Beauty in Math: A Fable

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.

howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.


FAQ: Homeschool Burnout

“Spring cleaning has made my desk look worse than before. Nobody feels like studying. The kids would rather be outside, and their mom would rather take a nap. If I line everyone up on the curb in the morning, do you think the yellow bus will take them?”

Homeschool burnout — it’s a perennial problem. If you’re suffering from lethargy and can’t face another day of school work, here are some ideas that kept me going long enough to graduate almost-five kids (my “baby” finishes homeschooling this spring!):

(1) Re-read the homeschooling books on your shelves, or get some new ones from the library. Write down your favorite quotes as you read. Try to read about one a month, to help get your enthusiasm back. And then read at least one new homeschooling book per year to help you stay inspired.

(2) Connect with other homeschoolers. Meet with friends for tea, or have a Mom’s Night Out while Dad babysits. Talk about substantive things, like educational philosophy — what you like about homeschooling, and what you’d like to change. Share your dreams for your children. Remind each other why you’re doing this.

(3) Attend support group meetings. I find that after so many years, I let the meetings slide. I think, I already know everything they are going to say. But being with other homeschoolers is encouraging. And if you find out that you can help a new homeschooler with advice, that gives you a boost, too.

(4) Find one or two forums where you can become one of the resident experts, and answer posts as often as you can. As with number 3 above, being able to give advice (and being appreciated for it) can give you the energy to keep on going.

(5) Go to a homeschooling convention, if you get the chance. The speakers are stimulating, and you may find some new book or tool that sparks your imagination.

(6) Do school anyway. It may seem impossible when you’re stuck in the doldrums, but once you get going, you may find it easier. The light of understanding in a child’s eyes can give Mom quite a lift!

(7) Try something completely different. If you have always used a textbook program, then set it aside for a month and just read library books. If you have read lots of great literature, then try some hands-on projects, or get out those science experiments you keep putting off, or visit all the museums within a two-hour radius, or… I’m sure you can think of something that has been lingering on your good-intentions list. I never could stand to teach the same old thing every year, and none of my five kids got exactly the same education. Happily, there is always another way to approach any homeschooling topic. How about Gameschooling?

(8) Figure out what your students are able to do on their own, and let them do it. Encourage them to develop as much independence as possible.

(9) Use some of your children’s independent time to learn something new for yourself. Have you always wanted to try painting, or crochet, or woodworking? Be an example of life-long learning.

(10) Start (or join in progress) a group class or co-op. You may be able to trade around with some other families: you teach history and others teach math or cooking, or whatever arrangement fits for you. This is especially helpful for those time-consuming projects that always seem to get put off, like art or science experiments.

(11) Try some of these intensely practical Tips For Coping With Homeschool Burnout.

(12) And are you a Christian homeschooler? Then pray! Your Father knows what you need, and Immanuel is with you always. Try praying your way through 1 Corinthians 13 (or this homeschooling version).

If you have any other ideas for beating the burnout blues, please share!

Homeschooling is not always peaches and cream. If anyone promised you that, they lied. But be assured that it homeschool burnout is not a terminal condition. You will recover your joy in sharing your children’s education.

I learned one thing from every story I’ve ever read: adventures never run smoothly.

And what greater adventure could there be than to introduce your child to all the wonderful things in God’s world?

homeschooling


CREDITS: “Scream” photo (top) by greg westfall via Flickr (CC BY 2.0).

howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.


Visualizing Word Problems with Bar Model Diagrams

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.

visualizing-word-problems

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?


CREDITS: Videos and quotations from Dr. Yeap Ban Har’s YouTube channel. “Girl doing homework” photo (top) by ND Strupler and “math notebooking equal fractions” by Jimmie via Flickr (CC BY 2.0).

howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.


Playful Math Carnival #105 via Mrs. E

mtap-105Check out the new carnival of playful math for all ages at Mrs. E Teaches Math. Each month’s carnival brings you a great new collection of puzzles, math conversations, crafts, teaching tips, and all sorts of mathy fun.

This month’s post features algebra tips, geometry proofs, Fibonacci rabbit trails, math art, and much more.

Click Here to Go Read the Carnival Blog!

Past carnivals are still full of mathy treasure. Check them out:


howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.


A Map of Mathematics

Pure mathematics, applied math, and more — all summarized in a single map! Watch the video by physicist and award-winning science writer Dominic Walliman:

Walliman says, “To err is to human, and I human a lot. I always try my best to be as correct as possible, but unfortunately I make mistakes…”

  • Can you find three mistakes in the map?

Check your answers in the description on Walliman’s YouTube page.

If you enjoy this video, you can purchase the poster (or T-shirt, coffee mug, tote bag, etc.) at Red Bubble.


Map of Mathematics poster by Dominic Walliman via Flickr (CC BY-NC-ND 2.0).

howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.


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

Quotable: Then and Now

“I used to think my job was to teach students to see what I see. I no longer believe this. My job is to teach students to see; and to recognize that no matter what the problem is, we don’t all see things the same way. But when we examine our different ways of seeing, and look for the relationships involved, everyone sees more clearly; everyone understands more deeply.”

Joe Schwartz
Then and Now


[Feature photo (above) by jenn.davis via Flickr (CC BY 2.0).]

howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.