Counting

“Counting can be used to find out “how many” in a collection.”

-Erikson Institute’s Early Math Collaborative

One afternoon I sat with a little boy looking at pods we collected from a vine growing on a nearby fence. I asked him to tell me how many there were.

Boy: 1, 2, 3, 4, 5

Me: Are you sure about that? How many are there?

Boy: 1, 2, 3, 4, 5

Me: Okay, so there are how many?

Boy: 1, 2, 3, 4, 5.  

This interaction showed that the boy did not fully understand the purpose of counting. He knew the number names in order, however, he did not know that the last number counted was the quantity of all the items in the group.  

He did not have what is called Cardinality.

“Although adults take it for granted because it is so familiar, the connection between the list of counting numbers and the number of items in a set is deep and subtle. It is a key connection that children must make.” -National Research Council, 2009 p25

Children who recite the number names in order may appear knowledgeable, however, a true understanding of counting occurs when children use counting to find out how many.

Playful invitation

1. Prepare: Gather about 12 small natural items and a few large leaves. For this activity I will use acorn tops and leaves.

Observe the child throughout the interaction. Use the Invitation to Play Documentation Tool to collect data.

2. Invite: Freddy Frog is looking for a pile of five! Will you help him find the perfect pile with five things?

3. Play: Place a leaf with 1, 2, or 3 items on it in front of the child.

Point to a pile and ask: How many do you think we have in this pile? (*Asking the child to predict provides practice labeling the set with a number at the beginning and allows you to assess their ability to subitize.)  After they say a number- Let’s see if you’re right!

Model counting: Tap each item with your finger as you count OR count each item and set it to the side (*both strategies help to maintain order so the child does not get confused about which item was already counted)

After counting say: So that group has three! We counted, 1, 2, 3 and there are three in that group.

So how many did we have in that group? (assess their understanding that the last number counted is the amount of the group)

Is this the pile for Freddy Frog?  Remember he wants five.

Next, point to another pile with a different amount between 1-5. Continue asking how many until you reach the pile of 5.

When the child finds the pile of five celebrate!

4. Reflect and Assess: Did the child predict? If so, they are showing some ability to understand quantity. Were they able to count using one to one correspondence? Did they retell the amount after the items were counted? Did they know whether the pile was right for Freddy Frog?

*Think about how you can adapt the activity based on the child’s interest.

i.e. “The dinosaur needs to eat exactly 5 leaves. Is this five leaves?” or “The bird needs five sticks to build her perfect nest. Is this five? Let’s count and see.” Continue to vary the number of items. Count together. Decide if it is the right amount or not.

Ready: In order to count to find out “how many” the child needs to know the count sequence in order and must understand that each number counted corresponds with one, and only one, item. This is called Rational Counting.

Ready to move on: The child easily counts a group of up to five items and then tell how many were counted.

Extend: Count higher numbers of items up to 10 to determine “how many.”

Practice the count sequence to 20. Also, select a number and practice counting on (i.e. Let’s count starting with 6.)

Math language

Cardinality- understanding that the last number counted represents the quantity of the

entire set.

Predicting-   to make an educated guess.

Rote count- the ability to recite the count sequence in the correct order without understanding.

Rational counting: the ability to count in sequence and use one to one correspondence to determine the number in a set.

Subitizing- ability to quickly perceive a number without counting.   

Resources

Erikson Institute’s Early Math Collaborative (2014). Big Ideas of Early Mathematics. Pearson Education.

National Research Council , 2009, Mathematics learning in Early Childhood*

Clements, D.H., & Sarama, J. (2017)*

Mix, K.S., Sandhofer, C.M., Moore, J.A., & Russell, C. (2012)*

*full citation can be found on the Research Page

Spatial Sense

“Relationships between objects and places can be described with mathematical precision.”

-Erikson Institute’s Early Math Collaborative

In a kindergarten class, a group of four children were excitedly planning to act out the story The Three Billy Goats Gruff.  I sat nearby and watched as the children marked out space for the stage, then dragged props from around the room. For 15 minutes I watched as the children negotiated the perfect placement for the bench (the bridge), a blue princess dress (the creek), chairs (hillside), and finally, a spot for all the characters to stand. Without any adult interaction, the children negotiated using language, “on top of, under, over there, next to, up, down” and many more.  After organizing every last detail, it occurred to me that without experience with these spatial terms the play would not have been a success.

“Children explore spatial concepts through play from an early age. For instance, building with blocks and playing with puzzles have both been linked to stronger spatial skills. Additionally, young children who hear more spatial words from their parents tend to talk about space more themselves, and these same children perform better on later spatial reasoning tasks.”

-Eason & Levine, 2017

The language we use to describe an object’s position in space helps children to learn and understand spatial relationships. This activity provides an opportunity to use this rich spatial language.

Playful Invitation

1. Prepare: For this activity gather an assortment of natural materials and paper/writing tools or a camera (like many people have on their phone). 

Arrange the materials in any way. Be sure some items are placed in what could be described as, “next to, on top of, under, above, below, in front of or behind.” Once materials are arranged, snap a photo or draw a quick sketch.

Observe the child throughout the interaction. Use the Invitation to Play Documentation Tool to collect data.

2. Invite: Show the child the drawing/photo and ask if he/she would like to try and recreate the picture.

3. Play: Work alongside the child, observing as he/she positions objects to recreate the picture. If necessary, guide the child using spatial relationship language (next to, on top of, under, above, below, in front of and behind).

Ask open-ended questions: How did you know to put that one there? OR Where does this belong?

Model thinking: Talk out loud as you place an item. I see the leaf belongs under the lily pod, so I’ll put it down first and then I’ll put the lily pod on top of the leaf!

4. Reflect and Assess: Did the child show an understanding of spatial relationships? Which position words did they know and which did they struggle with?  Did the child follow directions when asked to place an object in a certain position? Did the child use spatial relationship words on their own to describe their thinking?

