How the Many Worlds Interpretation of Quantum Mechanics Almost Cost Me my Job

End of the Year

Time permitting, I like to end the academic year with Special Relativity with an emphasis on how it allows time travel. We then watch the Nova episode Time Travel with Kip Thorne. One segment uses the Many Worlds (MW) interpretation of Quantum Mechanics to suggest that the single-photon double slit experiment shows that time travel is possible between parallel universes.

DAVID DEUTSCH: The photon that we can't see is a photon in a parallel universe which - a nearby parallel universe which is interacting with the photon in our universe and causing it to change its direction. The result of these single photon inference experiment is the strangest thing I know. It is conclusive evidence that reality does not consist of just a single universe because that result could not come about unless there were another nearby universe interacting with ours.

NARRATOR: If true, this idea has profound implications for time travel.

DAVID DEUTSCH: When one travels back in time one does not in general reach the same universe that one starts from. One reaches the past of a different universe.

To illustrate MW (that when an action has different possible outcomes, the universe splits so that each possible outcome happens in a different universe), I grab a ball from my desk, wind up, and "throw" it at a student (I actually drop it behind my back). There is a collective gasp and then relief that no ball was thrown. I then tell them that according to MW, there is a universe where I did throw the ball. We all giggle a little and move on.

This year, in my last class, the ball didn't drop behind me. Fortunately, I narrowly missed a student. After a moment of shock and then profuse apologies, I got back on track and said "In another universe, I didn't miss. (pause) And I am now being lead away in handcuffs. (pause) And because I can't grade your finals, you all get 100's."


Next year, I'll use a foam ball.

Not Taylor, but Cordic

There are certain "Wow" moments in the education of a mathematics student. For me, they include Cantor's discovery of different levels of infinity, Gödel's Incompleteness Theorem, and the Lancaster War Model. But on most student's list will be the Taylor Series.

 Geometry and Arithmetic

To see that a geometric quantity (the sine of an angle of a right triangle), the ratio of two lengths, can be calculated with an arithmetic formula is astounding. And to implement this formula with only a few lines of computer code boggles. It is then tempting to assume that this is how your calculator does its calculations. And it probably would be except for one fact; your calculator is dumb.

Addition is easy, Multiplication is Hard

Since a calculator at its core uses binary arithmetic, it can easily do addition. You probably learned how to do it grade school. But multiplication is harder, with one exception. It is easy to multiply by 2 (10 in binary). Let's look at a simple example, 5 x 2 = 10. In binary, it is 101 x 10 = 1010. To multiply a binary number by 2, all you need do is shift the digits to the left, and shifting bits is very easy for a calculator's CPU to do.

The problem with using the Taylor series is that calculating just the first 5 terms of the sine expansion requires more than 50 multiplications and divisions. However, there are other ways of calculating the sine of an angle that, while harder for humans to do, are easier for computers.

COordinate Rotation DIgital Computer

An alternate method of calculating sine (and other transcendental functions) was developed in the 1960's that uses multiplications and divisions by 2. The details are beyond the scope of this post but can be found using the links below. At its heart is the decomposition of the angle into the sum of a set of specific angles.

Going from one angle to another is thought of as a rotation in space. The angles are picked so that the rotations are easy for the computer to sequence and calculate with. While the CORDIC algorithm was developed for early binary computers with limited core functionality, it has been adapted for calculators as evidenced with this from Texas Instruments

 Past threads on Graph-TI asked about the internal methods
     used to compute trigonometric and other transcendental functions.
     We wanted to add some specific information to this dialog so that
     someone could perhaps develop a short module for the classroom if
     they would like.  This topic should be interesting for background
     on calculators or as a good example of a math application.

     Most practical algorithms in use for transcendental functions
     are either polynomial approximations or the CORDIC method.  TI
     calculators have almost always used CORDIC, the exceptions being
     the CC-40, TI-74 and TI-95 which used polynomial approximations.
     In the PC world, the popular Intel math coprocessors like the
     8087 use CORDIC methods, while the Cyrix 83D87 uses polynomial
     methods.  There are pros and cons to both methods.

The details of CORDIC can be found at various sites with a few below. 

