On Productive Failure

Here is a 7 minute video of a young math major, Elly Schofield. She reflects on her K-12 mathematics education, and the disconnect between that and the mathematics she encountered in college.

Fibonacci and Fractals Links

I had the pleasure of being the co-presenter of a workshop called “Fibonacci, Fractals, and the Common Core Math Standards” at the VCTM conference at Saint Michael’s College on Friday. My partners were Professor Tim Whiteford of Saint Michael’s College and Laura Sommariva, a math teacher at Colchester High School.

The three of us shared what we’d done with students during a fifth grade field trip to Tim’s famous Penny Arcade. Here is Tim’s original write-up about that.

As promised, for the participants in our workshop and others, here are the resources we used to teach about fractals and Fibonacci numbers.

The Fractal Foundation Fractivities. How to draw Sierpinski triangles, do fractal cutouts and more.

Worksheetworks. Create your own graph paper and triangle paper.

On Being a Plant, Part 1, the 6 minute Vi Hart video we watched. She explains Fibonacci numbers, and demonstrates how to count the spirals on pinecones, draw realistic pinecones, and use graph paper to make a golden spiral.

Laura teaches summer math classes for elementary students in Switzerland. She has photos of her students here doing wonderful, related activities like building giant tetrahedrons out of mini marshmallows.

Have fun!

Summer Gems

Here are two articles you shouldn’t miss this summer, and they make a nice pair.

“Why Do Americans Stink at Math?” by Elizabeth Green, July 23, New York Times, and “Most Math Problems Do Not Have a Unique Right Answer” by Keith Devlin, August 1, 2014, Devlin’s Angle.

I’ve come to the conclusion that our old ideas and stereotypes about math and education are hard to shake. It really means reading articles like these as they come out every month or so, in order to truly change our minds and our practice.

To whet your appetite, here are a few quotes:

One of the first math classes he observed gave him such a jolt that he assumed there must have been some kind of mistake. The class looked exactly like his own memories of school. “I thought, Well, that’s only this class,” Takahashi said. But the next class looked like the first, and so did the next and the one after that. The Americans might have invented the world’s best methods for teaching math to children, but it was difficult to find anyone actually using them. - from “Why Do Americans Stink at Math?”

Having earned my living as a mathematician for over 40 years, I can assure you that the belief is false. In addition to my university research, I have done mathematical work for the U. S. Intelligence Community, the U.S. Army, private defense contractors, and a number of for-profit companies. In not one of those projects was I paid to find "the right answer." No one thought for one moment that there could be such a thing. - from “Most Math Problems Do Not Have a Unique Right Answer”

The last summer gem is this youcubed video about using Number Talks in math class. How would you solve 18 x 5 mentally?

Latest and Greatest in a Nutshell

Dan Meyer

There are some very good things happening right now in the world of mathematics education. Being at the NCSM Conference gave me the opportunity to connect with many of the greatest minds in the field and hear about their current projects.

Here are my top picks for books to buy, courses to take, blogs to read...right NOW.

Books

NCTM’s Principles to Actions: Ensuring Mathematical Success for All. The book Steve Leinwand keeps waving around. I read a draft in the fall. Educators need to start turning to this when figuring out how to do professional development, how to craft school goals, what to look for in classrooms. It’s radical.

Visible Learning by John Hattie. This book is being enthusiastically talked about by greats like Tim Kanold and Bob Laird. Subtitled “A Synthesis of Over 800 Meta-Analyses Relating to Achievement”, it’s the new go-to resource for knowing what truly works in education.

Smarter Together: Collaboration and Equity in the Elementary Math Classroom by Featherstone, et al.

Blogs

Tim Kanold. Check out his most recent post about math homework.

Max Ray. I went to an excellent workshop for teachers by Max and also saw his engaging Ignite talk. He’s with the Math Forum @ Drexel. There are problems, videos and more.

Dan Meyer. Prolific blogger, thinker. Recently published Great Modeling Tasks in Three Acts (for NCSM members only, except for one freebie).

Annie Fetter. At Math Forum with Max. Great Ignite talk about her artist mother and the math she used.

YouCubed. Not a blog but go there and sign up for updates. Watch some of the videos.

Courses

There are two online Stanford courses taught by Jo Boaler that are a must-do. Last summer, I took How to Learn Math for Parents and Teachers. It was fabulous. It was free then, now it costs $125. Worth it.

This year, Jo Boaler has added a brand new course called How to Learn Math for Students. It’s free. I haven’t taken it yet but I recommend trying it with a young person you know. It sounds like it is appropriate for children ages 10-18, but for children younger than 13, a parent or teacher should register and share the material with the child. Go to the site and register. It begins in May.

People to Follow on Twitter

@jgough, @steve_leinwand , @SkipFennell, @maxmathforum, @tkanold, @MFAnnie, @TuvaLabs,@Trianglemancsd, @wahedahbug, @lambchop1998 (I met this guy at NSTA Boston, but he was great), @IllustrateMath, @joboaler

Who I missed

Uri Treisman. But here’s the audio of his talk.

Hour of Code Video

This was so much fun!

