What might you see or hear if children are thinking mathematically about attribute? They might be matching objects, describing objects or sorting objects. They might be paying attention to color or shape or size or texture.  If you notice children noticing and using attributes, you might ask them: Why do these go together? Why do these not go together?

When adults are comfortable talking about math, children will share ideas without prompting. Here’s an example from a preschool classroom:

Chris & Tracy approach their teacher with excitement: “Look, our shoes are the same so they are a group! There are 4 shoes, 1, 2, 3, 4 … It is a group of shoes with holes”

Teacher: “I see you have a group of 4 shoes with holes. My shoes have little holes on the strap where I buckle the shoes. Can my shoes be part of your group?”

Chris & Tracy: “No, teacher, you have to have big holes all over to be in our group”

Teacher: “I see … you have made a group of shoes with big holes all over. Does anyone else have shoes that belong to your group?”

In a kindergarten classroom, a child runs up to the teacher and says, “A triangle equals a square!” Some teachers might say, “What do you mean? Triangles have 3 sides, and squares have 4. They’re not the same.” However, this mathematically sensitive teacher says, “What do you mean?” The child answers, “Come see!” The child leads the teacher over to the block area, where there are a lot of unit blocks for the children to play with. (Do blocks give you an idea of what the child might be thinking?) This is what the child showed the teacher:

Two square blocks can be put together to make the same shape as one longer rectangle block.

Two long triangle blocks can also be put together to make the same size rectangle shape.

Therefore, one of the squares takes up the same space as one of the triangles. In other words, they are worth the same, or, as the child says, they are “equal.”

This child was thinking hard about both the attributes of these shapes and the relationships between them. One might even say that the child was doing algebra, because they were using equivalences:

2S = R   and   2T = R           Therefore, 2S = 2T               Therefore, S = T

In both of these scenarios, the teachers are building the children’s understanding of foundational mathematical ideas and their confidence in using math to make sense of their world.

I hope that you will be open to the joy of finding math any time, anywhere with the children in your life!

It is hard to know what is going on in anyone’s brain. Even when asked to explain ourselves, we cannot always express our ideas clearly. Young children, who are still developing both their communication and reasoning abilities, have an especially hard time explain their own thinking in words or “showing their work” when they are solving problems. Why does it matter to understand what children are thinking about? It helps us to respond to them in ways that nurture competence and confidence.

So, how do we figure out what’s happening in children’s minds? We need to watch what they are doing and listen to what they are saying for clues, then interact with them intentionally. Here is an example from a preschool classroom:

From nearby the teacher has been watching Jenny and Samantha work with the materials at the sorting station.

The teacher has noticed the children taking the straight items from the collection without talking and putting them on the black paper. The teacher also noticed that Samantha takes each item and rolls it with her hand before she places it on the paper. After the paper is almost full with items, the teacher decides to approach the children.

Teacher: Tell me about your group.

Jenny: Son Palitos (These are sticks)

Samantha: Uh… no..no

Teacher: Samantha you don’t seem to agree with Jenny. What is your group then?

Samantha: A group that rolls

Teacher: Are you saying that what makes this a group is that all the things in it roll?

Samantha: yes, see (Samantha shows how different items in her group roll)

Teacher: Could this be part of your group? (The teacher offers a circular wooden object)

Samantha: (Samantha takes the wooden objects and tests if it rolls) Well, it kind of rolls.

(Samantha then adds the object to her paper)

Teacher: What should we call the group then?

Jenny: Cosas que ruedan (things that roll)

Teacher: I can see why you are naming your group: “things that roll.” There are sticks, pens, brushes, markers, pencils, straws and round pieces of wood but they all roll.

This teacher is drawing from a repertoire of intentional responses that help to surface children’s thinking:

• Stop & Look – Take the time to observe what a child is doing. Try to figure out what the child’s ideas or goals are.
• Say what you see – Use precise, descriptive language to describe what you notice. Provide labels to actions or structures that are mathematical.
• Re-voice what you hear – Use precise, descriptive language to echo and expand on what a child says.
• Check with the child – Always ask child to confirm your understanding of their ideas and intentions.
• Wait – Allow child the time to react or respond to an adult prompt
• Use comments / questions to invite / provoke children’s thinking

We will never be able to be inside children’s brains, but we can closely attend to their actions and words, and then intentionally respond in ways that draw out their ideas. The more we understand about what children are thinking, the easier it will be to help them love, understand and use math.

