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.

- 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.

- 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.

- 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.

- I recognize joining, separating, comparing, and sorting situations and can describe them with mathematical expressions or equations using addition or subtractions.
- I recognize proportional, grouping, or sharing situations and can describe them with mathematical expressions or equations using multiplication or division, or with fractions.
- In algebra or geometry, I know how to recognize typical function or shape relationships.
- I can make assumptions or approximations to simplify a complex situation.
- I always ask myself, “Does this make sense?” and try to make my mathematical models better.

- I can make a chart, graph, data table, or diagram.
- I can use a ruler, protractor, or compass.
- I know how to use a calculator when I need it. I never copy down all the digits on my calculator, but round numbers to the appropriate degree of precision.
- I like to experiment with online graphing tools.
- I know how to look up information online and how to recognize a trustworthy website.

- I know how important it is to define my words and symbols.
- I don’t misuse the equal sign.
- I’m careful about units of measurement.
- I label my graphs and diagrams.

- I know that patterns can make math easier to work with.
- I use common number patterns to simplify arithmetic calculations.
- I use common algebra patterns to simplify equations.
- I use common shape patterns to simplify geometric and trigonometric puzzles.

- If I see a new pattern, I don’t automatically trust it. I always ask, “Does it make sense?”
- I ask myself, “Will the pattern always work? Or does it only work in special cases?”
- I look for ways to explain the pattern in general terms.
- When I find a true general pattern, I use it to help me solve new problems.

[Based on the Standards for Mathematical Practice, translated into conversational English.]

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Only dead mathematics can be taught where competition prevails: living mathematics must always be a communal possession. —Mary Everest Boole

Wednesday Wisdom features a quote to inspire my fellow homeschoolers and math education peeps. Today’s quote is from Mary Everest Boole. Background photo courtesy of State Library of Queensland, Australia (no known copyright) via Flickr.

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.

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*[Photo by Olga Berrios (CC BY 2.0) via Flickr.]*

Do you have a favorite blog post about math activities, games, lessons, or hands-on fun? The *Math Teachers at Play* (MTaP) math education blog carnival would love to feature your article!

We welcome math topics from preschool through the first year of calculus. Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

- Click here to submit your blog post.
- Browse all the past editions of the
*Math Teachers at Play*blog carnival

Have you noticed a new math blogger on your block that you’d like to introduce to the rest of us? Feel free to submit another blogger’s post in addition to your own. Beginning bloggers are often shy about sharing, but like all of us, they love finding new readers.

**Don’t procrastinate:** *The deadline for entries is this Friday, February 20.* The carnival will be posted next week at CavMaths.

**Click to tweet about the carnival.**

(No spam, I promise! You will have a chance to edit or cancel the tweet.)

Hosting the blog carnival is fun because you get to “meet” new bloggers through their submissions. And there’s a side-benefit: The carnival often brings a nice little spike in traffic to your blog. If you think you’d like to join in the fun, read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

While you’re waiting for next week’s *Math Teachers at Play* carnival, you may enjoy:

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.

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The ball is traveling around a shape that can’t exist in our real world: the Penrose triangle. This illusion is the basis for some cool art, like Escher’s Waterfall. And I’m using it in my *Math You Can Play* books as a design on the back of my playing cards:

Here’s a few links so you can try it for yourself:

- How to Draw an Impossible Triangle
- Build a Penrose Triangle with Legos
- How the Lego Triangle Works (Impossible triangle in Perth, Australia)
- Or Make a Paper Model
- How I Made the Image for My Playing Cards

I’ve sent the first two *Math You Can Play* books to a copy editor (she edits the text part), so my focus this month is on finishing the illustrations and downloadable game boards. And designing the book covers — I think I’ll call this latest iteration done.

If everything stays on schedule, both *Counting & Number Bonds* and *Addition & Subtraction* should be available by mid- to late-spring. Fingers crossed…

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I noticed a couple of people who joined the mailing list but neglected to ask for either the math or fantasy fiction updates — and we won’t send you any updates unless you ask for them! If you thought you signed up, but you didn’t receive this morning’s email (and it’s not in your spam folder by mistake), then leave me a comment here or just go sign up again.

If you’re not on the mailing list, you can still join in the fun:

Math Snack: Fractal ValentinesWhat better way to say “I love you forever!” than with a pop-up fractal Valentine? My math club kids made these a couple years back, and they turned out great.

To make your card, choose two colors of construction paper or card stock. One color will make the pop-up hearts on the inside of your card. The other color will be the front and back of the card, and will also peek through the cut areas between the hearts. Fold the papers in half and cut them to card size.