Machines to Materials

I just watched two PBS NOVA shows that I must share. One is called Making Stuff: Smaller. The other is Making Stuff: Stronger. There are other Making Stuff shows I haven’t watched yet.

Check these out! You can watch them for free online (at least right now you can).

As an educator and a parent, these videos get me all fired up. Young people should be learning about cutting edge science and math, not just the stuff of textbooks and standards. I want to show these to students and then I’d expect to hear them talking about wanting to be a chemist, material scientist, or nanotechnologist when they grow up.

My favorite part of Making Stuff: Smaller was the concept of starting by using a machine to complete a task, and then evolving to using a material to accomplish the same task. This is a key feature of miniaturization, which is what allows us to have laptops and cellphones, among other things, today. Computer processing went from giant rooms of vacuum tubes to silicon chips, but the story I liked the most was the journey from behemoth pendulum clocks to quartz watches. If this makes no sense to you, watch the video.

Another awesome story in Making Stuff: Smaller is about driving a tiny robot around inside someone’s eyeball to deliver medicine. Holy cow!

Estimation Jars

Today I’m creating Estimation Jars for Math Night at John J. Flynn Elementary School. This is always a highlight at Math Night and it’s fun to watch kids at work on their estimate. I have one jar full of Rolos (there is something appealing about the gold wrapping), a smaller jar containing Skittles, and a very small jar with unpopped popcorn.

This year, I plan to accept all estimates within a reasonable range for each jar and then randomly draw a winner. That’s different than past Math Nights when the person who estimated the number closest to the actual number won the jar. I am hoping this helps promote a more correct understanding of estimation, with the goal being to produce a reasonable estimate, not to land on or closest to the exact number.

It is a good idea to have a few jars containing different sized objects so students can think about how the relative size of the objects affects the number that fit in a space. Ongoing math explorations can be done using estimation jars outside of an event like Math Night. Here is an excellent write-up and video about how classroom teachers can use estimation jars with their students, with a focus on the all-important concepts of doubling and halving.

Sugata Mitra: School in the Cloud

Sugata Mitra recently won the TED Prize for 2013. His wish is to build a virtual school where children learn from each other.

Well, I bumped into this whole thing completely by accident. I used to teach people how to write computer programs in New Delhi, 14 years ago. And right next to where I used to work, there was a slum. And I used to think, how on Earth are those kids ever going to learn to write computer programs? Or should they not? At the same time, we also had lots of parents, rich people, who had computers, and who used to tell me, "You know, my son, I think he's gifted, because he does wonderful things with computers. And my daughter -- oh, surely she is extra-intelligent." And so on. So I suddenly figured that, how come all the rich people are having these extraordinarily gifted children? (Laughter) What did the poor do wrong? I made a hole in the boundary wall of the slum next to my office, and stuck a computer inside it just to see what would happen if I gave a computer to children who never would have one, didn't know any English, didn't know what the Internet was. - Sugata Mitra

The Future of STEM Education

Hear Professor Roni Ellington’s inspirational talk from TEDxBaltimore, 2013.

Learning to code

http://www.code.org/

Watch this video. I completely agree that learning to code is learning to think. I think it is getting easier to teach, too, because there are lots of resources like Scratch and the Lego Mindstorms, plus tons of other links on the code.org. We are running out of time and excuses when it comes to our kids.

Marjorie on Math Munch

There’s a great blog called Math Munch. I mentioned it in a previous post, but I have to revisit it here. I’ve signed up to receive new posts from Math Munch via email, something I don’t usually do. But I’ve found that this is worth the space in my inbox. Each post has a few different topics, everything from cool paper folding projects to high quality videos and online games. You must check it out!

The current Math Munch post includes a piece about a woman named Marjorie Rice who became interested in tesselations and discovered some new ones on her own. These pictures are from her website. What’s great about this story is that a) she’s a woman, b) she didn’t have a math background beyond high school, and c) she was curious, worked at something, and found enjoyment and success.

I won’t go on and on here, but the other two stories with Marjorie’s in this issue of Math Munch are equally worth reading.

The pictures I’ve included here are some of Marjorie’s tesselations.

How much time do you have?

Teachers: Do you ever find yourself with a little time on your hands and nothing planned? There are zillions of videos out there these days - many are awful and many are wonderful. If you have a short space in the day, you can treat your students to something fun and thought-provoking. Here are some videos I like, with running times, to keep up your sleeve for just the right moment.

Eratosthenes (2:16)
This guy figured out the size of the earth a very long time ago in an elegant way using his knowledge of geometry.

Richard Tapia (2:21)
The kid loved race cars and became a mathematician.

Frostie Dancing to Shake a Tail Feather (2:43)
What can I say? It’s a surefire mood improver.

OK Go, This Too Shall Pass (3:54)
A band makes a music video ala Rube Goldberg. A TV smashes, people are shot with paint.

Doodling in Math Class: Infinity Elephants (4:36)
Makes you want to grab a pencil and start drawing fractals.

Artist of the Floating World (4:47)
An artist creates a giant pair of floating dice and sets them adrift in the ocean.

Brooklyn’s Rube Goldberg (5:34)
A real page turner.

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.

