How & Why (Understanding Math)

Math Debate: Adding Fractions

Posted on May 18, 2018December 30, 2024 by Denise Gaskins

Cover image by Thor/ geishaboy500 via Flickr (CC BY 2.0)

I’ve been working on my next Playful Math Singles book, based on the popular Things to Do with a Hundred Chart post.

My hundred chart list began many years ago as seven ideas for playing with numbers. Over the years, it grew to its current 30+ activities.

Now, in preparing the new book, my list has become a monster. I’ve collected almost 70 ways to play with numbers, shapes, and logic from preschool to middle school. Just yesterday I added activities for fraction and decimal multiplication, and also tips for naming complex fractions. Wow!

Gonna have to edit that cover file…

In the “Advanced Patterns” chapter, I have a section on math debates. The point of a math debate isn’t that one answer is “right” while the other is “wrong.” You can choose either side of the question — the important thing is how well you support your argument.

Here’s activity #69 in the current book draft.

Have a Math Debate: Adding Fractions

When you add fractions, you face a problem that most people never consider. Namely, you have to decide exactly what you are talking about.

For instance, what is one-tenth plus one-tenth?

Well, you might say that:

$\frac{1}{10}$ of one hundred chart
+ $\frac{1}{10}$ of the same chart
= $\frac{2}{10}$ of that hundred chart

But, you might also say that:

$\frac{1}{10}$ of one chart
+ $\frac{1}{10}$ of another chart
= $\frac{2}{20}$ of the pair of charts

That is, you started off counting on two independent charts. But when you put them together, you ended up with a double chart. Two hundred squares in all. Which made each row in the final set worth $\frac{1}{20}$ of the whole pair of charts.

So what happens if you see this question on a math test:

$\frac{1}{10}$ + $\frac{1}{10}$ = ?

If you write the answer “ $\frac{2}{20}$ ”, you know the teacher will mark it wrong.

Is that fair? Why, or why not?

CREDITS: Feature photo (above) by Thor/geishaboy500 via Flickr (CC BY 2.0). “One is one … or is it?” video by Christopher Danielson via TED-Ed. This math debate was suggested by Marilyn Burns’s blog post Can 1/3 + 1/3 = 2/6? It seemed so!

Playful Math Education Carnival 115—Women of Mathematics

Posted on March 26, 2018September 9, 2023 by Denise Gaskins

Welcome to the 115th edition of the Playful 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.

In honor of Women’s History Month, this carnival features quotes from fifteen women mathematicians.

If you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

Let the mathematical fun begin!

The Women of Mathematics

They came from many countries and followed a variety of interests.

They conquered new topics in mathematics and expanded the world’s understanding of old ones.

They wrestled with theorems, raised children, published articles, won awards, faced discrimination, led professional organizations, and kept going through both success and failure.

Some gained international renown, but most enjoyed quiet lives.

They studied, learned, and lived (and some still live) as most of us do — loving their families and friends, joking with colleagues, hoping to influence students.

I think you’ll find their words inspiring.

“What I really am is a mathematician. Rather than being remembered as the first woman this or that, I would prefer to be remembered, as a mathematician should, simply for the theorems I have proved and the problems I have solved.”
—Julia Robinson (1919–1985)

“All in all, I have found great delight and pleasure in the pursuit of mathematics. Along the way I have made great friends and worked with a number of creative and interesting people. I have been saved from boredom, dourness, and self-absorption. One cannot ask for more.”
—Karen Uhlenbeck (b. 1942)

Continue reading Playful Math Education Carnival 115—Women of Mathematics

Cultivate Mathematical Curiosity

Posted on January 17, 2018September 9, 2023 by Denise Gaskins

“Cultivating thinking skills is the main reason for teaching math. It is the mind’s perfect playground for shaping up.
…
To begin developing thinking, you must first have a child who is curious. For without curiosity, there is only forced thinking.
…
The problem with traditional math is it jumps to the punchline.
…
Absolutely no mystery or suspense is developed in traditional math books. Why? Apparently, someone thought math was without mystery. That math is a definitive subject of rules and algorithms that all have been discovered.
…
We must persuade children that math is a worthy pursuit through interesting stories, examining quirky math properties, and asking good questions.”

— Lacy Coker
5 Tips to Cultivate Math Curiosity

The Mind’s Perfect Playground

My K-2nd-grade homeschool co-op math class will be following many of the tips in Lacy’s article.

