A Book: Thinking Mathematically

I'm re-reading Thinking Mathematically, an assigned book from a math course I took years ago in Portland. I was teaching science at the time but signed up anyway because I've always loved math.

Thinking is still so good and resonates much more now that I've been teaching mathematics.

In the Introduction, under "How to use this book effectively!":

Recalcitrant questions which resist resolution should not be permitted to produce disappointment. A great deal more can be learned from an unsuccessful attempt than from a question which is quickly resolved, provided you think about it earnestly, make use of techniques suggested in the book, and reflect on what you have done. Answers are irrelevant to the main purpose of this book. The important thing is to experience the process being discussed.

... our approach rests on five important assumptions:

You can think mathematically Mathematical thinking can be improved by practice and reflection Mathematical thinking is provoked by contradiction, tension and surprise Mathematical thinking is supported by an atmosphere of questioning, challenging and reflecting Mathematical thinking helps in understanding yourself and the world

These assumptions need to live in our classrooms.

The problems in Thinking are mostly brief and simply stated -- yet each one has the potential to make you linger a bit longer because you want to savor your own thinking. Not even productive struggle, this is sweet struggle.

How many rectangles are there on a chessboard? [Page 43]I have just run out of envelopes. How should I make myself one? [Page 35]A certain village in Jacobean times had all the valuables locked in a chest in the church. The chest had a number of locks on it, each with its own individual and distinct key. The aim of the village was to ensure that any three people in the village would amongst them have enough keys to open the chest, but no two people would be able to. How many locks are required, and how many keys? [Page 176]

I'm finding out that the 2nd Edition came out in 2010. Amazon does not have it in stock currently, but when it does become available, we can rent it for $54.77 or buy it new for $91.29. What??

More Rectangles and One Good Read

I just started reading this book on Ramanujan, and I highly recommend it.

Curmudgeon just posted this Painter's Puzzle yesterday on Christmas Day — what a nice gift for us!

A painting contractor knows that 12 painters could paint all of the school classrooms in 18 days.They begin painting. After 6 days of work, though, 4 people were added to the team. When will the job be finished?

Students typically read this as a proportion problem: 12 painters can do it in 18 days, so 1 painter can do it in 1.5 days. Except... hmmm, no.

Edward Zaccaro uses what he calls the "Think One" strategy in his book to solve this type of problem. I guess mine is the same idea, except I draw rectangles. Shocking. :)

Kids are terrified of fractions already. Teaching them to solve this problem — or any of the work problems — using rational equations will only confirm how much they dread the blessed fractions. Sure, I'll get to the equations, but I just wouldn't start with them.

Another common problem — that I'll use rectangles to help my kids — goes something like this:

In a state with 10% sales tax, someone buys an article marked "50% discount.” When the price is worked out, does it matter if the tax is added first, and then the discount taken off, or if the discount is taken off, and then the tax added?

Tax, then discount:

Discount, then tax:

Heya, I'd love to send the Ramanujan book (from Amazon) to the first person to email me at fawnpnguyen at gmail dot com.

[1:49, Robin S. from PA will be receiving the book!]

[3:42, Elaine W. from VT is also getting the book.]

Hope you're enjoying your break.