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 farming than teaching.

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, it is a focus for external pressure upon you and the little ones who might be better off having a childhood playing and discovering themselves and their communities. This page is the the doorway to what I have discovered about mathematics as play fitting in with my view, at least, of being a child.

If I were you, I would be thinking “Why would I be interested in Tom’s treatise on math for kids?” There are soooooo many books on this with lessons all ready with measured steps to follow. It’s STEM. It’s covered.

I figure I have about one minute to intrigue you — hopefully, you’ll get down to the “Is this teaching?” header.

For many educators math hasn’t been fun. 90% of the students I have had agree, “I’m no good at math.”  You might be one, finding yourself being told to make sure to pour arithmetic into children’s brains — like concrete into a form — guaranteeing a strong foundation for school success. If you’ve tried that, I’m sure you have found awkwardness, at least. There must be a better way, right? Yes, there is, and it’s fun.

What you will find here is playful and absolutely natural. You might expect my treatment of mathematics for preschool children to have challenging point of view. Ahead:

  • 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 ones 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. One noticing energizes further representations and relationships.

languagesIt 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. At school 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. Let’s for the moment assume this child intentionally made this arrangement 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.

What if, after school, 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. Maybe they thought that the arrangement exposed not only the child’s intention but also the child’s aesthetic sensibility: the arrangement is indeed beautiful.

What if the educators considered these four 1″x3″ unfinished wooden blocks as a language like I discussed above. If flat wooden blocks were a language, and if the educators expected them to represent a meaning, they might translate this block-speak into mathematics-speak in several ways. In the language of mathematics this could be a mental comprehension of a perception grasped directly as “four-ness.” In the lingo, that is called subitizing. Partitions are another meaning visibly presented as one — three. This could be expressed in English as “one plus three” or expressed in symbols as 1+3 or 4/3 or 3/4 or 3:1. This block-speak might express a story, too, as one wolf pushing on the house of the three little pigs. That’s language. A representation changes how one regards phenomena; language calls up connections which become alternate perspectives to tickle inventiveness. When others are there, too, we have conditions for synergy.

The sequence of this event:

  1. The educators offered an elegant, aesthetic material.
  2. The child arranged four of them and paused.
  3. The educator responded by commenting on any of the mathematical, aesthetic, or story line attributes she thought was appropriate for this child at this moment.
  4. The educator took a photograph that documented this event for the child, other children, the other educators, and the child’s family.
  5. The child, whose initiative arranged them this way, now has the opportunity present to reconsider that result through the lens of what it might represent, which in turn could have connected other experiences she has had with that reframing. Whatever way she reconsidered her creation as a representation of a mathematical or design concept, she has experienced a provocation to use that idea in a new intention, either at this moment or in the future.
  6. A child across the table could have looked at the child and her four arranged blocks, seen the camera, and heard the comments. This event becomes a provocation for him to reconsider arrangements of blocks as being more than what first met the eye.

Snow ballOne event of initiative, engagement and intentionality — commented upon, recorded, and shared — became a model of discovery not only supporting the inventiveness of this one child but also marking the expectation that this community of children discovers clever things — clever beings that they are. Cleverness, 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 educator 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 would be the impossibility of finding a “readiness point.” Therefore, another inescapable fact is that any creation of a standardized readiness point cannot be proclaimed free from racism, cultural bias, or white privilege. People with power call it. That is a fact.

I have watched children of many abilities and economic adversity lovingly and richly grow, each in their unique way. I would like us all to call out the coercive pretense of readiness. Readiness is a construct to label those that fail as failures, hardly educative. What really matters is how these particular children are engaged with whatever they have right now, how they relate to their peers in their discoveries, and how the community fosters the energy of inquiry rather than destroying it. I know that if you have read this far, you know it’s pretty cool what young children do with their naturally curious brains when they have chance and opportunity to discover, with others, what they want to figure out.

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. It kind of separates 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