Our topic is “Math Storytime,” so we’ll be starting with picture books, exploring the ideas they bring up, and finding things to notice and wonder about.

I’m looking forward to it.

But picture books aren’t just for little kids. They can be great discussion-starters at any age. Have you enjoyed math books with your students?

I’d love to hear your suggestions!

CREDITS: Background photo courtesy of Bekah Russom on Unsplash.

Learning Math Requires Imagination

Posted on January 3, 2018September 9, 2023 by Denise Gaskins

“Teach mathematics the way we learn any other subject: Make it visual, make it concrete, not dependent on meaningless, abstract symbols, employ all the senses!
…
If math is such an important subject (and it is) why teach it in a way that is dependent on a child’s weakest mental ability: memory, rather than her strongest mental ability: imagination?”

— Geoff White
The Grade 10 Math Crunch, or Hitting the Wall at Grade 10

Mathematics and Imagination

How can we stir up our students’ imagination?

Teachers have struggled with this question for years — perhaps since the beginning of the profession.

Consider these comments by W. W. Sawyer in Mathematician’s Delight:

“Earlier we considered the argument, ‘Twice two must be four, because we cannot imagine it otherwise.’ This argument brings out clearly the connexion between reason and imagination: reason is in fact neither more nor less than an experiment carried out in the imagination.

“People often make mistakes when they reason about things they have never seen. Imagination does not always give us the correct answer. We can only argue correctly about things of which we have experience or which are reasonably like the things we know well. If our reasoning leads us to an untrue conclusion, we must revise the picture in our minds, and learn to imagine things as they are.

“When we find ourselves unable to reason (as one often does when presented with, say, a problem in algebra) it is because our imagination is not touched. One can begin to reason only when a clear picture has been formed in the imagination.

“Bad teaching is teaching which presents an endless procession of meaningless signs, words and rules, and fails to arouse the imagination.”

CREDITS: Background photo by Mehmet Kürşat Değer on Unsplash.

Holiday Math and More: Playful Math Education Carnival 114

Posted on December 6, 2017September 9, 2023 by Denise Gaskins

Do you enjoy math? I hope so! If not, the links in this post just may change your mind.

Welcome to the 114th edition of the Math Teachers At Play math education blog carnival — a smorgasbord of articles by bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college.

If you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

By the way, I found a cool, semi-self-referential trivia tidbit about our carnival number: 2⁷ − 14 = 114. And if you put 114 dots into a 1←7 Exploding Dots machine, you’ll get the code 222. Pretty neat!

As you scroll through the links below, you find several puzzle graphics from the wonderful Visual Patterns website. Use them as conversation-starters with your kids: What do you notice? How does each pattern grow? For older students: Can you write a formula to describe how each pattern? What will it look at stage 43?

A BIT OF FUN

Setting the mood: Enjoy this bit of seasonal fidgeting from Vi Hart (@vihartvihart).

If you don’t understand some of the references, that’s normal! Pick a phrase, Google it, and enjoy the fun of learning something new.

And now, on to the main attraction: the blog posts. Some 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.

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

Let the mathematical fun begin!

Continue reading Holiday Math and More: Playful Math Education Carnival 114

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

Posted on September 21, 2017February 8, 2024 by Denise Gaskins

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: “Math Phobia” photo by Jimmie (blog post title added) via Flickr (CC BY 2.0). Phil Daro video by SERP Media (the Strategic Education Research Partnership) via Vimeo.

New Book: Word Problems from Literature

Posted on June 5, 2017December 30, 2024 by Denise Gaskins

The posts on my Let’s Play Math blog are, for the most part, first-draft material. Of course, I’ve proofread each post — many times! because I’m a perfectionist that way, and yet I still miss typos — but these articles haven’t gotten the sort of feedback that polishes a book manuscript.

Well, now I’m taking some of the best of my old blog posts, expanding them with a few new games or activities, and giving them that book-quality polish. Let me introduce my newest series, the Playful Math Singles.

Under Construction …

The Playful Math Singles from Tabletop Academy Press will be short, topical books featuring clear explanations and ready-to-play activities.

I’m hoping to finish up two or three of these this year. Watch for them at your favorite online bookstore.

The first one is done …

Word Problems from Literature: An Introduction to Bar Model Diagrams

You can help prevent math anxiety by giving your children the mental tools they need to conquer the toughest story problems.

