Hexaflexagons

Alex Reutter, C.P. Smith Math Night Parent Extraordinaire, got me thinking about hexaflexagons. The other day, he mentioned that he thought kids would enjoy making them at Math Night. Hexaflexagons are folded paper hexagons that do a special flip. I decided I’d better try to figure these out ahead of time on a nice calm weekend at home.

I began by watching Vi Hart’s videos about hexaflexagons. Of course, Vi is fond of speeding things up, but I still thought I should be able to fold a strip of paper into equilateral triangles and make a hexaflexagon. Vi didn’t seem to be doing any measuring, like other websites recommend, so I was resistant to using a ruler. But after a bunch of mangled paper strips, I knew I needed a different strategy.

I found some pre-made, printable PDFs of hexaflexagon patterns. Some of the best are on a website called Aunt Annie’s crafts, on her Flexagons page. Print them, preferably in color. My brother and I were going to drive all the way down to Thetford, so I packed up some flexagon patterns, a scissor, and some double-sided tape for the trip.

I had plenty of time in the car to cut, tape, and fold several hexaflexagons, but I still didn’t know how to get them to do their special flip. I handed one to my brother while we stood around watching the Thetford cross country races, and he was able to figure it out.

These hexaflexagons are really cool. It’s worth trying one yourself. They were discovered by Arthur H. Stone in 1939, then popularized by Martin Gardner in his Scientific American column called “Mathematical Games” in 1956. Try pre-made patterns at first. I think starting students this way, then asking them how they might create their own if given a blank piece of paper, pencil, ruler, glue, and scissors, would be an excellent math activity, perfect for differentiation. We'll see how it goes at Math Night!

There is a free iPad app from IBM called Minds of Modern Mathematics. It is a beautiful timeline of the history of mathematics and world events, and there are 9 vintage videos to watch as well. Charles and Ray Eames, a famous, husband-and-wife design team created the original interactive mathematics exhibit back in 1961, which provided the content for this app. The Eameses films are wonderful, simple, short. I enjoyed Something About Functions, and the non-verbal Exponents.

I also happened upon a great blog called MathMunch. MathMunch calls itself “a weekly digest of the mathematical internet”. They posted something about Minds of Modern Mathematics, and have a fabulous page of math videos. If you miss the old Schoolhouse Rock videos and you know a young person who could use some work on counting by 5s, MathMunch has the link. Their video collection is exhaustive, and includes videos about solving a Rubik’s Cube, paperfolding, Pi, M.C. Escher, and more.

Teachers’ Expectations

When I arrived at work last week, several people were excitedly talking about the NPR piece they’d heard on the radio during their drive in. Here is a link to the Morning Edition show, Teachers’ Expectations Can Influence How Students Perform, which aired September 17.

I recommend listening to the audio, but you can also read the transcript. Back in 1964, Harvard professor Robert Rosenthal began studying how teachers’ expectations influence student achievement.

[Rosenthal] found that expectations affect teachers' moment-to-moment interactions with the children they teach in a thousand almost invisible ways. Teachers give the students that they expect to succeed more time to answer questions, more specific feedback, and more approval: They consistently touch, nod and smile at those kids more.

"It's not magic, it's not mental telepathy," Rosenthal says. "It's very likely these thousands of different ways of treating people in small ways every day."

It is difficult to truly change our beliefs, but there is a way. Recent studies have shown that teachers who actively worked on their teaching through videotape analysis and targeted work with coaches in their classrooms to change their behavior also experienced a significant shift their beliefs about students.

What's Math Got to Do with It?

A man is on a diet and goes into a shop to buy some ham slices. He is given 3 slices which together weight ⅓ of a pound, but his diet says that he is only allowed to eat ¼ of a pound. How much of the 3 slices he bought can he eat while staying true to his diet?

There is a lovely book in the John J. Flynn Parent Resource Center these days. It’s called What’s Math Got to Do with It? How parents and teachers can help children learn to love their least favorite subject, by Jo Boaler. The book was published in 2009, but I have only recently discovered it.

