Playful Math Carnival #106

Do you enjoy math? I hope so! If not, browsing this post just may change your mind.

Welcome to the 106th edition of the Math Teachers At Play math education blog carnival — a smorgasbord of links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college. Let the mathematical fun begin!

By tradition, we start the carnival with a puzzle in honor of our 106th edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

Try This Puzzle

If you slice a pizza with a lightsaber, you’ll make straight cuts all the way across. Slice it once, and you get two pieces.

If you slice it five times, you’ll get a maximum of sixteen pieces. (And if you’re lucky you might get a star!)

  • How many times would you have to slice the pizza to get 106 pieces?

Click here for all the mathy goodness!

Confession: I Am Not Good at Math

I want to tell you a story. Everyone likes a story, right? But at the heart of my story lies a confession that I am afraid will shock many readers.

confessionPeople assume that because I teach math, blog about math, give advice about math on internet forums, and present workshops about teaching math — because I do all this, I must be good at math.

Apply logic to that statement.

The conclusion simply isn’t valid.

Continue reading Confession: I Am Not Good at Math

Quotable: Then and Now

“I used to think my job was to teach students to see what I see. I no longer believe this. My job is to teach students to see; and to recognize that no matter what the problem is, we don’t all see things the same way. But when we examine our different ways of seeing, and look for the relationships involved, everyone sees more clearly; everyone understands more deeply.”

Joe Schwartz
Then and Now


[Feature photo (above) by jenn.davis via Flickr (CC BY 2.0).]

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Math Inspirations: Why Study Mathematics?

why-study-math

What teacher hasn’t heard a student complain, “When am I ever going to have to use this?” Didn’t most of us ask it ourselves, once upon a time?

And unless we choose a math-intensive career like engineering, the truth is that after we leave school, most of us will never again use most of the math we learned.

But if math beyond arithmetic isn’t all that useful, then what’s the point?

If you or your student is singing the “Higher Math Blues,” here are some quotations that may cheer you up — or at least give you the strength of vision to keep on slogging.

We Study Mathematics…

To Understand Creation

I don’t want to convince you that mathematics is useful. It is, but utility is not the only criterion for value to humanity. Above all, I want to convince you that mathematics is beautiful, surprising, enjoyable, and interesting. In fact, mathematics is the closest that we humans get to true magic. How else to describe the patterns in our heads that — by some mysterious agency — capture patterns of the universe around us? Mathematics connects ideas that otherwise seem totally unrelated, revealing deep similarities that subsequently show up in nature.

— Ian Stewart
The Magical Maze

That vast book which stands forever open before our eyes, the universe, cannot be read until we have learnt the language in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

— Galileo Galilei
quoted by Clifford Pickover, A Passion for Mathematics

To Train Our Minds

The investigation of mathematical truths accustoms the mind to method and correctness in reasoning, and is an employment peculiarly worthy of rational beings.

— George Washington
quoted by William Dunham, The Mathematical Universe

I told myself, “Lincoln, you can never make a lawyer if you do not understand what demonstrate means.” So I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what “demonstrate” means, and went back to my law studies.

— Abraham Lincoln
quoted by William Dunham, The Mathematical Universe

To Understand History

In most sciences, one generation tears down what another has built, and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure.

— Herman Henkel
quoted by Noah benShea, Great Quotes to Inspire Great Teachers

Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals — the flotsam and jetsam of historical currents. The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all.

— Martin Gardner
quoted by G. Simmons, Calculus Gems

I will not go so far as to say that constructing a history of thought without profound study of mathematical ideas is like omitting Hamlet from the play named after him. But it is certainly analogous to cutting out the part of Ophelia. For Ophelia is quite essential to the play, she is very charming. . . and a little mad.

— Alfred North Whitehead
quoted in The Viking Book of Aphorisms

To Appreciate the Beauty

The mathematician does not study pure mathematics because it is useful, he studies it because he delights in it, and he delights in it because it is beautiful.

— Henri Poincaré
quoted by Theoni Pappas, More Joy of Mathematics

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful. The ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.

