
I tell stories with mathematical concepts - anything involving money (selling and buying things) is popular around here. And the Boolean Logic Baby Gate was a big hit (you have to put in just the right filter function if you want to let your friends in but keep the disruptive babies out).
https://bancosparenting.wordpress.com/2021/05/08/pickle-and-...
I sing Baa Baa Black Sheep but with lots of different numbers of wool distributed in different ways (this also keeps the 1yo entertained).
https://wordpress.com/posts/bancosparenting.wordpress.com
Guess My Secret Number gets lots of requests: "If you have my secret number and you add four more, you get seven. What's my secret number?" (We take turns guessing and giving clues.)
https://wordpress.com/posts/bancosparenting.wordpress.com
Another good game is Tiny Polka Dot - we've come up with so many of our own variations on the games.
https://wordpress.com/posts/bancosparenting.wordpress.com
In terms of actual books, we really enjoy Life of Fred.
https://www.rainbowresource.com/category/9813/Life-of-Fred.h...
Learning how to use an abacus was something the kids enjoyed and helped with ways of thinking about numbers quite a bit.
You may find researching what home schoolers are doing for math programs helpful.
It's quite common for a child growing up in an anti-science household to be convinced to change their views on science by devout Christians writing about evolution, dinosaurs, astronomy and so on. Fundamentalist parents are often quite surprisingly happy for their children to read those kinds of books because they are written from the perspective of the in-group. There's a reason that Life of Fred is popular.
If you want to write a book about the theory of science and what truth is, go ahead, but you aren't going to make much of an impact on that particular target audience.
For multiplication, King Domino is great.
I like that neither game requires reading. Century Spice Road is another one like that we enjoy.
Agricola is bigger and has arithmetic more in the background, but it has a lot of affinity with math. My oldest is decent at it. My youngest aren't competitive at all but they love just collecting big flocks of sheep. You can play without cards (the only reading), but personally I like to deal them a hand and let them ignore them if they like.
Another beautiful no-reading game with less arithmetic but plenty of affinity is Carcassonne.
My kids also get super excited about Robo Rally. It's kind of a programming/planning game. It's way more fun than that turtles game. It starts to bog down past four players though.
So I think one answer to "casual math" is playing games. You're giving them a chance to exercise their brain while having fun (and spending time with you!). No need to mention that it's for the sake of math.
Besides board games there are also puzzles. Just Saturday I brought home some books of brain teasers from Costco. These:
https://www.amazon.com/Mindworks-Brain-Training-Right-Brain-...
https://www.amazon.com/Mindworks-Training-puzzles-Bind-up-Pa...
They are much higher-quality than a lot of the math puzzle books out there. So far they seem to be really enjoying them.
For older kids, maybe read a Smullyan book together.
You can also just talk about math a lot, and if you love it you will infect them. One bedtime I explained the Bridges of Königsberg to a then-5-year-old, drawing pictures and letting her try to find a path. Even she could understand a casual proof that only the start & end locations could have an odd number of bridges.
If you tell them about negative/complex numbers a bit too early they feel like you've let them in on a big secret. The fact that high schoolers spend a year studying nothing but triangles blows their mind. I'm sort of notorious for fetching a pencil & paper in the middle of dinner to draw pictures about a thing from geometry/trig/calc/whatever. . . .
http://www.lifeoffredmath.com/
books for children about mathematics, and had newly joined my local mathematics class. On only the second or third week of class, she came up to me after class and said, "I've discovered a proof by contradiction for the parallel postulate." As you can imagine, I found this quite amazing. (I knew her mother, and thus knew the daughter a little before she joined my class, but I would say that's rather precocious behavior even in the social circle I keep.) Her "proof," of course, was really Saccheri's flawed proof
http://www.jimloy.com/geometry/saccheri.htm
http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean...
that assumed the postulate to show the "impossibility" of any quadrilateral that didn't fit Euclidean geometry. But most of us have minds that begin study of mathematics with a stubbornly Euclidean set of presuppositions, so that was all right. The girl eventually advanced from my mathematics class to my colleague's more advanced class, and then did a summer at Epsilon Camp
in that program's first year of existence.
