!Toddler math games --- agk's diary 3 March 2025 @ 02:20 UTC --- written on Evy's GPD Mini in my bedroom after putting daughter to bed --- Before preschool math, my first daughter and I played toddler math games. We started before her second birthday and continued through her third. Evy remembers childhood math as pleasant, a site of care from mentors, self-efficacy, and pleasure. For me, reading was like that but math just made me feel stupid. Evy wants daughter to have the oppor- tunity to love math like she did. I didn't keep a notebook until I started preschool math, so these notes are incomplete, and ages and dates lost, like tears in rain. When daughter was a toddler, she and I were often in the woods, in a cave, on a bicycle, or reading books, especially folk stories. She ate a lot. She learned to play solo. I held her through church services. She grew longer and longer hair. Daughter was learning to speak, learning to not fall down, learning about blood and vulvas, learn- ing to recognize her need to pee and run to the bathroom yelling "I'm peeing!" She started to tell wild stories with simple words. She loved to play with dry beans. She poured them from one thing into another, separated kidney from black from pinto beans, dug her hands in them, hid them around the house, put them in little piles. When she started imaginative play, she pretended to cook them. QUANTITY One book stood out for teaching numbers, because she loved it, so did I, and it opened her ability to perceive quantity everywhere and name it: Christie Matheson (2020), The Hidden Rainbow. It's an active book, like Eric Carle's From Head to Toe (which gets kids to kneel like camels and thump chests like gorillas!). Each page of Matheson's book is a gorgeous new flower in a new hue to brush dry or tickle or find. Bees increase, challenging kids to find and count them. The rhyme and rhythm are as heady as burgeoning spring. Fruits burst out in every rainbow hue. The habit of counting must be discarded shortly after it's got in order to recognize quantities holistically, but numbers are essential concepts, essential vocabulary. Gaining object permanence, noticing quantities differ and can be recreation- ally compared, these are important tasks! A young mother hides behind a curtain, then emerges, smiling, with a "Peek-a-boo," then disappears again. Her toddler greets these appearances by clapping his hands and shrieking with glee. Both are utterly happy. And both have no idea that what they are doing is mathematics. ...up to the age of eighteen months, the most important intellectual problem...a child faces is to discover the law of object permanence. This means that the objects out of sight...still exist somewhere, although we can't see it. The child discovers that such an important object as "mother," even if she has disappeared behind the curtain, is still there and will soon reappear from behind that same curtain. The above is from p.5 of Alexander Zvonkin's (2010) Math from Three to Seven: The story of a mathemat- ical circle for preschoolers. I love his overview of what is happening in toddler math acquisition. His account continues: As a child grows, so does his understanding of the world. A very small girl is playing an excit- ing game: she picks up blocks scattered on the floor and hands them, one by one, to her dad accompanying each gift with a jubilant cry: "Here!" Her dad takes each block and she laughs happily. She has learned the word "Here" only recently and tries to use it as much as possible. But suddenly her little clumsy hands seize two blocks at once. She thinks for a few moments and then---"Eureka"---she offers them to her dad say- ing: "Here-here!" To paraphrase Pushkin, "It is a most fascinat- ing pastime to follow a little one's thoughts." On p.6, Zvonkin tries to drive home how extraordin- ary a task is learning to count: Certainly, it is important to know how to count. but adults don't always fully realize what is at stake here. Let's try to get into the child's shoes and learn to count, but in Japanese. Here are the first ten numbers: ichi, ni, san, shi, go, roku, shichi, hachi, kyu, ju. The first assignment is to learn this sequence by heart. You will see: it is not that easy! When you have managed to do that, ...learn how to count backwards, from ju to ichi. Once you have done that, try and calculate. How much is roku and san? Shichi minus go? Hachi divided by shi? And now a sum for you: Mother bought at the market kyu apples and gave each of shi children ni apples. How many apples are left? There is just one rule: you are not allowed to translate into English, even mentally. PATTERN Some other books we read that I think, in retro- spect, were important for daughter's awareness of pattern and sequence: Ezra