“Let’s do math!” I often say at the start of a workshop. This may lead to some panicked looks or trips to the bathroom. If I say, “no pencils or calculators involved,” then a few people will laugh, and most will look more willing to try what I suggest …

What are these people doing? The Counting Calisthenics! If you need a break from the computer, you can do it, too! Stand up and count while you move:

Touch toes – “1”

Touch knees – “2”

Touch hips – “3”

Touch shoulders – “4”

Throw hands up – “5!”

Continue counting while repeating movements

(toes, 6; knees, 7; hips, 8; shoulders, 9; hands up, 10 …)

Keep going until you want to stop.

After you sit back down, consider this question: If you kept doing the toes, knees, hips, shoulders, hands up counting calisthenics, what movement would you be doing at “456”?

It would be quite exhausting to actually keep doing the counting calisthenics all the way to 456, but I expect that, if you think about it a little, you will be able to figure out that you would be touching our toes at “456.” You probably noticed that your hands were up in the air every 5^th number, and then the cycle started again. So, your hands would be up in the air for “455” and back at your toes at “456.”  You are using what you know about the structure of our base-10 number system and the pattern of the calisthenics movements to arrive at an answer. You are doing math!

We at the Early Math Collaborative want to encourage you, and all adults who spend time with young children, to build new math ideas and new math associations by engaging in fun and meaningful math activities. Why? If adults are going to engage children in doing real math and constructing authentic mathematical understanding, then the adults need to exercise their own “math muscles.” Because many adults have bad memories of math in school, they often avoid doing math with children. It can be hard to engage adults in exploring foundational math if they don’t feel good about math and don’t think they can do math. Just as learning math facts from flashcards is not an effective way for children to become fluent and flexible problem solvers, blindly following activity directions will not help adults understand the math in the activities or respond effectively to children’s comments and questions.

“Counting Calisthenics” is just one of our Adult Learning Activities. While they involve basic math concepts, Adult Learning Activities are not children’s math that we are asking adults to do; nor are they activities to repeat with young children. We have designed these activities so that they:

• pose a puzzle or problem;
• are interesting enough to capture and retain adult attention;
• are easy to implement;
• may have more than one solution or route to solution;
• clearly focus thinking on foundational mathematical idea(s).

We hope that having fun doing math will help convince you that the world is not divided into those who are good at math and those who aren’t. We can all be doers of math, and we can build the same confidence and excitement in the children in our lives.

When you hear or see the word “math,” what do you think of? Your high school algebra class? Balancing your checkbook? A geeky engineer with pocket protectors? When you add “early childhood” to “math,” what do you think of then? A little one learning to say, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10”? A bright poster with a circle, triangle and rectangle neatly labeled? All of these are common ideas about what math is and how math starts, but none of them are what I mean when I say “foundational math.” Before I tell you what I do mean, I want you to try something.

Look at this image:
Consider this question:

Which of the figures are the same?

Often when I ask this, a person says, “They are all different from each other.” Another says, “They are all the same; they are all shapes.” Both of these answers make sense, but I often ask people to keep looking to see if anyone can come up with another answer. Usually, people then generate these six answers:

• top two shapes are both orange
• bottom two shapes are both green
• left two shapes are both striped
• right two shapes are both solid
• top left and bottom right are both circles
• top right and bottom left are both triangles

In fact, although none of the two shapes are identical to each other, any two of them are “the same” in some way. Figuring this out involves logical thinking about the attributes of the shapes.

This shape activity demonstrates one definition of mathematics – a logical way of thinking that allows for increasing precision. We can use math to make sense of the world. We can use math to solve problems. To use math in these ways, though, we cannot just memorize facts. We must build our own understanding, so that we can think flexibly in different situations. Without a strong foundation, a tall building would not stand for long. Likewise, without a strong foundation in mathematical concepts, children can struggle to understand the more complex mathematical thinking they need later in life.

At the Early Math Collaborative, we have developed a set of 26 “Big Ideas” – key mathematical concepts that lay the foundation for life-long mathematical learning and thinking. While these concepts can be explored at any early age, they are powerful enough that children can and should engage with them for years to come. As you engaged in the shape activity earlier, you were using two of the Big Ideas:

• Attributes can be used to sort collections into sets.
• The same collection can be sorted in different ways.

Most likely, you were not thinking about these ideas consciously; rather, you were looking at the shapes and thinking about them. You were using math to make sense of the puzzle I posed and to come up with a solution. This type of math may not match your prior notion of math as quickly-recalled facts and properly executed procedures. You may need to set aside some of those notions in order to develop a deep understanding of foundational math that will help you have fun doing math with children.