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.

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, May 20.* The carnival will be posted next week at **Math Misery?** blog.

Help! I can’t keep the carnival going on my own. Hosting the blog 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:

**
**

- Browse past editions of the
*Math Teachers at Play*blog carnival - Carnival of Mathematics
- Carnaval de Matemáticas

Claim your **two free learning guide booklets**, and be one of the first to hear about new books, revisions, and sales or other promotions.

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On Monday, Emily Grosvenor (author of the ** Tessalation! picture book**) asked me how parents who are insecure in math could help their children learn through play, and I responded with this quote from my

If you are intimidated by numbers, you can look for patterns of shape and color. Pay attention to how they grow. Talk about what your children notice.

But I wasn’t entirely satisfied with that answer. So many adults have come away from their own school experience thinking math is only numbers. Even with shapes, isn’t it the numbers about them — how many sides, what size of angles, calculate the the area or perimeter — that are important? That’s what school math tends to focus on.

Those of us who are comfortable with math know that there are many more things to notice and think about than just numbers. We know that it’s this noticing, thinking, and wondering that is at the heart of math. And that just *playing* with shapes can build a powerful foundation for future math learning.

And then yesterday, Malke Rosenfeld posted **a beautiful article** about **a paper manipulative** created by Paula Krieg. Which included this video:

The ability to create, and maintain, and manipulate shapes mentally — that’s the goal. Just like kids who can put numbers together in their heads, kids who can rotate, flip, and think of how shapes fit together in their heads have a powerful tool to analyze not only simple shape puzzles, but dividing up an area that’s a more complex room shape … to look at a piece of artwork … or look at a building … For these kids, all the world around becomes a playground to use mathematical ideas.

— Doug Clements

Problem Solving Development: Composing Shapes

Of course, pattern blocks are good for much more than just filling in worksheet pictures. But I love this peek into how a child’s understanding grows, in bits and spurts — without any numbers at all — until the world itself becomes a playground for mathematical ideas.

Want more?

You know what? Children like mathematics. Children see the world mathematically … When we do a puzzle, when we count things, when we see who’s got more, or who’s taller … Play and mathematics are not on opposite sides of the stage.

— Doug Clements

Why Early Childhood is the Right Time to Start Learning Math

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Welcome to the 97th edition of the ** Math Teachers At Play** math education blog carnival: a monthly 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.

A few articles were submitted by their authors, but most were drawn from the immense backlog in my rss reader. If you’d like to see your blog post featured next month, **be sure to send it in yourself**. Our hosts are busy parents and teachers who have limited time to scour the Internet for goodies.

To add a bit of color, I’ve thrown in several favorites from my newly updated **Math with Living Books** pages. Some (affiliate) links go to Amazon.com, where you can read descriptions and reviews — but there’s no need to buy. Most of these books should be available through your local library.

If you’d like to skip directly to your area of interest, click here:

**
**

- Talking Math with Kids
- Elementary Exploration and Middle School Mastery
- Adventures in Basic Algebra and Geometry
- Advanced Mathematical Endeavors
- Puzzling Recreations
- Teaching Tips

**Please:** If you enjoy the carnival, would you consider **volunteering to host** sometime this year? Classroom teachers, homeschoolers, unschoolers, or anyone who likes to play around with math (even if the only person you “teach” is yourself) — if you would like to take a turn, please speak up!

And now, let the mathematical fun begin!

When the queen of her bugs demands that her army march in even lines, Private Joe divides the marchers into more and more lines so that he will not be left out of the parade.

- Tracy Zager (@TracyZager) and her daughter are
**Talking Math in Ghirardelli Square**: “Wait! It’s the same thing again! It’s going to go on forever!”

- David Wees (@DavidWees) and his son explore
**Inductive Reasoning**: “You know, Dad. There are an infinite number of solutions…”

- Joshua Greene (@JoshuaGreene19) writes
**Ode to a bead string (a non-poem poem)**: “One of the cool things about open play with math manipulatives is that it provides a lot of easy entry points into short math chats.”

- Dan Finkel (@MathforLove) explains
**How to help your kids fall in love with math: a guide for grown-ups**: “Think out loud. Your child is imitating everything you do, and the more you can narrate your thoughts, the better a model of thinking you can be.”

