Mathematics and Design for Young Children

Leadership for Cleverness

My goal always as been to transform young children’s lives by developing competent, reflective, and authentic early childhood teachers/care-givers/parents who become increasingly committed to the possibility of continuous personal development and the creation of democratic learning communities.

I have learned from a river of children and early educators and listened to what they have learned about being well-regarded teachers of math for children. Now I am putting these pages together, so this doesn’t get lost to an amiable world for humanity’s children. I’d like to highlight that Math and Design addresses these aspirations for leaders in spaces for young children.

1. To distinguish the way we lead in free play from the way we lead in group times. These are different worlds. In free play, the children are responsible for their choices and whether they find joy or not. In group times, the leader has the responsibility for the children’s engagement and joy.
2. To focus in free times upon our personal presence: the Being-with role. A leader is attentive to what is happening, curious about the children’s intentions behind what they have done, and able to stop to offer time for the children to reflect upon what happened either now or later.
3. To acquire more understanding of the domains that work in math and design: (1) the early skills and concepts of arithmetic, (2) the breadth of mathematical languages, and (3) the human yearning for beauty and elegance.
4. To foster the participation of those children who really need a teacher. I have no reason to identify them as a group for we know our children. I tune group times to spark whoever is less engaged and work behind the scenes for them to take initiative in some way.

A Greenhouse, Not a rowboat

Instead of teaching math directly, we build healthy conditions—the air. light, and water. We build community energy, where children realize their own cleverness; it grows over time. A year of mathematics and design offered with precision, appearing to them as common daily life, children look around and discover they have become smart cookies. When their families see it, too, they glow.

Oddly, mathematics may be the easiest way to build a personal dispositions to figure things out.

These pages grow a community of energized children, who, like the girl in dots below, who has created, at her own initiative, both beauty and mathematics this morning with yellow, blue, and red pattern blocks.

Math and Design together

It’s been known since Frederich Fröbel called it the “free expression of the child’s soul.” When we link mathematics and creative design, young children play in the skills and concepts of arithmetic as delightfully as they learn to sing, dance, and draw. Because they become the inventors, their dispositions to learn last a lifetime.

Those dispositions are visible, too; no need to test for it. We can see math dispositions at work when dot girl happens to count as she regards her arrangement. We can see it, too, when a child comments upon what is missing or choses to invent order out of chaos. I see it when children laugh when I call them “math heads.”

Mathematics Education

This page you are reading is an overview of how this works. I use the image of a snowball on a slope to mark the start and the end. I invite you to persist through the words to the next snowball picture. I will try to illuminate what is going on for children and what the educator is doing in the background.

Beyond this snowball bit are 9 pages to poke through, an entire course on what I learned. Hundreds of teachers have taken that course, built the materials, and led group times using these words. They know it works, because they have successfully created a culture of engagement and play for children up into the early years of common school.

The pages beyond offer materials you can download and use.

Counting Cards With PDF Counting Cards Without PDF

You can practice the insightful guides of Friedrich Fröbel to guide what you say to the five-year-old child who created this arrangement shells.

I share a bunch of successful group time games, too, so when it’s not free play and I am the entertainer, I give that snowball a nudge.

You will find a mathematics curriculum where mathematics is play—truly play, without any adult direction inserting itself

The leader uses group times to engage the community by leading games

You will find a way to offer mathematics and design experiences to young children in free play, in small group settings, and in the full class group.

You will find how to create a classroom culture that .starts slowly at first and then gathers speed, like a snowball rolling downhill. The energy is amazing. 

You will find a structure for reconceptualizing simple life events, like the shoes in the hall and crackers on a plate.

When children become engaged, they start making further discoveries and intentions, evolving one year to elementary level representations and abstractions at whatever level they enjoy.

A Language of Materials

Let us begin. We learn a language, like Spanish, theater, blocks, or clay, because we use it to communicate. Because of its expressiveness of ourselves and our off-hand impulsive discoveries and connections, children try new possibilities, which lead to further understandings and attentiveness.

Here is an example. Here the language isn’t English. The language is Unfinished Wood Blocks.