Links

A paper on the CORDIC algorithm for the general student (from the site that hosts WinPlot which was used to generate the opening graphic)
http://math.exeter.edu/rparris/documents.html
http://math.exeter.edu/rparris/winplot.html
Sadly, Mr. Parris passed away a few years ago.  The school recently de-activated his account.  The paper can now be found here.

A paper from the Texas Instruments Calculator division about how their calculators use CORDIC
ftp://ftp.ti.com/pub/graph-ti/calc-apps/info/cordic.txt

How CORDIC was implemented on the HP-35 with details on the bit-shifting operation
http://www.jacques-laporte.org/Trigonometry.htm

The ubiquitous Wikipedia entry
http://en.wikipedia.org/wiki/CORDIC

A Kvetch

The Technology Treadmill

 

And so another major change is coming my way. For years, we have been using Moodle as our classroom management system. Most teachers in our school just used it as a way for students to get materials they missed in class while some teachers were more advanced and used it to collect and grade some assessments.

A handful of us used Moodle to administer (and grade) multiple choice quizzes and tests. It allowed the randomization not only of the answers within each question, but also the randomization of the question order for each student.  Administration encouraged that use because it would reduce the use and purchase of the "bubble sheets" we use (at about a nickle each). 

Knowing that I would have such questions available for years in the future, I willing spent 5 to 15 minutes crafting each appropriate multiple choice question (see a previous blog post on that). Some of those questions had graphics in the question text and also in the answer choices. I also spent hours figuring out the Moodle way to include a semi-random number generator in my question text and how to craft appropriate formulae in the answer text, so that each student would see the same question, but with different values. For example, I could ask a typical optics question of calculating the image distance given a focal length and an object distance where the givens are random multiples of 10 or 20. I now have a growing collection of MC questions appropriate for my classes.

Just recently, administration has let loose through the grape-vine that we will be switching CMS's to Schoology. No teacher input. While it is claimed that the switch-over will be gradual (over the next academic year) and painless (supposedly we can just copy over our material from Moodle directly into the new), veteran teachers know that such major changes rarely go as smoothly as is claimed.  

I have had a brief chance to try the new CMS, Schoology. My first impression is that Schoology is designed to be easy for the 95% of teachers who use only the basic features. And that is where they are spending most of their development time. That makes sense for a commercial product. 

However, our old CMS (Moodle) is open-source and community supported. There are a large number of outside developed modules that vastly extend the basic product. For example, one person decided that it would be nice to be able to insert Geogebra interactive graphics easily. So that person spent some time, wrote a module to do so, and then offered it to the general public on Moodle. The Schoology is closed-source. So any new feature has to be begged-for and the company has to decide if it makes economic sense to work on that feature.

It is like the difference between Linux and Microsoft Windows. MS is easier to use, however, Linux allows you to do what you want to do if you are willing to spend some time modifying things. Windows is nice for the 95%, but Linux is better for ultra-power-users. And Linux users like to show off and share. Moodle is like Linux. Schoology is like Windows (or even worse, Apple).

For example, there is a very powerful feature in Moodle that allows a teacher to make a quiz with questions using random numbers or numbers taken from a dataset. Here is an example. "How far does a car go when it has a speed of 'x' m/s and travels for 'y' minutes?" I can chose to let 'x' and 'y' be completely random numbers, or random numbers in a set range, or numbers chosen randomly from a list I specify. Obviously, this feature is of interest almost exclusively to teachers of the mathematical sciences. It took me a while to master the syntax needed for these questions, but I found that feature a helpful one. I could tailor my online quizzes according to my educational philosophy on such assesments. Several people have asked for such a feature on Schoology for over a year, but there has been no response from the company. So, for me, a critical feature is no longer available.

Since it seems that the switch-over is going to happen, I tried the new site. Trouble. Quizzes transfered over with graphics only in the question text. No graphics in the answer choices. I also have several practice quizzes that were made as websites. As such, the files for question pages and answer choices have to have a specific file structure for the links to work. All the quizzes and all their supporting files were dumped into one file folder. The links are borked. So I will have to waste several hours recreating content instead of enhancing it. I haven't even looked at the questions where the answers are generated from a random value given in the question and an appropriate formula in the answer text.

