September 2023
Welcome to EFM's September Newsletter!
EFM: Supporting families to play, explore, and love math
It is essential that every family and caregiver in the world read books and do math with their young children!
News
Activities for Educators – We have been busy adding games and puzzles to our collection. There are now 100+ games and activities for educators, and 200+ puzzles of the week. Check these out at the Activities for Educators page.
Games for Focused Practice – Given the educational setbacks from the pandemic, EFM thought we would help out. On the Activities for Families page we have added a selection of games for families to play that provide focused practice for children with skills they’d like to strengthen. In the introductory pages, the games are grouped by math topic to make it easy to identify the games to use for each skill.
Project Zero
One of the things I enjoy about writing these newsletters is that it forces me to take time to look closely at some aspect of early math education. Taking time to observe is central to some of the ideas I want to highlight from the collection of wonderful ideas at Project Zero(PZ) at the Harvard Graduate School of Education.
PZ has many thought provoking ideas to use in classrooms, and several of those ideas are appropriate for families to use in their home. PZ’s Pedagogy of Play is one of their many projects, and it looks deeply at the value of play in education. In addition to their projects, they have a collection of thinking routines collected together in their Thinking Routines Toolbox. I have gathered together several of their routines into topics I find particularly useful.
Pedagogy of Play (POP)
Let’s start off with an often quoted line from Dan Finkel’s Ted Talk: “What books are to reading, play is to math.”
Play is essential to a healthy relationship with mathematics. An attitude of experimentation, exploration, wonder, and joy in discovery is so important, and those attitudes are impeded if you are grinding through uninteresting exercises and your first concern is whether you got the right answer and whether you did it quickly enough.
Defining play precisely is hard, and doing it exactly is not essential. Dr. Stuart Brown of theNational Institute For Play describes play as having seven properties: 1) done for its own sake, 2) voluntary, 3) inherent attraction, 4) freedom from time, 5) diminished consciousness of self, 6) improvisational potential, and 7) a desire to keep going. Many characterizations of play are reminiscent of the work of psychologist Mihály Csíkszentmihályi on Flow, and for good reason. A person caught up in play will have feelings of losing track of time and being caught up in the activity.
The Lego Foundation has a website dedicated to playful learning: Learning Through Play. They describe five ways to spot playful learning: Joyful; Meaningful; Actively Engaged; Iterative; Socially Interactive.
The POP project began in 2015, and has received major support from the Lego Foundation. POP has spent a lot of effort characterizing what playful activity is and what it looks like. They have done this in several cultures on several continents. In their book, which is free to download, “A Pedagogy of Play: Supporting playful learning in classrooms and schools!” they describe a great many features of play, which I will loosely summarize and somewhat condense here.
From POP’s research, playful learning is described as having three components:
1) Exploring (Wonder, Curiosity, Meaning)
Feels Like – engagement, novelty, surprise, challenge, inspiration, eagerness, positive frustration, authenticity, connections to interests, sense of purpose, audacity, resourcefulness
Looks Like – risk taking, focusing attention, asking questions that further learning, pretending, reflecting on mistakes, improvising, inventing, imagining, creating, experimenting, trying, discussing and debating
2) Student Led (Choice, Ownership, Empowerment)
Feels Like – intrinsic motivation, responsibility, courage, pride, freedom, commitment, achievement, belonging, error acceptance, competence
Looks Like – setting goals, being spontaneous, making and changing rules, choosing collaborators and roles, choosing how long to work/play, confidence, valuing own and others’ ideas, seeing peers as resources, voicing opinions, asking for help, dialogue, leadership, initiative, negotiation
3) Joyful
Feels Like – satisfaction, belonging, excitement, inspiration, pride, flow, trust, safety, fun, togetherness, figuring it out, empathy, satisfaction, concentration, familiarity
Looks Like – celebrating, discovering, working through a challenge, competing, singing/humming, anticipating, being silly, smiling/laughing, joking, hygge, buzz of activity, friendship, enthusiasm
In brief, playful learning is where students have a lot of autonomy to explore and create, with wonder and curiosity, activities that feel meaningful and bring them joy. However, there are lot of interesting details in all those extra descriptors listed above. Also, not all play will involve every one of these characteristics, and some characteristics may be stronger than others in various playful activities. Notice that play is not just being silly and aimlessly moving from one thing to the next. Among many other things, there is a lot of purpose, goal setting, social engagement, and effort in these words.
Despite the importance of student autonomy, the teacher or parent is not removed from the equation – this person becomes more of a guide than an instructor, guiding students on how to choose and explore. There is art to being a good guide, supplying enough guidance to keep students productive, but not so much as to remove the students’ sense of agency, independence, and discovery.