Jack Keats (1962), The Snowy Day We read this book last winter and this winter. A little boy walks in the snow, leaving prints, knocks snow off fence posts, and creates other patterns and sequences as he plays, gloriously unsupervised, in urban snow. Lynn Kertell, John Maslen, and Sue Hendra (2008), My First Bob Books: Pre-reading skills The adventures of Sally the circle, Tanner the tri- angle, and Seth the square built on daughter's shape recognition ability, and really challenged her to sort, classify, identify symbols, recognize patterns, and practice problem-solving, sequencing, and predicting. These booklets were the first thing we did together that really felt like tutoring. She wanted to unlock the mysteries in these 12 booklets, and took them to Evy and especially our roommate to puzzle through them with different adults with different approaches. My approach won me a nickname bestowed by her, "the mean one," which has become permanent, and half- silly. We usually spent more than thirty minutes doing challenging stuff in the Bob books. She'd get tired and frustrated, then figure something out, be delighted, and *then* we'd stop and celebrate. REASONING The books we read the most and the longest (she asked me to read from one tonight at bedtime) are Anno's three Math Games books. We've been reading them for a year and a half. As she's grown in know- ledge and reasoning ability, they're still the right mix of accessible and challenging, and we still haven't finished any of them. Mitsumasa Anno (1982, 1989), Math Games II The second Math Games book was the most accessible when we started and remains our favorite. Each chapter in each book introduces a machine the characters Kris and Kross wield to transform things into other things, or a problem they're puzzling out, like how to count the sand on the beach, the differences between two dogs or two dolls, or directions to get somewhere in a picture from some- where else. Most chapters start with a challenge---a "game"--- that may be tough but doable for a toddler who loves books, and interesting but not daunting for a preschooler. With each new page the challenge is applied to new situations or new tools for solving that category of challenges are introduced, esca- lating the difficulty til it gets beyond the kid's ability. Help her reason it out or drop it and move on. Don't solve it for her when it gets too tough. The beginning of any chapter will be accessible. You can keep coming back to a tough challenge over as much as years, until she hits a developmental shift or develops a strategy to solve it. The reasoning is more important and often more interesting than the solution. We feel so close, sharing with each other with the language available to us how we are thinking. Sometimes I see and accept her logic, but still prefer mine. As the games get tougher, *her answer may not have to match mine to be correct!* These books are beautifully, brightly illustrated, use very few words, and have notes to parents in the back that shaped my approach to early child- hood math. Because of these books, she calls our preschool algebra/geometry tutoring math games. Other than dry beans and brightly colored stacking shapes, both of which she mostly played with inde- pendently, our time doing toddler math games didn't use dedicated manipulables. We brought awareness to patterns, quantities, and our reasoning about the rich sensory world in which we live. We talked, with the words and concepts available to us, about our developing awareness. One more story from Zvonkin (p.6) puts my reflect- ions on learning toddler math with my daughter in context. By the age of two many things have already been mastered. A boy of two tries to wake up his dad in the morning. "Papa, are you asleep?" "No I am not," answers his dad rubbing his eyes. "I am in the kitchen, having tea." His son is bewildered: this contradicts every- thing he knows. Just in case, he runs off to the kitchen to check. His return is triumphant: "No, you are *not* in the kitchen. You're here, you're here!" Next time he won't be fooled that easily. But I would like to insist on this moment of indepen- dent research when he ran off to the kitchen to check.... I feel it to be a very important qual- ity and wish children to preserve it as long as possible. ...If I taught kids anything at all, it was the idea that they should experience their world with openness and interest. Not that kids need to be taught that. But there's a lot I did, as a mother and a rank amateur at educating toddlers, to bring her to awareness of objects, places, and relationships that came to interest her and me. Through math games, my only daughter, in fact, teaches *me* to experience the world with more openness and greater interest.