Ready: The child understands position words (in, on, under, up, down) and follows directions related to proximity (next to, above, beside).

Ready to move on: The child recreates the arrangement with the assistance of an adult, describing the position of objects using spatial terms.

Extend: Lessen the assistance you provide and encourage the child to recreate on their own. When the child finishes ask him to use spatial language to describe where he placed the items OR while the child works have her talk about where she places objects and why.

Challenge the child to create another arrangement that you will recreate.

Math language:

Precision- exact and accurate

Relationships- the state of being connected

Spatial Positions- an objects position in space in relation to other object/s (up, down, above, below, next to, under, etc.)

Spatial Relationships-  how an object is located in space in relation to another object.

Resources:

Erikson Institute’s Early Math Collaborative (2014). Big Ideas of Early Mathematics. Pearson Education.

Clements, D.H., & Sarama, J. (2017)*

Eason, S.H., & Levine, S.C. (2017)*

National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence.*

*full citations can be found on the Research page

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Counting Large Numbers

Lately my child is obsessed with counting to 100. At school, the class is gearing up for the “100th day of school” and teachers are asking the children to try and count as high as they can. A simple counting book from the library with a page that shows 100 baby chicks is a favorite. Right before bed she pulls out the book and insists on counting all 100 chicks. I am both proud and impatient as I stare longingly at the clock and hope she counts fast.  

“Providing opportunities for children to count collections greater than 20 (even as they may still be struggling with the teens) can help them to engage with its underlying structure. In attempting to count further, children’s emerging understanding of the base-ten number system is often revealed.” 

–Carpenter, Franke, Johnson, Turrou and Wager (2017)

Children are not expected to count large numbers before entering Kindergarten, however, some preschool children are ready and excited to explore higher numbers. Some children may enjoy counting collections and feel proud as they count higher and higher. Often, listening as children count higher numbers reveals what they know, and do not know, about our number system. For instance, a child who counts 1-29 and then says twenty ten, twenty eleven… reveals an opportunity to talk about decades, and how the count sequence changes after a 9. (i.e. 9-10, 19-20, 29-30 and beyond). Watching children count physical objects also reveals counting strategies including organizing or grouping a large collection before counting. 

PLAYFUL INVITATION

1.     Prepare: Gather 100 or more objects from nature. Also, consider creating a 100 grid using chalk on the sidewalk or a piece of poster board.

Observe the child throughout the interaction. Use the Invitation to Play Documentation Tool to collect data. 

2.     Invite: Will you help me count this collection of things? I want to see if I have 100!

3.     Play:  Ask open-ended questions: What should we do first? 

Observe, does the child immediately begin counting, organize/sort the group, or place items on each square of the grid?  There is no right or wrong way. This is all about seeing what children know. At first, simply provide the materials and be a “watcher.”

Consider asking guiding questions: Is there a way we can get organized so we don’t count the same things more than once? This grid has one-hundred squares, maybe we could use this to help us count? OR Look each row has ten boxes. Every time we start a row we say something a little bit different!  OR Let’s count together (practice the count sequence and deliberately stress 10, 20, 30, 40…etc.)

Continue observing, offer guidance when necessary, and ask open-ended questions. 

 4.    Reflect and Assess: While counting, did the child skip any numbers, forget their place, or make any errors? What numbers were challenging to the child? What strategies does the child use to keep track of what he/she already counted? Does he/she line the items up, organize them in some way, or even put them into groups of 10?

Ready: The child can accurately count large numbers of items (10-20) using one to one correspondence and the child knows the count sequence beyond 20.

Ready to move on: If the child easily counts a collection of 100 items.

Extend: Count higher numbers beyond 100. Introduce counting by twos, fives, or tens. Count on from a number. Say let’s start at 29, what comes next?

MATH LANGUAGE

Base Ten Number System- The base-ten number system consists of ten digits (0,1,2,3,4,5,6,7,8,9) and groups number into tens.

One to One Correspondence- When counting a set, each item should be counted only once matching one number word to each item.

RESOURCES 

Clements, D.H., & Sarama, J. (2017)*

Carpenter, T.P., Franke, M.L., Johnson, N.C., Turrou, A.C., and Wager, A.A. (2017). Young children’s mathematics: Cognitively guided instruction in early childhood education. Portsmouth, NH: Heinemann.

*full citation can be found on the Research Page

Counting

“Counting has rules that apply to any collection”

-Erikson Institute’s Early Math Collaborative

Like most parents and teachers we make up rules. Some typical: Do not throw toys inside.  Some unusual: Do not eat the dog food! Whatever the reason, rules serve a purpose.

In preschool math, there are also rules. Counting, for example, has four rules. And, incredibly, most children learn the rules of counting without ever hearing them stated.

The Rules of Counting:

1. Number words are said in the same order every time. (stable order)

2. Each item should be counted only once, matching one number word to each item.

   (one to one correspondence)

3. A set of items can be counted in any order and the result will be the same.

   (order irrelevance)

4. The last number counted is the quantity of the set. (cardinality)

How to use the Blog Posts

Playful Invitation

1. Prepare: Cut an egg carton so it has ten cups. (Working with ten is important because our number system is based on a system of tens) Provide an assortment of small natural loose parts, or ask the child to walk around outside to gather items on their own.

Observe the child throughout the interaction. Use the Invitation to Play Documentation Tool to collect data.

2. Invite: Hand the child the container and say: Here is a container that I want you to fill with a few special things you find. Put only one item in each cup. Off you go!

3. Play: When the child returns, say: Let’s look at what you found! Notice whether the child used one to one correspondence and placed only one item in each cup.  If not, then work with the child to move or remove items so there is only one item in each cup. Modeling and practicing one to one correspondence will support the child’s ability to count.

Once items are in the cups ask How many did you gather?  Explicitly point to each item as the child or adult counts using the stable order principle.

Ask: So how many was that? to assess their understanding of cardinality. Hold up the same number of fingers to symbolize the number of items in their container.

