Geogebra, Beats and Tone Generators

For one reason or another I was looking over my IB Physics site. I haven't taught IB Physics is 6 years now and so I don't spend much energy updating the site. I happened onto my page about "Beats" and found that I'd written one of those "clearly, it's obvious" type justifications with zero real justification. Oops.

I got to thinking about tone generators to build up some level of justification, but I couldn't (quickly) find a web-based one. So I turned to Geogebra. Lo and behold they have a PlaySound command. Awesome! An hour later I was able to put together the an applet which could certainly be expanded to create a more complete web-based tone generator. You've got to love them Geogebra folk.

Geogebratube: http://www.geogebratube.org/student/m21179

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.

What would you do?

I work at a small, independent and expensive school. We work in a (international) community that does not have many other educational options for students. So here's my dilemma...

In one of my math classes I have 7 of the 8 seniors and there is some very difficult classroom dynamics and chemistry that comes with such a small group that also lacks academic leaders. At this point I see 3 students that are sources of great distraction or negativity. They seem almost proud of their dysfunction. I also see 3-4 students who are quieter, more introspective and who are ready to learn and succeed, but struggle to overcome the distractions of the other 3. I have had individual conversations with this second group and they have expressed great frustration with their classmates (in all classes).  These 3 distracting students are, in the words of a grade 12 student, “hindering our education.”

For me that is a big fat line in the sand that you don't cross. To mix, confuse and create my own metaphor... You can lead a horse to water and maybe you can't make the horse drink, but you sure as hell better not let one horse stop the rest of the herd from getting a drink.

This is not a new problem, but with IB exams coming all too soon there is a sense of urgency I feel more each time I wake up. The school has failed to solve the problem and many teachers seem content to simply chuckle and shake their heads.

So. What would you do?

What Got Me Thinking - Sept 14

Whiteboarding Mistake Game: A Guide - A great addition to whiteboarding in class. A long post, but worth the read.

Clever Hans - A story of a clever horse that "could do math" and a reflection on how it could and should change what we do as (math) teachers.

"The Exeter (Math) Series" - A series of posts on Exeter Math. Includes many resources, discussions of pedagogy and much more. A must read for those interested in Exeter Math.

The World of Mathematical Reality - via Keith Devlin

Robots & Calculus - Part 1

As part of SL Math I have to teach Kinematics, last year I sort of rushed through it. I was too easily convinced by the smiles and nods that learning was happening. In calculus, I've never felt like my students accomplished much more than simply learning a series of steps to get the right answer. Sure they can solve the problems and most of my kids got very good IB scores last year. But! I wanted more.

Too often in SL Math kinematics is window dressing and an excuse for context on a calculus problem and rarely more than that. As a physics guy I see kinematics as the driving motivation behind differential calculus. So this year I swore I'd actually walk the talk. In come the robots!

My school, like any other, has its problems, but we are blessed with dreamers and an administration that are willingly to fund those dreams. 4000 euros later we have lego robots coming out our ears. Our school already supports two FTC teams, one middle and one high school team, so it wasn't a stretch to find more funding for robots.

The students were given about 40 minutes to create their "car" with the two stipulations that it had one motor and allowed for easy removal of the NXT brick (so other classes could use the brick).

And now for the calculus...

Part 1 - Constant Velocity

I pre-programmed 7 functions for the students. The first 3 are constant velocity functions. Student's used Logger Pro's video capture and analysis features to create position and velocity graphs and equations for each function.

Students wrote each set of equations on the board so as a class we had more data from which to make conclusions. I proceeded  to math-teacher-ninja them to see that the slope of the velocity equations was virtually zero when compared to the other parameters. Then the jump to see the similarity in the slope of the position and the constant value of the velocity was quick and easy. Leaving the students to conclude (written in my 2-grade hand writing):

If s(t) is linear then v(t) is constant.  Velocity is the slope of position. 

What more could a teacher ask for?

After these conclusions the students worked some very simple problems. Working with equations, graphs and most importantly writing rules to go from linear position to constant velocity and vice versa. I dangled the idea of the integration constant in front of them, not looking for full understanding, but trying to raise awareness that information was lost or is missing when moving back and forth between position and velocity.

Next up: Linearly increasing velocity in Part 2. Coming soon...

The Class I Always Wanted

In high school and college, being the nerd that I am, I always dreamed of a fantastic course where I and other students simply got to ask cool questions and where we were given the time, resources and space to find the answers. I dreamed of learning to program something that I wanted create - not being told what to create. I dreamed of building a robot or a contraption that did nothing useful but was fun and challenging. I dreamed of a true nerd elective where problems came first and solutions came second, where problems came from creativity and solutions came from a desire to learn and not from a lecture.

