Structuring School Opportunities

Mathematics and Design

Group Math Games

I offer my favorite math games, formally at a group time and informally waiting for the bus.

I present these games for children ages 3 to 5/6 with the caveat to not kill the children with them. Nor do I want to imply that these ought to be more than a rare event. It seems reasonable that one can play games like these with children once a week or once every two weeks and have a good time throughout their early years. It would be more than sufficient to take five minutes at a Monday meeting time to play one of the games. I like my prep easy on Mondays.

If you try some of these, I think you will be pleasantly surprised about how much fun you and the children have. [I would like to have some stories of your experience to add to the bottom of the page. People might enjoy hearing what happens when they try this kind of thing.] One result is seeing families relax about “readiness” and tell me about the children’s interest in number at home. In my experience, the many children who have not had these experiences benefit the most. I have found it fascinating to see the ripple effect in free play, too; changes occur in peer interactions when the children share common experiences.


People Counting

Children learn to count by counting and seeing counting happen. Duh. So I count many things: crackers, stairs, people, coats, rocks, toes, whatever.  Being my silly self, I count the children who come to group time. “You are one. You are two. You are three. You are four.” The children often think of their age as their number label, which makes counting them this way a hoot. “You are one. You are two.” “No! I am four!” It’s fun to count the children who are here and later, sometime, try to count the children who are not here. It’s interesting to see how they figure that out.

Wall to Wall

Outside or in a gym I run from one wall or fence to the other with whomever wants to join me. (I want my aerobics time.) We line up with our backs to one wall or fence and count to a number before we go. “OK, we’re going to go on six. Ready. Count. One, two, three, four, five, six!” Then we go any silly way we can: backwards, sideways, skipping, spinning, laughing, etc. The children choose a number and the silly way to get to the other wall. Since the most useful counting skill for arithmetic is rote counting down, I like using Wall to Wall for this. It’s easy to count down from ten and more useful to count down from the teens, such as starting at thirteen, fifteen, or nineteen. etc.

Pendulum Count

pendulumThis is from Mary Baratta-Lorton in Mathematics Their Way. I made a pendulum weight from one of those plastic eggs you see everywhere at Easter time. I drilled a hole in the narrower end, opened a large paper clip to insert through the hole to leave a wire hook extending out, and filled the both halves with plaster or concrete patch compound. When it cured I possessed a colorful weight that I could hang from a short length of string. I hooked another paper clip onto a longer string that stays hanging down from the ceiling in the group meeting area. It is one of the set of game materials I stored in a handy place to use when it seemed to fit.

I begin the Pendulum Count by choosing one number to use that day as the weight swings. I usually start with three. I use four as an example here. If I know a child in the group is on the edge of counting up to four (when everyone else is counting to higher numbers), I choose four. That is the child I watch. No need to push any higher. (You can choose to do this in other languages, too, by the way.)

One thing I appreciated in Mary Baratta-Lorton’s book was the specificity of exactly what to say and do as the leader. Here is what to say in the Pendulum Game. The order is important:

Start the pendulum swinging in a short arc, say, two or three feet in travel. Too big an arc makes it too slow.
At some point start counting aloud clapping only on four.
Watch me. I can do it. One. Two. Three. Four. Clap only on four.
I did it.
The pendulum stays swinging without interruption, unless it loses energy.
Do it with me.
One, two, three, four. Clap only on four.
Can you do it by yourselves? Watch with interest.
Now I am going to whisper. Same clap/four but counting in a whisper voice.
Do it with me. Whisper with loud clap at four.
The pendulum continues to swing with everyone whispering the count. The challenge is to see the group clap simultaneously on four.
Now I am going to mouth the words. Demonstrate: mouth forms the words and makes no sound; clap at four.
Do it with me. Silence with loud clap at four.
Can you do it by yourselves?
The pendulum continues to swing with everyone whispering the count for long enough to get the claps somewhat in unison.
Tricky people.

This may be difficult to imagine. The children are watching the pendulum swing rhythmically back and forth trying to clap at the end of the arc when they think four.
The leader steps that performance in by demonstration, then do together, then children do alone.
It easy to see who needs more experience with counting or keeping a beat.
Once a month? At the most, never taking it higher than six.