Using the same collection, or another collection gathered by a peer, count again.  This time count from the bottom row of the container or spin the container and count vertically. This shows the child that the order is irrelevant and the collection could be counted in many different ways.

4. Reflect and Assess: Make a list of the four counting rules.  Place a check next to each one as you work with the child. What do you notice about the child’s ability to count?

Ready: The child is ready for this activity if he/she can count to 10 with understanding (most of the time), and can label small quantities of items.  If the child struggles then try the activity using only three - five cups. Also, continue counting 1-10 using songs, counting body movements (stomps or jumps) and counting items.

Ready to move on: If the child easily counts to 10, can gather 1-10 items when asked, uses one to one correspondence when counting, and determines the quantity in their container.  

Extend: Specify the number of items the child should find. (Can you find five items?)

Increase rote counting to 20.  

Use two containers for gathering items and then add them together.  Write the final number on the ground with chalk.

This activity also supports the development of concepts like full, half full and empty (precursors to fractions) and spatial relationships like on and in.

According to the National Research Council- For now focus on whole items rather than looking at features of an item. For instance, points on a leaf, leaves on a tree branch, or branches on a tree, may seem like good ideas for counting, however children need time and practice before they learn to count parts of a thing. For now just count the “thing” as a whole. NRC, 2009, 136

Math language:

Cardinality- understanding that the last number counted represents the quantity of the entire set.

One to one correspondence- when counting a set, each item should be counted only once matching one number word to each item.

Order Irrelevance- items can be counted in any order and the result will be the same.

Rote count- the ability to recite the count sequence in the correct order without understanding.

Stable order- number words are said in the same order every time

Resources:

Erikson Institute’s Early Math Collaborative (2014). Big Ideas of Early Mathematics. Pearson Education.

National Research Council (2009)*

*full citation can be found on the Research Page

Subitizing

“The quantity of a small collection can be intuitively perceived without counting.”

-Erikson institute’s Early Math Collaborative

When my daughter and I started playing games with dice I noticed her ability to quickly call out the number without actually counting the dots.  She rolled three dots, and without counting, said THREE! How did she learn to do that?

Later, I attended a conference session all about Subitizing- the ability to quickly perceive a number without counting.  Subitizing is very important in early childhood and something we all use in our day to day life.

The primary focus in most preschools is to teach children to count, however research has shown that another crucial skill is attaching number words “labels” (one, two, three) to small sets of objects. This is an essential stepping stone to developing an understanding of quantity.

-Nguyen, Laski, Thomson, Bronson, & Casey, 2017

In this playful invitation focus on developing the ability to label small groups without counting. Preschoolers begin to use subitizing with 1, 2 and 3 items.  As they grow in their abilities they see 4, 5, and 6. You can practice your child’s ability to quickly perceive an amount by playing simple games.

How to use the Blog Posts

Playful Invitation

1. Prepare: 5 shells, acorns, or other natural material, and a piece of paper, cardboard, or fabric. Sit facing the child with the materials on the table/ground and the paper/fabric/cardboard hiding the materials from the child’s view.

Observe the child throughout the interaction. Use the Invitation to Play Documentation Tool to collect data.

2. Invite: I have a game to play with you! When I move this paper you will see some shells/rocks/etc.. I want you to take a quick peek and tell me how many you saw.

3. Play:

Secretly place two shells on the table or ground and cover with the paper. Quickly reveal the item for about 3 seconds and then cover again.

Ask: How many did you see? and How do you know?

Model thinking and counting: Wow, so you knew the number without even counting? You’re right! There were two. See, 1, 2! Deliberately point and count.

Vary the amount (no more than 5 to start) to challenge the child.

4. Reflect and Assess: How did the child do with this game? Was the child able to instantly name the amount without counting? What number of items posed a challenge?

Ready: The child is ready if he/she shows an understanding of one, two and more. Also the child knows the number words one, two, three and beyond.

Ready to move on: The child quickly calls out the number when shown 1, 2, 3 items.

Extend: Work on subitizing to 5.

Also, try combinations of materials or colors. For example, two rocks and two shells. Ask How many did you see? Then ask How many were rocks? and How many were shells?

Noticing smaller groupings within the set is the beginning of Conceptual Subitizing- when you quickly perceive smaller groups within the set and use these to determine the amount. Check out this video at mathisvisual.com for a fun example based on the ideas of Jo Boaler.

Another idea I play with my children is “Fingers, Fingers,” developed by Kristen Reed and Jessica Mercer Young. Check out their great article about games: https://www.naeyc.org/resources/pubs/tyc/oct2017/play-games-learn-math-explore-numbers; Youtube video

*One additional thought. It is recommended that children primarily practice Subitizing with simple dots. Consider how you can incorporate dice and dot cards into your daily routines and math games.

MATH LANGUAGE

Quantity- the amount of something. The “how many or how much.”

Perceived- to understand or realize something.

Subitizing- ability to quickly perceive a number without counting.   

Conceptual Subitizing- when you quickly perceive smaller groups within the set and use these to determine the amount.

Set- items that are grouped together in a meaningful way.

Number sense- the ability to understand the quantity of a set and the name associated

with that quantity.

Resources

Erikson Institute’s Early Math Collaborative (2014). Big Ideas of Early Mathematics. Pearson Education.

Clements, D.H., & Sarama, J. (2017)*

Reed, K., & Young, J.M. (2017)*

Nguyen, Laski, Thomson, Bronson, & Casey, 2017*

Mathisvisual.com

* full citations can be found on the Research Page

SHARE YOUR EXPERIENCES WITH OTHERS IN THE COMMENTS SECTION

Collecting Data

“The purpose of collecting data is to answer questions when the answers are not immediately obvious.”

-Erikson Institute’s Early Math Collaborative

This past week I had the opportunity to walk along the beach with my preschool-aged children and marvel at the abundance of seashells. My older child is very interested in big numbers and often combines numbers she’s heard into unusual and enormous combinations. She asked if I knew how many shells were on the beach. “I don’t know?” I replied. Excitedly she said…“I think there are thirty two hundred thousand and sixty thousand!”  This made me wonder, could she be right?

