Math education for young children
(agk) Links – 2025 March
This page is focused on preschool math education, with peripheral attention to toddlers and American Kindergarten to Grade 1.
Toddler math
My notes on haphazardly doing math education with my child from the ages of one through three are here: Toddler math games (plaintext, 3/3/25). Below are resources I used:
- Christie Matheson (2020), The Hidden Rainbow. You can see the book read by the author on youtube. I don't link to youtube, it invites bad bots. It's also probably at your public library.
- Lynn Kertell, John Maslen, and Sue Hendra (2008), My First Bob Books: Pre-reading skills. This is probably at your public library, and there's a pdf at annas-archive.org.
- Mitsumasa Anno (1982, 1989), Math Games II. All Anno's Math Games books are easy to get with Inter-Library Loan (ILL), about 5 USD used, and downloadable as pdfs from annas-archive.org.
Preschool math
My kid became a preschooler at some point after 3 1/2. As I type my handwritten notes on doing preschool math games with her I'll link them here:
Introductory notes on preschool math (plaintext, 3/10/25).
Some of what we heavily use is below.
- Anno's Math Games books.
- Cuisenaire rods were invented in the 1920s in France by a musician and math educator and popularized in the 1950s for algebraic teaching of arithmetic and higher math by Caleb Gattegno, who wrote a series of textbooks on using them. I got a wooden "small group set" of them from learningresources.com for 30 USD, as well as a Big Time mini clock for 4 USD (which we also use daily) and a 2-row rekenrek counting frame for 7 USD we used for a couple weeks before it was mostly supplanted by the Cuisenaire rods. To see what they can do (my experience can confirm), see on Educational Solutions' youtube channel the three-part series of Gattegno using them to help Canadian first-graders learn complex arithmtical manipulation of fractions, and fourth graders learn permutations. It's called Mathematics At Your Fingertips.
- We spend more than an hour a day most days working through Shevrin and Zhitomirsky's math books for 4-8 year olds. My kid loves these books, and often demands them at bedtime. L.N. Shevrin & V.G. Zhitomirsky (1978, 1985), Let's Play Geometry and V. Zhitomirsky & L. Shevrin (1980, 1987), Maths with Mummy, each about 200 pages. These books are treasures and hard to find. You can get them Inter-Library Loan, or download pdfs from annas-archive.org. Maths with Mummy begins by introducing the number one, and proceeds by stories and exercises through topics in arithmetic (including multiplication and division), geometry, fractions, sets, relationships between sets, permutations, and basic algebra. Both books, early on, also introduce left and right, comparisons of larger or smaller in height, length, and angle, and observing daily life to apply mathematical thinking.
- Tools: Crayola Short Colored Pencils, 8-count pack (1.25 USD, Dollar Tree), pencil sharpener (>1 USD), nice HB pencil with eraser broken to the size of the colored pencils (>1 USD), ream of photocopier paper.
- Mail-ordered tools, from jetpens.com: Rhodia graph paper memo pad No. 11 (A7, 80 sheets, 5mm graph, 3.15 USD), RayMay Study Mate Zero Edge Ruler Set (15cm ruler with no inches to confuse us, compass, protractor, and 2 triangles in a sturdy plastic case, 11.75 USD). We already had a LAMY ABC fountain pen (20 USD) and some ink cartridge refills (4.50 USD/5), commonly used in Germany to learn to write because it is much easier and forces holding a pen properly. My kid can do much more with it than with pencils or other pens.
- Improvised tools: Weight on string for plumb bob, dry beans for counting, scissors and watercolors for making and altering paper shapes, play-doh (vertices) and cut-down wooden skewers (sides) for shapes, coins in her piggy bank, alarm clock and wristwatch with hands, little plastic magnetic numbers from the refrigerator, a receipt from the grocery, measuring cups and spoons, twine and parachute cord to tie knots, etc.
- Puzzles: Dissection puzzles (like tangram or egg of Columbus) -- We have a set of Melissa & Doug pattern blocks (~30 USD). The public library has something like the Shapes Puzzle wooden tangram in a tin with cards (25 USD on etsy, 7 USD on temu). We used to play with these a lot. As we get past triangles in geometry, we might get them out again to do some explicitly geometric tasks instead of representational ones (make a triangle with 3 triangles and 1 hexagon/2 trapezoids, a hexagon with 2 rhombus and 2 triangles, etc). We also might add narrative, with Ann Tompert's (1990) Grandfather Tang's Story. I imagine we might end up with a storebought or homemade Tower of Hanoi.
Preschool math circle
I plan to start a weekly math circle in my home, at the gazebo at the base of the mountain, or at the public library in the next six months with four to eight children between the ages of 4 and 6. Math circles present problems, not exercises, that participants figure out strategies to solve, some of which may occupy the circle for a year as they solve other problems and return again and again to unsolved ones. They generally involve the participation of at least one academic mathematician, and transmit mathematical folklore as well as being a forum for thinking like a mathematician. Math circles often become competitive in upper grades, where they're associated with Olympiads.
- The essential book is Alexander Zvonkin's (2010) Math from Three to Seven: The story of a mathematical circle for preschoolers. A pdf is on a University of Sao Paulo website or at annas-archive.org. It's worth having a paper copy to dogear and scribble in. It's published by the American Mathematical Society, who sell it for just under 30 USD.
- I'm working to build a group of leaders and advisors including the early childhood programming librarian at my public library, math department faculty at a nearby private liberal arts college, and an innovative occupational therapist at our rural critical access hospital who our kid saw when she was a toddler. I might also be able to collaborate with my kid's daycare.
- The format I imagine:
- Begin with a period of one or several problems drawn from Zvonkin or similar. If kids solve or are uninterested in one, we move on to another.
- Next, shift to work with Cuisenaire rods using Gattegno's textbooks and stuff adapted from Anno and Shevrin & Zhitomirsky's books to instruct. We can't asssume kids can recognize or manipulate quantities, do basic arithmetic, tell left from right, compare greater and smaller, tell time, or use basic geometric concepts. This is a short class to introduce these building blocks.
- Finally, learn and play structured games kids can take away and play independently that engage math skills: playing cards (go fish, memory, maybe golf), checkers (Russian rules, with toys on felt boards they assemble before playing), marbles (on felt with a circle of string or outside in a chalk circle on pavement), dominoes (traditional and domine), phutball, sprouts, nine men's morris, mancala, clapping rhyming games. When they're a little older, jacks, jumprope, mafia, dice games, and chess (I learned from my great-grandpa at 7!). Games allow relaxation, increase independence from teacher, apply math knowledge to build fluency, and build a repertoire of means to self-structure play and boredom.
Elementary math
Topics in elementary math can be dragged back into preschool math home education or circles as appropriate.