OK, subtilizing is a strange word. If you want to be an expert on it, you can try this link. When I first heard of subitizing — the idea of instantly recognizing or naming a randomly arranged small set — I thought it was irrelevant or meaningless like saying “Look at me; I can recognize my mom.” However, I have learned from presenting this to all sorts of children that recognizing or naming random arrangements of small sets, directly in the brain without counting, is intriguing for them, so playing around with this may have a value. I still am not sure if subtilizing activities have any value other than increasing the disposition to have fun with numbers. It’s value for me lies in having another way to perceive a child’s mental construction of number simply by watching. No testing needed.

middle complexity cube

Dice Toss

Dice toss is one way to play with this. I bought 3 foam cubes, say 3″ on a side, and drew dots on the faces with a permanent marker. I used three different colors to differentiate three levels of difficulty, beginning to advanced.  The six faces of my three cubes have these dot sets;   Beginning: 1, 2, 0, 3, 2, 3; Middle: 0, 3, 3, 4, 4, 2; Advanced: 4, 4, 5, 3, 2, 0.

Selecting the die that is at the right level, I throw it in the center of the circle or table and say, “Say it fast!” About 5 tosses is all I do, and the game is over. I think one experience every two to three weeks is sufficient. This can be done with objects, also. I put 5 spoons in a box, shake the box, hold some of the spoons back, and toss the remaining into the circle.

Usually adults can subitize quantities to six; beyond six they tend to group the random array into subsets and combine them. How about that? We are back at that magic number six.  I keep the largest random array at the children’s age, e.g., sets of five for five-year-olds.


Here is an activity that I invented, at least I haven’t seen it elsewhere. With this equipment at hand it is easy to do. A 18″ x 24″ magnetic white board, magnetic numerals for buses, 3/4″ magnetic circles, one color or flip-over is best, a tray, and a covering cloth exactly the right size.



estboardHere is what the board looks like when the children first see it. The numerals start beyond the children’s subitizing level, 6, 7, or 8 and go up into the teens to whatever level you like. My goal is for children near age six to be able to estimate fairly accurately the quantity 10, because I have seen them look at larger quantities and imagine the number of tens it might have. So 13-15 seems fine to me as an upper level for this activity.


In preparation I place a set of items on a tray, covered with a hiding cloth. I show toy cars here; other ideas are spoons, blocks, scissors, paint brushes, toy animals, etc. At this point the children see the magnetic board numbers and a covered tray.

trayflashEstimation Graphing

“The question is how many cars are on this tray under this cloth. I am going to show you, but not long enough for you to count them. You have to estimate how many without counting. You make a guess.”

I remove the cloth for about a second and replace it. I give them a brief view one more time., leaving the tray covered. I ask each child in turn how many cars they think are on the tray. As each person guesses I place a magnetic circle under that number, trying to keep them a uniform distance apart like in this picture. I am making a graph of the children’s estimates. When it is complete we can count a few of the columns. We can make comparisons among the columns as a graph. We can talk about how many more one column has than another. The level of challenge you wish to present to the children is flexible. The goal is to present many ways to describe these comparisons as possible over time, not all at once.

estgraphingThen I remove the hiding cloth and we count the cars. This is the order of cues for group counting; “Ready? Count. One, two, three, four, five, six, seven, eight, nine! How many? (pause) Nine. Estimates only have to be close, you know, so 8, 9, and 10 are awesome guesses.”

This final step completes the package: I take the cars off the tray one at a time. “OK. We’ve got nine cars, right? I am going to take one away. Now how many do we have?”  If the group doesn’t say “eight”, we count the cars again. Over time the children become experts in the skill called Subtracting One.

Estimation Graphing provides a group experience in estimation, counting how many, comparing, and subtracting one. Not bad for one prep, huh? In my experience children can do this every other week, with varied objects, and not lose their enthusiasm for the game.


As you may have gathered from the Arithmetic page, the internal representation of a quantity, constructed somehow in the child’s brain, is the most important early childhood acquisition. The Hand Game and Bowl Game are directly from Mary Baratta-Lorton’s Mathematics Their Way. Hand Representation makes envisioning visible.