“The collection of data should ideally start with a question of interest to children.” –NRC 2009 

This playful invitation is a basic start to data collection, asking and answering questions that involve math. The idea for the blog was inspired by my child’s spontaneous questions as we looked at shells on the beach. “I wonder which one has the most?” she asked. 

While each child’s questions about the world are different, one way to encourage interest in data collection is to pose a question yourself and work together to figure out the answer.  

PLAYFUL INVITATION

1.     Prepare: Gather a large collection of natural loose parts. Think about having multiple items that are the same within the collection. Ideas might include: various colored fall leaves, different materials that fall from trees, shells, etc.

Observe the child throughout the interaction. Use the Invitation to Play Documentation Tool to collect data. 

2.     Invite: Display the materials in a random order. Say, Let me show you something….I have this big  collection of shells and I’m wondering which type of shell has the most. I really want to figure it out. Will you help me answer my question? 

3.     Play:  Ask the child What do you think I should do first? Follow the child’s lead and really listen to their reasoning. 

If he/she seems stuck scaffold through questions and suggestions. i.e. Perhaps we could sort all the shells? OR I see a lot that look like this (point to one type) first let’s find all of this kind.

Once the shells are in groups ask the child, What do you notice about these groups? Which one has the most? How do you know?

Using a piece of paper, or a stick in the sand, write numbers/tally marks/dots.  Talk about your findings. Which had the most/least/same. Why is collecting data useful?

4.     Reflect and Assess: What did you learn about the child’s strategies for problem solving? Was the child able to develop a plan to answer the question?  Did his/her suggestions lead you down a different path than you thought? Did you see the child using skills like sorting, classifying, counting, estimating, measuring, etc. 

What other questions about the world do children ask?  Pay attention. Take advantage of the authentic opportunities for math learning that stem from their inquiry.

Ready: The child is ready for this activity if he/she can count with understanding, knows ordinal numbers (1st, 2nd and 3rd) and has basic sorting skills.

Ready to move on: If the child uses problem solving skills to gather relevant data, and determines the answer to simple math questions involving small numbers, then he/she is ready to extend.

Extend: Represent findings using simple graphs. Gather data in other ways, for example, simple surveys of peers. Use larger numbers when exploring data.

MATH LANGUAGE

Sort- organize items based on characteristic/attribute.

Reasoning- thinking about something in a logical way. Children are asked to explain their answers so an adult and/or other children can hear the thought process or reasoning that went into solving a problem.

Problem Solving- finding a solution to a question, issue, or problem. 

Ordinal Number- tells the position of an object when it is part of a list. 1st, 2nd, 3rd, 4th, etc.

RESOURCES 

Erikson Institute’s Early Math Collaborative (2014). Big Ideas of Early Mathematics. Pearson Education.

National Research Council , 2009, Mathematics learning in Early Childhood*

Clements, D.H., & Sarama, J. (2017)*

*full citation can be found on the Research Page

 HAPPY NEW YEAR!

Operations

“Sets can be changed by adding items (joining) or by taking some away (separating)”

-Erikson Institute’s Early Math Collaborative

My children are very different. One is a rule follower, structured and orderly. The other is more physical and free spirited. One day we were raking the leaves in our yard into two piles. Child #1 carefully added leaves to a pile...which quickly grew!! Child #2 added leaves to a second pile, then stopped, jumped on top of the pile, and threw the leaves around. Whenever her pile diminished (due to so much joyful play) she would sneak over and steal leaves from the larger pile.  This back and forth, adding and taking away, putting together and separating was real life math in action!

“Young children’s lives are filled with situations and with beloved tales that pose mathematical problem situations. If we want them to understand how joining/separating...numbers makes sense, we must mathematize those stories. Just as counting is meaningful only if we are counting something, children need to be given many opportunities to experience that every operation tells a story.“ -Erikson Institute’s Early Math Collaborative, 2014

In this activity use storytelling to work on beginning number operation skills.  There is no right way to tell the story, just find ways to add or subtract characters then talk about how many.

Playful Invitation

1. Prepare: For this activity I will use a set of rocks with eyes. (these are well loved, and were a favorite of my class and children). You can use any natural material. Draw eyes with a sharpie marker or paint pen if you like.

2. Invite: So I have these little rock guys and I want to tell you a story. Would you like to hear a story? Help me count as I tell the story?

3. Play:  

One day there was one little rock guy playing all alone. He was so sad.

Then he saw two friends walking by.

Ask the child: Should we ask them to play?

Friends would you like to play? Yes!

Ask the child: Now how many do we have?

The three friends were having SO much fun and some other rock guys could hear them…so along came two more friends.  

Can we play too?

Yes! You can play too!  

Ask the child: Now how many do we have?

But then it started to get late and the first rock guy had to leave. Goodbye.

Ask the child: Now how many do we have?

Continue telling the story. Adding and taking away characters.

4. Reflect and Assess: Did the child count to find out “how many?” Was the child able to figure out “how many” without counting (this is called subitizing)? How many characters/materials did you use? Is the child ready to add more characters/materials?

Ready: The child can rote count to five and above. The child uses one to one correspondence when counting.

Ready to move on: The child accurately counts “how many” each time a character is added or subtracted.  You may see the child using subitizing (ability to perceive the number without counting) or you may see the child “counting on” from the original number of items when adding. (1 is added to a group of 5. Child counts 5, 6!)

Extend: Ask the child to tell you a story using the materials.

Add additional characters to increase the number being used to 8 and then 10.

Math Language

Number Operations- addition, subtraction, multiplication and division.

One to one correspondence- when counting a set, each item should be counted only

once matching one number word to each item.

Rote count- the ability to recite the count sequence in the correct order without understanding.

Subitizing- ability to quickly perceive a number without counting.  