Next year I get to create that course. I get to teach a Nerd Elective! The course was created on the premise that many students at my school were not being challenged with the current math and science structure. We have the beginnings of a robotics program and the resources to support it. The stage is set. Now all I have to put the structure together and create the environment for the students to thrive in...

The Plan So Far...

• I will give a brief intro to some ideas: Lego robots, iPad app development, Flash programming, Kodu Game Lab (maybe) and anything else I can find in the next 5 weeks. 
• Students will create their own question or project including time frames with goals, deadlines and expected challenges. 
• Students will work alone or hopefully in groups of 2. 
• Final choice of projects and deadlines will be supervised and negotiated. 
• Projects MUST be interesting to the STUDENT. 
• Student will create a record (blog?) of there work including discoveries, progress and challenges.  

What I need...

• Framework to foster creativity and maintain accountability. 
• Rubrics to grade student work?
• Reasonable expectations for students. 
• More project platform ideas. 
• General advice. What am I not thinking about? 

Piece-wise Function

My IB students are working on a portfolio project that will requires some piece-wise functions to model. I used the function below as a tongue-in-cheek example of a piece-wise function. Including the idea that changes in function type ideally have a contextual reason or event that changed the system or scenario being modeled.

Oddly enough they all understood what I was talking about...

What Got Me Thinking - June 1st

Mindset - Carol Dweck's book discussing the ideas of "closed vs. growth" mindsets. A very good (and easy) read. If you haven't read it you need to. It was the summer read at my previous school and for my current school.

Grit - Character is a better measure of student success than most anything else. Similar idea but maybe with more resolution than Mindset.

Our Buggy Moral Code - Dan Ariely's TED talk. Not really math/physics related, but pretty interesting research on irrational behavior. He touches on the ideas of pain but mainly focuses on cheating and what influences when and how much we cheat.

Flipped Classroom? - An article on the ever more talked about "flipped classroom." Why does a video lecture seem so much better than asking the kids to go home and read a chapter? I can't help but think that a flipped classroom is just passing the buck...

Building a New Culture of Teaching and Learning - Great commentary on schools and learning. Worth the time investment to listen.

People Stuck on an Escalator - Was used by a former head of math to set the stage for an all faculty discussion on Mindset. He also used it to start his classes each year.

Inspired by Coffee

The other day I was sitting in a coffee shop and thought to myself, "Self, this is a nice place to be. I like sitting here." To which I responded, "Yeah, why don't our classrooms look more like this?"

Not the actual shop, but you get the point...

Internal monologues aside, I have honestly begun to wonder why we don't draw more inspiration from places that people enjoy spending time in? What if our students actually enjoyed being in a classroom? Next year, for the first time in 7 years, I will have my own classroom... I desperately want to make it a space that feels good. I might even spring for an espresso machine. 

Today in 9th Grade Math

There is something to be said for the gamification of education. I couldn't believe how motivated my students were to win little plastic rectangles.

Today we started playing Blackjack!

I had no real curriculum with my 9th graders so I've used the year to explore ideas... I've had some great successes and some lackluster results. Last year the previous 9th grade teacher had taught some probability so I figured I'd follow his footsteps, but I wanted to do it by playing games not by marching page by page through a textbook.

Today, we started playing blackjack. I'm not 100% sure where we are going math-wise, but I know there is more math in Blackjack than my students (or I) can handle. There is a lot of great thinking in developing a strategy and plenty students can learn through simulations and gathering experimental data. I can't wait to see where this goes!

Death by Definition?

In the world of standards and the increasing need to document and define, what does Robert Pirsig's thoughts on Quality mean for us as teachers and for our students?

"Quality cannot be defined. If we do define it we are defining something less than Quality itself." – Wikipedia

I can't get this quote/statment out of my head. Wondering if we are better off to lead by example than by definition or standard. Isn't that the danger of math that we boil everything down to well-defined rules and turn our classes into exercises in completing algorithms?

Introducing Radians

I usually introduce Radians via the unit circle making an theatrical argument that degrees come from Babylonian religious views regarding numbers with lots of factors and their base-60 number system (all true or at least plausible as far as I know). This begins to make the argument that maybe a better system could be devised, one based on logic and geometry, but this approach has always fallen short of allowing the students to make the jump to radians.

So last year I began to experiment with sectors to build up arguments that arc-length and radius determine the angle. I saw some success, but I had left it too unstructured for the students to reach a meaningful conclusion. When I tried to make a jump to radians I got a lots of looks that said, "I got what we just did, but what does THIS have to do with THAT?"

