## What Are Mixed Numbers?

I just discovered a fascinating fact: In some places in the world, mixed numbers apparently don’t exist.

So that made me curious about my blog readers:

• Did you learn about mixed numbers in school?
• Do you ever use mixed numbers in daily life?
• Are your children learning to work with them?

And if you DO know mixed numbers, can you simplify this mess:

[If you enjoy dry math humor, the answer is worth the work.]

## Make Sense of Math

So, I decided to rewrite the Standards for Mathematical Practice into student-friendly language.

Here’s the final installment…

### Math Tip #8: Make Sense of Math.

• Use the patterns you discover to help you solve problems.
• Don’t get lost in the details of a problem. Look for general truths.
• Apply common sense to math situations.
• Think about how different things are similar.
• Think about how similar things are different.
• Remember that your mind is your most important math tool.
• Pay attention to your thinking process. What patterns do you find there?

## The Professor of Legend

The traditional mathematics professor of the popular legend is absentminded.

He usually appears in public with a lost umbrella in each hand.

He prefers to face the blackboard and to turn his back to the class.

He writes a, he says b, he means c; but it should be d.

Some of his sayings are handed down from generation to generation.

• “In order to solve this differential equation you look at it till a solution occurs to you.”
• “This principle is so perfectly general that no particular application of it is possible.”
• “Geometry is the science of correct reasoning on incorrect figures.”
• “My method to overcome a difficulty is to go round it.”
• “What is the difference between method and device? A method is a device which you used twice.”

If you’re not familiar with Polya’s work, here’s a 4-page summary of his problem-solving method.

Or check out David Butler’s wonderful Solving Problems Poster, which encapsulates Pólya’s system in a visual, easy-to-follow way that works with younger students, too.

CREDITS: “Professor” cartoon (top) by André Santana via Pixabay.
THE FINE PRINT: I am an Amazon affiliate. If you follow the book link above and buy something, I’ll earn a small commission (at no cost to you).

## Discern Patterns

I’m almost done rewriting the Standards for Mathematical Practice into student-friendly language.

They say mathematics is the science of patterns. So here’s…

### Math Tip #7: Discern Patterns.

• Look for patterns in numbers, shapes, and algebra equations.
• Notice how numbers can break apart to make a calculation easier.
• Number patterns morph into algebra rules.
• Adapt math situations to make the structure clear. (For example, by adding new lines to a geometry diagram.)
• Step back from a situation to see it from a new perspective.
• Try to find simpler patterns within complex equations or diagrams.
• Not all patterns continue forever. Test your patterns. Can you trust them?

## Say What You Mean

Continuing my project of rewriting the Standards for Mathematical Practice into student-friendly language.

Here’s my version of SMP6…

### Math Tip #6: Say What You Mean.

• Words can be tricky, so watch your language.
• Label drawings and graphs to make them clear.
• If you use a variable, tell what it means.
• Care about definitions and units.
• Pay attention to rules (like the order of operations).
• Use symbols properly (like the equal sign).
• Understand precision. Never copy down all the digits on a calculator.

## Math as a Verb

Here’s the full quote:

I like to play games. Almost any type of game.

I also like to play math.

If you’ve known enough mathematicians, you may have noticed that this isn’t unusual. I’m not sure if a love of games and puzzles among mathematicians exceeds a love of music among mathematicians, but both are strong and intersect.

Math in play is also a way of teaching mathematics. I think that as a metaphor, it best describes how I want to teach math.

I am constantly seeking ways to get my students thinking about math as a verb. It is about doing, not just about having right answers or the end product.

Games help set the culture I want to develop: Teaching students that multiple approaches and strategies are valued; trying is safe; and conversations about why, how, and discovery are the goals.

—John Golden
Yes, Playing Around

CREDITS: “Football outside Jakarta” photo by Robert Collins on Unsplash.

## Master Your Tools

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

Here’s my version of SMP5…

### Math Tip #5: Master Your Tools.

• Collect problem-solving tools.
• Practice until you can use them with confidence.
• Classic math tools: pencil and paper, ruler, protractor, compass.
• Modern tools: calculator, spreadsheet, computer software, online resources.
• Physical items: dice, counters, special math manipulatives.
• Tools for organizing data: graphs, charts, lists, diagrams.
• Your most important weapon is your own mind. Be eager to explore ideas that deepen your understanding of math concepts.

## Look Beneath the Surface

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

Here’s the fourth installment…

### Math Tip #4: Look Beneath the Surface.

• Notice the math behind everyday life.
• Examine a complex situation. Ignore the parts that aren’t relevant.
• Pay attention to the big picture, but don’t lose track of the details.
• Make assumptions that simplify the problem.
• Express the essential truth using numbers, shapes, or equations.
• Test how well your model reflects the real world.
• Draw conclusions. Explain how your solution relates to the original situation.

## 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!

## 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.”