Hand Game

You can do this with anything handy at any time, such as waiting for the bus or waiting to wash hands. Being a guy, I always have pocket change. I take out a number of coins, usually the age of the children or their age plus one. Let’s say four. I show them my open hands. “I have four pennies in this hand and zero pennies in this hand.” Then I close my hands, hiding the contents.

I say, “Here is the game. When I open my hand, you say the number.” I open my right hand showing the four coins. The children say, “Four.” I say, “If you want me to open my other hand you have to say, ‘And.'” Children say, “And.” I open the hand with nothing in it. I say, “Zero.” Children say, “Zero.” I do it again to rehearse the game by closing my hands again and not saying anything. I simply open my right hand, pause, and open my left. The children do the talking from then on.

Then I openly move one coin from my right hand to my left. Openly is important: they are supposed to see the movement. Then I open my right. Children say, “Three.” “And.” The pause before I open my left to give them time to think. I open the second hand. Children say, “One.” I procede the same way one object moved at a time until we have zero and four. Then I do it back, one at a time. That is the end of the game.

When I first tried this with children, I thought it was simplistic or possibly useless. Then I kept doing it at odd times of the day. I soon saw the children “cheat”. They worked hard at trying to recall the quantity in the second hand as they see the perceptually compelling first hand before them. Does that make sense? They see me move one coin, let’s say increasing the quantity from 2 to 3 in my left hand. They see my hand close around three pennies. Then their attention changes to the opening of the first hand. They see one penny. They are challenged to recall the image of the three pennies in their mind while seeing one penny dominating their perception.

Nobody is wrong. Ever. There are no demands. The extroverts say things. The introverts watch things.  I laugh. This doesn’t  have to be done very often. Once a week or once every couple of weeks is enough. Representations in the brain are bio-chemical and take time to develop. No hurries. They have an entire year to envision their age, one number per year.

Hand Representation of Number

Another way to represent number instead of shouting “Three!” is to hold up three fingers. Now if silent responding were to be part of the Hand Game or other large group games, such as Word Problems below, children would have another access point for multiple intelligences to work. Children might more engaged (The Learning Frame) by actively representing the number themselves. Note this way of representing number I just learned about on YouTube. It’s from China. It seems obvious when you see children use it that this way of symbolizing number offers a lifetime of benefits.

I call your attention to five and nine in the illustration below. I invite you to change over to this in everything you do with children.

As discussed on the Arithmetic page, “really understanding” a quantity means a person has an idea of that quantity firmly represented somehow in their brain. They can immediately know the missing as well as the visible. For example, if something is supposed to be ten and they see eight, two imagined holes come to mind.

This way of representing number by the fingers of one hand is like an abacus. Five has its own indicator to which one, two, three, and four can be added and subtracted. In this way of fingering, the thumb represents five. It stays out there for six, seven, eight and nine. Imagine the problem of adding two to five. Raise two fingers and there is your old friend seven.

Now the deep bit: we have another hand. This hand represents the tens, counting in the same way. With two hands, children can represent any number from zero to ninety-nine. With that ability they can do all sorts of addition and subtraction problems later in elementary school as well as understand not only place value, but also think because they are making their own hands do it. Number is linked to brain motor centers as they run their fingers. Doesn’t this seem a better way to approach representation of number than forcing children to use symbols to understand 5 + 2 = ___ as traditional math instruction does in the USA?

The way this leads on from preschool into the rest of arithmetic is transformational. If you were fluent here, you probably couldn’t understand how the old (the current) way would ever be used.

Bowl Game

This has to take place at a regular group time. I prepare the materials: one bowl for each child containing specified quantity of non-rolling objects. Small paper cups or yogurt containers can work. Let’s say we’re counting with fava beans, because fava beans are beautiful and don’t roll away. Let us say we have older children, so we are working on five. I give each child one bowl and a workspace. Containers of beans are spread around. The teachers have a set, too.

margarine“We start by putting five fava beans on our workspace.”
“Now I cover them up with the bowl. Good bye!”
“You can do that, too. Goodbye, five favas. So long. Sometimes I cry, because I hate to see my fava go.
“OK, here is the game. We look at the cup and talk the beans on the top. “There it is: ‘Zero’. You can say it with me. ‘Zero.’ Then we say, ‘And.’ All together, ‘And’.
“Then we lift the bowl and talk what we see underneath. Hah! Five!”
“I take one out and put it on top. Goodbye again, dear fava. Now we talk it. One.”  “And,” as I lift the cup. “Four.”

