Tag Archives: Teaching

The Math Student’s Manifesto

[Feature photo above by Texas A&M University (CC BY 2.0) via Flickr.]

Note to Readers: Please help me improve this list! Add your suggestions or additions in the comment section below…

What does it mean to think like a mathematician? From the very beginning of my education, I can do these things to some degree. And I am always learning to do them better.

(1) I can make sense of problems, and I never give up.

  • I always think about what a math problem means. I consider how the numbers are related, and I imagine what the answer might look like.
  • I remember similar problems I’ve done before. Or I make up similar problems with smaller numbers or simpler shapes, to see how they work.
  • I often use a drawing or sketch to help me think about a problem. Sometimes I even build a physical model of the situation.
  • I like to compare my approach to the problem with other people and hear how they did it differently.

(2) I can work with numbers and symbols.

  • I know how numbers relate to each other.
  • I’m flexible with mental math. I understand arithmetic properties and can use them to make calculations easier.
  • I’m not intimidated by algebra symbols.
  • I don’t rely on memorized rules unless I know why they make sense.

(3) I value logical reasoning.

  • I can recognize assumptions and definitions of math terms.
  • I argue logically, giving reasons for my statements and justifying my conclusion.
  • I listen to and understand other people’s explanations.
  • I ask questions to clarify things I don’t understand.

Continue reading The Math Student’s Manifesto

Ruth Beechick on Teaching

[Feature photo above by Samuel Mann (CC BY 2.0) via Flickr.]

Here’s one more quote from homeschooling guru Ruth Beechick. It applies to classroom teachers, too!

Everyone thinks it goes smoothly in everyone else’s house, and theirs is the only place that has problems.

I’ll let you in on a secret about teaching: there is no place in the world where it rolls along smoothly without problems. Only in articles and books can that happen.

you can

— Ruth Beechick
You Can Teach Your Chile Successfully (Grades 4-8)



Tabletop Academy PressGet monthly math tips and activity ideas, and be the first to hear about new books, revisions, and sales or other promotions. Sign up for my Tabletop Academy Press Updates email list.


Math Debates with a Hundred Chart

Euclid game
Wow! My all-time most popular post continues to grow. Thanks to an entry from this week’s blog carnival, there are now more than thirty great ideas for mathematical play:

The latest tips:

(31) Have a math debate: Should the hundred chart count 1-100 or 0-99? Give evidence for your opinion and critique each other’s reasoning.
[Hat tip: Tricia Stohr-Hunt, Instructional Conundrum: 100 Board or 0-99 Chart?]

(32) Rearrange the chart (either 0-99 or 1-100) so that as you count to greater numbers, you climb higher on the board. Have another math debate: Which way makes more intuitive sense?
[Hat tip: Graham Fletcher, Bottoms Up to Conceptually Understanding Numbers.]

(33) Cut the chart into rows and paste them into a long number line. Try a counting pattern, or Race to 100 game, or the Sieve of Eratosthenes on the number line. Have a new math debate: Grid chart or number line — which do you prefer?
[Hat tip: Joe Schwartz, Number Grids and Number Lines: Can They Live Together in Peace? ]


Tabletop Academy PressGet monthly math tips and activity ideas, and be the first to hear about new books, revisions, and sales or other promotions. Sign up for my Tabletop Academy Press Updates email list.


Math Teachers at Play #79

79

[Feature photo above by Jimmie, and “79” image (right) by Steve Bowbrick via flickr (CC BY 2.0).]

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

Welcome to the 79th edition of the Math Teachers At Play (MTaP) 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, game, or trivia tidbits. If you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

Since I’ve been spending all my free time working on my upcoming Math You Can Play book series, I’m in the mood for games. So I found a few games featuring prime and nonprime numbers [which category is #79 — do you know?], and I’ll sprinkle some of my best-loved math game books throughout the carnival.

TRY THESE NUMBER GAMES

Students can explore prime and non-prime numbers with two free classroom favorites: The Factor Game (pdf lesson download) or Tax Collector. For $15-20 you can buy a downloadable file of the beautiful, colorful, mathematical board game Prime Climb. Or try the following game by retired Canadian math professor Jerry Ameis:

Factor Finding Game

FactorFindingGame

Math Concepts: multiples, factors, composites, and primes.
Players: only two.
Equipment: pair of 6-sided dice, 10 squares each of two different colors construction paper, and the game board (click the image to print it, or copy by hand).

On your turn, roll the dice and make a 2-digit number. Use one of your colored squares to mark a position on the game board. You can only mark one square per turn.

  • If your 2-digit number is prime, cover a PRIME square.
  • If any of the numbers showing are factors of your 2-digit number, cover one of them.
  • BUT if there’s no square available that matches your number, you lose your turn.

The first player to get three squares in a row (horizontal/vertical/diagonal) wins. Or for a harder challenge, try for four in a row.

