The pandemic forced schools to close in March 2020. Like many parents, I found myself suddenly homeschooling two children. The eldest was pretty much fine on her own, being sixteen and always very self-sufficient and self-directed in her learning. But the youngest, age seven, needed more help. As he has had a long-standing interest in mathematics, we decided to explore this by following his interests rather than through the slow school curriculum. We used Khan Academy to do, among others, fractions, algebra, negative numbers. We also watched Numberphile videos and read several mathematics-related books for children. A list with of four of these books, with short descriptions, follow at the end of this post.
These children’s books tend to focus on the gems of mathematics, such as the Fibonacci series, the Königsberg bridges, fractals, Pascal’s triangle, and so on. They do not teach in depth how you should do a long division, or multiply fractions, or solve an equation. Still, when working through these books or following these videos, you cannot help but be struck with amazement and awe at the beauty of mathematics. My son was so taken with fractals, he came up to me with several original origami designs that were fractal (if you could keep on folding before things got too tiny), see title picture.
Here’s a quick example of something we saw/did today. We looked at Mersenne primes, which are prime numbers of the form 2n – 1. For example, 3 is a Mersenne prime (namely, 24 – 1); 31 is a Mersenne prime (it’s 25 -1), but 26 – 1 is not a Mersenne prime, as 63 is not prime. Perfect numbers are numbers of which the divisors, excluding itself, sum to itself (you could also say that perfect numbers are numbers of which the divisors, including itself, sum to 2 times that number). For example, 6, 28 and 496, and 8,128 are perfect numbers. For 28, the divisors are 1 | 2 | 4 | 7 | 14, which sum to 28.
We do not know if there are infinitely many perfect numbers or if there are infinitely many Mersenne primes. We know that Mersenne primes go far. The highest Mersenne prime (the 51st) is 282589933 – 1 and was found in December 2018. Mersenne primes are used to discover new primes through the The Great Internet Mersenne Prime Search (GIMPS). Remarkably, mathematicians don’t know if there are any odd perfect numbers–none have been discovered so far.
Here’s the cool thing: perfect numbers have as one of their divisors always a Mersenne prime, for instance, with 28, it’s 7. With 496, it’s 31. More generally, we have this theorem: If 2n-1 is a prime number, then 2n-1(2n-1) is a perfect number and every even perfect number has this form. The proof for this theorem is short, elegant, and understandable for anyone who has a basic grasp of algebra (see here).
I have written a fair bit of philosophy of mathematics, because I enjoy it, but I had (like many people) a teacher who stamped the joy of mathematics out of me for a long time. People shuddered whenever they had to go up to the board to solve a mathematics problem. I recall one occasion when I was 12 years old and I had to solve an equation seated at my desk. I made a mistake and the teacher then walked up to my desk, she started yelling at me, and then went on to grab my folder, ripped out several pages, and tore them up in front of everyone’s eyes, telling me all the while how useless I was at math. I went on to have that teacher for four more years.
In my final two years of high school, I finally had a different teacher; he studied for a PhD in mathematics and did some part time teaching to supplement his income. It was only then that I realized how marvelous mathematics was–next to the curriculum we would do fun things such as constructing Pascal triangles or using alternative methods to solve large multiplications and divisions. My math grades shot up, but by then the damage was done, and I had become so fearful of making mistakes through many years of math terror, that I certainly did not want to take any math in college.
When I was in my final year of high school, some other student asked our teacher, “Why must we learn this?” (we were learning integration). She continued, “I understand you told us it’s useful but why must each of us learn to do this?” “Well, it’s to exercise your mind,” he answered. While that is a good response, once you get into mathematics and can start to see the beautiful patterns, you see another reason: you do math for its own sake. Mathematics is useful, and a great way to exercise the mind, but is also is the way we think of patterns and find intricate, surprising and deep connections. It’s a vast world where those connections continue to turn up unexpectedly, a world one can explore for a lifetime even as one is mostly house-bound.
The children’s books on mathematics my son and I explored focus on the beautiful stuff–the exercises are fun, short, they pick out the gems. But they instill a sense of joy of mathematics, an appreciation of the intricacy of patterns that underlie the universe (I have written some papers defending mathematical platonism, see here and here, and while it’s not a philosophical hill I want to die on, it’s still a position I am pretty committed to). Even though these children’s mathematical novels pick out just some select pieces of math, they help to keep one’s eye on the prize.
The beauty of mathematics should thus be more centered in pedagogy. My sense of American education is that the awe and wonder-driven way teach math is restricted to gifted programs–in regular math, kids just trudge through learning multiplication tables and rules for solving equations and so on, but they don’t generally get to meet the whacky Fibonacci numbers and their interesting connection to the Golden Ratio. Why only convey a sense of wonder at math for children who are deemed gifted? The sense of wonder, enjoyment and aesthetic delight in mathematics are epistemic emotions. They motivate children (and adults) to do more mathematics, and thus help them to advance and to learn more about it. Everyone can benefit from this.
As we are moving into the fall semester, and due to premature reopenings the chances of safely opening schools in the midst of a pandemic are dwindling, I recommend the following books on mathematics for children (roughly, I think this is for elementary school and middle school).
Lilac Mohr, Math and magic in wonderland (2016)
Two sisters, Lulu and Elizabeth travel to a wondrous land reminiscent of Alice’s wonderland, and solve lots of mathematical problems along the way. You get to solve the problems along with the girls. The nice feature in this book is that you get to understand certain mathematical procedures, for example, you don’t just learn that 1 + 2 + 3 +… + n can be solved with n(n+1)/2 but also why this formula works. The book covers a wide range of mathematical fields, including graph theory. If you want a book with exercises that are doable (still tricky sometimes!), including some logic puzzles, this one is excellent.
Hans Magnus Enzenberger, The number devil: a mathematical adventure (1997)
This is a classic in the genre (originally in German). It tells of Robert, a schoolboy with fear of math, who gets 12 dreams in which a number devil, called Teplotaxl visits him. Robert grows interested and more confident about his mathematical abilities over time. The explanations of the mathematical concepts in this book are always very clear in spite of the weird vocabulary the author uses such as rutabaga for square roots and vroom! for factorials. It goes quite advanced; it is not so much a workbook but a book to marvel and delight at the intricacies of number theory.
Malba Tahan (real name: Júlio César de Mello e Souza), The Man Who Counted (1938).
Since its publication, this has been an immensely popular work. This is the most literary of the four books I review here. This story weaves an elegant narrative of a man with extraordinary mathematical abilities who travels to Baghdad, and his friends and potential foes. The problems are hard but fair, including many logic problems, and with a seven-year-old I had to work along to work them out, and could not solve them all (but the Man who Counted will give you the answer if you can’t find it). Here’s a fun problem from the book. With four fours you can make all numbers from 0-10, you need to use all 4s (so 4 should appear exactly 4 times in your calculation), you can use multiplication, division, addition, subtraction, and you may also stick them together e.g., 44 – 44 = 0, 44/44 = 1. Can you find the other numbers (1-10)?
Theoni Pappas, The Adventures of Penrose the Mathematical Cat (1997)
This is a collection of short stories featuring a cat called Penrose and his adventures with mathematics, as he stumbles on the work of his mistress who is a mathematician. The chapters are short covering topics such fractals, how to use a soroban (Japanese calculator), or how to convert any number into a binary number easily, as there is always a short (and usually not too hard) exercise to complete. Mathematical concepts are anthropomorphized: bubbles, polygons, tangram figures, and numbers talk to the cat as they explain what they entail.