Communicating Ideas with Blocks

Here it is: a single image of a child’s hand resting on a blue table and four flat blocks. With nothing more than this, let’s dive in.

These are the facts, the evidence.

• This is a photo someone took and shared with me.
• The photo shows a table, a child’s hip in a chair, and four 1″ x 3″ wooden blocks.
• Three of the blocks are identically oriented and fully adjacent; one block is butted against the ends, at right angles to the three, slightly offset.

These are subsequent conclusions, that we might or might not agree with.

• We can assume, I think, that this child intentionally placed them this way.
• Further, we could conclude from the placement of the child’s hand that the child has paused to have the blocks photographed.
• Building further, we could assume that the act of pausing and having work photographed was normal in this particular setting.
• Because the child’s face isn’t included in the shot, we could assume further that this photo addressed the block arrangement rather than the child.
• Stepping into our imagination, we might be willing to assume the child expected, at times, that someone might stop, notice, comment, and even photograph their work.

Now let’s consider the setting and the teachers. Imagine you were one of them. What might result if you and your group of colleagues shared your own opinions of what might be going on for this child, whom you know, whose hand you see.

Your thoughtful group might share your best guess not only about the child’s intention but also about the child’s aesthetic sensibility. Because you all get to discuss it, you could agree that this arrangement conveys information not only about number but also about beauty.

The arrangement itself communicates much more than what one can easily say in words. That’s blocks as a language.

Representation

If you want to read these flat wooden blocks as an expression containing number, your community could treat them as manifestations of how four-ness works. Four has consistency no matter what the arrangement. That constant, if recognized rapidly, without the need to count, is an understanding in arithmetic called subitizing. Another basic concept, partitions, is represented as one-three, which could be expressed in English as “one plus three” or expressed in symbols as 1+3 or 4-1 or 4/3 or 3/4 or 1/4 or 3:1 or 25% and 75%.

If you want to read these flat blocks as a story instead, you could say the wolf attacks the house belonging to three fearful pigs.

If you want to read these blocks in terms of their aesthetics, you could note the arrangement of wood grain, perhaps, how it flows across the faces, the knots flowing and breaking, imagining the history of those trees.

Imagine being the child hearing these responses to their work. Words like these shift awareness. Those comments might change how this child places blocks next week or what the child says to their friend while they build a unicorn corral perhaps. If that friend had heard that talk of beauty or number or story, they would be more likely to comment to each other about the wood or arrangements of animals.

Seeds Grow

The common experience adds a play connection. Children who might not usually play together share a common interest, which energizes as described in The Stewardship of Play . A watered seed grows up and down. If they keep at this over time, they might become interested in representing what they have done—the passages of natural learning described in the The Learning Frame.

In schools where this kind of leadership intent exists, children remain free to play and activities build upon their interests. From the very beginning their interest starts it. In connections with each other and with materials they discover who they are.

It’s delicate, however. It’s easily messed up by good intent, such as thinking this is a ‘teachable moment’ for counting or assuming the children need ‘encouragement’ to further their learning.

This is a different kind of school.

Imagine a school where

1. The educators offer elegant, aesthetic materials with mathematical affordances.
2. A child happens to arrange four of them as above and pause.
3. The educator notices the arrangement, because she was looking for it, and comments using any of the mathematical, aesthetic, or story attributes she thinks might intrigue this child at this moment.
4. The educator takes a photograph that records this event. She prints it to stick on the wall near that table, which the children might notice tomorrow. She emails the image home for the family and the child to discuss if they want. She shows it to the whole group of children because she is interested in what discussion might take place. She is most eager to show to the other teachers, because she values hearing their perceptions of its meaning and their ideas about what possibilities could be invented.
5. The child, whose impulse arranged them this way, gets to reconsider that event through the lens of what it conveys, which in turn gives thoughts of other experiences she has had. In whatever way she found interest, mathematics or design, she incorporates into her life an expanded noticing and freedom in intentions.
6. A child across the table looks at the child being noticed, sees the camera emerge, and hears the comments. They, too, may offer a comment or reconsider his own work in a new light.