Resources

Erikson Institute’s Early Math Collaborative (2014). Big Ideas of Early Mathematics. Pearson Education.

National Research Council , 2009, Mathematics learning in Early Childhood*

*full citation can be found on the Research Page

Shapes

“Shapes can be defined and classified by their attributes”

-Erikson Institute’s Early Math Collaborative

My preschool class loved to play I-spy.  Sometimes we played with color, “I spy something red” and sometimes we played with shapes, “I spy a triangle.”  Playing this game really helped the children to notice the shapes all around. I believe one of the most engaging ways to learn about shape is exploring our everyday environments. Books like Shapes, Shapes, Shapes by Tana Hoben and , Circles, Stars, and Squares: Looking for Shapes by Jane Brocket inspire us to explore shapes in our surroundings. However, researchers recommend parents and teachers do more than name shapes, they need to talk about the key features of each shape.

In this activity I encourage you to go beyond simply finding shapes. Use sticks and other natural materials to create shapes and talk about their features. Counting sides and corners helps children to understand what makes a triangle, square, rectangle and other shapes. This sheet created by Reed & Young, 2007 is a wonderful guide.

How to Use the Blog Posts

Playful Invitation

1. Prepare: Gather sticks or other long (and fairly straight) natural items of varying sizes. Cut or break some sticks to be the same length (this is important for making squares and equilateral triangles).  Also use chalk, cardboard, paper or other materials as shape guides.

Observe the child throughout the interaction. Use the Invitation to Play Documentation Tool to collect data.

2. Invite: I would really like to see what shapes you can make with these things. OR Show me how you could use the materials to make a triangle/ square/ rectangle etc.

3. Play: While the child works ask, Tell me about the shape you’re making?  How do you know its a triangle/square/rectangle?

Model thinking: I know that triangles always have three straight sides, so let’s count the sides. 1,2,3..so it is a triangle.  OR Wow, this looks a lot like a square, I know squares have four sides, four corners, and all the sides are all the same length. Let’s count the sides. 1,2,3,4. Are they all the same length?

Continue creating shapes and talking about their features.

4. Reflect and assess:  Was the child able to identify and create shapes? What shapes will you continue to work on? What new shapes can you introduce?

Ready: Children are ready to talk about the attributes of shapes once they can match two identical shapes and are beginning to identify basic shapes like circles, squares, triangles and rectangles in their environment.

Ready to move on:  The child easily identifies circle, square, triangle and rectangle. Also, the child can create shapes using a guide, or independently.

Extend: Continue to present shapes in different sizes and orientations. Continue to talk about the attributes of each shape. Ask the child why it is a certain shape and listen to his/her reasoning.

Try and make a circle with small items, then try and make it with sticks. Talk about the difference and how circles can’t be made with straight lines.

Begin looking for 3-dimensional shapes in the environment and notice how squares, circles, triangles and rectangles make up the faces of these shapes.

IMPORTANT: Creating accurate shapes relies on precision.  Most natural materials are slightly imperfect making it difficult to create perfect shapes. At this point in learning, focus on the sides and corners, counting, and learning the attributes of each shape.  A follow up to this invitation might include a discussion about what a “straight line” is and comparing shapes with non-shapes that have curved sides. i.e. a triangle vs. a slice of pizza.
*Also a square is a special kind of rectangle with four square corners and four sides that are the same length.”

Math language

Classify- to put into categories

Attributes- characteristic of an item. Examples include color, shape, texture, size, type, number, etc.

Equilateral triangles-a triangle where all three sides are exactly the same length.

Resources

Erikson Institute’s Early Math Collaborative (2014). Big Ideas of Early Mathematics. Pearson Education.

National Council of Teachers of Mathematics. (2006). Curriculum focal points*

Reed, K.E. & Young, J.M. (2017)*

*full citation can be found on the Research Page

Number Sense

“Quantity is an attribute of a set of objects, and we use numbers to name specific quantities”

-Erikson Institute’s Early Math Collaborative

Did you know...infants notice a difference between small groups of 1, 2 and even 3 objects? This ability makes them ripe for learning math.

“Learning the number words and relating them to a set of objects is a major new kind of learning done by toddlers and preschoolers at home and in care and education centers...children who acquire this knowledge at earlier ages are provided with a distinct advantage.” -National Research Council, 2009

“Distinct Advantage” As a parent and teacher, I want the children I work with to have a distinct advantage, don’t you?

This activity is about developing a sense of number and how numbers are used. Very simply, we want children to learn the amount of something can be an attribute of the set (just like color, shape, etc.). When children label small sets with a number word they learn that number words can be used to describe the set.

Hand me the yellow leaves. Hand me the three leaves.    

I want round crackers. I want two crackers.

Practice counting and labeling sets of items. Begin with 1, 2, 3. Don’t rush to higher numbers.  Start small, build, and have fun.

How to Use the Blog Posts

Playful Invitation

1. Prepare: a small assortment of natural materials and a piece of chalk. I will use sweet gumballs and magnolia seed cones.

Observe the child throughout the interaction. Use the Invitation to Play Documentation Tool to collect data.

2. Invite: Use 2 seed cones to begin. Look at what I found! Aren’t these amazing! Tell me, how many do I have?  If necessary-Allow time for the child to think.

3. Play: The child may immediately call out a number, count each item, or guess.

Ask: How do you know I have that many? Observe their response.

Model thinking if needed:   Lets count together 1, 2.  There are 2 seedcones! See how I count the gumballs and the last number I say is how many we have.

Continue the game gathering new materials and counting how many. Use chalk to write the number on the ground next to the items to show a connection between the materials, the amount, and the numeral.

Ideas for keeping it playful:

Hide materials behind your back and say, “Now I’m going to challenge you! How many do I have this time!” Reveal a new amount each time.

Let the child take the lead, Now it’s your turn! You choose a few things and I’ll tell you how many!”  Play back and forth as long as the child is interested.

