# The Math Student’s Manifesto

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

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.

# 2015 Mathematics Game

[Feature photo above by Scott Lewis and title background (right) by Carol VanHook, both (CC BY 2.0) via Flickr.]

Did you know that playing games is one of the Top 10 Ways To Improve Your Brain Fitness? So slip into your workout clothes and pump up those mental muscles with the Annual Mathematics Year Game Extravaganza!

For many years mathematicians, scientists, engineers and others interested in math have played “year games” via e-mail. We don’t always know whether it’s possible to write all the numbers from 1 to 100 using only the digits in the current year, but it’s fun to see how many you can find.

## Rules of the Game

Use the digits in the year 2015 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.

• You must use all four digits. You may not use any other numbers.
• Solutions that keep the year digits in 2-0-1-5 order are preferred, but not required.
• You may use +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and parentheses, brackets, or other grouping symbols.
• You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
• You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.

#### My Special Variations on the Rules

• You MAY use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal. But students and teachers beware: you can’t submit answers with repeating decimals to Math Forum.
• You MAY NOT use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. Math Forum allows these, but I’ve decided I prefer my arithmetic straight.

# Math Debates with a Hundred Chart

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? ]

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# Math Teachers at Play #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!

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 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. Click to tweet: Share the carnival with your friends. (No spam, I promise! You will have a chance to edit or cancel the tweet.) # Horseshoes: A Place Value Game [Feature photo above by Johnmack161 via Wikimedia Commons (CC BY 2.5).] I first saw place value games on the old PBS Square One TV show (video below). Many teachers have posted versions of the game online, but Snugglenumber by Anna Weltman is by far the cutest variation. Anna kindly gave me permission to use the game in my upcoming Math You Can Play book series, and I added the following variation: ### Horseshoes Math Concepts: place value, strategic thinking. Players: two or more. Equipment: one deck of playing cards, or a double deck for more than three players. Separate out the cards numbered ace (one) through nine, plus cards to represent the digit zero. We use the queens (Q is round enough for pretend), but you could also use the tens and just count them as zero. Shuffle well and deal eleven cards to each player. Arrange your cards in the snugglenumber pattern shown here, one card per blank line, to form numbers that come as close to each target number as you can get it. Score according to horseshoes rules: • Three points for each ringer, or exact hit on the target. • One point for each number that is six or less away from the target. • If none of the players land in the scoring range for a target number, then score one point for the number closest to that target. For a quick game, whoever scores the most points wins. Or follow tradition and play additional rounds until one player gets 21 points (40 for championship games) — and you have to win by at least two points over your closest opponent’s score. ### But Who’s Counting? Get 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. # Fractions: 1/5 = 1/10 = 1/80 = 1? [Feature photo is a screen shot from the video “the sausages sharing episode,” see below.] 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? # Fraction Game: My Closest Neighbor [Feature photo above by Jim Larrison, and antique playing cards below by Marcee Duggar, via Flickr (CC BY 2.0).] I missed out on the adventures at Twitter Math Camp, but I’m having a great time working through the blog posts about it. I prefer it this way — slow reading is more my speed. Chris at A Sea of Math posted a wonderful game based on one of the TMC workshops. Here is my variation. Math concepts: comparing fractions, equivalent fractions, benchmark numbers, strategic thinking. Players: two to four. Equipment: two players need one deck of math cards, three or four players need a double deck. ## How to Play Deal five cards to each player. Set the remainder of the deck face down in the middle of the table as a draw pile. You will play six rounds: • Closest to zero • Closest to 1/4 • Closest to 1/3 • Closest to 1/2 • Closest to one • Closest to two In each round, players choose two cards from their hand to