Know How to Argue

You may remember, I decided to try my hand at rewriting the Standards for Mathematical Practice into student-friendly language.

My kids loved to argue. Do yours?

Math Tip #3: Know How to Argue.

  • Argue respectfully.
  • Analyze situations.
  • Recognize your own assumptions.
  • Be careful with definitions.
  • Make a guess, then test to see if it’s true.
  • Explain your thoughts. Give evidence for your conclusions.
  • Listen to other people. Ask questions to understand their point of view.
  • Celebrate when someone points out your mistakes. That’s when you learn!

Continue reading Know How to Argue

W.W. Sawyer’s Rules of Mathematics

“In the beginnings of arithmetic and algebra, the main purpose is not to get the pupil making calculations. The main purpose is to get him into the habit of thinking, and to show him that he can think the problems out for himself.

“Pupils ask ‘Am I allowed to do this?’ as if we were playing a game with certain rules.

“A pupil is allowed to write anything that is true, and not allowed to write anything untrue!

“These are the only rules of mathematics.”

—W. W. Sawyer, Vision in Elementary Mathematics

[THE FINE PRINT: I am an Amazon affiliate. If you follow the link and buy something, I’ll earn a small commission (at no cost to you). But this book is a well-known classic, so you should be able to order it through your local library.]

Inspired by Sawyer’s Two Rules

I love this quote so much, I turned it into a printable math activity guide. I hope it helps inspire your students to deeper mathematical thinking.

Here’s the product description…

Join the Math Rebellion: Creative Problem-Solving Tips for Adventurous Students

Take your stand against boring, routine homework.

Fight for truth, justice, and the unexpected answer.

Join the Math Rebellion will show you how to turn any math worksheet into a celebration of intellectual freedom and creative problem-solving.

Help your students practice thinking for themselves as they follow the Two Rules of the Math Rebellion: “A pupil is allowed to write anything that is true, and not allowed to write anything untrue! These are the only rules of mathematics.”

Find Out More

Don’t Panic

As I mentioned last Saturday, I decided to try my hand at rewriting the Standards for Mathematical Practice into student-friendly language.

Here’s the second installment…

Math Tip #2: Don’t Panic.

  • Don’t let abstraction scare you.
  • Don’t freeze up when you see complex numbers or symbols.
  • Break them down into simpler parts.
  • Take each problem one step at a time.
  • Know the meaning of the math, how it relates to the “real world.”
  • But if it gets in your way, ignore the “real world” situation. Revel in the abstract fantasy.

Continue reading Don’t Panic

Never Give Up

Have you read the Standards for Mathematical Practice? Good idea in theory, but horribly dull and stilted. Like math standards in general, the SMPs sound as if they were written by committee. (Duh!)

I’ve seen several attempts to rewrite the SMPs into student-friendly language. Many of those seem too over-simplified, almost babyish.

Probably I’m just too critical.

Anyway, I decided to try my hand at the project. Here’s the first installment…

Math Tip #1: Never Give Up.

  • Fight to make sense of a problem.
  • Think about the things you know.
  • Ponder what a solution might look like.
  • Compare this problem to those you solved in the past.
  • If it seems too hard, make up a simpler version. Can you solve that one?
  • If one approach doesn’t work, try something else.
  • When you get an answer, ask yourself, “Does it truly makes sense?”

Download the poster, if you like:

What do you think? Would this resonate with your students?

What changes do you suggest?

You can find the whole SMP series (eventually) under the tag: Posters.

Update: I Made a Thing

I had so much fun making these posters that I decided to put them into a printable activity guide. It includes the full-color poster shown above and a text-only version, with both also in black-and-white if you need to conserve printer ink.

Here’s the product description…

Join the Math Rebellion: Creative Problem-Solving Tips for Adventurous Students

Take your stand against boring, routine homework.

Fight for truth, justice, and the unexpected answer.

Join the Math Rebellion will show you how to turn any math worksheet into a celebration of intellectual freedom and creative problem-solving.