This year I gave the sector thing another shot, but this time narrowing what I asked them to look at. I defined arc-length and a sector and got them working on tasks like:

At this point I asked them to create an equation relating arc-length, radius and the angle. A few arguments broke out (Awesome!), a few questions were asked and virtually all of the students wrote down something akin to:

From this I asked the question "if you know the arc-length and the radius can you tell me the angle?"  I had the students draw a sector whose arc-length and radius ratio was 2 and then measure the angle... Each student had a different sized sector but all had the same angle. This lead to a conversation about ratios where a student pointed out that: 

Awesome! 

With a little math teacher magic, and no mention of radians, by the end of the 65-minute period I had most of the students agreeing that:

The next class we talked about the "arc-length" of a full circle. We worked our way around the unit circle talking about half and quarter circles before jumping into smaller angles and coming up with radian equivalents for the typical 30-45-60 angles. Only at the end did I finally mention the name "radian." 

I've followed this up with a few "Pop Quizzes" that were not graded, but just a bit of practice estimating and sketching angles in radians. Quiz 1 and Quiz 2. 

The result is the least confused class I've ever had. Dealing with radians still takes some effort, but they seem to get it.

My handout with the sectors can be found here via Google Docs. The rest of my trig bits (a work in progress - isn't everything?) are in a Google Doc/Drive collection. 

Can You Guess When The IB Exam Was?

A few years back I posted my physics class notes on a wiki after an email request to keep the notes publicly available. As all good nerds do I watched the traffic with Google Analytics. I'm not here to brag, my site has seen less traffic in 4 years then a successful commercial site sees in a week, but I noticed something interesting the first May that the site was up and running... There was a huge spike in traffic for 1 or 2 days. The same happened the next year. Can you guess when the IB Physics exams are?

If we think our students aren't cramming, we're wrong.

Update: David posted the image to Dan Meyer's 101qs.com. Man, was I wrong, it generated all kinds of questions...

What Got Me Thinking - May 18

Student's Brain Flatlines in Class - Differences in brain activity between class and labs is amazing. One more argument that what we are (often) doing is not engaging our students.

Estimating Gravitational Field Strength

Saw this on Dan Meyer's 101qs.com this evening. What a great photo to clearly show the gravitational field strength on Earth.  I don't like the term "acceleration due to gravity" as its wrong or at best misleading and incomplete, but that's the subject of another post.

This could make a homework assignment and or a great introduction to Logger Pro's video and photo analysis. Plenty to talk about in terms of estimation (of lengths), center of mass, horizontal velocity, etc.

BTW TinEye Reverse Image Search is a great way to find similar photos or in the case above same photo but different resolution. 

Free Dropbox Space - Am I Cheap or What?

I hate monthly bills, I just despise them. So the thought of paying a monthly fee to store my files with Dropbox frustrated me - no matter how much I appreciate the service. IB students are required to do written work - even in math. They turn their work in via Moodle and I return it via Dropbox in a shared folder. This had two results:

1. They will now have an digital portfolio of all their work.  
2. I got free space! 

So far 2/3's of the students have completed the necessary steps for me to get extra space! Not too shabby and cheaper than the Lifehacker solution.

If you haven't tried Dropbox and want to here's a link (wink wink) Dropbox.

What Got Me Thinking - May 9th

TEDX - Gaming for Understanding: Not the usual take on learning through gaming.

Mathematics Education for a New Era: Video Games as a Medium for Learning - I am enjoying the arguments regarding the validity of games (and gamers) the book it continues to shed light on why and how games can and should connect to education.

Toilet Breaks Land Speed Record - At least BBC doesn't make this headline news like CNN might...

Google Apps is Everywhere - US DoD is adopting Google Apps for 90,000 employees. If the US government can do something then any school should be capable too. -- My school is in the process of adopting Google.

Changing Assessment - Talking points from David Martin (along with his Prezi). Good Stuff.

3D Vector Magnitude

After watching my current IB year 2 students struggle with vectors (I took over the class this year) and it was the 1's turn to push slog death march through vectors I was determined to slow down and make sure they understood the basics... Or at least give them as much time as possible to create some conceptual understanding (certainly not the same thing).

We started vectors by talking about different notations and their meanings. We talked about the magnitude of a 2D vector. They could see the clear connection with Pythagorean Theorem, a few even suggested its use, but the 3D formula was not clear to them. Not at all. Students suggested we could use a cube root or cube the coordinates. Both potentially valid extensions of the pattern for sure. I ended class by giving them the formula with no proof no justification - time was running out and I had problems I wanted them to finish. Oops.