make a fraction that is as close as possible (but not equal) to the target number. Draw two cards to replenish your hand. The player whose fraction is closest to the target collects all the cards played in that round. If there is a tie for closest fraction, the winners split the cards as evenly as they can, leaving any remaining cards on the table as a bonus for the winner of the next round. After the last round, whoever has collected the most cards wins the game. Get 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. By Denise Gaskins Posted in Games # Quotable: Math Connections It turns out that the people who do well in math are those who make connections and see math as a connected subject. The people who don’t do well are people who see math as a lot of isolated methods. — Jo Boaler Math Connections If you or your children struggle with math, Boaler’s non-profit YouCubed.org may help you recover your joy in learning. Get 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. # Reblog: Solving Complex Story Problems [Dragon photo above by monkeywingand treasure chest by Tom Praison via flickr.] Over the years, some of my favorite blog posts have been the Word Problems from Literature, where I make up a story problem set in the world of one of our family’s favorite books and then show how to solve it with bar model diagrams. The following was my first bar diagram post, and I spent an inordinate amount of time trying to decide whether “one fourth was” or “one fourth were.” I’m still not sure I chose right. I hope you enjoy this “Throw-back Thursday” blast from the Let’s Play Math! blog archives: Cimorene spent an afternoon cleaning and organizing the dragon’s treasure. One fourth of the items she sorted was jewelry. 60% of the remainder were potions, and the rest were magic swords. If there were 48 magic swords, how many pieces of treasure did she sort in all? [Problem set in the world of Patricia Wrede’s Enchanted Forest Chronicles. Modified from a story problem in Singapore Primary Math 6B. Think about how you would solve it before reading further.] How can we teach our students to solve complex, multi-step story problems? Depending on how one counts, the above problem would take four or five steps to solve, and it is relatively easy for a Singapore math word problem. One might approach it with algebra, writing an equation like: $x - \left[\frac{1}{4}x + 0.6\left(\frac{3}{4} \right)x \right] = 48$ … or something of that sort. But this problem is for students who have not learned algebra yet. Instead, Singapore math teaches students to draw pictures (called bar models or math models or bar diagrams) that make the solution appear almost like magic. It is a trick well worth learning, no matter what math program you use … [Click here to go read Solving Complex Story Problems.] Don’t miss any of “Let’s Play Math!”: Subscribe in a reader, or get updates by Email. # Playing with Pinterest: New Math Boards Do you like Pinterest? I’ve enjoyed exploring the site lately, so I set up a few boards where I can pin the goodies I find. It may take awhile before I get all the old games and posts from this blog loaded up, so save the links and come back often… ### Playful Math Games & Activities As our children (and their parents!) play around with mathematical ideas and the relationships between them, we develop deep understanding that is strong enough to support future learning. Playful math links include math games, activities, and interesting lesson plans. ### Math Doodling Making abstract math visual: Math doodles let us see and experiment with a wide range of mathematical structures — and even to feel them, if we include hands-on 3D doodles in clay or other media. Links include art projects, geometry constructions, and physical models to explore. ### Math Teaching Tips & Resources A variety of math teaching ideas for homeschool families or classroom teachers. Learning mathematics is more than just answer-getting: help your students make conceptual connections. These links are more “schooly” than on the other boards, and they include conceptual lessons that build your own understanding of mathematics as well as that of your students. And math notebooking resources, too. ### MTaP Math Education Blog Carnival Archive Since early 2009, the Math Teachers at Play (MTaP) blog carnival has offered tips, tidbits, games, and activities for students and teachers of preschool through pre-college mathematics. Now published once a month, the carnival welcomes entries from parents, students, teachers, homeschoolers, and just plain folks. If you like to learn new things and play around with ideas, you are sure to find something of interest. ### Math-Ed Quotes Inspiration for homeschooling parents and classroom teachers. This is where I’m posting my Wednesday Wisdom quotes. And that’s the end of my Pinterest boards (so far). What are