- Crystal Wagner (@Tri_Learning) shares several
**Math Games to Play in the Car**: “Or maybe you are waiting in line at the grocery store or doctor’s appointment. Turn these times of waiting into learning opportunities.”

- Caroline Mukisa (@MathsInsider) lists
**11 Award Winning Math Books to Share With Your Child**: “Whether you want to introduce a young child to their very first math concepts or supplement an older child’s math curriculum, this list is for you.”

- Christopher Danielson (@Trianglemancsd) shows how
**The sequence machine**can launch math conversations with older students: “Now you can generate number sequences, without being distracted by the multiplication facts.”

Help inspire your kids to try writing their own unique problems. Includes a wide range of math topics and concepts: money and time, fractions, percentages, geometry, logic, and multi-step problem solving.

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- Jamie Duncan (@JamieDunc3) presides over
**First Grade Math Fight… Fractions, Proportional Reasoning, and Algebra, Oh my!**: “The uproar was on the border of out of control, but it’s in the name of math, so…”

- Joe Schwartz (@JSchwartz10a) goes
**Dot Crazy**investigating number patterns with third-graders: “It was like we had turned on a fire hose. Math started*gushing out all over the place.”*

- Mike Jacobs (@MsbJacobs) uncovers an unexpected misconception in
**Measuring Up**: “They see something like length as a*count*and not a*measure*.”

- John Stevens (@Jstevens009) wants kids to justify their decisions in
**Would You Rather…?**: “Ask them what they would rather have that many of? Dollars? Shoes? Pimples?”

- For my own contribution to the carnival, I’ve refined my explanation of multiplication in
**Multiplication Is Not Repeated Addition: Update**: “A strange, new concept sits at the heart of multiplication, something students have never seen before.”

- Erick Lee (@HRSBMathematics) crosses a piece of rope with a tangram and gets
**Fraction Talks Clothesline**: “This was a really fun activity and there was lots of engagement.”

- Margie Pearse (@Pearse_Margie) wonders
**How Do I Teach Numeracy in My Class… Tomorrow?**: “Now write another fraction that is closer to 1 than the first one you picked. Explain how you know this fraction is even closer to one.”

This collection of puzzles, games and activities is designed to stimulate and challenge people of all ages who enjoy puzzles with a mathematical flavour. Many of the puzzles have a long history, while others are original.

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- Paula Beardell Krieg (@PaulaKrieg) creates a beautiful
**Puffy Pentagon Box**template: “It was simple to do, delightful, and just the sort of hands-on relief I needed. But then I got to thinking…”

- “Area v. Perimeter” is just one of the goodies in the Des-Blog’s (@Desmos)
**Friday Five for April 22**: “What happens when you turn Dan Meyer loose…?”

- Patrick Vennebush (@Pvennebush) and sons apply mental algebra in
**All Systems Go**: “Why can’t these poor boys look at a pizza menu without perceiving it as a system of equations?”

- Lisa Bejarano (@LisaBej_Manitou) is
**Analyzing Triangle Congruence with AngLegs**: “Wait! I made two different triangles with all three angles the same and one side the same. Does this count? Look!”

- Elizabeth Statmore (@Cheesemonkeysf) is dumbfounded by
**Volume of a Pyramid: Proof by Play-Doh**: “This is the best idea I never had.”

- Michael Pershan (@Mpershan) harnesses the power of a non-specific question in
**Cognitive Load Theory and Why Students Are Answer-Obsessed**: “Attention is a zero-sum game. There’s only so much that a person can notice. A person focused on finding the solution is unable to focus on much else.”

- Jo Morgan (@MathsJem) links to a great Sum of Exterior Angles demonstration and other treats in
**Gem Awards 2016**: “I’ve never thought of doing this lovely paper demonstration.”

- Keith Devlin (@ProfKeithDevlin) dips into math history in
**Algebraic roots – Part 1**: “Instead of algebra being a codification of human logical thinking that emerges from within, it becomes a set of externally imposed, and often arbitrary-seeming rules to be mastered by repetitive practice. The natural, relevant, and empowering becomes the artificial, pointless, and tedious.”

Renowned mathematician Ian Stewart gives math buffs and non-technical readers the perfect guide to today’s mathematics. He shows us not only that math can be explained in everyday language, but that it can be downright fun as well.

