Tiny Polka Dot is a game developed by Dan Finkel and the crew at Math for Love. It is essentially a set of nice, pleasing cards with dots on them, spanning from 0 to 10. Some of the dots are ordered, while others are intentionally disorganized. The deck comes with a set of cards that give the rules for simple games that kids can play. Most of the games are perfect for kids between the ages of 3 and 7, although there are some activities that were fun and challenging for me!

This morning, my son and I decided to play a game. I got all the cards with numbers from 1to 6 and placed them face-down in rows. Our goal was to pick two cards. If the cards add to 7, you keep the pair and pick again. J has been playing all sorts of matching games at school recently, so this felt like a mathy spin on his current favorite game.

During the game, I noticed that J had some of the pairs memorized. When he flipped over a 6, he would say "Now I just need a 1." When he flipped over a 5, he knew he needed a 2.

So during one turn, J flipped over a 2 and a 6, counted to 8, and flipped them back over. I asked him "When you flip over a 2, what card do you need to make seven?" J paused, then flipped the cards back over and looked at them again...

At first, this seems like a redundant question. After all, J knows that five and two make seven, right? But that's not actually what J knows. He knows that if he *already* has five, he needs two more to make seven. I'm asking him a different question. I'm saying "If you *already* have two, how many do you need to make seven?"

This feels like the same thing to a parent, but to a young kid it's a very different question. He doesn't yet know that 5 + 2 = 2 + 5. He hasn't internalized the commutative property. So he is starting this question from the beginning.

Anyway, J is pondering over these two cards, and he decides to count them again. He counts to eight, then pauses before saying "Seven" I ask him "Two and seven make seven?" and he scrunches his nose to indicate that yeah, that doesn't sound exactly right. So he counts again, gets to eight, and this time says "One." Again, I prompt: "Two and one make seven?" And he says "no, they make three." I think that in each case, J is noticing that his total of eight dots is one too many, but he doesn't know how that affects the answer to my question. First he compensates in the wrong direction, then he answers "one" because he is thinking about the one extra dot.

So he counts **again,** gets to eight, pauses and then says "4?" I can tell he's starting to lose track of the problem, so I cover up two dots with my thumb and say "Ok, let's check. Does two and four make seven dots?" He counts and quickly says "No, five!" So I cover up a single dot and he counts to confirm. Success! Two and five make seven!

Later on, we finish the game and are comparing how many matches we each made. I've already counted my eight matches, but J mixed all his cards together, so he can't easily count them. He starts matching each pair again, painstakingly counting dots as he assembles his line Again, from my perspective he could just match the cards up randomly and then count the pairs, but J is still working on how addition works. So he's going to match them all up first and then count.

I love a lot of little things in this video: the way you can tell when he is confident in his matches and the moments when he has to count to confirm. The way he counts the first row of his cards, then swipes his hand and says "Twenty!" The way he guesses that I have 18 cards. After all, he had ten pairs and twenty cards, which is just ten more. I have eight pairs so I must have eighteen cards. It's a fantastic hypothesis, and it reminds me that kids have ideas about math. Those ideas make sense, even if they are wrong. Joel used a shortcut that worked, but then applied it in a different scenario where it no longer worked. And he'll keep doing that over and over for the rest of his mathematical life.

Oh yeah, my review. Tiny Polka Dot is absolutely fantastic. It gives J a structure that he enjoys exploring, and when he tires of the Match 7 game, we will move on to Match 8, Match 9, or any of the dozens of games and iterations we can create from these cards. These cards are going to be part of our playtime for years to come.