I continue moving one bean at a time to the top of the cup. The children do the remainder of the talking. Then one at a time, we move them back under. That is the end of the game.

Later, one can move two at a time or three. The Bowl Game is physical: the children move the objects and lift the bowl at their own rate. This game is unique because the hidden portion of the set is labeled when it is not visible. Once we understand the concept of envisioning being constructed, we can do it anywhere. I can imagine doing this at snack, putting three crackers on one napkin, covering it with another, them moving them out one at at time.

Once a month seems often enough to me.

Set Games in Song

frogsThe common number songs can be modified by attending to the divided set. I made a felt board for Five Little Speckled Frogs with frogs, a bumpy log, and a lake. The five frogs start on the log and one at a time jump into the lake, which keeps all the frogs visible. Some remain on the log, and some are in the lake. I name the divided quantity, “four and one,” before I sing the next verse. The components of the set are visible all the time.

Word Problems

workspacesHere children represent quantities with markers as a story, told in words. Each child follows along using counters and a workspace.  I cut 9″ x 12″ black construction paper in half and put those 9″ x 6″ rectangles in a box near where I conduct group time. The box is labeled “workspaces”. I can easily pull them out and offer one to each child for the Bowl Game, too. The counters you already may have seen on the Mathematics Materials page here. I do a unique work problem with unique materials about once every two or three weeks.


All of the members of our community have birthdays. I have a birthday. Families, moms, dads, siblings, have birthdays, too. The other staff, the cooks, the maintenance people, the neighbor, and the delivery person who comes by every now and then all have birthdays. When a birthday comes up, we can represent the number of years with any kind of handy counter to represent the age, using shallow portion cups (also in a handy box) for tens when needed. The black workspaces represent the birthday cake. If we have tens to represent, I model counting them into portion cups to be counted as tens as the children follow along with their own counters and cups. “Two tens and three: twenty-three.” The experience exists of representing a socially significant number; the children can follow along doing their own set alone or together as they wish.

Eyes in the Dark

Of all the word problems I have made up, this is my favorite. I buy a hundred of those plastic google eyes you can get at fabric stores. I keep a bunch of them at the group time site in a small box. The 20 portion cups are in another handy box. (I like not having to get up to get things when some great idea emerges from the children.) For this word problem I give each child a cup of eyes and a black workspace.

“I didn’t want to tell you this story, because I was afraid you would be scared. Maybe as scared as I was. So, I won’t say anymore about it. It might be too scary for you.
“OK. OK. I’ll tell you the story. I went to bed last night. You probably go to bed at night, too. One of the problems is that it gets really, really dark at night. I mean, really dark. If you close your eyes right now you can see what I mean about really, really dark. If you shut your eyes tight and put your hand in front of your face, you can’t see your hand. You know it is there, but you can’t see it. That is what I call really, really dark.eyesonblack
“Well, anyway, it was that dark last night. I was in bed, but I wasn’t sleeping. I don’t know why, but I was still awake looking into all that blackness, when SUDDENLY two eyes appeared. Really! YIKES!
“You can put those two eyes on your black workspace.
“YIKES! I was really scared. You can see it too right there in front of you. See how scary that is?
“YIKES! I said. We can say it together, ‘YIKES!’
“Well, not a moment later, you won’t believe it, two more eyes appeared.
“You can add those to your workspace.
“Oh my. I said — you might guess what I said — ‘YIKES!’
“Well, look at those eyes. Ooooooo scary.
“We can count them using two fingers, like this, ‘two, four.’
“Then… you won’t believe this, two more eyes appeared.
“You can show that in your space. You know what I said? ‘YIKES!’
“We can count them. Two, four, six. SIX EYES!!! Oh my heavens.
“I was shivering like this. You can shiver, too.
“Then — you won’t believe this — but two more eyes appeared. (pause) I said…
“We can count them now. Two, four six, eight. EIGHT! Eight? Eight? In my room? I really shivered then.
“Then I had a thought. I can deal with this. I don’t have to be afraid. I said, as loud as I could say it. ‘Go away!’ You can say it with me, too. ‘Go away!’
“Bam! Just like that two eyes disappeared.
“You can take them off your workspace. That is so much better, right?
“We can count them now. Two, four, six.
“Then I said again, you can say it with me, ‘Go away!’ Two more eyes disappeared.
“Now we can count again.
“Then I said, ‘Go away!’ and two more disappeared. Then I said it again.
“Ah. That’s better. Then I turned over, fluffed my pillow, and went to sleep.
“And that’s the story of the eyes in the dark.”


opinion1I do this activity more often than any of the above, because it can be about anything that is important to the children. They have opinions, of course, and love to share them. When something comes up we simply graph what people think. That’s it.

