The basic challenge as I see it is that a young child who is enthusiastic about math is generally enthusiastic about numbers. The natural response to such enthusiasm from any reasonable adult is to give that kid more numbers. But this very quickly puts the kid on the path towards accelerating through the school curriculum, making it harder for them to have a good time at school and narrowing their mathematical experiences.
And the other part of the problem is the kid’s maturity and independence. If you have a teen who is enthusiastic about math, that’s fine, they’ll do their schoolwork and then noodle around on Numberphile or Desmos. But younger children frequently lack the ability to direct their own mathematical lives. They want an adult to lead.
If you put those two pieces together — number-centricity and the need for adult guidance — it makes it very hard to give a young child what they want without accelerating them through the K-5 math curriculum.
I maintain that it’s worth trying to avoid as much of the school curriculum at home as you can. If you can get beyond the fun of “hey my six-year old is doing ten-year old math” you’re at risk of making it harder for schools to keep your kid happy. And while I’m not opposed to straight-up acceleration or grade-skipping in some cases, in general I think it’s best to avoid these with very young children, who however advanced they are in math often have totally age-typical emotional needs. (This is why, for all my love of Beast Academy, I don’t typically recommend it to parents.)
I’m trying to speak generally here, but without being super-weird about it I’ve encountered this both at school and at home. My oldest son is very enthusiastic about numbers, and I think it’s in his best interests to not race ahead too far of his classmates. And at school I have tried to express the perspective above to parents of my 3rd and 4th Graders, to mixed success. (Thankfully, most parents get it.)
OK, but all this leads to a question: what do you do with a kid who wants to talk about numbers during breakfast? and on the subway? and over dinner? and after dinner? and at many other times, especially when he’s bored and moody?
It’s in that spirit that I share some of the things that we’ve been talking about in my own home lately. Some of these I proposed to my oldest. Others he came up with on his own. I’ll add that in my teaching life I sometimes encounter accelerated students who totally lack the ability to ask questions independently. That doesn’t make sense to me, since often as kids become more independent they can get more out of their grade-level classes.
Anyway, here are some of the number-centric ideas my son and I have been talking about lately.
Switcheroo Numbers: Take a two-digit number, and swap its digits. 35 and 53. 72 and 27. 19 and 91. Then take the difference of those numbers. You will find that all those numbers are _________________. Why is that? Does it work with three-digit numbers? Four-digit numbers? Consider representing these numbers as sums — 30 + 5 and 50 + 3 — to play with some of these ideas.
Writing a Scratch Program for Counting Factors: My son is most eager to talk about factors, so we often end up drifting towards factor math. Together we made a Scratch program that finds all the factors of a number. I let him tinker around with it and he was able to show some indpenedence there, which I was gratified to see — coding is something unrelated to K-5 math that he can do outside of school. Together we added a factor counter to this program, so he could try to “set records” and find numbers with the most number of factors.
Factor/Number Fraction: Conversations about which numbers had a lot of factors led to discussions about the relationship of the number to its factors. My son loved talking about perfect, abundant, and deficient numbers. I, on the other hand, find this stuff dull as dirt, so I proposed a different question. 3 has two factors, so its factor to number fraction is 2 out of 3, or 2/3. 4 has three factors, so its fraction is 3/4. What numbers have the largest of these sorts of fractions?
Building Bigger Factors: This one also came from me, I thought he would like seeing how you could use multiplication to find factors of bigger numbers. For instance, the factors of 10 are 1, 2, 5, and 10. The factors of 9 are 1, 3 and 9. What are the factors of 9 x 10 = 90? How many factors does it have? And (if this is new territory for you) you might discover that there is a pattern that seems to hold for some, but not all numbers. (For instance, compare the factors of 9 x 10 and 5 x 10.)
Triangular Square Numbers: I don’t know if my boy got into triangular numbers from Marilyn Burns’ books or from Numberblocks, the TV show. But it might have been Numberblocks, with their annoyingly catchy “step squad” chant. You know the one I’m talking about.
Anyway, are any triangular numbers also square? That made for a few days of fun.
The Difference Between Nearby Multiplications: I don’t know why, a lot of my son’s math has to do with finding the difference between slightly different things. For instance, he came into the kitchen the other day eager to share a “trick.” Start with two numbers, like 2 and 7. Make 2 x 8 and 3 x 7. Find the difference between them. He thought there was a pattern but it took him a few false before he figured out what was going on, and then it was interesting to try to explain what was happening and why.
The Sum of Consecutive Powers: This one came from him, I think. Or maybe it was a book he was reading? I’m not sure. Either way, add consecutive powers of 2, like 4 + 8 = 12 or 32 + 64 = 96. His claim was that these numbers are always divisible by 3. Really cool, right? Try it with other bases besides 2. Why is this happening? Fun times.
The “One and a Half” Sequence: This one is very much kid-math, but I think it’s interesting and it occupied my son for a while. He spent a few days (or weeks? what is time?) looking for numbers that were “one and a halfs.” For example, for him 3 is a one and a half (because it is 2 and half of 2) but 4 is not. The full sequence of “one and a half” numbers for him was: 3, 6, 9, … right! He noticed that these were multiples of 3. And then what about “one and a third” numbers? “One and a fourth”? And so forth. An interesting regularity for him to notice and a challenging one for him to try to explain.
These are just the little number investigations that have landed at home. There are other things to recommend: Prime Climb, Math for Smarty Pants, The I Hate Mathematics Book are the biggest winners in our house at the moment. And of course it partly depends on your child’s taste.
But for parents of enthusiastic young children, I think the most important work at home is to broaden those tastes. Because even if what we love most is calculation and numbers, mathematics is big. Kids, especially those self-professed lovers of math, deserve the chance to explore it.