Boaler includes many more problem-solvers like the one above. Don’t worry if you don’t know a formula to figure out how many turkey slices the man can eat. Start drawing pictures and think about what a whole pound would look like. Go slow and use your intuition. Says Boaler, “Children begin school as natural problem solvers and many studies have shown that students are better at solving problems before they attend math classes.”

…People don’t like mathematics because of the way it is misrepresented in school. The math that millions of Americans experience in school is an impoverished version of the subject and it bears little resemblance to the mathematics of life or work or even the mathematics in which mathematicians engage.

In addition to the prompt I shared in my last post “What do you think you should try next?”, Boaler shares more good prompts to use with children:

• How did you think about the problem?
• What was the first step?
• What did you do next?
• Why did you do it that way?
• Can you think of a different way to do the problem?
• How do the two ways relate?
• What could you change about the problem to make it easier or simpler?

Enjoy.

Teach your children well

 

Here’s a nice article from the Wall Street Journal for parents, with an accompanying video. 

This short piece hits the important points when it comes to math and parenting. 

The one thing I would add is my favorite question for my daughter when she asks for help with her algebra homework: “What do you think you should try next?”

Mindset

Charles Coiner, 1961, from Smithsonian American Art Museum

I am a huge fan of Carol Dweck’s work on people’s beliefs about the nature of intelligence. A student’s mindset has a measurable impact on his or her success in math, among other things.

We praised the children in one group for their intelligence, telling them, “Wow, that’s a really good score. You must be smart at this.” We praised the children in the other group for their effort: “Wow, that’s a really good score. You must have worked really hard.” That’s all we did, but the results were dramatic.

Doesn’t that statement make you want to read more? Here is a condensed version of Dweck’s research (you can read the book, Mindset, and find more articles here).

And here is a student-friendly version of the article. Perfect non-fiction reading for fourth and fifth graders!

The Necessity of Algebra

On Sunday, the New York Times published a piece by Andrew Hacker with the provocative title “Is Algebra Necessary?”. Several people have emailed me the link, so it seems like it has garnered some attention. It is worth reading.

While I disagree with most of Hacker’s anti-advanced-math argument, the statistics about the barrier students face when they are not proficient in high school math is important to consider. Failure in math does profoundly affect a young person’s options when it comes to college and career choices. How many people do you know whose math ability was a significant factor in their choice of college major and career? Just the other day, a friend admitted that she’d always dreamed of being a doctor, but she wasn’t good at math so she went into journalism instead.

Rather than lowering the bar for students, we should look at how we can improve mathematics education. Algebra is about abstraction, which is one of the most fundamental ideas across disciplines. When I was a software engineer, much of my day-to-day work was about designing abstract functions with variables to hold unknowns. It was about finding clean, elegant solutions to complex problems through thought, collaboration, trial and error. We conjectured, tested, proved. I couldn’t have learned concrete “job skills” ahead of time. I had to keep learning, inventing and working with big ideas. Great algebra and other math classes are excellent preparation for students, as are great science, literature, writing, and history classes.

Hacker’s article prompted many thoughtful responses. I’ve collected some here along with excerpts for your reading pleasure. I’ll end with a nice video of Paul Lockhart talking about the wonderfulness of math.

The New York Times Letters to the Editor: A National Conversation on Math
Andrew Hacker is right: most students will never need to use algebra. Many will struggle to learn it nonetheless. But the answer is not to let students quit as soon as they begin to struggle.

I myself hated mathematics for many years. Through algebra, geometry and trigonometry, I cursed a system that compelled me to take such “useless” courses. Eventually, I was required to take calculus, the most dreaded of all math courses. I prepared for the worst.

It came as a surprise, then, that I quickly found myself enjoying the class. The reason was that I had finally encountered a talented math teacher with a passion for the subject. His passion proved infectious, and now, a year later, I’m looking to study mathematical biology at an Ivy League university.

It’s an outcome I would have never predicted just a few years ago. It could have never happened if I had been allowed to quit when I first struggled with math.

ZACHARY MILLER
Kitty Hawk, N.C., July 29, 2012

Motherlode: Adventures in Parenting by Jessica Lahey
...I know precisely where I lost my battle with math, the moment I was informed clearly and unequivocally that I simply wasn’t “a math person.” My seventh-grade math teacher, an otherwise lovely man, called each of his students up to his desk one by one in order to write a “1” (for the honors track) or “2” (for the standard track) on the school’s official math placement forms. As I watched from over his hunched and courduroyed shoulder, he wrote a beautiful, decisive and neat “1” on my form.