Hat tips: Jimmie Lanley.



TABLE OF CONTENTS

And now, on to the main attraction: the blog posts. Many articles were submitted by their authors; others were drawn from the immense backlog in my rss reader. If you’d like to skip directly to your area of interest, click one of these links.

Tweet: Math Teachers at Play #76: a smorgasbord of great ideas for learning, teaching, and playing around with math. http://ctt.ec/fU9Z2+

Click to tweet: Share the carnival with your friends.
(No spam, I promise! You will have a chance to edit or cancel the tweet.)

Continue reading Math Teachers at Play #79

Fractions: 1/5 = 1/10 = 1/80 = 1?

[Feature photo is a screen shot from the video “the sausages sharing episode,” see below.]

Fractions: 1/5 = 1/10 = 1/80 = 1?

How in the world can 1/5 be the same as 1/10? Or 1/80 be the same as one whole thing? Such nonsense!

No, not nonsense. This is real-world common sense from a couple of boys faced with a problem just inside the edge of their ability — a problem that stretches them, but that they successfully solve, with a bit of gentle help on vocabulary.

Here’s the problem:

  • How can you divide eight sausages evenly among five people?

Think for a moment about how you (or your child) might solve this puzzle, and then watch the video below.

What Do You Notice?

Continue reading Fractions: 1/5 = 1/10 = 1/80 = 1?

Math Teachers at Play #76

76[Feature photo (above) by U.S. Army RDECOM. Photo (right) by Stephan Mosel. (CC BY 2.0)]

On your mark… Get set… Go play some math!

Welcome to the 76th 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.

By tradition, we start the carnival with a puzzle in honor of our 76th edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

PUZZLE: CRYSTAL BALL CONNECTION PATTERNS

K4 matchings

In the land of Fantasia, where people communicate by crystal ball, Wizard Mathys has been placed in charge of keeping the crystal connections clean and clear. He decides to figure out how many different ways people might talk to each other, assuming there’s no such thing as a crystal conference call.

Mathys sketches a diagram of four Fantasian friends and their crystal balls. At the top, you can see all the possible connections, but no one is talking to anyone else because it’s naptime. Fantasians take their siesta very seriously. That’s one possible state of the 4-crystal system.

On the second line of the diagram, Joe (in the middle) wakes up from siesta and calls each of his friends in turn. Then the friends take turns calling each other, bringing the total number of possible connection-states up to seven.

Finally, Wizard Mathys imagines what would happen if one friend calls Joe at the same time as the other two are talking to each other. That’s the last line of the diagram: three more possible states. Therefore, the total number of conceivable communication configurations for a 4-crystal system is 10.

For some reason Mathys can’t figure out, mathematicians call the numbers that describe the connection pattern states in his crystal ball communication system Telephone numbers.

TheWizardBySeanMcGrath-small

  • Can you help Wizard Mathys figure out the Telephone numbers for different numbers of people?
    T(0) = ?
    T(1) = ?
    T(2) = ?
    T(3) = ?
    T(4) = 10 connection patterns (as above)
    T(5) = ?
    T(6) = ?
    and so on.

Hint: Don’t forget to count the state of the system when no one is on the phone crystal ball.

[Wizard photo by Sean McGrath. (CC BY 2.0)]


TABLE OF CONTENTS

And now, on to the main attraction: the blog posts. Many articles were submitted by their authors; I’ve drawn others from the immense backlog in my rss reader. If you’d like to skip directly to your area of interest, here’s a quick Table of Contents:

Continue reading Math Teachers at Play #76

Playing With Math — the Book

body_Book_cover_for_upload

Update: The crowdfunding campaign is now closed and the book is in the final stages. It should be headed to the printer soon. Check the Playing With Math homepage for publication and ordering information.


There are only a few days left to reserve your copy of Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. I don’t have time to finish the review I hoped to write, so instead I’ll share some of my favorite quotes from the book:

What do mathematicians do? We play with math. What are little kids doing when they’re thinking about numbers, shapes, and patterns? They’re playing with math. You may not believe it yet, but you can have fun playing with math, too.

— Sue VanHattum, editor

We had a discussion at the end of the club on how we are all confused now, but pleasantly so, and how important it is to rejoice in confusion and to be comfortable with it. Adults often strive very hard to get rid of any and all possible traces of confusion for kids, making things dreadfully boring.

— Maria Droujkova, after a math circle exploration of infinity

All it talkes to do mathematics is opportunity, a frustrating problem, and a bit of stubbornness.

— Ellen Kaplan, math circle leader

Our own school experiences can make it hard for us to teach without being tempted to “help them master” a concept that they may or may not be ready to master. What we never learned in school was the concept of playing around with math, allowing ideas to “percolate,” so to speak, before mastery occurs, and that process may take time.

— Julie Brennan, homeschooler

Continue reading Playing With Math — the Book