One event of initiative, engagement and intentionality—commented upon, recorded, and shared with the community—becomes a model of discovery not only supporting the inventiveness of this one child but also marking this community of children as the clever beings they are. Being tricky, after all, is what we do at school.

Children Play Together

An event when captured and reviewed from a mathematical and design perspective stimulates connections to pattern, symbols, sequences, and so on, changing expectations in approaching a pile of plain old wood blocks.

Chairs, benches, tables, displays, comments in group time are ways to intentionally design spaces to make connections more likely among the children.

Workstations

I evolved the schedule to create a special time of the day, which I call workstations, adding energy to the mathematic discoveries. A separate free play time.

In large group time I offer simple demonstrations, like counting visible buttons on clothes or graphing flavors of ice cream or playing The Hand Game. Since all the children know the games from those shared times, they bring attentiveness to mathematics in their play, enabling freer communication and cooperation among children, connecting with others who may not necessarily be best friends.

These integrated free times and group times build the culture of the school.

The Snowball

Returning to the image of the four blocks and the numbered list of events above, would you say the educators in this setting taught mathematics?

I would say the educator had clear intentions in what she was looking for—an intention to be present and to listen at the right moment and an intention to document with a camera. Clever-as-a-fox leadership capitalized on a small event to build a culture-of-clever in the classroom, allocating time, sourcing particular kinds of materials, and building the physical spaces where children could see and relate to each other at the same time.

Goal for Mathematics and Design

I have watched children of many abilities (and unfortunate adversities) lovingly and richly grow, each in their unique ways, into powerful, capable, and generative human beings when they are treated well, fed, listened to, and valued. All children deserve to participate in a community of generative energy and joy.

We want children to develop effective habits of mind (wondering, figuring it out, predicting, and challenge-seeking) and the disposition to participate in cooperative inquiry for extended periods of time.

Deep Understandings About Learning Math

If you want to see how this is done and even be able to do it with little time and few resources, I recommend you first invest in yourself by taking on the challenges presented in the slide show below. It is worth it because you can see and experience something that you can’t really convey in words.

It is a sequence of hands-on explorations for adults borrowed from Marilyn Burns. If you participate you will explore predicting length and volume. Please do this with a partner, because alone you miss the point. Understanding what happens with others is essential to a healthy facilitation of experiences in mathematics for young children. Because you construct these deep understandings yourself, you’ll never forget them. If you do this, you will be in a place where you wish others understood, too: telling them doesn’t work.

To do the activities you will need some materials ready.

These experiments are ideally undertaken in pairs or trios, so each person contributes to the discussion and the group can discover how interaction with others becomes essential to the encounter. The picture below lists what you and a partner have to gather to do the experiments.

Once you have the equipment, you can click play.

After you have done the activity, you can peek at what others before you thought at this link. Experiment Reflections

Ahead:

Arithmetic Skills

Arithmetic — one aspect of mathematics — includes, at this age, counting, quantity, and the basic symbols. These acquisitions can be classified as skills (practiced abilities), concepts (mental understandings that are constructed over time), and making connections to symbols once those understandings are strong. Included in the Arithmetic Skills portion is a discussion of assessment, something I have found essential to address. The goal here is to be able to see the skills and concepts as they emerge and communicate that to families.

Independent Activities

The heart of this work is the creation of a time for children to inquire, discover, and practice what they are interested in individually and cooperatively. Having the children engaged allows educators the opportunity to document in order to reflect upon the children’s work and begin to build the culture of inquiry. Independent means free. Adults listen, watch, and respond — and hopefully never ask another question! In this section are videos of children at work, pages that offer ways you might construct a similar space, and how to respond to children’s work. Both Fröbel and Malaguzzi figure big in this discussion.

Group Games

Educators can lead brief games and challenges, both formal and informal, that create a community of children who are excited to experience their mathematic brains. This is not free play time; it’s a routine gathering or something to do while in a group waiting for the bus. Since the children all have these same experiences, they bring a shared background into their independent activity time. I have had so much fun leading these.