I don't have high hopes that these problems will be fixed when we officially switch over. I am the only one at my school using these advanced features of our current CMS. I am sure the company at Schoology will look at the problem and then issue a shrug of the shoulders. It will be left to me to sort things out. So, instead of writing new questions that try to get students to think before reaching for a calculator, I will have to rewrite the old ones and try to figure out the Schoology way of doing things.

There is one concession to my plight. Through a grant, my school is hosting a weeklong summer work session for a core of teachers to pilot this new CMS. We will get together and try things in a workshop setting, and we will get a stipend. So at least I will get some monetary compensation for my pain. I hope it works out to at least one-tenth minimum wage for all the time I will have to spend implementing the change-over.

  

Addendum

The new CMS company does have a support site. I looked there to see if other users had some of the same concerns as I. Some do, however, it has been over a year since the first request was made for the random number use in quiz questions. Nothing yet. I have no hopes for the future. I

Optics and the Caveman Brain

You see with a Cave Man Brain

Ray-tracing in optics is one of those skills that is enjoyable to teach in class. A simple procedure, but with powerful results. Every year, I see a couple of students who struggled with the algebra of motion analysis who get a sense of accomplishment when mastering this technique. However, a few years ago, I noticed that many students seem to believe that the rays stopped at the image location. That is when I introduced the Giant Eye and the Caveman Brain.

I now start optics with a simple description of how the eye-brain combination works, that our brain has evolved to instinctively know that light has traveled a straight-line path from object to our eye. To illustrate, I get two volunteers to play a caveman and a fuzzy bunny.

We discuss the role of sunlight in seeing, how light reflects off the fuzzy bunny and goes in a straight line to the eye of the caveman. What would happen if the caveman's brain told the caveman that, while the light from the fuzzy bunny came from in front of him, the fuzzy bunny is over to the right? I have the caveman then throw a "spear" over to the right. I ask the class if this caveman will get to eat tonight. What if the brain told the caveman that the bunny was in the same direction that the light came from? The class caveman then throws the "spear" and "kills" the fuzzy bunny (a chance for some hammy acting). Caveman gets to eat tonight! And since I was on a roll, a couple of years ago, I added "Caveman makes fuzzy bunny slippers. Caveman gives to to cavewoman. (slight pause) Caveman pass on genes!"

While the actors are getting back to their seats, I emphasize that evolutionary pressure has made it so that we instinctively know that the light that enters our eye has traveled a straight line from object to eye. That lesson seems to help students understand lens and mirror optics later.

If spring starts Thursday, why does Tuesday have 12 hours of daylight?

17 Mar has 12 hours of daylight, but spring is 3 days after. Why?

Why is the Vernal Equinox confusing to students of Latin.

Ask most people what happens on the first day of spring and you will get "12 hours of daylight and night" (i.e. equinox), which is close but not quite right. Some might say the northern hemisphere of the Earth is neither tilted away or toward the Sun. A few might even state that on that day, the equatorial plane of the Earth crosses the center of the Sun. However, these last two explanations, while correct, don't explain why the first day of 12 hour-daylight happens a few days earlier.

The way I use in class to explain what is going on is to say that on the day of the vernal (or the autumnal) equinox is when it takes 12 hours for the center of the Sun to cross the horizon in the morning to when the center crosses the horizon in the evening. Since we start counting daylight when the first part of the Sun peeks above the horizon to when the last part of the Sun goes below the horizon, it becomes obvious that the day of the "equinox" will not really have equal day and equal night.

Please note that 12 hours of daylight happens on different days depending on how close you are to the equator (the closer, the more ahead of the the Vernal Equinox you will get the true equinox).

His contribution to the world, WinPlot

Richard Parris, someone who made a difference

As educators, we have a list of colleagues we admire, ones we know from experience make a difference.  Your list probably includes some teachers from your school and maybe from your school district. If you are fortunate enough to be able to attend many professional opportunities, that list includes some people you see only occasionally. And we may admire ones we know only from blog posts or tweets. For me, Rick Parris was in that last category.