some of your favorite Pinterest sites? Please share a link in the comments! Get 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. # The Linear Inequality Adventures of Ohio Jones Last week, Kitten and I reached her textbook’s chapter on graphing linear equations, and a minor mistake with negative numbers threw her into an “I can’t do it!” funk. It’s not easy teaching a perfectionist kid. Usually her mood improves if we switch to a slightly more advanced topic, and luckily I had saved these worksheets on my desktop, waiting for just such an opportunity. Today’s lesson: • Some fun(ish) worksheets “For tomorrow, students will be graphing systems of inequalities, so I decided to create a little Ohio Jones adventure (Indiana’s lesser known brother)…” I offered to give her a hint, but she wanted to try it totally on her own. It took her about 40 minutes to work through the first few rooms of the Lost Templo de los Dulces and explain her solutions to me. I’m sure she’ll speed up with experience. So far, she’s enjoying it much more than the textbook lesson. It’s fascinating to me how the mere hint of fantasy adventure can change graphing equations from boring to cool. Thanks, Dan! Get 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. # 2014 Mathematics Game [Feature photo above by Artis Rams (CC BY 2.0) via flickr. Title background (right) by Dan Moyle (CC BY 2.0) via flickr] Have you made a New Year’s resolution to spend more time with your family this year, and to get more exercise? Problem-solvers of all ages can pump up their (mental) muscles with the Annual Mathematics Year Game Extravaganza! For many years mathematicians, scientists, engineers and others interested in mathematics have played “year games” via e-mail and in newsgroups. We don’t always know whether it is possible to write expressions for all the numbers from 1 to 100 using only the digits in the current year, but it is fun to try to see how many you can find. ## Rules of the Game Use the digits in the year 2014 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable by age: Young children can start with looking for 1-10, or 1-25. • You must use all four digits. You may not use any other numbers. • Solutions that keep the year digits in 2-0-1-4 order are preferred, but not required. • You may use +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and parentheses, brackets, or other grouping symbols. • You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number. • You may use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal. • You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them. • You may use a double factorial, but we prefer solutions that avoid them. n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. [Note to students and teachers: If you want to take part in the Math Forum Year Game, be warned that they do not allow repeating decimals.] By Denise Gaskins Posted in Puzzles # Things To Do with a Hundred Chart #30 Here’s one more entry for my 20+ Things to Do with a Hundred Chart post, thanks to David Radcliffe in the comments on Monday’s post: (30) Can you mark ten squares Sudoku-style, so that no two squares share the same row or column? Add up the numbers to get your score. Then try to find a different set of ten Sudoku-style squares. What do you notice? What do you wonder? [Suggested by David Radcliffe.] ## Share Your Ideas Can you think of anything else we might do with a hundred chart? Add your ideas in the Comments section below, and I’ll add the best ones to our master list. Get all our new math tips and games: Subscribe in a reader, or get updates by Email. # Things To Do with a Hundred Chart #29 Here’s a new entry for my 20+ Things to Do with a Hundred Chart post: (29) Blank 100 Grid Number Investigations: Challenge your students to deduce the secret behind each pattern of shaded squares. Then have them make up pattern puzzles of their own. [Created by Stuart Kay. Free registration required to download pdf printable.] ## Share Your Ideas Can you think of anything else we might do with a hundred chart? Add your ideas in the Comments section below, and I’ll add the best ones to our master list. Get all our new math tips and games: Subscribe in a reader, or get updates by Email. # Parents, Teachers: Learn about Teaching Decimals Many children are confused by decimals. They are convinced 0.48 > 0.6 because 48 is obviously ever so much bigger than 6. Their intuition tells them 0.2 × 0.3 = 0.6 has the clear ring of truth. And they confidently assert that, if you want to multiply a decimal number by 10, all you have to do is add a zero at the end. What can we do to help our kids understand decimals? Christopher Danielson (author of Talking Math with Your Kids) will be hosting the Triangleman Decimal Institute, a free, in-depth, online chat for “everyone involved in children’s learning of decimals.” The Institute