (ex: ruler for length, scale for weight, etc)
Measure with multiple copies of the same unit, laid end to end (such as sticky notes measuring
the table length).
Use repetition of a single unit (such as 1 meter stick) to measure something larger (such as a
room)
Use tools to measure
Develop a common referent when measuring 2 or more items for comparison and estimating
(using a 3rd object to measure 2 or more objects. For example, using a paper clip to measure two
objects)
Data Analysis and Probability Standard
Young children are naturally curious about their surroundings and opportunities abound for data
collection.
Collecting data - Questions such as: “What is your favorite color?” “Do you like candy?” “Who is your
favorite super hero?” can be answered by collecting data from friends and organizing it in a graph with
pictures or objects. Additionally, objects such as colored candy can be sorted and and graphed.
Organizing and representing data – once data is collected, it needs to be organized and “shown” in
some way. Common representations include graphs, objects and pictures.
Describing data – after the data is organized in a graph or other format, children can “read” their data
and make statements about it. In the colored candy example, questions such as “what color do we have
the most of in our bag?” “Which two colors have the same amount?”
Probability –. Describing events as likely or unlikely and possible, certain or impossible are explored.
Pre-K – 2nd grade Expectations in Data Analysis and Probability (NCTM, 2000)
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Pose questions and gather data about themselves and their surroundings
Sort and classify objects according to their attributes and organize data about the objects (ex:
sorting fruit snacks by color or type of fruit and graphing the data)
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Representing the data using concrete objects, pictures and graphs
Describe parts of the data and the set of the data as a whole to determine what the data shows
(ex: “How many more people like vanilla ice cream than chocolate?”)
Discuss events related to student’s experiences as likely or unlikely (ex: “Are we likely to get a
neon candy from this bag of colored candy?”)
Copley, J.V. 2000.The young child and mathematics. Washington, DC: NAEYC
National Council of Teachers of Mathematics (NCTM). 2000. Principles and standards for school
mathematics. Reston, VA: Author.
National Research Council. 2009. Mathematics Learning in Early Childhood: Paths Towards Excellence
and Equity. Committee on Early Childhood Mathematics, Christopher T. Cross, Taniesha A. Woods, and
Heidi Schweingruber, Editors. Center for Education, Division of Behavioral and Social Sciences and
Education. Washington, DC: The National Academic Press.