— Godfrey H. Hardy
A Mathematician’s Apology

And Most of All, to Play

Mathematics is a world created by the mind of men, and mathematicians are people who devote their lives to what seems to me a wonderful kind of play!

Constance Reid

At age eleven, I began Euclid, with my brother as tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world.

— Bertrand Russell
The Autobiography of Bertrand Russell

I love mathematics … principally because it is beautiful, because man has breathed his spirit of play into it, and because it has given him his greatest game — the encompassing of the infinite.

Rózsa Péter
quoted by Rosemary Schmalz, Out of the Mouths of Mathematicians

Did you enjoy these? You can find plenty more on my Math & Education Quotations page.

  • I would LOVE to hear YOUR favorite mathematics, education, or inspirational quote. Please share in the Comments section below!

 photo exploreMTBoS_zpsf2848a9a.jpgNever Ending Math Problem photo (above) by Danny via Flickr (CC BY 2.0). This post is part of the #MTBoS #MtbosBlogsplosion blogging challenge.


howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.


Prof. Triangleman’s Abbreviated List of Standards for Mathematical Practice

How can we help children learn to think mathematically? Live by these four principles.

PTALSMP 1: Ask questions.

Ask why. Ask how. Ask whether your answer is right. Ask whether it makes sense. Ask what assumptions you have made, and whether an alternate set of assumptions might be warranted. Ask what if. Ask what if not.

PTALSMP 2: Play.

See what happens if you carry out the computation you have in mind, even if you are not sure it’s the right one. See what happens if you do it the other way around. Try to think like someone else would think. Tweak and see what happens.

PTALSMP 3: Argue.

Say why you think you are right. Say why you might be wrong. Try to understand how someone else sees things, and say why you think their perspective may be valid. Do not accept what others say is so, but listen carefully to it so that you can decide whether it is.

PTALSMP 4: Connect.

Ask how this thing is like other things. Try your ideas out on a new problem. Ask whether and how these ideas apply to other situations. Look for similarities and differences. Seek out the boundaries and limitations of your techniques.

— Christopher Danielson

And a Puzzle

Practice applying Professor Triangleman’s Standards to the puzzle below. Which one doesn’t belong? Can you say why someone else might pick a different one?

wodb


multfrac-300An expanded version of the standards originally posted in Ginger ale (also abbreviated list of Standards for Mathematical Practice). Feature photo by Alexander Mueller via Flicker. This post is an excerpt from my book Multiplication & Fractions: Math Games for Tough Topics, available now at your favorite online book dealer.


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The Value of Math Games

From Peggy Kaye’s classic book Games for Math:

Kaye-Games4Math

“Games put children in exactly the right frame of mind for learning difficult things.

“Children relax when they play — and they concentrate. They don’t mind repeating certain facts or procedures over and over, if repetition is part of the game.

“Children throw themselves into playing games the way they never throw themselves into filling out workbook pages.

“The games solidify the achievements of children who are already good at math, and they shore up children who need shoring up. They teach or reinforce many of the skills that a formal curriculum teaches, plus one skill that formal teaching sometimes leaves out — the skill of having fun with math, of thinking hard and enjoying it.

“If you play these games and your child learns only that hard mental effort can be fun, you will have taught something invaluable.”

Peggy Kaye
Games for Math

Sample Peggy’s Games for Math


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Making Sense of Arithmetic

Homeschoolers have an advantage in teaching math: As our students grow, our own understanding of math grows with them because we see how the ideas build on each other.

This is especially true for those of us with large families. We pass through the progression of concepts with each student, and every pass lays down another layer in our own minds.

If you’d like to short-cut that process, check out Graham Fletcher’s Making Sense of Elementary Math video series. He’ll walk you through the topics, showing how manipulatives help build early concepts and gradually give way to abstract calculations.

“Understanding the vertical progression of mathematics is really important in the conceptual development of everyone’s understanding. This whole Making Sense Series has truly forced me to be a better teacher.”

— Graham Fletcher

Continue reading Making Sense of Arithmetic