Realizing the obvious error of my ways, I started the next class with a model of a 3D vector (I used a wire coat-hanger as the vector and large graph paper to form the X-Y plane). My model was just to show them what I wanted them to build. Their task was to use a meter stick as their vector - which required some engineering - and find the coordinates of the vector in 3D. A few students suggested aligning the vector with a coordinate axis... I nixed that idea.

Once they had the coordinates I asked them to prove the formula given in the last class or better yet  to ignore the formula and use only the coordinates to find the length of the vector - which they already knew from the meter stick. 10-15 minutes later folks had drawn right triangles and created a solution!

Once everyone was finished, I pulled the whole class over to one of the more neatly created models so I could point and formalize the ideas.Towards the end several students commented "oh that makes sense," or "now I get it - that formula does make sense." Voila! A simple but effective way to spend 40 minutes of class.

How to Teach Math Anxiety

Maybe this one has been bumping around the interwebs for awhile... Today was my first encounter with it thanks to David Wees. Too good not to share.

Spring Break!

It has finally arrived. Two weeks to eat, sleep and ride my bike. I'm looking forward to the night I don't wake up thinking about school... I love teaching but this break is much needed and hopefully much deserved.

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.

• 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 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.

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. 

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 http://youtu.be/zxJgPHM5NYI

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.

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.

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

Pseudo Context Minus the Context?

Saw this gem in my textbook today. It's maybe the third or fourth time this year I have cracked the book. This was a good reminder why I do it so infrequently...

The only thing that is decaying is our student's attention or their hope that math class might be remotely interesting.

Digesting a Math Conference

I am trying to digest and process my experience from an IB Math conference this past weekend. This was also my first time in a small room with so many other math teachers (Scary!).

Comments, ideas and the like that surprised me or caught my attention:

• Teachers are most scared of "Inquiry" because of time constraints. 
• We discuss "speed" or "efficiency" in a classroom with no clear definition of those terms. 
• We need (successful) Inquiry and Student-Centered classrooms to be modeled, not simply written about.
• Many teachers are concerned about "cheat sheets" and students NOT memorizing.
• Teachers seem to attribute significant importance to memorizing.
• Many teachers talk negatively about student abilities. 
• The IB has goals for students beyond getting high marks yet, many teachers seem heavily focused on (internal/external) assessment results. 
• Teachers seemed hungry for a recipe to "write labs" or to "teach an idea" suggesting discomfort with vagueness. 
• Teachers have a knee-jerk reaction to "why are we doing/learning this" and to "when will we ever use this?"
• While the IB is very aware, I was not aware of the apparent divide or conflict between the IB Learner Profile and the way IB Math is (often) getting taught.

Well there you have it. I will readily admit that I am reading into many comments and conversations making quasi-conclusions based on my own biases and preferred lingo. I have no way of knowing what these teachers classrooms are really like, but still I left feeling discouraged.

If nothing else the conference got me thinking. 

Filing Cabinet

In the spirit of sharing and maybe getting some feedback and at the risk of overwhelming others I have shared my collection of files. So far just the ones in Google Docs... There are a few in Dropbox but they are making their way in to GDocs as PDFs.

Filing Cabinet - Click Here

Teaching to the Test: IB Fever and Numb Face

It has been a busy, busy year. I have been cranking out new ideas and new content left and right... Some good, some barely okay and maybe the occasional "damn that worked pretty good". I dream of my old schedule with only two classes to prep and so much more time to think, to process and to research new ideas. Many teachers have it worse than me, but I am finding four preps the limit of what I can do and feel good about. I have yet to fumble too badly, I haven't forgotten what class I have next or forgotten to prep a class, but there have been a few close calls.

I have never taught in a public school and or truly been a slave to a standardized test, I count myself lucky on that score... This year my wife and I returned to teaching overseas and I deeply wanted to return to teaching IB. I loved the rigor and I loved the depth of understanding needed to receive the highest marks. Yet my experience this year with "IB Fever" has made me want to have nothing to do with the IB. I want to grab the wonderful resources of past IB exams and the syllabus and run away to craft my own course free of the score driven and uber-IB focus that my school has managed to cultivate.

A colleague conducted an informal poll asking other IB teachers "We do so much here, what do you think about getting rid of the extras and just teaching IB and IB prep?" He was referring to sports, mandatory after school co-curricular (such as robotics, math competition, ping pong or rock climbing), ski week and anything else not directly prescribed on an IB syllabi. The overwhelming response was "yeah, sounds great, lets do it."

It made my face go numb to hear it.