starts tomorrow, September 30 (sorry for the short notice!), but you can join in the discussion at any time: Past discussions stay open, so feel free to jump into the course whenever you can. Here is the schedule of “classes”: # Puzzle: Algebra on Rectangles Gordon Hamilton of Math Pickle recently posted these videos on how to make algebra 1 puzzles on rectangles. As I was watching, Kitten came in and looked over my shoulder. She said, “Those look like fun!” They look like fun to me, too, and I bet your beginning algebra students will enjoy them: # Multiplying Negative Numbers with Rectangles I love using rectangles as a model for multiplication. In this video, Mike & son offer a pithy demonstration of WHY a negative number times a negative number has to come out positive: Get all our new math tips and games: Subscribe in a reader, or get updates by Email. # How To Master Quadratic Equations feature photo above by Junya Ogura via flickr (CC BY 2.0) A couple of weeks ago, James Tanton launched a wonderful resource: a free online course devoted to quadratic equations. (And he promises more topics to come.) Kitten and I have been working through the lessons, and she loves it! We’re skimming through pre-algebra in our regular lessons, but she has enjoyed playing around with simple algebra since she was in kindergarten. She has a strong track record of thinking her way through math problems, and earlier this year she invented her own method for solving systems of equations with two unknowns. I would guess her background is approximately equal to an above-average algebra 1 student near the end of the first semester. After few lessons of Tanton’s course, she proved — within the limits of experimental error — that a catenary (the curve formed by a hanging chain) cannot be described by a quadratic equation. Last Friday, she easily solved the following equations: $\left ( x+4 \right )^2 -1=80$ and: $w^2 + 90 = 22 w - 31$ and (though it took a bit more thought): $4x^2 + 4x + 4 = 172$ We’ve spent less than half an hour a day on the course, as a supplement to our AoPS Pre-Algebra textbook. We watch each video together, pausing occasionally so she can try her hand at an equation before listening to Tanton’s explanation. Then (usually the next day) she reads the lesson and does the exercises on her own. So far, she hasn’t needed the answers in the Companion Guide to Quadratics, but she did use the “Dots on a Circle” activity — and knowing that she has the answers available helps her feel more independent. # How to Recognize a Successful Homeschool Math Program photo by Dan McCarthy (cc-by) After teaching co-op math classes for several years, I’ve become known as the local math maven. Upon meeting one of my children, fellow homeschoolers often say, “Oh, you’re Denise’s son/daughter? You must be really good at math.” The kids do their best to smile politely — and not to roll their eyes until the other person has turned away. I hear similar comments after teaching a math workshop: “Wow, your kids must love math!” But my children are individuals, each with his or her own interests. A couple of them enjoy an occasional geometry or logic puzzle, but they never voluntarily sit down to slog through a math workbook page. In fact, one daughter expressed the depth of her youthful perfectionist angst by scribbling all over the cover of her Miquon math workbook: • “I hate math! Hate, hate, hate-hate-HATE MATH!!!” Translation: “If I can’t do it flawlessly the first time, then I don’t want to do it at all.” photo by Jason Bolonski (cc-by) # 2013 Mathematics Game feature photo above by Alan Klim via flickr New Year’s Day Now is the accepted time to make your regular annual good resolutions. Next week you can begin paving hell with them as usual. Yesterday, everybody smoked his last cigar, took his last drink, and swore his last oath. Today, we are a pious and exemplary community. Thirty days from now, we shall have cast our reformation to the winds and gone to cutting our ancient shortcomings considerably shorter than ever. We shall also reflect pleasantly upon how we did the same old thing last year about this time. However, go in, community. New Year’s is a harmless annual institution, of no particular use to anybody save as a scapegoat for promiscuous drunks, and friendly calls, and humbug resolutions, and we wish you to enjoy it with a looseness suited to the greatness of the occasion. For many homeschoolers, January is the time to assess our progress and make a few New Semester’s Resolutions. This year, we resolve to challenge ourselves to more math puzzles. Would you like to join us? Pump up your mental muscles with the 2013 Mathematics Game! By Denise Gaskins Posted in Puzzles # Who Killed Professor X? ## What a Fun Book! Who Killed Professor X? is a work of fiction based on actual incidents, and its heroes are real people who left