Music for the Eyes

Joe Garofalo, C.P. Smith music teacher and captain in the Lyric Theater’s recent production of Titanic, kindly shares his room with me. Often, before the students arrive in the morning, we find ourselves discussing the math-music connection and how we might help bring this to life for students. Last week, Joe showed me some animated music he’d discovered. I had a hard time tearing myself away. See for yourself. Here is Johann Sebastian Bach’s Toccata and Fugue in D Minor  by a company called Musanim. Musanim says it will send a free DVD of these videos to public libraries and schools. We’ve got ours on order.

Caine's Arcade

My dear friend, Rachel Wynne, sent me the link to this wonderful video. Nine-year-old Caine designed and constructed an entire arcade out of cardboard. He taped calculators onto the front of the games so he could validate pin numbers using the square root function. So cool. Rachel, thank you for thinking of me and Jim when you saw this. Caine is an inspiration to us all.

Fly with math

This is my new favorite TED Talk. It's called Vijay Kumar: Robots that fly...and cooperate. It is one of the best I’ve seen. UPenn researchers have created mini helicopters so elegantly programmed that they function autonomously and can work together. You must watch the entire 16 minutes. Use this to inspire and motivate students, and to help keep math relevant.

Being Less Helpful

Less helpful. That is Dan Meyer’s tag line on his blog. He is a high school math teacher who says we must give students more time and less help in solving real math problems.

Here is an excerpt from Dan’s talk entitled Math Class Needs a Makeover:
David Milch, creator of "Deadwood" and other amazing TV shows...swore off creating contemporary drama, shows set in the present day, because he saw that when people fill their mind with four hours a day of, for example, "Two and a Half Men," no disrespect, it shapes the neural pathways, he said, in such a way that they expect simple problems. He called it, "an impatience with irresolution." You're impatient with things that don't resolve quickly. You expect sitcom-sized problems that wrap up in 22 minutes, three commercial breaks and a laugh track. And I'll put it to all of you, what you already know, that no problem worth solving is that simple. I am very concerned about this because I'm going to retire in a world that my students will run. I'm doing bad things to my own future and well-being when I teach this way. I'm here to tell you that the way our textbooks -- particularly mass-adopted textbooks -- teach math reasoning and patient problem solving, it's functionally equivalent to turning on "Two and a Half Men" and calling it a day.

Right now I am taking a fantastic class with Sandi Stanhope about math problem-solving in elementary school. She frequently mentions the need to “sit on our hands”. We’re so used to teaching, guiding, instructing, that we deprive students of the opportunity to develop into competent problem-solvers and mathematicians. Our homework for Sandi’s class is to pose appropriately challenging problems to our students and then to encourage them to solve them in whatever way makes sense to them. We work on our ability to ask good questions and to facilitate high-quality dialogue among students rather than to front-load students with algorithms and strategies we hope to see.

So far, it has been incredible to work with students this way and to hear others in the class describe their experiences doing the same. Our students tend to pleasantly surprise us with when we let them. They also demonstrate the ability to work long and hard on a single problem when we allow and encourage that.

Be Less Helpful can be applied to homework as well. If you find yourself very involved in a young person’s homework, it might be time to take a step back and rethink things. I have applied this principle to my involvement in my own children’s homework. I find I am still helpful and supportive, but just in different (better) ways.

In Sandi’s class, we’re using a book by Larry Buschman called Share & Compare. I highly recommend it. Buschman has also written many great articles in NCTM’s Teaching Children Mathematics magazine.               

I'll close with Buschman's two most important discoveries as a teacher:
Teachers give themselves teachable moments by carefully listening to children.
Teachers give children magic moments by letting go.

Math > Calculation

This morning I found a New York Times article about using calculators on exams like the SAT, GRE, etc. It led me to Conrad Wolfram's TED Talk entitled "Teaching kids real math with computers", which I have embedded below. It is well worth taking the 17 minutes to watch it.

The title is misleading; it seems like you'll be hearing about doing our usual classroom stuff with technology instead of paper. Don't be fooled: Wolfram's talk is about boldly rethinking and reinventing mathematics education.

Fractions and My Brother

Man changing stock quotes on a chalk board. NYPL Digital Gallery

My brother, Zack, was here visiting the other night. My 12 year old son had a question about his fraction homework, and this sparked a conversation between Zack and me about how fractions are taught in school. 

Zack said, “Fractions are a useless implementation of decimals.” 

Hmm. Zack worked in finance for many years as a derivatives trader. I never really understood what he did besides the fact that it entailed making split second decisions involving lots of money and using insane spreadsheets. Zack is still my go-to person when I need help writing a really complicated spreadsheet formula or macro. Not only does he understand what I’m saying, but he has the answer in .9 of a second or less.

Here is a video of Zack sharing his thoughts on fractions. I hope it sparks some interesting conversation for readers of this blog. I brought this up with my colleague, Penny Stearns, who expressed a completely different perspective from Zack. Perhaps she will star in a video rebuttal at some point. Professor Tim Whiteford weighed in with still more to think about. What an excellent topic for students to question and debate in order to deepen their understanding of the application of fractions and decimals in different contexts.