Structuring School Opportunities

Mathematics and Design for Young Children

We can educate children in a beautiful way when we realize that the cultural transmission of any language, such as French or watercolor or number, is more like growing plants than imparting understanding.

oneononeHere you’ll find methods for developing in 3, 4 and 5 year-old children a sense of competence and enjoyment in exploring cognitive-mathematical activities. Mathematics for this age group is more than numbers: it includes writing symbols, creating designs, counting stuff, the comprehension of number, sequencing, ordering, patterning, beginning graphing, problem solving, and discovering how clever they are in reframing experience through the affordances of logical-mathematical thinking.

Because math is academic, like reading, a huge external pressure is put upon early educators to teach it, while the little ones might be better off having a childhood playing and discovering this stuff for themselves within their communities. Here is a doorway to mathematics as a natural part of learning a language by using it.

You might be thinking “Why would I be interested in Tom’s huge bunch of pages on math for kids?” There are soooooo many books on this with lessons all ready with measured steps to follow. It’s STEM, for heavens sake, the inside track to academics and success in life.

Hopefully you will stick around beyond the snowball picture and read IS THIS TEACHING?

For many educators math has never been much fun. 90% of the students I have had agree with the statement, “I’m no good at math.”  (Many fewer agree with “I’m no good at reading.”) Preschools start in a hole when educators who never enjoyed mathematics are told to make sure to pour arithmetic into children’s brains — like concrete into a form — to guarantee a “strong foundation for school success.” There must be a fun way to get out of that hole, right? Yes. Really fun.

What you will find here is playful and absolutely natural, yet a challenging point of view.

  • 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 also will find challenges to your own expectations about what kinds of math one is looking for as children explore and discover together.
  • You will find how to create a classroom culture that builds slowly at first and then gathers speed, like a snowball rolling downhill. Peers are the gravity rolling the snow balls, expanding what everyone notices and shares.
  • You will find a structure that engages young children in the language of mathematics, which offers new ways of regarding simple life events, like the shoes in the hall, with a different noticing. One noticing energizes further noticing on into representations and abstract relationships.

It took me some years before I began to understand how mathematics for young children is a language. We learn to communicate with language because it conveys meaning in our relations with others. Mathematics is a language, too, because it involves others. It’s a way share discoveries, connect ideas, and explore possibilities.

4blocksFor example, this image shows an arrangement of four blocks and the hand of a child sitting on a chair. I don’t know any more than you do about what’s going on, but we can see a photo someone took and kept for a reason. For the moment we can assume this child intentionally made it and that she paused to have the arrangement photographed. We might also assume pausing and photographing was normal. Probably the child expected, at times, that the teacher would stop, notice, comment, and even photograph children’s work.

Imagine, then, what might result if a group of educators looked at this picture together and talked about what each of them thought might be going on for this child at that moment. The group might agree that the block arrangement conveys information not only about the child’s intention but also the child’s aesthetic sensibility. I think most people would agree that this arrangement has beauty as well as conveying information about number and space beyond what one can easily say in words.

If four 1″x3″ flat wooden blocks were considered a language, educators would expect them to represent meaning in several ways. The mental comprehension “four-ness” has consistency no matter what the arrangement. That constant, if recognized rapidly, without the need to count, is an understanding in arithmetic called subitizing. Partitions are represented here, too, 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%. This arrangement might be considered a story, too, as one wolf pushing on the house of the three little pigs. Each of these meanings changes how one regards the possibilities present. As a language it surprises us with the breadth of what it can represent, which tweaks our perspectives and may spark inventiveness in such things as getting the blocks lined up around a unicorn corral. The true magic happens when peers are playfully engaged at the same time; that mutual interest brings synergy (The Stewardship of Play) and invents opportunities (The Learning Frame).

Imagine this school:

  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 her workmates, because she values the different perceptions of its meaning and what possibilities could be invented.
  5. The child, whose impulsive initiative arranged them this way, gets to reconsider that event through the lens of what it conveys, which in turn may raise thoughts of other experiences she has had. In whatever way she found interesting, mathematics or design, she takes forward into her life an expanded noticing and altered intentions.
  6. A child across the table looks at the child being noticed, sees the camera emerge, and hears the comments. He may offer a comment, too, or reconsider his own materials in a new light.

Snow ballOne event of initiative, engagement and intentionality — commented upon, recorded, and shared — 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. The snowball gains speed.

Building a Relational Culture

An event when captured and reviewed from a mathematical perspective stimulates connections to pattern, symbols, sequences, and so on, changing expectations in approaching a pile of plain old wood blocks. Relational spaces make connections more likely among the children. I use a special time of the day, which I call workstations, adding energy to the mathematic discoveries. In group times 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 to workstations, enabling freer communication and cooperation among children who are not necessarily best friends. Integrated free times and group times change the culture of the school.

Is this teaching?

Returning to the image of the four blocks and the six events above, I have a question: Would you say the educators in that imagined school taught mathematics that day?

I wouldn’t. I would say the educator had clear intentions in what she was looking for, intention to be present and to listen at the right moment, and intention in having her camera ready. Clever-as-a-fox leadership capitalized on a small event to build a culture-of-clever in the classroom. Of course, behind it all, was an intentionality of allocating time, sourcing particular kinds of materials, and building the physical spaces where children could see each other work. Is that teaching mathematics?

Academic Expectations

Gotta get ’em ready for kindergarten? Have to show “readiness” in your work? How can one be ready for significant change? Were you ready for marriage? Ready for a baby? Ready for high school? Ready for graduation?

If a reasonable person looked at the broad diversity of children, the inescapable fact is the impossibility of finding a “readiness point” for an imagined future. Another inescapable fact is naming a standardized readiness point cannot happen free from racism, cultural bias, or white privilege. People with power over other people are naming an imaginary point.

I have watched children of many abilities and unfortunate adversities lovingly and richly grow, each in their unique way, into powerful, capable, and generative human beings. The idea of readiness is coercive and stultifying. Readiness is an arbitrary construct that labels those children who don’t do what they are told to do as needy in some regard that labels them as needy without a known way to invest in remediation. This is hardly educative. We know what matters is how the particular children you know well are engaged in whatever they have at hand. We care how they relate to their peers in their uniqueness and how the community builds the energy and joy everyone loves rather than binding them up with external prescription and control.

We know it’s pretty cool what young children do with their naturally curious brains when they have chance and opportunity to discover, with others, how the world works.

Goal for Mathematics and Design

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.

(Not stuff like “counts how many to twenty and beyond” which is one example in our state’s Learning Pathways to Counting and Cardinality.)

Deep Understandings About Learning Math

If you want to proceed further, I recommend taking the time right at the beginning to explore the challenges presented in the slide show below. It is worth it, really. 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 it is then more possible to discover the fundamental underpinnings to a healthy facilitation of experiences in mathematics for young children. Because you construct those underpinnings yourself, you’ll never forget this lesson. These underpinnings separate the sheep from the goats.

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. Experiments Reflections


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.

Next Arithmetic