their mark on the history of mathematics. The murder takes place in Paris in 1900, and the suspects are the greatest mathematicians of all time. Each suspect’s statement to the police leads to a mathematical problem, the solution of which requires some knowledge of secondary-school mathematics. But you don’t have to solve the puzzles in order to enjoy the book. Fourteen pages of endnote biographies explain which parts of the mystery are true, which details are fictional, and which are both (true incidents slightly modified for the sake of the story). I ordered Who Killed Professor X? from The Book Depository (free shipping worldwide!), and it only took 5 days to arrive here in the middle of the Midwest. My daughter Kitten, voracious as always, devoured it in one sitting — and even though she hasn’t studied high school geometry yet, she was able to work a couple of the problems. Get all our new math tips and games: Subscribe in a reader, or get updates by Email. # Rate × Time = Distance Problems I love how Richard Rusczyk explains math problems. It’s a new school year, and that means it’s time for new MathCounts Mini videos. Woohoo! # Build Mathematical Skills by Delaying Arithmetic, Part 4 To my fellow homeschoolers, While Benezet originally sought to build his students’ reasoning powers by delaying formal arithmetic until seventh grade, pressure from “the deeply rooted prejudices of the educated portion of our citizens” forced a compromise. Students began to learn the traditional methods of arithmetic in sixth grade, but still the teachers focused as much as possible on mental math and the development of thinking strategies. Notice how waiting until the children were developmentally ready made the work more efficient. Benezet’s students studied arithmetic for only 20-30 minutes per day. In a similar modern-day experiment, Daniel Greenberg of Sudbury School discovered the same thing: Students who are ready to learn can master arithmetic quickly! ## Grade VI [20 to 25 minutes a day] At this grade formal work in arithmetic begins. Strayer-Upton Arithmetic, book III, is used as a basis. The processes of addition, subtraction, multiplication, and division are taught. Care is taken to avoid purely mechanical drill. Children are made to understand the reason for the processes which they use. This is especially true in the case of subtraction. Problems involving long numbers which would confuse them are avoided. Accuracy is insisted upon from the outset at the expense of speed or the covering of ground, and where possible the processes are mental rather than written. Before starting on a problem in any one of these four fundamental processes, the children are asked to estimate or guess about what the answer will be and they check their final result by this preliminary figure. The teacher is careful not to let the teaching of arithmetic degenerate into mechanical manipulation without thought. Fractions and mixed numbers are taught in this grade. Again care is taken not to confuse the thought of the children by giving them problems which are too involved and complicated. Multiplication tables and tables of denominate numbers, hitherto learned, are reviewed. — L. P. Benezet The Teaching of Arithmetic II: The Story of an experiment # Multiplication Challenge Can you explain why the multiplication method in the following video works? How about your upper-elementary or middle school students — can they explain it to you? Pause the video at 4:30, before he gives the interpretation himself. After you have decided how you would explain it, hit “play” and listen to his explanation. # Thinking (and Teaching) like a Mathematician photos by fdecomite via flickr Most people think that mathematics means working with numbers and that being “good at math” means being able to do (only slower) what any$10 calculator can do. But then, most people think all sorts of silly things, right? That’s what makes “man on the street” interviews so funny.

Numbers are definitely part of math — but only part, and not even the biggest part. And being “good at math” means much more than being able to work with numbers. It means making connections, thinking creatively, seeing familiar things in new ways, asking “Why?” and “What if?” and “Are you sure?”

It means trying something and being willing to fail, then going back and trying something else. Even if your first try succeeded — or maybe, especially if your first try succeeded. Just knowing one way to do something is not, for a mathematician, the same as understanding that something. But the more different ways you know to figure it out, the closer you are to understanding it.

Mathematics is not just memorizing and following rules. If we want to teach real mathematics, we teachers need to learn to think like mathematicians. We need to see math as a mental game, playing with ideas. James Tanton explains: