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Thinking Mathematically

Prentice Hall
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Some books try to convince the reader that she should go and pick up a piece of paper and a pencil before sitting down to read. Many times I have found that I could simply skip the parts that needed the heavy lifting that this type of advice warns about, and though at the end I would not have a perfect understanding of the whole book, I would get just what I needed and that was that.

Not so with Thinking Mathematically. The authors recommend that you start keeping a journal where you write down what you are working on and follow certain steps as you attack each and every problem they offer you. Of course I nodded as I read this, knowing that I would, once again, simply skip over the steps where this might have come in handy. Now here is the punchline: I could not keep myself from getting up several times to get yet another piece of paper to work on just this little problem here and there. It is true that I did not start a whole problem-solving journal, but I think that would be what I would have done if I were a student reading this book. The offerings of Thinking Mathematically are so delightful that many times I found myself simply flipping through the pages and looking for yet another new problem to amuse myself. The text in between the problems is also very much worth reading however, so I always came back to where I was, and continued to read, until the end. Then I just started from the beginning again, looking for problems to share with my students…

Perhaps a review should contain a plot summary, so, if it is not clear what this book is about, let me try again. Thinking Mathematically is a problem solving book; through solved problems and many others offered as practice, the book takes the reader through a whole program on how to attack mathematical problems. The goal may be to solve a problem assigned in a homework set, but the authors expect the reader to have more intellectual curiosity and higher aspirations than that. They hope and encourage the reader to make conjectures, extend, generalize the problems, and think of alternative frameworks that the particular problem may be relevant to.

For a problem solving class, this would be an ideal book. The authors divide the problem solving process into a few steps and expect the student to record her individual process in her journal in this particular format. In other words, they propose a multi-step approach to problem solving and demonstrate this with many examples. Just like Pólya’s simple scheme (see his How To Solve It), this sounds much simpler than it actually turns out to be once we attempt it for a particular, nontrivial problem. I do not think that the formal process proposed by the authors will work for everyone, but it is a good starting point. The format, though seemingly pedantic, will provide a gentle scaffold for beginners.

More experienced problems solvers, too, may appreciate what is new here, in particular the emphasis on retrospection, on the review stage. After all is said and done, and when many tired students might prefer to simply move on and forget the problem, we are encouraged to stop and think about the overall process and what we learned from it. What we might end up with in this review stage might be a simple observation along the lines of “On problems involving chess boards, it might be useful to consider a smaller scale version of the problem first.” Or we might have an epiphany and might wish to hold on to that for posterity.

An experienced reader might ask why such a book exists. Didn’t Pólya already teach all of us “how to solve it”? But the answer is clear. Going into any mathematics classroom, we can find several students who are excellent problem solvers, and many others who are ready to learn, but it’s clear that Pólya certainly has not reached everyone.

In addition, I see Pólya’s as a theoretical treatise, one that describes a basic outline of how to attack a mathematical problem and a few hand-picked examples. (Apparently Pólya’s methodology is not limited to the mathematical realm; see [2]). But students learn by doing. And doing. And doing yet again.

Another difference that I should point out is that Thinking Mathematically aims to teach a methodology to expand and strengthen one’s problem solving skills as one looks introspectively deep within oneself to see what one’s mindset is and how it contributes to the process.

The education background of the authors is clear. Their emphasis on the pedagogical aspects will make this a wonderful opportunity for those of us who wish to instill in our students curiosity of mind, perseverance and confidence that hard work pays off, comfort dealing with ambiguity and uncertainty and frustration of being stuck. As motto one can always take on theirs: “Being stuck is an honorable state” for that is when and where breakthroughs come.

The authors guide us through one problem after another. On many others they leave us at the edge of a cliff, so that one just has to stop and solve that little problem before moving on. We can clearly see the fruits of this good pedagogical approach. The problems get more layered and more complex as we move through the book, and yet the authors always find new perspectives to note for even the simpler ones.

I should note here that this book is in its second edition, and earlier editions have been influential in bringing many into the world of mathematics (see, for instance, [3]). In its new edition it is even more attractive. Younger students might find the hand-drawn imagery amusing, older ones will appreciate the typesetting which is clear and neat. The colloquial style of the authors pulls the reader in and accompanies her along this beautiful path. This book will not be put back on the shelf for a while, as I intend to continue flipping through it for just a little longer.


[1] George Pólya, How To Solve It, many editions.

[2] Susan D’Agostino, 2011. “A Math Major, Pólya, Invention, and Discovery”. The Journal of Humanistic Mathematics, Volume I, Issue 2 (July 2011), pages 51–55. Available at:

[3] Noel-Ann Bradshaw, 2010. “Review of Thinking Mathematically”, MSOR Connections, Volume 10, Number 3 (Autumn Term), pages 49–50. Available at:

Gizem Karaali is assistant professor of mathematics at Pomona College and an editor of the Journal of Humanistic Mathematics.

Date Received: 
Thursday, January 27, 2011
Include In BLL Rating: 
J. Mason, L. Burton, and K. Stacey
Publication Date: 
Gizem Karaali
BLL Rating: 

1.    Everyone can start                                                                                               

2.    Phases of work                                                          

3.    Responses to being STUCK                                                                                

4.    ATTACK: conjecturing    

5.    ATTACK: justifying and  convincing

6.    Still STUCK?   

7.    Developing an internal monitor  

8.    On becoming your own questioner                                                                      

9.    Developing mathematical thinking 

10.  Something to think about                                                                                     

11.  Thinking mathematically in curriculum topics                                                     


Subject Index                                                                                                               

Index of questions                                                                                                       


Publish Book: 
Modify Date: 
Monday, September 5, 2011