I love it when a plan comes together!

I was working my way through right angle trig. It was going pretty alright, but I was looking for an exit strategy that would tie in RAT to the new topic... Polygons, Surds, Exponentials. I saw connections to all of them and I was flip flopping between topics unable to choose. Seventeen ideas running through my head... Where's my Ritalin?

Then Kate Nowack of f(t) fame tweeted a link to a video:

There is genius in this video. Maybe not original math genius, but genius all the same. The connection was obvious. Pythagorean theorem and maybe some trig. Best of all students would need to think about lines, angles and triangles. Perfect.

I didn't really like the way the question was framed, felt like he gave away a bit too much (even in the original video, the one above presents the solution), but it does make for a good video. Before class I drew a series of 4 dots on all of the whiteboards, providing spaces for the students to present their results with some semblance of order. Each time an improved result was written up the class was drawn to it and they madly checked each other's math.

I heard things like, "that side is a hypotenuse and the other side is 1, so that side (the hypotenuse) is more than 1!" Or, "you used 0.333 instead of 1 over 3 that's why it didn't work!" Freaking awesome.

30 minutes in, I stopped class and asked how many wanted more time? Almost unanimously they wanted more time. I gave them the option of seeing the answer, but they didn't want it. What more can a teacher ask for? Finally, I ended the 65 minute period with the video and a brief discussion. During which one student asked, "Do mathematicians really play with soap bubbles?" With a twinkle in my eye, I replied, "Yep."

Math is real. Soap bubbles are real. The learning was real. No pseudo-context needed.

So close...

I am teaching Geometry for the first time this year. Which means I get to teach right angle trig from scratch! In the past I have taught trig (circular) functions, but never the more basic bits.

I was supposed to teach "triangles" which they already knew and had for some time (my school seems to have accepted that math will be boring and repetitive). So rather than belabor the point, I did a quick review of terms that led to the idea of similarity. I had them apply the idea of similarity to bigger things, like the school building... or the playground equipment... or the local mountain top.

Within a class period most if not all understood the idea that the ratio of two sides of a triangle can only be changed if the angle changes, in fact when I brought it up half the class gave me that look of "duh mister, we figured that out." This was all done with no mention of sine or cosine.

With the real learning done, I started into a bit of formalization. With a short back and forth discussion that required students to draw a few more triangles to convince themselves of a the details we had the 3 main trig functions on the board in just a few minutes, but still unlabeled.

With nothing more to teach I finally labeled the functions...  I gave them some typical problems to practice with, only one of which I did on the board. Then they were off and running. I felt so close to a masterful introduction of trig. So very close.

Next on my list was inverse trig functions...

I posed the question (with a drawing on the board), given two sides of a right triangle how do we find the angle? I figured there would be lots of drawing and struggle to draw the correct triangle in order to measure the angle. So close, I could almost taste it. Somewhere somehow during this time someone shared that they could use the calculator to find the angle, i.e. inverse trig functions. They don't understand inverse functions, but they have buttons to push...

I was so close. Sometimes I despise the power that buttons seem to have over students.

Should I have given them a table of trig functions instead of allowing them to use their calculators? That seems so contrived, so fake, so false. How do I avoid this next year?

Transformations Take 2

This year my usual transformation of function investigation didn't seem to work. I suspect it was not the investigation itself, but rather my students adjusting to a new classroom culture, one where the teacher doesn't tell you everything...

We took a break from functions and function notations to work on an investigation with some toy cars (post of that may be coming if the results are half decent), but I promised the students that we would revisit the topic of transformations. Its simply too core of a topic to move on with half the students not grasping the idea.

Searching for a new way to tackle transformations, I found one it was brilliant! Then I forgot the idea and came up with this:

http://www.geogebratube.org/material/show/id/708

Using Geogebra I created a simple transformation applet (link just below the pic) allowing them to graph any function and adjust the 4 parameters, do reflections, plus the nasty bits of absolute values. Paired up with the applet was a worksheet to guide them through.

The results seem to be far more positive and constructive. They seem to understand that they will need to ask questions and do their best to organize the answers and their understanding. I will of course do a debrief afterwards and to try and scoop up one or two lost students, but "take 2" seems to be working

Throwing Stones...

The day after getting totally riled up by Sal Khan  I feel frustrated with myself. I find that what I am doing in my IB classes looks more and more like lecturing. Why is this happening?