There it was, in permanent ink. I was good at math.

“Jess, could you come back up here for a minute?” he asked as I floated back to my seat.
He reclaimed my form, and carefully overlaid that beautiful “1” with a dark, clumsy “2,” pressing hard with his black pen in order to make sure the ink obliterated any evidence of his indecision.

And from then on, I wasn’t good at math anymore....

Scientific American: Abandoning Algebra is Not the Answer by Evelyn Lamb
Mathematicians are recruited by hedge funds, consulting firms, and technology companies not because they already know how to balance portfolios, what the best corporate strategies are, or how to optimize user interfaces, but because their mathematics degrees indicate experience and acuity at problem solving. It’s easier for companies to teach someone with a strong mathematics background how to do their specific work than to teach someone who knows the company business how to solve problems. And, like it or not, algebra is one of the first places students start to learn these problem solving skills....

Math education needs to improve, but if illiteracy were on the rise, I don’t think we’d be talking about eliminating reading from the curriculum.

Rob Knop
Algebra is fundamental to nearly all of "higher math". Even if you want to do more than the most basic of things with statistics, you need to know some algebra. To give up on that would be right on par with the giving up on the teaching of history as anything other than memorizing the occasional date, and to give up on the teaching of English literature as anything other than being able to read a short document for simple surface content and to put together a simple declarative sentence. If you want people to be educated beyond elementary school and beyond "job training", then algebra is one of the intellectual foundations of our civilization that simply cannot be neglected.

Harvard University Press Blog: Is Mathematics Not Beautiful?
For all his focus on the pain and fear of mathematics, Hacker has little to say about its beauty. He does suggest that we should treat mathematics as a liberal art, “making it as accessible and welcoming as sculpture or ballet,” but in the service of rationalizing its marginalization rather than encouraging its embrace.

What must be and what can't be

Have you heard of KenKen puzzles? They are logic puzzles like Sudoku but with an extra element of math calculation. They are great challengers for people of all ages and abilities, since there is such a great range of levels of difficulty. I’ve given them to students, teachers, and friends. 


My dad first introduced them to me. He must have discovered them when they showed up in the New York Times on the same page as the crossword puzzle. He does them in pen, with many cross-outs. The answers to the previous day's puzzles are always there, but they are unnecessary. You know when you've got it right.

I have figured out how to do KenKen puzzles and can do an OK job, especially with the easier puzzles. I want to be better than I am now, so I know I need practice. My dad and mom are visiting me in Vermont this week, so I decided to see if I could get some help with my KenKen chops.

Here is my dad’s advice to me: “First you look at the arithmetic, put in some candidate numbers. Then you start looking for what must to be and what can’t be. Remember to see the whole puzzle even as you work on the pieces. Don’t quit. Keep looking for what must be and what can’t be. If you look at it a little bit you will see it. If you get really stuck, go pour yourself a glass of Pinot Noir or something and the answers will become evident.”

Dad is all about creative problem-solving perseverance, which means he is down with the Math Common Core without knowing it. I was stuck on the harder of the two New York Times puzzles today but he freed me with one little tip. I filled in the rest of the numbers almost as fast as I could write. The tip went something like, “This has to be a 5 and that can’t be a 5 so it must be a...”.

KenKen puzzles are great for young mathematicians. My youngest KenKen student was in first grade. I think most first graders are too young, but she nailed it. I like giving out packets that start with the easier 3x3 grids and are limited to addition and subtraction, and then progress to larger grids with multiplication and division. Most students can quickly find a just-right puzzle and dive in.

The best part of this story is that the marvelous Phia S., a student entering fifth grade, loves KenKens so much she decided to write her own. Here is a photo of the treasured book of KenKen’s given to me by Phia. 

Phia's amazing KenKen book

So, enjoy your summer and take your KenKens (and Pinot if you are of age) to the lake or wherever you might be relaxing. The New York Times publishes two a day on the crossword page (they always include the directions) and more online. Cheers!