This 42-page printable activity guide features a series of Math Tips Posters (in color or ink-saving black-and-white) that transform the Standards for Mathematical Practice to resonate with upper-elementary and older students.

Available with 8 1/2 x 11 (letter size) or A4 pages.

Check It Out

Math for Star Wars Day

May the Fourth be with you!

Here is a math problem in honor of one of our family’s favorite movies…

Han Solo was doing much-needed maintenance on the Millennium Falcon. He spent 3/5 of his money upgrading the hyperspace motivator. He spent 3/4 of the remainder to install a new blaster cannon. If he spent 450 credits altogether, how much money did he have left?

Stop and think about how you would solve it before reading further.

Continue reading Math for Star Wars Day

How to Succeed in Math: Answer-Getting vs. Problem-Solving

You want your child to succeed in math because it opens so many doors in the future.

But kids have a short-term perspective. They don’t really care about the future. They care about getting through tonight’s homework and moving on to something more interesting.

So how can you help your child learn math?

When kids face a difficult math problem, their attitude can make all the difference. Not so much their “I hate homework!” attitude, but their mathematical worldview.

Does your child see math as answer-getting? Or as problem-solving?

Answer-getting asks “What is the answer?”, decides whether it is right, and then goes on to the next question.

Problem-solving asks “Why do you say that?” and listens for the explanation.

Problem-solving is not really interested in “right” or “wrong”—it cares more about “makes sense” or “needs justification.”

Homeschool Memories

In our quarter-century-plus of homeschooling, my children and I worked our way through a lot of math problems. But often, we didn’t bother to take the calculation all the way to the end.

Why didn’t I care whether my kids found the answer?

Because the thing that intrigued me about math was the web of interrelated ideas we discovered along the way:

  • How can we recognize this type of problem?
  • What other problems are related to it, and how can they help us understand this one? Or can this problem help us figure out those others?
  • What could we do if we had never seen a problem like this one before? How would we reason it out?
  • Why does the formula work? Where did it come from, and how is it related to basic principles?
  • What is the easiest or most efficient way to manipulative the numbers? Does this help us see more of the patterns and connections within our number system?
  • Is there another way to approach the problem? How many different ways can we think of? Which way do we like best, and why?

What Do You think?

How did you learn math? Did your school experience focus on answer-getting or problem-solving?

How can we help our children learn to think their way through math problems?

I’d love to hear from you! Please share your opinions in the Comments section below.

CREDITS: “Maths” photo (top) by Robert Couse-Baker. “Math Phobia” photo by Jimmie. Both via Flickr (CC BY 2.0). Phil Daro video by SERP Media (the Strategic Education Research Partnership) via Vimeo.

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

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-300CREDITS: An 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.

New Book: Avoid Hard Work

I’ve loved James Tanton’s How to Be a Math Genius videos for years. He offers great problem-solving tips like:

  • Visualize: think of a picture.
  • Use common sense to avoid grungy work.
  • Engage in intellectual play.
  • Think relationally: understanding trumps memorization.
  • Be clear on what you don’t know — and comfortable enough to admit it.

Seriously, those are wonderful videos. If you haven’t seen them before, go check them out. Be sure to come back, though, because I’ve just heard some great news.

Natural Problem-Solving Skills

Avoid Hard WorkTanton has joined up with the NaturalMath.com team of Maria Droujkova, Yelena McManaman, and Ever Salazar to put together a book for parents, teachers, math circle leaders, and anyone else who works with children ages 3–10.

It’s called Avoid Hard Work, and it takes a playful look at ten powerful problem-solving techniques.

Join the Crowdfunding Campaign

For more details about Avoid Hard Work, including a 7-page pdf sample with tips and puzzles to enjoy, check out the crowdfunding page at Natural Math:

Read the questions and answers. Try the activities with your children. And donate to support playful math education!

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.

Continue reading The Math Student’s Manifesto