The Most Valuable Lesson I've Learned in 10 Years of Teaching

When I began teaching I knew something was wrong with how I had been taught and how I was currently teaching. I did what I could to adjust my style, but the change was slow. I couldn't put my finger on the problem - so I couldn't really solve the problem. Each year I'd try something new... then in my 5th year of teaching I took over a math position from a friend and former colleague. She gave me some of her math materials to give me a head start. I took her materials and (with minor adjustments) just ran with them. In the process I discovered or at least was able to verbalize what had been missing for me all these years. The simple "Joy of Discovery. "

During that year; I watched kids get excited about making connections. They cheered when they began seeing patterns. My math classes in school were never like this... I wasn't using flashy multimedia, or a modern textbook, kids were simply working together, making observations and drawing conclusions. Magic.

Why are people willing to spend hours working out Sudoku puzzles? Why are kids excited to spend hours with a video game? Why do mathematicians and scientist devote their lives to research?

There is inherent joy in the act of discovery. As a species that evolved intelligence at the cost of physical strength it shouldn't surprise us that we have a built in reward system for learning and discovery.

It seems to me that modern math education is not in need of more technology or new standards. Rather as math educators we need to focus our energy on returning the joy of discovery to our math classes. Isn't that what is so good about Dan Meyer's 3 Act approach and so wrong with the Khan Academy approach?

This topic came up again as I was describing the Exeter Math program to an administrator  I found that I was using language very similar to as if I was teaching a textbook driven course - which was a bit troubling. Yet there is a key difference. While Exeter is paper based and not all the problems are amazing - it forces the joy of discovery back into the classroom. The burden to discover is on the students. It is a problem first solution second approach. The complete opposite of most (math) classrooms.

If I reflect on all the PD, all the TED talks, all the blog posts, all the books and all the long conversations over a beer or two three the most valuable lesson I have learned in 10 years of teaching has be the recognition of the value, the motivation, and the sheer joy of discovery.

Math Teacher Ninja - The Unit Circle

I have been trying to teach the unit circle for 5 or 6 years. Each year I think that I've tweaked the process to make it clear(er). Each year, at some point in the process, I have been met with blank stares of "what in the world are you talking about?"

This year I finally had success! In my 9th grade Geometry class of all places.

The Setup:
Earlier in the year I introduced trig through similar triangles. Despite some success, the intro had left many students with murky feelings of "so what" or "seems hard." So a few weeks ago I began planning a second trip through trig-land. I drew heavily from the Exeter Math problem sets and (contrary to Exeter's intentions) put together 3 problem sets each focused loosely around a different trig function.

• Tangent - I loved starting here to make connections to slope/gradient
• Sine
• Cosine

The students were coming around and felt better using and choosing trig ratios to solve problems. I loved the mix of the standard "how far/long is _____" combined with questions seeking deeper conceptual understanding. 

Feeling good about right angle trig I shifted the focus of class towards circles, in particular I wanted/needed to look at the equation for a circle. To start, I gave the students the equation:

In groups they found points that satisfied the equation (using any method they liked) and then used Geogebra via my computer and projector to plot the points. As a class they slowly watched the circle take form as more points were added. We then used (previously learned) geometric construction tools/concepts to find the center and the radius of the circle. I made no mention of how the radius or center were related to the equation, that would come later. 

Wanting to continue the exploration I reached out, once again, to the Exeter problems for inspiration and pulled together a bunch of problems on Circles. We spent 2-3 days working, solving and sharing solutions. Some students grabbed graph paper, some fired up Geogebra and others tackled the algebra head on... 

Today - The Unit Circle:

Then today rolled around. It's the day before spring break, we're all fried - teachers and students. I had my doubts, but I went all in and laid my cards on the table. I started my 65 minute class by putting the following image on the board or at least a hand drawn rendition of this image.

