Thinking Mathematics 1: Arithmetic = Gateway to All, by James Tanton

I wrote about Math Without Words, one of James Tanton's textbook alternatives, about a month ago. He had kindly sent me a box of seven of these books back in the spring, and I was too overwhelmed by the magnitude of it all to manage to review them, until I realized it would be much better to do it one by one. Although I haven't yet delved into Thinking Mathematics 1: Arithmetic = Gateway to All as thoroughly as I'd like, I've seen enough to be delighted.

He's just posted a video about his favorite puzzle, in which he describes the Victorian ceiling in his childhood bedroom.  I loved that story when I read it in this book, and shared it once before. Here's an excerpt from his print version of the story:

My career as a mathematician began at age ten. I didn't realize this at the time, of course, but in retrospect it is clear to me that my journey into the rich world of mathematical play - and I use the word play with serious intent - was opened to me thanks to a pressed-tin ceiling in an old Victorian-style house.

I grew up in Adelaide, Australia, in a house built in the early 1900s. The ceiling of each room had its own geometric design and each night in my bedroom I fell asleep staring at a 5x5 grid of squares above me, lined with vines and flowers.

I counted squares and rectangles in the design. I traced paths through its cells and along its edges. I tried to fit non-square shapes onto the vertices of the design. In short, I played a myriad of self-invented games and puzzles on that grid of squares as I fell asleep.

The puzzles he gave himself when he was a child have found a home in Math Without Words. That book was out of print for a while, but it's now back in print, and I hope one day it will be considered a classic.

The story above goes on in his six-page introduction to the Thinking Mathematics series. He had a high school teacher who had the students each draw 3 right triangles and measure the sides to verify the Pythagorean Theorem (sounds good so far, but...). When James asked "How do we know this isn't just coincidence?", the teacher just said "Go back and draw another three right triangles." He knew at that point that he was on his own!

James, on the goal of the Thinking Mathematics series:

... to simply re-examine the standard K-12 mathematics curriculum, starting with matters of arithmetic and algebra, moving on from there, to revel in the delight of intellectual play and not knowing. Is zero even or odd? Is negative zero (if that makes sense) the same as zero? Why is 3⁰=1 and 0!=1? Why does the divisibility rule for 3 work? What's a divisibility rule for 7? Why is negative times negative positive? What does Pascal's triangle really tell us? Should we trust patterns? What is infinity? What is this thing called synthetic division and what is it really doing? Is there such a thing as base one-and-a-half? How many prime numbers are there? Why are primes interesting?

Chapter one, on the counting numbers, includes history, activities, and explorations related to the basic properties. My favorite activity is:

Chapter two, on figurate numbers (square, triangular, ...) includes this challenge:

As I turned to chapter three, on factors and primes, I thought he might include the locker problem. He does, but he adds some new twists:

Anyone up for these research questions?


And chapter four is what decided me to buy a download version of the book, so I can offer some of the delights in this chapter to students of mine who struggle with negative numbers. James' favorite model for negative numbers is holes and piles in sand.

Alas, he hasn't addressed subtracting a negative, which may be my students' biggest challenge. Let's see if I can improve on my 'taking away a debt makes you richer' rule with his piles and holes in the sand...

^-5-^-3 means that we have 5 holes and we are taking away 3 holes. Well, that clearly leaves us with 2 holes. James says there's no such thing as subtraction - we can add the opposite instead. Yes, taking away 3 holes is the same as adding 3 piles (into the holes!), we still have 2 holes left.

Now let's try a harder one: ^-3-^-5 means that we have 3 holes and we are taking away 5 holes. Hmm... Well, we just saw that the way to take away a hole is to add a pile, so that taking away 5 holes amounts to adding 5 piles. And we know what to do with 3 holes and 5 piles already. Now I'll go test this out with my students.

This book has 15 chapters, so I can't go through it all in this one review. But I trust that you'll find gems in every chapter. The table of contents is here. The print version costs $27.50 and the download version costs $20.

Enjoy!