4. Reflect and Assess: Was the child able to verbally label small groups (1, 2, 3) or larger groups (4, 5, 6) with a number? Did they count each item using one to one correspondence? Did they restate the total amount when asked, or did they recount the items each time (repeat the count errors occur when the child does not understand the last number counted is the amount of the group)?

Ready: The ability to label small quantities with a number develops over time with counting and one-to-one correspondence. Practicing one skill reinforces the other two. Counting and labeling sets can be done with infants and toddlers, but grows in importance as the preschool child shows an interest in numbers and counting.

Ready to move on: If the child easily labels a group of items 1, 2, 3 and beyond.

Extend: Ask the child to find 1, 2 or 3 items.  Hand me three rocks?  Now hand me two leaves? Do they produce their own sets?

Next, draw a circle and write a number. Have the child practice finding different small sets of materials.

Finally, count larger groups of items. When you finish counting ask, So how many were in that group? If the child restates the number without recounting, it shows the child is developing cardinality and understands the amount counted is the quantity of the set.

Math Language

Attribute-characteristic of an item. Examples include color, shape, texture, size, type, number, etc.

Set- items that are grouped together in a meaningful way.

Sort- to organize items based on characteristic/attribute.

Quantity- the amount of something. The “how many or how much.”

Cardinality- understanding that the last number counted represents the quantity of the entire set.

Rote count- the ability to recite the count sequence in the correct order without understanding.

One to one correspondence- when counting a set, each item should be counted only once matching one number word to each item.

Resources

Erikson Institute’s Early Math Collaborative (2014). Big Ideas of Early Mathematics. Pearson Education.

Nguyen, Laski, Thomson, Bronson, & Casey, 2017*

Cordes & Brannon, 2009*

NRC, 2009 pp 60-66  “Evidence for early understanding of number”*

*full citation can be found on the Research page

Measurement

“All measurement involves a “fair” comparison”

-Erikson Institute’s Early Math Collaborative

One morning my child was digging around in the pantry when she noticed a potato beginning to sprout. We decided to conduct an experiment and placed it in a glass of water to see if it would grow.  Every morning for about a week, we drew a line measuring it’s height. After awhile we found an onion and started the process again. This time we used a ruler and talked about numbers. This took about 2-3 minutes each morning, but provided a powerful experience with measurement.

“Children identify objects as “the same” or “different,” and then “more” or “less,” on the basis of attributes that they can measure. They identify measurable attributes such as length and weight and solve problems by making direct comparisons of objects on the basis of those attributes.” -NCTM Preschool Focal Points”

For this activity use one consistent material to practice preschool measuring skills. Using one material will allow the child to focus on math talk related to “same, different, bigger, smaller, taller, shorter, heavier and lighter.”

How to Use the Blog Posts

Playful Invitation

1. Prepare: Gather any material in an assortment of sizes. I will use pinecones.  

Observe the child throughout the interaction. Use the Invitation to Play Documentation Tool to collect data.

2. Invite: Look at all these pinecones! There are so many sizes. Lets compare and put them in order from largest to smallest?

3. Play:

Ask open-ended questions: Tell me about the pinecones. How did you know this one goes here and this one goes here?

Model thinking: Pick up another pinecone and say, I wonder where this one would go? It’s smaller than this one so it must belong over here. But this one is even smaller than the one I’m holding! It must go in between.

Variations might include: using a small number of pinecones, placing a few ahead of time and asking the child where to add pinecones, or putting them into groups of small-medium-large.

4. Reflect and Assess: Did the child use words related to measurement (same, different, bigger, smaller, taller, shorter, heavier and lighter)? Did the child order the materials based on size or length? What else did you notice while observing?

Ready: The child compares between two items, and can indicate which is bigger and smaller. When materials are arranged in order, the child compares to figure out where to place an object.

Ready to move on: Compares and orders a set of objects (size, length or weight).

Extend: Continue comparing and ordering materials based on other ways of measuring, like volume and area. Use various containers and talk about which one holds the most or has the most space inside.  

Practice nonstandard measurement using materials. (How many pinecones tall are you?)

Talk about ways we measure using scales, tape measures, rulers, thermometers, etc.

Math Language

Measurement- the process of measuring the size, length, weight, volume and other attributes of an object.

Attributes- characteristic of an item. Examples include color, shape, texture, size, type, number, etc.

Comparison- noticing similarities or differences between two or more items.

Resources

Erikson Institute’s Early Math Collaborative (2014). Big Ideas of Early Mathematics. Pearson Education.

Clements, D.H., & Sarama, J. (2017)*

National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence.*

*full citations can be found on the Research Page

Look for the extra large sugar pine cones at your local garden store. They cost about a dollar or two.

Measurement

“Many different attributes can be measured, even when measuring a single object”

-Erikson Institute's Early Math Collaborative

About a month ago I purchased an assortment of pumpkins to decorate our porch. The pumpkins were a great delight to my children who enjoy stacking them one on top of the other. On a few occasions, I watched as they attempted to stack the littlest one on bottom with no success. Then, they tried the medium one on bottom, which worked, but made the stack very unsteady. They both agreed the best placement was biggest on bottom, medium in the middle and smallest on top. “Like a snowman!” they said.

Lucky for us, it snowed today, giving us the chance to make real snow people.

Children’s understanding of measurement has its roots in infancy and the preschool years…preschoolers can be guided to learn important concepts if provided appropriate measurement experiences. They naturally encounter and discuss quantities. They initially learn to use words that represent quantity or magnitude of a certain attribute. Then they compare two objects directly and recognize equality or inequality.  –National Research Council, 2009

How to use the Blog Posts

Playful Invitation

1.    Prepare: Gather small, medium, and large bowls, an assortment of natural materials, and head outside for snow. 

Observe the child throughout the interaction. Use the Invitation to Play Documentation Tool to collect data. 

2.    Invite: Will you help me make a snowman? Lets try to make three snowballs that are different sizes.

3.    Play: Begin shaping snowballs using the bowls as a mold. Make one, any size, then encourage the child to help you create the next size. Work together, scooping, packing and rolling the snowballs into shape. 