A little background: For my own organization I write up notes and post them on a website, I began doing this years ago in physics and got more intent on publishing my notes at the request of another teacher. A request that came in via email...

The other day I assigned some reading from my site. I did this simply because I couldn't figure out how to convene the information in a student-centered way. Judging from their understanding it seemed to work or at least no worse than if I had lectured. Judging from Google Analytics they spent about 13 minutes reading, which is considerably less time than if I'd tried to lecture.

I can make excuses; I am teaching 4 new classes at a new school in a new country, I can't seem to get a grasp on what my students understand or the class dynamics of having 5 students makes discussions almost impossible. This last excuse bothers me the most as it is a result of my school's desire to "track" students in math. But in the end my IB classrooms are not the dynamic interactive classrooms that I used to have or want to have.

On the flip-side. If having my students read was (relatively) successful does that mean that reading is a good option? If I can't, for whatever reason, find a better way to expose students to material than to lecture is reading then good teaching practice? 

So if I start to use this tool more and more then how different am I from Sal Khan? 

Sometimes I feel like I'm throwing stones in my own glass house. 

Sal Khan did not Invent the Personal Classroom

As he so often does, Dan Meyer got me curious with his latest post on Sal Khan. I wanted to see and hear Sal Khan for myself (again). Several quotes were troubling, one of them was regarding schools and teachers who have been using Khan Academy videos:

They [Teachers] took a fundamentally dehumanizing experience - 30 kids with their fingers on their lips, not allowed to interact with each other. A teacher, no matter how good, has to give this one size fits all lecture to 30 students - blank faces, slightly antagonistic - and now its a human experience. Now they are actually interacting with each other.

Taken from Sal Khan's TED Talk  (starting at roughly  7:00).

I take major issue with the implication that no matter how good a teacher is he or she must lecture to their students. Being good has nothing to do with it. Its a matter of philosophy and approach. Its a matter of implementing best practice, which should involve as little lecturing as possible. Lectures are dehumanizing when the target audience is 30. How much more so is a video that is targeted at thousands or millions? Where is the ability to adapt to student needs? Where is the ability for students to ask questions?

The Khan Academy might be an improvement over teachers who don't teach (they lecture), but lets not pretend that a 20 minute video has created a sense of humanity in the classroom. Lets not pretend that videos watched at home allow students to talk and work together in class. It is the teacher who must realize that their time spent with students is best spent actually talking with and listening to students not talking at students from the front of the room.

I am not of the opinion that Sal Khan has it all wrong, but I think his perspective is skewed. He sounds a bit like Al Gore claiming he invented the internet. Sal Khan did not invent or allow the creation of a personal classroom he created a set of (boring) videos.

Mutual Humanity in a Classroom

A week or two back, I made a few large mistakes when I was beyond tired. (A new school and new country will do that to you.) They weren't mistakes that I could easily recover from as they were in print and I was struggling to function let alone think clearly enough to wright the ship mid-class. My students were shocked that I admitted to a mistake! 20 minutes later I heard them talking in the halls that the other math teachers never admitted to mistakes...

Last time I checked students and teachers, even math teachers, are all human. To ignore our mutual humanity (I  have yet to fully understand what that means to me) and pretend somehow that I as a teacher am superior or significantly different is to lie to our students and will likely make our jobs in the classroom even harder.

Homework as Review?

This year I am exploring the idea of using true homework solely as a review tool. I feel strongly that typical homework is rarely useful and 80% of the time (or more) it falls into one of two major categories:

• Student's know how to do it, so its boring, emphasizing that math is boring and repetitive. 
• Or student's don't know how to do it, so they spend hours being frustrated, emphasizing that "they are not good at math" or that "math doesn't make sense." And often the next class period is spent "explaining" the homework.

Yet I do see the benefit of students "practicing" and working through problems on their own.

So what if homework was just a review tool? What if students were given a week (or more) to grasp material and work through their questions in class? What if homework was assigned only on content that is not currently being covered in class? 

To me this looks something like this:

Where each color represents a different topic or sub-topic. Week 1 in class is spent working of Topic A. Then Week 2's homework focuses on Topic A. Finally, during Week 3 there is some sort of quiz or assessment on the topic. 

Things I like about this:

• Students are forced to work with a concept for a longer period of time, without giving up more class time.
• Students are given time to struggle and learn with the support of their classmates and the teacher rather than trying to learn at home by themselves.

Things I don't like:

• It doesn't begin to deal with the "boredom" factor. It might even emphasize it more...