Set the outer card aside and focus on the inside. The fractal cutting pattern is simple: press the fold, cut a curve, tuck inside, repeat…

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Wednesday Wisdom features a quote to inspire my fellow homeschoolers and math education peeps. Today’s quote is from @Mr_Harris_Math, via Twitter. Background photo courtesy of Forrest Cavale (CC0 1.0) via Unsplash.

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The January math education blog carnival is now posted for your browsing pleasure, featuring 23 playful ways to explore mathematics from preschool to high school:

**Highlights include:**

Young children making bar graphs.

A wide variety of math games.

Fractions on a clothesline.

Quadrilaterals on social media.

Non-transitive dice.

Writing in math class.

Negative number calculations made physical.

Inverse trig graphing.

Function operations.

And much more!

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Wednesday Wisdom features a quote to inspire my fellow homeschoolers and math education peeps. Today’s quote is from Raoul Bott, via The MacTutor History of Mathematics archive. Background photo courtesy of Swedish National Heritage Board (CC BY 2.0) via Flickr.

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The monthly *Math Teachers at Play* (MTaP) math education blog carnival is almost here. If you’ve written a blog post about math, we’d love to have you join us! Each of us can help others learn, so in a sense we are all teachers.

Posts must be relevant to students or teachers of school-level mathematics (that is, anything from preschool up to first-year calculus). Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

- Click here to submit your blog post.
- Browse all the past editions of the
*Math Teachers at Play*blog carnival

Have you noticed a new math blogger on your block that you’d like to introduce to the rest of us? Feel free to submit another blogger’s post in addition to your own. Beginning bloggers are often shy about sharing, but like all of us, they love finding new readers.

**Don’t procrastinate:** *The deadline for entries is this Friday, January 23.* The carnival will be posted next week at Mrs. E Teaches Math.

**Click to tweet about the carnival.**

(No spam, I promise! You will have a chance to edit or cancel the tweet.)

Help! I can’t keep the carnival going on my own. Would you volunteer to host the MTaP math education blog carnival some month this year? Hosting the carnival can be a lot of work, but it’s fun to “meet” new bloggers through their submissions. And there’s a side-benefit: The carnival usually brings a nice little spike in traffic to your blog.

If you think you’d like to join in the fun, read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

While you’re waiting for next week’s *Math Teachers at Play* carnival, you may enjoy:

- Last month’s MTaP 81 at Life Through A Mathematician’s Eyes
- Carnival of Mathematics
- Carnaval de Matemáticas

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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.

— Ruth Beechick

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

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An * algorithm* is a set of steps to follow that produce a certain result. Follow the rules carefully, and you will automatically get the correct answer. No thinking required — even a machine can do it.

This photo shows one section of the first true computer, Charles Babbage’s Analytical Engine. Using a clever arrangement of gears, levers, and switches, the machine could crank out the answer to almost any arithmetic problem. Rather, it would have been able to do so, if Babbage had ever finished building the monster.

One of the biggest arguments surrounding the Common Core State Standards in math is when and how to teach the standard algorithms. But this argument is not new. It goes back at least to the late 19th century.

Here is a passage from a book that helped shape my teaching style, way back when I began homeschooling in the 1980s…

Understanding this item is the key to choosing your strategy for the early years of arithmetic teaching. The question is: Should you teach abstract notation as early as the child can learn it, or should you use the time, instead, to teach in greater depth in the mental image mode?

Abstract notation includes writing out a column of numbers to add, and writing one number under another before subtracting it. The digits and signs used are symbols. The position of the numbers is an arbitrary decision of society. They are conventions that adult, abstract thinkers use as a kind of shorthand to speed up our thinking.

When we teach these to children, we must realize that we simply are introducing them to our abstract tools. We are not suddenly turning children into abstract thinkers. And the danger of starting too early and pushing this kind of work is that we will spend an inordinate amount of time with it. We will be teaching the importance of making straight columns, writing numbers in certain places, and other trivial matters. By calling them trivial, we don’t mean that they are unnecessary.

But they are small matters compared to real arithmetic thinking.

And when she does get to abstractions, she will understand them better. She will not need two or three years of work in primary grades to learn how to write out something like a subtraction problem with two-digit numbers. She can learn that in a few moments of time, if you just wait.If you stay with meaningful mental arithmetic longer, you will find that your child, if she is average, can do problems much more advanced than the level listed for her grade. You will find that she likes arithmetic more.— Ruth Beechick

An Easy Start in Arithmetic (Grades K-3)

(emphasis mine)

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I just sent out my first newsletter to everyone who signed up for math updates. If you’re not on the mailing list, you can still join in the fun:

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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.