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- Matt Vaudrey (@MrVaudrey) performs an
**Algebra 2 Math Makeover: Spencer’s Soccer Ball**: “It’s often the students who don’t know how to start that get stuck, and something like that will offer a low door for entry.”

- Bryan Anderson (@Anderson02B) poses a
**Matrix Multiplication — Open Middle Problem**: “Using the digits 1-9, each only once, fill in the blanks to create the smallest possible value…”

- And check out Gironda’s calculus lesson
**Volumes Performance Task: A Love Story**: “I left it totally open ended and they all took a different approach — but every group nailed it!”

- Dan MacKinnon (@MathRecreation) investigates
**dividing polynomials: the backwards reverse tabular method**: “If you are intrigued by the possibility of polynomial explosions, read on; for others: you’ve been warned.”

- Caitlyn Gironda (@Caitlyn_Gironda) challenges her students with
**Logarithm Clothesline**: “They actually had to think about a logarithm having a value — something which is usually a struggle for them.”

- Dave Richeson (@DivByZero) shares
**A Geometry Theorem Looking for a Geometric Proof**: “The heptagon is noteworthy because it is the regular polygon with the fewest number of sides that cannot be constructed with compass and straightedge alone.”

- And don’t miss the
**133rd Carnival of Mathematics**.

Puzzle-lovers of all ages will gobble up this smorgasbord of riddles, mysteries, and logic problems. When Smullyan tells stories, fun and wonderment ensue.

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- Recreational math is not just for the big kids. Sonya Post (@NoMathFears) helps four- and five-year-old children take on the
**Hundred Face Challenge**: “This exercise has caught on and now it’s a ‘thing’. You know an internet ‘thing’. Not like the ice-bucket challenge or bullet-proof coffee, but we’ll get there.”

- Rodi Steinig’s math circle students play with
**Parity #1: NIM, Sums, and Conjectures**(see also meetings**#2**and**#3**): “The students made predictions, some accurate and some not-so-accurate, then used a jumprope as the river and tested their conjectures.”

- Kate Owens (@KateMath) shares several activity links in her
**Fun with Paper Folding**workshop: “The scalene triangle puzzle stuck with me for several hours one day and I was almost unable to function in any capacity until I figured it out.”

- Ben Orlin (@BenOrlin) poses
**The Accidental Fraction Brainbuster**: “Sometimes you accidentally write a problem six times harder than you meant to.”

- A good recreational math puzzle is one that spawns new ideas every time you look at it — like the painted blocks. Sam Shah (@samjshah) looks at
**A New Insight on the Famous Painted Block Problem**: “When I finally figured it out, my mind was blown. So simple and elegant, yet so unintuitive for me.”

- David Butler (@DavidKButlerUoA) explores 100 ways to
**Quarter the Cross**: “Along the way I’ve learned a few brand new things; I’ve relearned a few things I’d forgotten; I’ve seen things I knew with new eyes and understood them better; and I’ve made brand new maths (for me anyway).”

- Tanya Khovanova mourns the ruin of a great puzzle in
**The Battle I am Losing**: “In the simplified adapted video, there are no longer any discoveries. There is no joy. People consume the solution, without realizing why this puzzle is beautiful and counterintuitive.”

- Mike Lawler (@MikeAndAllie) throws down
**A challenge / plea to math folks**: “This problem provides a great opportunity for people to see how people in math think, and, importantly, that the path to the solution of a problem isn’t always a straight line.”

The daughter of mathematician Theon, Hypatia grew up on the northern tip of Egypt in the great city of Alexandria in the 4th century A.D. Records of her fame as a teacher can be found in the writings of Socrates.

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- Jo Boaler (@JoBoaler) announced the opening of YouCubed’s free
**Visual Math Network**, which includes research links, activities, and a discussion forum: “We know that the very best learning comes from teachers discussing and sharing ideas with each other.”

- Malke Rosenfeld (@mathinyourfeet) shares
**Some thoughts on “hands on” math learning**: “Students need active and interactive experiences with math ideas in multiple learning modes to make sense of math.”

- Will Richardson (@WillRich45) examines
**9 Elephants in the (Class)Room That Should “Unsettle” Us**: “We ourselves have forgotten the vast majority of what we supposedly learned in school. Yet we continue to focus our efforts primarily on content knowledge.”

