Introduction ixWORLDLY WISDOM
There you are, you savvy sightseer, visiting new places and learning new things.
Watch for these sidebars that point out bits of information and insight about the world
of mathematics.MATH IN THE PAST
Over the centuries that people have studied mathematics, their way of writing num-
bers, performing calculations, and organizing their thinking about math have grown
and changed. Some of those ideas are still with us, some have faded away, and some
have led to important discoveries. These sidebars will highlight some of these histori-
cal developments that connect to your current studies.Acknowledgments
It’s always hard to know who to mention at this point in a book. You, my intended reader, may
have no idea who these people are, and you may skip over this section because of that. Or you
may read this and wonder if these folks are as strange as I am. The most important people may
never see the book, and yet they should be mentioned.
So I’ll begin by being forever grateful to E. Jones Wagner—Jonesy—who took a chance on an
eager but very inexperienced young teacher. Jonesy showed me that different students learn in
different ways and different teachers teach in different ways, and that to be a successful teacher,
I had to find my own way. She helped me look past a lifetime of “shoulds” to what actually
worked. She taught me, by her counsel, her example, her style, and yes, her eccentricity. Forty
years later, I still think back to what I learned from Jonesy and, when faced with a problem, won-
der what Jonesy would do. And to this day, if I see anything yellow or orange around the school
house, I still want to return it to Jonesy’s classroom.
My gratitude goes to Grace Freedson, of Grace Freedson’s Publishing Network, who not only
won’t let me get lazy but also offers me projects, like this one, that are satisfying and challeng-
ing, and help me to grow as a teacher and as a person. My thanks also go to Lori Hand and Ann
Barton for making this project an absolute delight, from start to finish, and for making my scrib-
blings about math look good and make sense.
One of the things I tell my students is that it’s normal, natural, even valuable to make mistakes.
It’s how we learn. Or more correctly, correcting our mistakes is how we learn. We all make
mistakes. I certainly do, which is why there is someone who reads this math before you do. The
Technical Reviewer’s job is to read everything I’ve written about the math and make sure it’s cor-
rect and clear. That job also includes checking all the problems and the answers and finding my
mistakes. Yes, I made mistakes, and I am grateful to my Technical Reviewer for finding them and