I described the circle as centered on the origin and with radius 1. As a group they all chanted the equation. I described the line segment, that it started at the origin, angled up at 30 degrees then ending when it intersected the circle. I posed the question "what are the coordinates of the point where the circle and line segment touch?" I mentioned that I could think of  3 ways to solve it and that there are probably more... I put them in groups and let them have at it. 10 minutes later 2 ways to solve the question were presented both with correct answers. 

Still, I made no overt connection to trig. At this point I emphasized that the students could use the equations for the line and circle (one group correctly pointed out that the tangent of 30 degrees is the slope of the line) and their GDC to find the intersection point. 

I put up the following spreadsheet and explained that I wanted them to repeat the process with more angles. Each group had a color that corresponded to different angles (see spreadsheet). When a group had an answer they quickly added it in on the spreadsheet. When all the results were up and the resulting patterns discussed and explained I quickly added the sin(x) and cos(x) columns to the spreadsheet. You could taste the learning that happened in those few moments. 

Now for the hard part... 

Graphing. This one took some work to explain. The idea of graphing "x" on the vertical axis made some heads spin. We talked about going around the circle more than once or going backwards and what that meant in terms of the graph... Here the GDC helped out wonderfully.

Class ended with the "non-mathy" students declaring, "I love it when you work hard and then it all makes sense" or "I really like it when I understand things." I felt like a freaking math teacher ninja! 

Reflections:

To be honest I don't know where the stage for this success was first set (what I described took 3 weeks - 8 to 9 hours in class). While having all the mathematical tools needed (circles, constructions and right angle trig) is necessary I don't think it's sufficient. Two more pieces were needed:

1. Students feeling the freedom to tackle a problem with different methods, thus allowing them to see problems in the context that is most natural to them.
2. Students being trained to solve new tough problems not simply repeating steps that the teacher has demonstrated on a whiteboard. 

What got me thinking this week - 3/15

Some of this stuff I agree with some I don't, but it's all bits that have made me think.

• The White Paper - Paper out of Stanford discussing blended learning. A bit on Chromebooks in school. 
• Fun Stuff - Post on some of the coolness in the Exeter Math problem sets.
• TED - How simple ideas lead to scientific discoveries.
• Khan't Ignore How Students Learn - More good stuff from Frank Nochese.
• Quantum Progress - The importance of teaching computational methods in physics.
• Case for Guided Learning - Kirschner & Sweller. Loved their ideas about Cognitive Load Theory, based my thesis on it. Can't say I love their thoughts on inquiry. 

Harkness Method

I am currently rather obsessed with Exeter Math and their Harkness Method/Philosophy. I am intriqued by how simple and different it is from the likes of the Khan Academy (or rather the philosophy that seems to guide KA).

I found this great short video. I'll let it speak for itself.

 

More Goodness:

This is what is meant by an educational philosophy, this is what KA is missing. A "good" explanation by a teacher does not equate to real student learning. 

Exeter Math

I recently (re)discovered the Exeter Math books. I had seen them once before but apparently didn't think twice about them. Now I can't get seem to get enough of them. I've been obsessed all week.

The books are simply problem sets. Designed to be done in order and more or less completely. No mindless context free pictures. No chapters. No definitions. No glossy pages with theorems. The questions are not even broken up by topic. The books are just filled with math problems.

Exeter runs them under/with their Harkness Philosophy. They describe it better than I can, but it's highly student centered, which in my book makes it worthy of more research if not emulation.

Some of the problems are not unlike word problems in a typical text, some are very tough and some are gems that can be solved half a dozen different ways. My favorite so far is:

This is a modified version I used in a exam, but its the same idea.

Rich Beveridge also describes his solution to the problems below that involved Fibonacci numbers and Phi.

The problem sets appear to be a potential backbone for true continuum of math classes. No more starting off on Chapter 1 of a new book just because its September. Topics truly spiral through the problems sets. No more boring the bejeebers out of the kids by doing 20 problems that are exactly the same. I see so much potential...

Take a look at the books, see what you think.

Update: Glenn Waddell posted a great series on Exeter math. He's got great insight into Exeter's pedagogy and has posted more resources than are available on Exeter's homepage.