- Katrina Schwartz (@Kschwart) explores
**How ‘Productive Failure’ In Math Class Helps Make Lessons Stick**: “But turning the difficult experience of failure into a positive isn’t as easy as telling students to change their mindsets.”

- Dan Meyer (@ddMeyer) shares
**Beyond Relevance & Real World: Stronger Strategies for Student Engagement**: “A very natural follow-up to the famous ‘Math class needs a makeover’ talk.”

- Marcus du Sautoy (@MarcusduSautoy) tries to change the public perception of math in
**Reckon you were born without a brain for maths? Highly unlikely**: “You might have trouble with your multiplication tables but actually be a great mathematician.”

- Jamie Duncan (@JamieDunc3) demonstrates that
**Teachers [Are Greater Than] Curriculum**: “This is where most, if not nearly all, of the publishers’ curriculum gets it wrong.”

- And finally, you may be interested in my new Frequently Asked Questions blog post series. Check out
**Let’s Play Math FAQs: Introduction**and my latest**FAQ: Lifelong Learning for Parents**: “When you struggle with a math concept and conquer it, it will make you free. You don’t have to be afraid of it anymore.”

More than thirty authors share their math enthusiasm with stories about their communities, families, or students. After every chapter is a puzzle, game, or activity to get you and your kids playing with math, too.

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- “97” photo by
**Steve Bowbrick**via Flickr. (CC BY 2.0) - Book links are from my
**Math with Living Books**pages.

And that rounds up this edition of the ** Math Teachers at Play** carnival. I hope you enjoyed the ride.

The May 2016 installment of our carnival will open sometime during the week of May 23-27 at **Math Misery? blog**. If you would like to contribute, please use this **handy submission form**. Posts must be relevant to students or teachers of preK-12 mathematics. Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Past posts and future hosts can be found on our **blog carnival information page**. Or browse all the **past editions of the Math Teachers at Play blog carnival** on Pinterest.

Claim your **two free learning guide booklets**, and be one of the first to hear about new books, revisions, and sales or other promotions.

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I’m so tired of being ignorant about math. I can memorize rules and do calculations, but if I miss a step the numbers make no sense at all, and I can’t spot what went wrong. Another struggle I have is keeping everything organized in my mind. When I learn a new concept or strategy, I easily forget it. My son is only a toddler now, but as he grows up, I don’t want to burden him with my own failures. Where should I start?

As a first step, convince yourself that math is interesting enough to learn on its own merits, because parental guilt will only carry you so far. Start with Steven Strogatz’s “Elements of Math” series from *The New York Times,* or pick up his book *The Joy of x.*

As a next step, reassure yourself that elementary math is hard to understand, so it’s not strange that you get confused or don’t know how to explain a topic. Get Liping Ma’s *Knowing and Teaching Elementary Mathematics* from the library or order a used copy of the first edition. Ma examines what it means to understand math and to clearly explain it to others.

Don’t rush through the book as if it were a novel. There are four open-ended questions, each at the beginning of a chapter, after which several possible answers are analyzed. When you read one of these questions, close the book. Think about how you would answer it yourself. Write out a few notes, explaining your thoughts as clearly as you can. Only then, after you have decided what you would have said, read the rest of that section.

Don’t worry if you can’t understand everything in the book. Come back to it again in a couple of years. You’ll be surprised how much more you learn.

To build up your own understanding of elementary arithmetic, the *Kitchen Table Math* series by Chris Wright offers explanations and activities you can try with your children.

If you want more detailed guidance in understanding and explaining each stage of elementary mathematics, you can pick up a textbook designed for teachers in training. I like the Parker & Baldridge *Elementary Mathematics for Teachers* books and the* Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction* series. The two series are completely different, but they complement each other well. Check out the sample chapters from the publishers’ websites to see which one you prefer.

**Elementary Mathematics**and**Elementary Geometry**for Teachers- Teaching Student-Centered Mathematics:
**preK-2nd**and**3rd-5th**and**6th-8th**grades

Discover more great books on my **Living Math Books for Parents and Other Teachers** page.

As you learn, focus on how the math concepts relate to each other. Then the more you learn, the easier you will find it to connect things in your mind and to grasp new ideas.

You might want to keep a math journal about the things you are learning. When you write something down, that helps you remember it, even if you never look back at the journal. But if your mind goes blank and you think, “I know I studied that,” the journal gives you a quick way to review. Make it even easier to flip back through by writing the topic you are studying in the top margin of each page.