I make small cards, approximately 2″ x 3″, with each child’s name that I keep in an envelope in the group meeting area. Also in that envelope are cards with the word ‘yes’ and the word ‘no’. I make two strips of card stock about 3″ wide and 12″ long with lines every 2″, or whatever the height of the name cards you have. I arrange it like this.

Opinion Poll

I bring out the yes and no cards and be silly. I hold up the “yes” card and ask, “Can you read this?” The children say, “Yes.” Then I hold up the “no” card and ask, “Can you read this?”  The children say, “No.” I say, “That’s too bad. Maybe you can read it when you’re older.”

opinionendNow I decide on a topic question that can be answered with yes or no. If we had strawberry yogurt for snack. I could ask, “Do you like strawberry yogurt?” If it rained hard today, I could ask, “Do you like to play in the rain?” There is always a current topic of interest.

Then I hold up one of the name cards, let’s say Mark’s card is first. “Mark, do you like to play in the rain?” Mark says, “Yes.” Then I place his card in the yes column. Moving quickly through all the children the display would look something like this.

Marshall said, “Sometimes.”

Now we can count the number of yes and number of no. The graphing questions are:

Which has more? Which has less?
How many more yes than no?


Sort and Classify

Sorting is the physical act of moving things. Classifying is the language we attach to the once-sorted set. Sort first. Name second.

Sorting by an adult-provided name, e.g., triangles here,  squares here, etc., destroys the opportunity for children to freely invent their own classifications. There’s no “right way” here.

Does this Belong?

sortI take a tray or a felt board and divide the space with tape into four quadrants, like in the picture. Does This Belong works just like on Sesame Street. I use variations of basic shapes in two sizes and several colors or patterns that I cut myself from origami paper to avoid primary colors.

shapesI put one shape in one of the quadrants. Then I pick up another object and hold it near the first one and ask, “Does this belong here?” The children decide yes or no. If no, I put it in another quadrant. I pick up another hold it near the first one again, “Does this belong here?” The children decide. On it goes.

The children don’t name the shapes or colors; they simply say yes or no. Once enough items are on display sorted by whatever criteria the children decided, then we name the sets. “These are all ____(pause)___” Children say, “Pointy.” “These are all ____(pause)___” Children: “Blue.” 

After several experiences, the materials go into workstations as the image shows.

Black and White

blackwhiteI place one white sheet of paper on the floor and call it “white.” Next to it I place a black sheet of paper, “black.” One can use many kinds of objects. I like things that are in the kitchen cabinets or drawers, but this day I had my Schleich African animal set of  gorilla, bear, giraffe, crocodile, elephant, rhinoceros, tiger, lion, hippo, and zebra.

I held up the black bear. “Should this go on the white or black paper?”  Most of the children said on the white paper, so I put it on the white. Then I held up the giraffe and asked the same question. They said black. Then I held up the black gorilla, and they said white. Then I held up the brown lion. They said black. I asked, “Why is the gorilla on the white paper and the giraffe on the black paper?” One of the children said, “Because the bear and the gorilla are black, and the giraffe and lion are orange.”

I put the animals and the black white papers in workstations and noticed one girl had the animals sorted. I asked her why that way, and she said, “All the animals with tails are on the black paper; all the animals without tails on the white paper.” The next thing she did was arrange five animals facing left on the white paper and five animals facing straight ahead on the black paper. I said, “Looks like some of the animals are headed for the door and some want to stay with you.”

I trust you are willing to try some of these ideas with children. They sure have worked for me. I was amazed what happened to children as they became intellectually engaged in these little games over the course of a year.

Children are never put on the spot, never required to answer, and enabled to simply watch.


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