**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.

- 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.

As usual, we will need every trick in the book to create variety in our numbers. Experiment with decimals, double-digit numbers, and factorials. Remember that dividing (or using a negative exponent) creates the reciprocal of a fraction, which can flip the denominator up where it might be more helpful.

Use the comments section below to share the numbers you find, but don’t spoil the game by telling us how you made them. You may give relatively cryptic hints, especially for the more difficult numbers, but be warned: Many teachers use this puzzle as a classroom assignment, and there will always be students looking for people to do their homework for them.

*Do not post your solutions. I will delete them.*

There is no authoritative answer key for the year game, so we will rely on our collective wisdom to decide when we’re done. We’ve had some lively discussions the last few years. I’m looking forward to this year’s fun!

As players report their game results below, I will keep a running tally of confirmed results (numbers reported found by two or more players). Today is Kitten’s birthday, however, so I won’t spend much time at my computer. Also, I may be traveling a lot this month, so this tally will probably lag a few days behind the results posted in the comments.

Percent confirmed: 96%

1-67, 69-81, 83-86, 88-93, and 95-100.

Reported but not confirmed: 3%

82, 87, and 94.

Numbers we are still missing: 1%

68.

Students in 1st-12th grade may wish to submit their answers to the Math Forum, which will begin publishing student solutions after February 1, 2015. Remember, Math Forum allows double factorials but will not accept answers with repeating decimals.

Finally, here are a few rules that players have found confusing in past years.

**These things ARE allowed:**

- You may use each of the digits 2, 0, 1, 5 only once in each expression.
- 0! = 1. [See Dr. Math’s Why does 0 factorial equal 1?]
- Unary negatives count. That is, you may use a “−” sign to create a negative number.
- You may use (
*n*!)!, a nested factorial, which is a factorial of a factorial. Nested square roots are also allowed.

**These things are NOT allowed:**

- You may not write a computer program to do the puzzle for you — or at least, if you do, please don’t ruin our fun by telling us all the answers!
- You may not use any exponent unless you create it from the digits 2, 0, 1, 5. You may not use a square function, but you may use “^2”. You may not use a cube function, but you may use “^(2+1)”. You may not use a reciprocal function, but you may use “^(−1)”.
- “0!” is not a digit, so it cannot be used to create a base-10 numeral. You cannot use it with a decimal point, for instance, or put it in the tens digit of a number.
- The decimal point is not an operation that can be applied to other mathematical expressions: “.(2+1)” does not make sense.
- The double factorial
*n*!! = the product of all integers from 1 to*n*that are equal to*n*mod*2*. If*n*is even, that would be all the even numbers, and if*n*is odd, then use all the odd numbers. We’ve allowed these the past couple of years, but I’ve decided I don’t really like them, so I’m putting them on the “naughty” list for this year. - You may not use the integer, floor, or ceiling functions. You have to “hit” each number from 1 to 100 exactly, without rounding off or truncating decimals.

- 2015 Mathematics Game Worksheet

For keeping track of which numbers you’ve solved.

- 2015 Mathematics Game Manipulatives

This may help visual or hands-on thinkers.

- 2015 Mathematics Game Student Submissions

For elementary through high school students who wish to share their solutions.

For more tips, check out this comment from the 2008 game.

Heiner Marxen has compiled hints and results for past years (and for the related Four 4’s puzzle). Dave Rusin describes a related card game, Krypto, which is much like my Target Number game. And Alexander Bogomolny offers a great collection of similar puzzles on his Make An Identity page.

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Number sense, measurement, place value, functions, calculus for kids, Christmas math activities, art, and much more — check out the December math education blog carnival:

**Math Teachers at Play (MTaP) Blog Carnival #81**

Welcome to the 81st edition of Math Teachers at Play (MTaP) Blog Carnival. I am extremely exited to host this post in my favorite month of the year, December…

Understanding 81:An interesting fact is that 81 is a tribonacci number (sounds a lot like Fibonacci) – the sequence of tribonacci numbers start with 3 predetermined terms (0,0,1) and each term afterwards is the sum of the preceding 3 terms. Thus the sequence starts like this: 0,0,1,1,2,4,7,13,24,44,81,… (you can go further if you want to see how fast the numbers go).Now the maths posts…

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