When you run into a new vocabulary word, draw a Frayer Model Chart and fill in all the sections. The Frayer Model provides a way to organize information about a new vocabulary word or math concept.

And if you read something that’s particularly helpful, you may want to turn to the back page of your journal and start a quick-reference section.

Find a fellow-learner to encourage you on your journey. Bouncing ideas off a friend is a great way to learn. You might want to join the parents and teachers who are learning math together at the Living Math Forum.

And here is the most important piece of advice I can offer. Your slogan must be the one used by the Chinese teachers Liping Ma interviewed: “Know how, and also know why.”

Always ask why the rules you learn in math work. Don’t stop asking until you find someone who can explain it in a way that makes sense to you. When you struggle with a concept and conquer it, it will make you free. You don’t have to be afraid of it anymore.

Know how, and also know why.

This post is an excerpt from my book * Let’s Play Math: How Families Can Learn Math Together—and Enjoy It,* as are many of the articles in my

**two free learning guide booklets**, and be one of the first to hear about new books, revisions, and sales or other promotions.

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It’s carnival time again. Activities, games, lessons, hands-on fun — if you’ve written a blog post about math, we’d love to have you join our ** Math Teachers at Play (MTaP) math education blog carnival**.

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

**Don’t procrastinate:** *The deadline for entries is this Friday, April 22*. The carnival will be posted next week at **Let’s Play Math**.

Hosting the blog 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:

**
**

- Browse past editions of the
*Math Teachers at Play*blog carnival - Carnival of Mathematics
- Carnaval de Matemáticas

**two free learning guide booklets**, and be one of the first to hear about new books, revisions, and sales or other promotions.

]]>

Some Internet topics are evergreen. I noticed that my old **Multiplication Is Not Repeated Addition** post has been getting new traffic lately, so I read through the article again. And realized that, even after all those words, I still had more to say.

So I added the following update to clarify what seemed to me the most important point.

I’d love to hear your thoughts! The comment section is open down below . . .

**Addition:** addend + addend = sum. The addends are interchangeable. This is represented by the fact that they have the same name.

**Multiplication:** multiplier × multiplicand = product. The multiplier and multiplicand have different names, even though many of us have trouble remembering which is which.

**multiplier**= “how many or how much”**multiplicand**= the size of the “unit” or “group”

*Different names indicate a difference in function.* The multiplier and the multiplicand are not conceptually interchangeable. It is true that multiplication is **commutative**, but (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.

In multiplication, we introduce a totally new type of number: the *multiplicand.* A strange, new concept sits at the heart of multiplication, something students have never seen before.

The multiplicand is a this-per-that ratio.

A ratio is a not a counting number, but something new, much more abstract than anything the students have seen up to this point.

A ratio is a relationship number.

In addition and subtraction, numbers count how much stuff you have. If you get more stuff, the numbers get bigger. If you lose some of the stuff, the numbers get smaller. Numbers measure the amount of cookies, horses, dollars, gasoline, or whatever.

The multiplicand doesn’t count the number of dollars or measure the volume of gasoline. It tells the relationship between them, the dollars per gallon, which *stays the same whether you buy a lot or a little.*

By telling our students that “multiplication is repeated addition,” we dismiss the importance of the multiplicand. But until our students wrestle with and come to understand the concept of ratio, they can never understand multiplication.

If you’re interested in digging deeper into how children learn addition and multiplication, I highly recommend Terezina Nunes and Peter Bryant’s book **Children Doing Mathematics**.

To learn about modeling multiplication problems with bar diagrams, check out the Mad Scientist’s Ray Gun model of multiplication:

And here is an example of the multiplication bar diagram in action:

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I noticed the “STEM Education” category at Amazon, so I updated my book’s keywords. And the *Let’s Play Math* paperback zipped into the Top 25!

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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.

—Dr. Ruth Beechick

*You Can Teach Your Child Successfully*

Learning math is an adventure into the unknown. The ideas we adults take for granted are a wild, unexplored country to our children. Like any traveler in a strange land, they will stumble over rocky places and meet with unexpected detours.

Whenever I visit a parenting forum, I feel compassion for the families who are struggling with math. No other school subject elicits such depths of frustration and despair:

- I’ve explained until I’m hoarse, and she still doesn’t get it. Help!! I want to pull my hair out.