Young children expect to look at a word problem and instantly see the answer. But as they get older, their textbook math problems also grow in difficulty, so this solution-by-intuitive-leap becomes impossible.

Too often the frustrated child concludes, “I’m just not good at math.”

But with guided practice, any student can learn to master word problems.

Word Problems from Literature features math puzzles for elementary and middle school students from classic books such as Mr. Popper’s Penguins and The Hobbit.

For each puzzle, I demonstrate step by step how to use the problem-solving tool of bar model diagrams, a type of pictorial algebra. For children who are used to playing with Legos or other blocks — or with computer games like Minecraft — this approach reveals the underlying structure of a math word problem. Students can make sense of how each quantity in the story relates to the others and see a path to the solution.

And when you finish the puzzles in this book, I’ll show you how to create your own word problems from literature, based in your children’s favorite story worlds.

Buy now at your favorite online bookstore.

If you’re using these word problems with your children, consider buying them the paperback companion Word Problems from Literature Student Workbook.

… and People Like It!

A screen shot from this past weekend:

“I found this method really clarified for me what was going on visually and conceptually. Particularly when it came to more complex questions, for which I would normally write out an equation, I felt that thinking about what was going on with the bars actually made more sense … This is a wonderful book for those who want to support their children in finding better ways to work on word problems.”

—Miranda Jubb, Amazon customer reviewer

Playful Math Education Carnival 106 with Math Art

Posted on March 28, 2017September 13, 2023 by Denise Gaskins

Do you enjoy math? I hope so! If not, browsing this post just may change your mind.

Welcome to the 106th edition of the Math Teachers At Play 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!

By tradition, we start the carnival with a puzzle in honor of our 106th edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

Try This Puzzle

If you slice a pizza with a lightsaber, you’ll make straight cuts all the way across. Slice it once, and you get two pieces.

If you slice it five times, you’ll get a maximum of sixteen pieces. (And if you’re lucky you might get a star!)

How many times would you have to slice the pizza to get 106 pieces?

Click here for all the mathy goodness!

Beauty in Math: A Fable

Posted on March 9, 2017August 6, 2019 by Denise Gaskins

Have you ever wondered what mathematicians mean when they talk about a “beautiful” math proof?

“Beauty in mathematics is seeing the truth without effort.”

—George Pólya

“There’s something striking about the economy of the counselor’s construction. He drew a single line, and that totally changed one’s vision of the geometry involved.

“Very often, there’s a simple introduction of something that’s not logically within the framework of the question — and it can be very simple — and it utterly changes your view of what the question really is about.”

—Barry Mazur
The Moral of the Scale Fable

CREDITS: Castle photo (top) by Rachel Davis via Unsplash. “A Mathematical Fable” via YouTube. Story told by Barry Mazur. Animation by Pete McPartlan. Video by Brady Haran for Numberphile.

Visualizing Word Problems with Bar Model Diagrams

Posted on February 27, 2017December 30, 2024 by Denise Gaskins

A friend emailed me, frustrated with her child’s math lesson on bar diagrams: “Why do they have to make it so complicated? Why can’t we just solve the blasted problem?”

I told her bar models themselves are not the goal. The real question for parents and teachers is:

What can you do when your child is stumped by a math word problem?

To solve word problems, students must be able to read and understand what is written. They need to visualize this information in a way that will help them translate it into a mathematical expression.

Bar model diagrams are one very useful tool to aid this visualization. These pictures model the word problem in a way that makes the solution appear almost like magic.

It is a trick well worth learning, no matter which math program you use.

Visualization

https://www.youtube.com/watch?v=HKsYDzQK8Zw

“Visualization is the brain’s ability to see beyond what the eyes can see, and we can develop visualization in many ways.”

The Bar Model Explained

https://www.youtube.com/watch?v=I6Ipio8JntU

“A bar model is a way to represent a situation in a word problem using diagrams — in particular, using rectangles.”

https://www.youtube.com/watch?v=i7LAHc1qvig

“This is one of the ideas that children learn in mathematics: the use of diagrams to represent quantities, especially quantities which are unknown.”

Word Problems from Literature

I’ve written a series of blog posts that explain bar model diagrams from the most basic through to solving multistep word problems. Check them out:

I’ve started working on a book about bar model diagrams, and I’d love to hear your input. Have you tried using them? Do they help your children? What questions do you have?

Update: My New Book