A sense of play is usually missing from math education, and it shows. I think most children ask the question “When am I ever going to use this?” not out of a sense of interest in applications, but from a general sense of not being drawn into the tiresome mathematics they are forced to do. As Grant Sanderson of 3Blue1Brown points out in his TedX Berkeley talk, no one reading Harry Potter ever asks “When am I ever going to use this spell?” Grant says that math needs a story you want to read, it needs wonder, it needs to establish a mystery you want to see resolved. Looking at the playful learning descriptors above, it looks a lot like Grant is saying math needs to be playful.
Example of Math Play: Number Shapes
Suppose your child is playing with different numbers of some small item. As the guide, you might ask what shapes can be made. As the exploration continues, maybe the child wonders which numbers can be split into two equal piles and which cannot – discovering even and odd numbers. Continuing the exploration, they might wonder what happens if these even and odd numbers are added together. Playing with the piles, they might discover the rule that an even plus even or an odd plus odd results in an even number, and that adding an even and an odd results in an odd number.
This is such a richer and more engaging experience than simply being told the rule for adding evens and odds!
Thinking Routine: What do you Notice? – What do you Wonder?
Taking the time to look intently at something for several minutes is a great habit to establish. There are two questions: What do you notice? and What do you wonder? The point is to really notice everything you can about a thing or situation and attempt to look at it afresh. After taking this time to notice, and perhaps taking notes, the next step is to go beyond what you can see and start wondering about things that are extensions of the situation.
PZ has three associated Thinking Routines that I feel are linked to Notice and Wonder. “Slow Looking” is a routine whose sole purpose is to slow you down and take a prolonged, thoughtful inspection of the thing. “See, Think, Wonder” is a similar routine that essentially combines Slow Looking with Notice and Wonder. “See, Feel, Think, Wonder” encourages you to take the Notice part of the exercise and separate it into describing what you Feel and what you Think.
Example of Notice and Wonder: KenKen Products and Sums
A math education friend of mine told me one day that he had noticed after playing a lot of rounds of KenKen that it seemed that, for a given number, pairs of numbers whose product is that number all had sums that differed by an even amount. For example, 2 x 8 = 4 x 4 and 2 + 8 is two more than 4 + 4. He wondered if there was any interesting mathematics going on.
It turned out there really wasn’t much of a pattern and the math wasn’t particularly interesting. And that’s fine! He noticed something, went exploring, played with the situation, and had a fulfilling experience. Not every investigation will lead to a new theorem or insight, but it did lead him on an interesting, playful journey.
Thinking Routine: Connect – Extend
Take a situation and ask how it connects to what you already know. Then, think about new ways your thinking was extended by the material. This feels connected to “I used to think, Now I think,” which asks you to consolidate your learning by reflecting on how the new material effects you. It also feels connected to “Imagine If … What If,” which asks you to think about interesting changes that could make something more effective / efficient / ethical / beautiful.
Example of What If: Magic Squares
As you may know, a 3 by 3 Magic Square is a 3 by 3 grid of numbers whose rows, columns, and two main diagonals each sum to the same thing. After doing these for a while, you might ask some What If? kinds of questions.
What if I tried larger squares, such as 4 by 4 or 5 by 5? How might those work? ANSWER: They are a lot harder, but people have done a lot of research on this and other configurations for magic squares.
What if I made a game out of this? 15 seems an important sum when using the numbers from 1 to 9 as the entries. What if I made a game where one person tries to make three entries in a line sum to 15, and the opponent tries to keep that from happening? ANSWER: The EFM game for this is called Math Tic Tac Toe.
What if I make a Magic Square using multiplication instead of addition? ANSWER: This new type of puzzle shows an interesting connection of multiplying numbers that are powers and adding their exponents. It also makes for a good exercise in prime factorizations.
Lot’s of wonderful What If explorations!
Thinking Routine: Think – Pair – Share
Take time to reflect and think before sharing ideas. There is a lot of pressure to start talking the moment a question has been posed. Encourage the taking of time to think deeply about a question. Of course, sitting around saying the first thing that comes to mind can be a good (and fun) way to brainstorm, but there is value in a more reflective approach as well.
Building thinking time into conversations allows everyone ample space for sense-making, and it gives more opportunities to those who are more confident sharing ideas after considering them carefully. It also encourages those who are often quick to answer questions the chance to practice extending contemplation time, which can be helpful for generating multiple ideas and may well lead to new insights.
Thanks
My thanks go to Alex Box of Maths Play for discussions and writings that led me to a number of the ideas and organizations covered in this newsletter.
If you have any questions or comments, please send them my way. I would enjoy the opportunity to chat with you. Also, if you are interested in collaborating with us or supporting us in any fashion, I would love to talk with you about ways we can work together!
Chris Wright
September 18, 2023
Chris@EarlyFamilyMath.org
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