March 30, 2012

What got me thinking - 3/30

Yet more of what got me thinking this week.

Coaching vs. Teaching - Lessons learned coaching applied to teaching. Good stuff.

Lessons in the Medium - How we teach is more important than what we teach. - Riley Lark

Now Hiring... - More good stuff from Riley Lark.

Relativity on an Escalator - Great account of using Dan Meyer's escalator videos in a physics class.

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.

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! 

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. 

March 26, 2012

What I read that got me thinking - 3/26

More bits and pieces I found on the interwebs that made me think. Some I agree with. Some I don't.

  • John Swelller - Interview by Derek Muller (of Veritasium frame) discusses Cognitive Load Theory and describes why "constructivism" doesn't work. I wrote my thesis on an application of CLT. I was shocked when I read about Sweller's view of constructivism.
  • The Relationship School - A shift in priorities for schools?
  • Khanversations - My first read on the blog Physics First Observations. I'm looking forward to reading more.
  • Minecraft Calculator - I'm convinced there are some great Design Tech type of projects that could be done with in Minecraft. Its cheap and easy to play - good potential tool for schools with tight budgets.

March 15, 2012

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.
  • TEDHow 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. 

March 12, 2012

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. 

Khan Academy: Good or Bad. It's not THE gold standard.

Like many other's I watched the recent Khan Academy piece on 60 minutes. Everyone has their opinion on the Khan Academy (I certainly do) ranging from thinking it's a God sent educational revolution to thinking quite the opposite.  Yet, might both be true?

Watch on YouTube at
If you think school and education should consist of talking at students for the better part of the period and then having them answer questions nearly identical to what you just showed them, then KA is potentially an improvement. No more lectures to prepare, no more examples to walk through, just time and energy to mingle giving help and advice as you go. If your classes have 30+ kids maybe this is a realistic (or only) way of spending some time with each student or having a better chance to individualize instruction. I remember my public school classrooms and full implementation of KA would have been a vast improvement.

Or if you think school and education should consist of students drawing conclusions in their own words and creating their own understanding then KA is nothing new its just more of the same albeit in digital/rewindable form. If you believe that learning is evidenced by transfer of knowledge not simply the recall of knowledge then the KA is a horrible idea. If you already spend the majority of class time sitting next to students engaging them in conversation about what they understand and what they don't understand then KA is a huge step backwards.

Sometimes beauty is not the only thing in the eye of the beholder...

However, after trying to digest all the fervor over the KA, I have one remaining complaint. The KA is being put out there as the gold standard as the best that can be done. The KA might be better than some are doing but it is not the top of the educational food chain. I would challenge Sal Khan (or Bill Gates, the $$$ for KA ) to go visit a truly student-centered classroom. Go watch Dan Meyer or Frank Noschese in action. What about the Exeter Math program? Before you declare you have THE solution go see what other solutions exist.

March 08, 2012

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.

March 06, 2012

How I Taught Geometric Transformations

Starting geometric transformations I had two goals. 1) Be as student centered as possible. 2) Get the kids to do a bunch of mental gymnastics.

I wanted my students to develop their own understanding of the ideas surrounding translation, dilation and rotation. I did not want to rush them from the concrete to the abstract. They worked in small groups, 3 or 4, a few choose to work more or less alone. I didn't explain a thing to them, at least not as a whole class. I simply showed them examples of transformations and asked them to describe the results and create the rules...
I continued on like this with each type of transformation. The students did not find them difficult, but I was also never asked "whats a _______ transformation?"

I gave them examples of transformations asking them to identify and describe the transformation that had occurred.
Here the questions began, but often they were addressed to another student and not to me, "the teacher." I kept marching on, letting students work in small groups at their own pace. I asked questions forwards and backwards, inside and out, anyway that I could think of to let students more fully explore and create their own understanding.

The result was a solid 2-3 week unit. The unit needs revision, but it worked well especially for a first iteration. All of my files can be found in a Google Doc folder, the naming convention is a bit non-conventional, but hopefully it is clear enough. Any comments or suggestions would be welcome.

To give credit where credit is due, many of the ideas and some of the questions came from Visual Math. All of the images were created by Geogebra

For those who prefer bullet points:

Topics addressed
  • Translation, rotation about a point, reflections across lines
  • Dilation in one direction and two, by negatives and by values less than 1
  • Vectors as a way of describing translations
  • Transformations of coordinate points
  • How to find the point of rotation
  • How to find the line of reflection
What was awesome?
  • It was nearly 100% student centered.
  • Students could work at their own pace resulting in varying amounts of homework
  • Homework was at a minimum, students worked and learned socially not in isolation
  • My students are solid on ideas of translation, reflection and dilation - all ideas they'll encounter again with functions
What was less than awesome?
  • Treatment of vectors was superficial - need quality problems to reinforce their use(fulness). 
  • Rotation bits were rough, but not terrible. Could be fleshed out more.
  • Finding the point of rotation and line of reflection were in the form of "follow these directions" not a structured or scaffolded inquiry 
  • I need more open ended investigative questions for the students to explore and extend their understanding - These could function as a final assessment
  • Ideas of symmetries could/should be added