- My child is not a mathy person at all. Now he’s convinced that he’s “dumb.”

- She says she can’t do it. She says she hates math. She says she can’t think. She hits her head and pounds her fists in frustration. I am so tired of fighting over math Every. Single. Day.

- The problem is not him … It’s me. I am a failure at math.

- I am sooooo struggling to teach my daughter math. Please, does anybody else deal with this? I will try anything!

Solving the problems of math education is not easy. Situations have built up over years, so they will take time to resolve.

But children are resilient, so improvement may not take as long as you fear.

No matter how much your family has struggled, there is hope. If children can get over the “I’m no good at math” mental block, they can learn all of elementary arithmetic in one school year of determined study.

Does that seem unbelievable? Consider Daniel Greenberg’s experience:

If math feels like a strange and dangerous wilderness to your children, you may need an experienced guide to lead you through the rough spots. For arithmetic, try Herb Gross’s Math as a Second Language webpages:

For upper-level math topics, explore Murray Bourne’s Interactive Mathematics pages or take a look at Kalid Azad’s Better Explained site:

- Ten Ways to Survive the Math Blues and Interactive Mathematics Index
- Developing Your Intuition For Math and Math Better Explained Index

The questions in this blog post series will be based on actual forum discussions, though I always change the details, removing anything that might identify the families involved.

We’ll look at a variety of struggles with math, such as:

- Lifelong Learning for Parents
- Primary Level Problems
- Middle Grade Mishaps
- The Agonies of Algebra
- Gaps and Standardized Testing

The questions will cover a wide range of common frustrations that resonate with anyone who has tried to explain an abstract idea to a confused child. Some questions apply specifically to homeschool math, yet non-homeschooling families can use many of the resources I recommend to supplement their children’s schoolwork or to keep skills sharp over the summer.

In my FAQ post answers, I will assume you are working with children of normal intelligence, facing the mental strengths and weaknesses that are common to us all. The human brain is not designed for working with abstraction, so most people find math difficult.

But some face additional hardship because their minds are unable to process numbers and related concepts. If you suspect one or more of your children may struggle with a learning disability, please have them tested and get advice from someone who can help you learn to deal with their special circumstances.

Auditory or vision problems, undiagnosed food allergies, and a family crisis or other emotional strain may also affect a child’s concentration. Sometimes, the best way to help your children learn math is to let it go and deal with other issues first.

*To be continued . . .*

This post is an excerpt from my book * Let’s Play Math: How Families Can Learn Math Together—and Enjoy It,* as are many of the articles in my

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The Farey Sequence was described in 1816 by English geologist John Farey, who was disparaged by the famous mathematical snob* G.H. Hardy as “at the best an indifferent mathematician.”

“I rather like the idea that the Farey Sequences are named after someone who noticed a pattern and asked a question — and not even the first person to notice the pattern, ask the question, or provide the answer. As math teachers, we teach plenty of indifferent mathematicians who wake up when they experience the joy of discovering something that is new to them, not necessarily new to the whole world.”

— Debra K. Borkovitz,

Farey Fraction Visual Patterns

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* See A Mathematician’s Apology Revisited by W.W. Sawyer.

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“Pieces of Math” poster from Loopspace (CC-BY-NC-ND).

If you like this, you may also enjoy the **Math That Is Beautiful** video.

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I am delighted to be hosting March’s MTaP Carnival! It’s late because I had the flu for 10 days – sorry for the delay, I know that we all look forward to reading the contributors’ posts each month. Without further delay, then, here are the great reads you won’t want to miss….

**Click here to go read the carnival post.**

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The most effective and powerful way I’ve found to commit math facts to memory is to try to understand why they’re true in as many ways as possible. It’s a very slow process, but the fact becomes permanently lodged, and I usually learn a lot of surrounding information as well that helps me use it more effectively.

…

Actually, a close friend of mine describes this same experience: he couldn’t learn his times tables in elementary school and used to think he was dumb. Meanwhile, he was forced to rely on actually thinking about number relationships and properties of operations in order to do his schoolwork. (E.g. I can’t remember 9×5, but I know 8×5 is half of 8×10, which is 80, so 8×5 must be 40, and 9×5 is one more 5, so 45. This is how he got through school.) Later, he figured out that all this hard work had actually given him a leg up because he understood numbers better than other folks. He majored in math in college and is now a cancer researcher who deals with a lot of statistics.— Ben Blum-Smith

Comment on Math Mama’s post What must be memorized?

