## Geometric Algebra

Another teacher and I have been exploring Geometric Algebra in application to geometry and physics together on most weekends using the online collaboration suite Elluminate (through LearnCentral, which allows one up to three collaborators for free).  I love David Hestenes’ Oersted Medal Lecture, but it was a bit deep for someone not steeped in the culture of physics.  So, I was delighted when I found a series of more introductory lectures by Chris Doran and Anthony Lasenby.  Occasionally they get into deep physics, but the bulk of the treatment is easy to follow for the mathematician, and I enjoy their style of content presentation.

Using the tools of geometric algebra I was able to prove (easily) a theorem that I have never before seen in physics.  Concerning a constantly accelerating model

$\begin{array}{rcl} a &=& \text{constant} \\ v &=& v_0 + a t \\ x &=& x_0 + v_0 t + \frac{1}{2} a t^2 \end{array}$

one can (and usually does) prove the following intermediate steps

$\begin{array}{rcl} v-v_0 &=& a t \\ v+v_0 &=& \frac{2}{t}(x-x_0) \end{array}$

on the way to proving

$v^2-v_0^2 = 2 a\cdot(x-x_0)$

but, by taking the wedge product instead of the dot product, one gets

$v\wedge v_0=a\wedge(x-x_0).$

In other words, the parallelogram formed by final and initial velocity has the same area as the parallelogram formed by acceleration and displacement. Not only this, but on the way to proving it, one finds that $(x-x_0)=\frac{v+v_0}{2}t$, so that the displacement lies in the same direction as the average velocity, which we proved intermediately. It makes for a great GeoGebra applet, but I can’t embed it with WordPress. It’s also the quickest way to prove (also using the first relation that $v^2=v_0^2$) that range is $\frac{v_0^2\sin(2\theta_0)}{|a|}$.

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## Reflections after the first week

I just finished Reluctant Disciplinarian by by Gary Rubinstein, which sums up my own inclinations toward being a “softy”.  The things I take away from this book include:

• Start the first few weeks of class with lessons that have clear expectations for students and, in my case, may be similar to what students are used to.  This sets up the students for a minimum of culture shock.  To this baseline I can slowly add more pizazz.  The time that I have spent “substituting” for my CT have been a convenient baseline.
• Be mindful about over-talking and over-sharing.
• Criticize one’s own lessons with a list of “cons”.

My tendency is to become so absorbed by the content and zany lesson that I forget to respond to student needs.  Any misbehavior is then so shocking that I deal with it too publicly, thereby alienating the rest of the students.  I have seen firsthand how a “softy” becomes the “mean teacher”.  I have also seen that it is possible to improve, and so I hold that hope going into the next week, the beginning of which my CT will again be gone.  I will have to prepare the students for an upcoming test.  Reviews are always hard for me to do well because every student has different needs.  I know that the students will be expecting a review game on the last day before the test, so I hope that my CT was able to put something together.  If not I will have to invent something.

This Monday is also the first day of official classes for my University.  I wouldn’t be so worried by this, but I am also studying to take the Praxis Secondary PLT this Saturday, and it has been quite some time since I took Educational Psychology.  I’m finding that the questions are IMHO so poorly written that the right answers are flawed, which leads me to reject them and try to find truth in another answer.  As a result, I’m not doing so well on the practice questions, so I will need to put in quite a bit of studying all week.  If anyone from ETS reads this, I suggest that rather than rewarding the top 15% (for Excellence!), you reward the top 5% by refunding their testing fees.  If you offer to refund even other materials (like your lousy online review course with terrible questions), you may even encourage more people to buy those.

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Due to an unexpected event, my cooperating teacher had to leave for a few days, putting a substitute in charge of the class.  Thus begins my student teaching!

On the plus side (opposite the minus of being unprepared), my CT has already prepared lessons, which will give me the chance to teach in a similar manner as I get started.  This should make the classroom management situation a little easier.

## First non-professional-development day

Yesterday was a semi-productive inservice day.  None of the inservice topics were new to me, but we were tasked with improving on a previously written lesson using differentiated instruction.  Seeing as I had no previously written lesson plan, I assisted one of my CTs with improving one.  I plan to try to write one later using differentiated instruction methods to satisfy the principal’s call, but I haven’t yet selected a topic.  Although I agree with differentiated instruction in principle, I find myself getting anxious over a tendency to cater to each student’s preferred modality rather than build on weaker ones.  Surely some balance in this dialectic exists?  Perhaps even synthesized into a harnessing of group diversity to advance individual learning à la Japanese mathematics?  I like the idea of improving stale lessons, but the process lacks the accountability and collaboration of lesson study.

Today was the first day with students.  I assisted only in a minor capacity while trying to learn student names, something at which I am exceedingly bad.  The day, however, passed without incident, so I shall use the remainder of my evening to make some flash cards to help me learn student names (and for later when asking questions).  The one thing I will note, which carries increased salience now that I have read The Teaching Gap and its comparisons with Japan, is that there were an aweful lot of interruptions to class today.  The teaching could not help but be fragmented, though the teachers did the best they could.

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## Lesson Study

I just finished reading the updated (2009) version of The Teaching Gap by James W. Stigler and James Hiebert.  I know it’s a bit old, but I really like the idea of lesson study.  Moreover, I think that in the absence of superintendent and principal support that it might be possible to subvert this power structure by using the Internet to conduct distributed lesson study.  It might go something like this:

1. Work collaboratively with other teachers interested in the same lesson topic.  Tag it with any state, CC, NCTM, ISTE, or any other standards that are relevant.
2. Create assessments that can test the effectiveness of the lesson.
3. Give the lesson.  Gather any in-class paper that the students used (to scan), video record the lesson, and otherwise obtain any data you can.
4. Repeat steps 1-3 until the lesson is as good as it can be.
5. Collect the materials on a website on which other teachers can comment, rate, and view the materials.  Do not publish it until at least three teachers have been involved in the development of the lesson and at least two have tried it out in a live classroom.

Blogs already provide a partial mechanism for Step 1, helping teachers to gather a wider range of ideas to use in lessons, but this alone will not help to create better lessons, especially if individual practitioners misinterpret the point of the materials (see the author’s notion of teaching as a cultural institution).

Technorati couldn’t find much on “lesson study” or “teaching gap”, so I wonder how much these ideas have circulated.  The University of Wisconsin La Crosse has incorporated lesson study into a project for college faculty.  There is also NSF- and IES-funded research into lesson study, but it is still conducted by researchers!  The Education Development Center also has an NSF-funded project going.