Ask open-ended questions related to measurement: What size is this snowball? Which one is the biggest? Which one is the smallest? Did we make one thats medium size?  

Stack them on top of each other to make a snowman. Is the snowman short or tall? Is it wide or thin?

Have the child stand next to the snowman to do a direct comparison. Measure how tall the snowman is on the child’s body. Model Measuring: Wow! This snowman is tall! It comes all the way up to your shoulders. OR This snowman is small, look, it only comes up to your waist. Use your hand to point to the spot where the snowman ends.  

Then, switch so the grownup stands next to the snowman and asks the child measure where it comes to on their body.

Continue decorating the snowman with your natural loose parts. (A perfect time to count eyes, nose, ears, arms and other body parts.) Take off a glove and feel the snow. Talk about temperature. Oh! The snow is so cold! My hands are much warmer in my gloves. Also talk about time. We’ve been out here a long time! How much longer should we stay outside?

4.    Reflect and Assess: Does the child use math language associated with measurement? Does the child accurately compare using small, medium and large? Did the child understand how to do a direct comparison to measure the height of the snowman?  What else did you notice?

Ready- The child is ready if they can make simple comparisons between two items. One is bigger or smaller than another.  

Ready to move on- The child is ready to move on if they easily make small, medium and large snowballs, then order them according to size. The child also is ready for more experiences with non-standard measurement if they easily compared their body, and the grownup’s to the snowman.

Extend- Measure “how many” tall the snowman is using a stick, leaf or hand. 

Talk more explicitly about capacity. This bowl holds a lot of snow. This one does not hold as much.  Ask the child if they would like to change the snowman to make it bigger or smaller. Use language related to adding and taking away during the process. 

Math Language

Measurement- when a number is used to show the size or amount of something.

Direct comparison-when you align two objects so they can be compared based on an attribute.

Resources

Erikson Institute’s Early Math Collaborative (2014). Big Ideas of Early Mathematics. Pearson Education.

Clements, D.H., & Sarama, J. (2017)*

National Research Council (2009)*

*full citations can be found on the Research page 

Patterns

“Patterns are sequences governed by a rule they exist both in the world and in mathematics”

-Erikson Institute’s Early Math Collaborative

Every morning on our way to school we drive by a car dealership with brightly colored balloons attached to shiny new cars. One day my daughter exclaimed, “Look it’s a pattern! Red, white, red, white!” 

Later that afternoon I asked her if ‘square, circle, square, circle,’ could be a pattern?  

“I think so,” she said. 

“What about this?” I said as I clapped, stomped, clapped, stomped.

“No! No!” she said. 

My daughter had a sense of what makes a pattern, but did not know patterns can be made of different things (objects and sounds).  Patterns are also found throughout our day-to-day lives. A pattern is simply, anything that repeats in a predictable way. 

“In addition to being a common topic for young children, patterns are considered a central idea in mathematics. Identifying, extending and describing predictable sequences in objects or numbers are core to mathematical thinking.” -Rittle-Johnson, 2015 

This activity is based on the National Research Council’s (2009) recommendation that preschool children learn abab, abcabc, and aabbaabb patterns.  This is only one small part of learning about patterns, but easy to do with materials found in nature.

How to use the Blog Posts

Playful Invitation:

1.    Prepare: Gather an assortment of colorful fall leaves.

Observe the child throughout the interaction. Use the Invitation to Play Documentation Tool to collect data. 

2.    Invite: Create a pattern ahead of time using two materials.  (start with abab- Red leaf, Green leaf, Red leaf, Green leaf) Show the child the pattern then ask: How could you make a pattern with these things?

3.    Play: Watch as they create a pattern.  

Ask open-ended questions: Tell me about your pattern? OR Read this pattern to me?  

Model thinking: A pattern is something that repeats or happens over and over again.

So, Yes!  This is a pattern! Look...Red, yellow, and then red, yellow again! 

 *If the child struggles to make their own pattern, guide the child to copy a pattern using the same materials OR start a pattern and work on extending.

Next, ask open-ended questions: Show me a pattern using three things? OR Show me a different pattern using the leaves?

Extend their thinking about other types of patterns (abab, abcabc and aabbaabb) For example: I saw you can make a pattern with two things, but did you know you can also make patterns using three things? Watch...red leaf, green leaf, yellow leaf, red leaf, green leaf, yellow leaf. 

OR

Watch this, sometimes you can use the same material two times.  Red leaf, red leaf, green leaf, green leaf, red leaf, red leaf, green leaf, green leaf.  See?  The pattern repeats and we can figure out what happens next!

4.    Reflect and assess:  Did the child create patterns? abab, abcabc and/or aabbaabb or something else? Did they copy or extend a pattern you made? How did they explain their pattern? What will you work on next?

Ready: The child identifies patterns in their life and surroundings. The child can copy a simple pattern (Can you make the same pattern I made?), and extends patterns that are already started. (What would come next?)

Ready to move on: The child creates simple patterns on their own without a model.

Extend: Encourage the child to continue creating new patterns using the same materials. How many different ways can you make a pattern using these leaves? Allow the child to be creative with patterns (not just abab, abcabc, and aabbaabb). Practice patterns that grow- use sounds or body (soft, medium, and loud! Soft, medium, loud!). Talk about patterns in your daily life (seasons: Fall, Winter, Spring, Summer).

MATH LANGUAGE

Pattern- A regularity in the world or created by a person. Patterns repeat in a predictable manner.

Resources

Erikson Institute’s Early Math Collaborative (2014). Big Ideas of Early Mathematics. Pearson Education.