Peanut Software

I first encountered Mr. Parris when, in the early 1990's, I was looking for a way to include digitally-drawn mathematical functions in tests and quizzes for a calculus class. I came across WinPlot. It had the nice compromise between small program size and ease of use, and number of features. It became a favorite tool when I was finishing up my graduate work and teaching as an adjunct at a local college, looking to impress the "real" professors. 

When I started my present job, I brought that tool with me, using it when I taught calculus and trigonometry classes. I noticed that Parris listened to feedback from users and made frequent updates. At one time, I wanted to create graphs where the axes are scaled in multiples of pi. So, with a few emails and a couple of weeks, that feature was added. Try doing that with a suggestion for a MicroSoft product. I wrote about using the program in an earlier post.

What is done regularly, is expected

I got used to the regular attention that Mr. Parris paid to WinPlot. I would check every so often to get the newest update. Recently, I noticed that he hadn't updated the program for a while. With a little bit of searching, I found that he has passed away. My heart sank. Someone who created a tool to help students, then noticed that fellow teachers could use it, then made the time over many years to make many changes that others wanted, is no longer with us. He will no longer be able to make WinPlot better. The education world is richer because he cared and poorer now because he is gone.
  

Addendum

I recently contacted a friend of mine who is now at Phillips Exeter where Rick Parris taught. There are no plans for anyone at Phillips Ex to continue with WinPlot or any of the other programs Parris wrote. I do not know how long they will be available (my guess is that the school's IT department will deactivate his account this summer). Even if you have no need now, you will want to download and try his programs here. There is much educational value here. You will find he will help you in your teaching mission.

Update

It has been a year since the death of Mr. Parris and the website is still active. I hope Phillips Ex keeps it going as a tribute to his life. 

My School has a Fantastic IT Department

  

Do they work with you or against you?

Networked computers are a fact of life in the modern classroom. However, many schools put barriers in the way of a teacher who wants to try new software or add new features. Most schools I know of have a "locked-down" policy for their school issued computers. Any new software or modifications of features requires official permission. Some schools will process requests with glacial speed and others schools will  so in a few days. One district in my area erases and then completely re-images the hard-drives each evening. I know of one teacher who was almost fired because, in his enthusiasm to share with his students the neat things that could be done with an iPad and Vernier's Video Physics software, downloaded a $3 app without prior approval (fortunately his principal came to bat for him, but I am sure that there is a "letter in his file").

Each time I hear about another school's restrictive policies, I am grateful for my school's IT staff.

Trust the Teachers

I have just finished installing Linux on my newly-issued MacBook. This is the third school-issued computer that I have done this on. Each time I have asked permission from the IT head, expecting a polite refusal, but the only restriction I have received is that they would not be able to give me support for it. It does help that the IT head is something of a penguin-head himself and knows that there is a world beyond the Win-Mac duopoly. 

Why Linux?

As someone brought up with the Unix ethos of hack-able software (in the original sense), I have valued FOSS (free and open-source software). Linux is built on that ethos. I try to show my classes that philosophy also. When we do linearization of data using spreadsheets, I demonstrate with Gnumeric. When we study sound, I show them what Audacity can do. If I have enough time, I like to use Audacity to show how to rebuild a sound using only sine-waves (a reverse FFT). To do that, I had to take the source code and make a minor modification. Had that code not been available, I would be at the mercy of what the programming team thought I wanted to do. Here is the ethos at work; here is my work, use it, improve upon it if you want. 

With Freedom comes Responsibility

Since I am trusted by my school's IT head, I do not want to betray that trust. So there are certain things that I could do that I will not do with school computers. I would love to jail-break my iPad to circumvent Apple's "do it our way or no way" mentality, but that might expose our school to legal problems. The same with using "abandonware". OK, once I used some cracking software on my laptop to recover a forgotten administrator password, but it was with an unofficial blessing, otherwise my computer would have to have been completely re-imaged, wiping out all the added programs and customizations.

It is more work for them

I am sure that a totally locked-down approach to school-issued technology would make my school's IT department's job much easier. But they have adopted the philosophy of letting the teachers deciding how they want to use their technology. We are then able to show what we have discovered with not only our students but also with fellow teachers. This policy lets us be more spontaneous and innovative teachers. For that freedom, I am thankful.