The entire discussion (article and comments) is well worth reading:

You may also enjoy:

- Math Facts Are like Learning to Type
- Math Facts: 5 Minutes a Day
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[Photo by City of Boston Archives via Flickr (CC BY 2.0).]

I’ve started collecting quotes about teaching math for the chapter pages in my next *Math You Can Play* book. Here are a couple snippets that don’t fit the theme of “Multiplication & Fractions,” but they struck my fancy anyway:

If teachers would only encourage guessing. I remember so many of my math teachers telling me that if you guess, it shows that you don’t know. But in fact there is no way to really proceed in mathematics without guessing.

You have to guess!You have to have intuitive judgment as to the way it might go. But then you must be willing to check your guess. You have to know that simply thinking it may be right doesn’t make it right.

[Photo by Nathan Russell via Flickr (CC BY 2.0).]

One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right answers to all of the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That’s so unlike the true nature of mathematics.

—

Leon Henkin

from “Round and Round at the Round Table”

Teaching Teachers, Teaching Students: Reflections on Mathematical Education

Do you have some favorite quotes on math and teaching? I’d love to hear them! Please share in the Comments section below.

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**Books for Parents and Other Teachers**

A bookshelf full of math activities for your family, classroom, or math club.

**Picture Books and Early Readers**

From counting books to math history, picture books offer a gentle introduction to a variety of topics. Elementary and middle school students will also enjoy many of these.

**Elementary and Middle School**

Patterns, puzzles, games, and activities — here are plenty of ideas to get your children playing around with math.

**Problem Solving and Math Circles**

From the elementary puzzles to Olympiad-level stumpers, the problems in these books will intrigue and challenge your students.

**High School and Beyond**

These histories, biographies, and explanations of mathematical concepts are written for an adult general audience, so most of them assume no mathematical knowledge beyond a vague memory of high school.

If you’re a subscriber but didn’t see your newsletter, check your Updates or Promotions tab (in Gmail) or your Spam folder. And to make sure you get all the future newsletters, add “Denise at Tabletop Academy Press” [Tabletop Academy Press @ gmail.com, without spaces] to your contacts or address book.

And if you missed this month’s edition, no worries—there will be more playful math snacks next month. Click the link below to sign up today, and we’ll send you our free math and writing booklets, too!

As a Bonus: Newsletter subscribers are always the first to hear about new books, revisions, and sales or other promotions.

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If you are a homeschooler or classroom teacher, student or independent learner, or anyone else who writes about math, now is the time to send in your favorite blog post for next week’s ** Math Teachers at Play (MTaP) math education blog carnival**.

**Don’t procrastinate:** *The deadline for entries is this Friday, March 18*. The carnival will be posted next week at **The Usual Mayhem**.

If you haven’t written anything about math lately, here are some ideas to get your creative juices flowing…

**Elementary Concepts:**As**Liping Ma**showed, there is more to understanding and teaching elementary mathematics than we often realize. Do you have a game, activity, or anecdote about teaching math to young students? Please share!**Arithmetic/Pre-Algebra:**This section is for arithmetic lessons and number theory puzzles at the middle-school-and-beyond level. We would love to hear your favorite math club games, numerical investigations, or contest-preparation tips.**Beginning Algebra and Geometry:**Can you explain why we never divide by zero, how to bisect an angle, or what is wrong with distributing the square in the expression ? Struggling students need your help! Share your wisdom about basic algebra and geometry topics here.**Advanced Math:**Like most adults, I have forgotten enough math to fill several textbooks. I’m eager to learn again, but math books can be so-o-o tedious. Can you make upper-level math topics come alive, so they will stick in my (or a student’s) mind?**Mathematical Recreations:**What kind of math do you do, just for the fun of it?**About Teaching Math:**Other teachers’ blogs are an important factor in my continuing education. The more I read about the theory and practice of teaching math, the more I realize how much I have yet to learn. So please, fellow teachers, don’t be shy — share your insights!

Hosting the blog 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:

**
**

- Browse past editions of the
*Math Teachers at Play*blog carnival - Carnival of Mathematics
- Carnaval de Matemáticas

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