Clements & Sarama, (2017)*

Rittle-Johnson, (2015)*

NRC (2009)*

*full citations can be found on the Research Page

Sets and Sorting

“Sets can be compared and ordered”

-Erikson Institute’s Early Math Collaborative

My preschool students loved picking flowers from the weedy areas around our playground.  Dandelions, Wild Violets and Clover. One afternoon I suggested putting the flowers into small jars to keep in our classroom. I held out two bunches, Dandelions in one hand and Clover in the other. Not missing a step, a four year old girl snatched the Dandelions. “Why did you choose those?” I asked, to which she quickly responded, “There’s more!” 

“Activities that ask children to identify “which is more” with small quantities will help them organize their thinking about magnitude or relative amount.” - Erikson Institute’s Early Math Collaborative (2014)

After playing with sorting and creating sets, compare the sets to see how they differ. One example is to look at two sets and see which has more and which has less.

How to Use the Blog Posts

Playful Invitation

1. Prepare: Gather rocks and pinecones. 

Observe the child throughout the interaction. Use the Invitation to Play Documentation Tool to collect data.

2. Invite: Place five rocks and three pinecones  in front of the child and ask: Will you sort these into groups? OR Put things that are the same into piles.

3. Play:

If the child sorts using an attribute (type, color, size) then keep going. If not, return to creating sets.

Ask: Which group has more?  less? OR Are there more ____ or more ____?

How do you know?

Model thinking: This pile is bigger, I think it has more. Let’s count...1, 2, 3, 4, 5, and only 1, 2, 3 in this group. There are more in this group and less in that group!  Really stress the math language.

Continue using different combinations of items (3 pinecones & 5 rocks, 1 pinecone & 4 rocks, 3 rocks & 3 pinecones, etc.) Vary the number of each item, compare, use the language "more" and "less."

4. Reflect and Assess: Did the child identify the group with more? less? Does the child estimate simply by looking or do he/she count each item individually?

Ready: The child is ready if he/she can sort items into sets based on an attribute. The child also must have a basic understanding of more and less.

Ready to move on: The child knows which set has more and which has less with ease when using a small number of items.

Extend: Ask how many are in each group and talk about how many more. Or match one item from each group in a line. Once all matches are made talk about the remaining objects. How many more were in that particular group?

Also, try using three different materials (rocks, pinecones, acorns) sort and then compare. Put the piles in order of most to least. 

*Remember- start with only a small number of items- then gradually increase. Children need to be successful working with numbers under five before advancing to higher amounts. 

Math Language

Attribute- characteristic of an item. Examples include color, shape, texture, size, type, number, etc.

Set- items that are grouped together in a meaningful way.

Sort- organize items based on characteristic/attribute.

Resources

Erikson Institute’s Early Math Collaborative (2014). Big Ideas of Early Mathematics. Pearson Education.

Development of Research in Early Math Education (2017)*

Ngyuyen, H.N., Laski, E.V., Thomson, D.L., Bronson, M.B., & Casey, B.M. (2017)*

*full citation can be found on the Research page

Sets and Sorting

“Attributes can be used to sort collections into sets”

-Erikson Institute’s Early Math Collaborative

The first time I noticed my daughter sorting she was about 9 months old. I placed the tiny finger foods on her high chair tray and she quickly gobbled up the cheese. When cheese was gone, she moved on to the fruit, leaving only the peas and carrots behind. On her tray were four items mixed together and quickly she used her ability to sort items into groups, cheese- her favorite, fruit- her second favorite, and the rest, leaving me with both disappointment (she didn’t eat her vegetables again!) and sheer amazement.

“Children are born learners who are naturally interested in math in their everyday experience. They’re very aware of math concepts that are important to them.” -Gennarelli, C. & DeBlasio, M. (2017)

This activity is all about creating sets, items that are grouped together or sorted in a meaningful way. Materials in a set share a characteristic or attribute, for instance, leaves or not leaves, green things, brown things, rough, smooth, etc.

Children show us what they know about creating sets when they sort.

How to use the Blog Posts

Playful Invitation

1. Prepare: Gather leaves and sticks. Start with only two different materials, then three or four as the child’s sorting skills increase. Draw two circles on the ground with chalk.

Observe the child throughout the interaction. Use the Invitation to Play Documentation Tool to collect data.

2. Invite: Make piles of all the things that belong together?

3. Play: Point to a pile. Why did you put these together? Allow the child time to think. Listen to the child's reasoning. Did they sort based on a common attribute?

Model thinking: Oh, so you put all the ______ in this group and all the ______ in this group! Those are the same and those are the same. Model sorting (if necessary): Look, I’m going to put all of the leaves here and all of the sticks there. Is this a leaf or a stick? A leaf! OK so that goes here...

4. Reflect and Assess: What attribute did the child identify when they sorted? Did they create two or more groups? Think about other attributes could they use? How will you challenge the child to continue sorting?

Creating sets based on similar attributes is a skill that contributes to the child's ability to count- because counting requires a set!

Ready:

• The child is ready for this activity if he/she can find an exact match. (Hold up a leaf and ask the child to find one that is the same.)

• Can place an object where it belongs. (i.e. Rocks are in one area and sticks in another area. Hand the child a rock. Ask- Where would you put this?)

• Can tell you about two piles that are already sorted. What is the same about these?

The child must be able to match before you ask them to produce sets with matches of their own.

Ready to move on: The child is ready to move on if he/she can sort into groups and explain why they sorted in a particular way.

Extend:  The same collection can be sorted in different ways. Ask: Lets look at these things again.  Can you sort the same things in a different way? or Do you see another way to sort these things?

Math Language

Attribute- characteristic of an item. Examples include color, shape, texture, size, type, number, etc.

Set- items that are grouped together in a meaningful way.

Sort- organize items based on characteristic/attribute.

Resources

Erikson Institute’s Early Math Collaborative (2014). Big Ideas of Early Mathematics. Pearson Education.

Development of Research in Early Math Education (2017)*

Clements, D.H., & Sarama, J. (2017)*

Gennarelli, C. & DeBlasio, M. (2017)*

*full citation can be found on the Research page 

Supplemental Materials

How to use the Blog Posts

Invitation to Play Documentation Tool