The math curriculum and department of my dreams

About 5 years ago, I got to spend a day visiting the Park School Math Department. At the time, they were just beginning a project to develop a math curriculum from scratch, based around 14 Habits of Mathematical Thinking. (I’ve written about these habits before, and even tried to create my own set of habits of scientific thinking). The parts of the curriculum I saw at the time were outstanding—organizing a curriculum around 14 habits of mind—Tinker, Visualize, Prove—seemed like a genius move, seeing them atop every blackboard in the department, and hearing students actively refer to these habits throughout the day convinced me that the were on to something big.

In the intervening five years, the teachers of the Park Math department have developed a complete 4-year high school math curriculum, and they are willing to share it with anyone who asks for it. You can read more about the curriculum and see sample lessons from the curriculum on the Park Math Blog (another awesome math blog you should be reading), and if you want to see the whole thing (you do), you can email parkmathblog@parkschool.net.

Not only is this curriculum filled with outstanding math thinking and problems, it looks like a similar amount of care was taken with its appearance (it was designed by a teacher in the department, Anand Thakker). The text is beautiful. Here are just a few images to whet your appetite and get you to send the email for your own copy.

The most amazing thing about this curriculum isn’t contained in the the pages of these pdfs, it’s in the collaboration itself. Just read their acknowledgements:

This curriculum was written by members (past and present) of the upper school math department at the Park School of Baltimore. This was possible thanks to Park School’s F. Parvin Sharpless Faculty and Curricular Advancement (FACA) program, which supports faculty every summer in major curricular projects. In addition to the support of FACA by the Nathan L. Cohen Faculty Enhancement fund and the Joseph Meyerhoff FACA fund, this project was also funded by grants from the E. E. Ford Foundation, the Benedict Foundation, Josh and Genine Fidler, and an anonymous donor.
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (http://creativecommons.org/licenses/by-nc-sa/3.0/).
© 2006-2010 Tony Asdourian, Arnaldo Cohen, Mimi Cukier, Angela Doyle, Rina Foygel, Tim Howell, Bill Tabrisky, Anand Thakker, members of the mathematics faculty of the Park School of Baltimore, Inc. 2006- 2010.

When I visited, teachers were literally writing this curriculum as they were teaching; I think they were only a few units ahead of the students. It seemed like an amazing collaboration—everyone in the department was enthusiastic and on-board with the biggest project any department could take on—rewriting the entire 9-12 curriculum from scratch. Can you imagine your department doing this? Can you imagine everyone in the department devoting summers and free time to this task? It gets even better—now the department is now co-authoring articles for the Mathematics Teacher.

So that’s what I want a copy of—how you create a collaboration between half a dozen or more teachers—how you get them to open up everything they do to the scrutiny of the department, to be willing to throw everything out, and then to fully invest themselves in writing and testing a new curriculum. Something tells me you can’t put that in a pdf.

But if you are a math teacher, you can join the team—they’re looking for a math teacher.

Still learning how to solve projectile motion problems after 20 years of study

To me, one of the most enjoyable things about teaching physics is the opportunity to continually learn new things. I’ve now been studying physics for over 20 years (with about half of that time in some formal setting), and so it might be easy to think that, especially when it comes the basic stuff like projectile motion, there literally is nothing left to learn. Not true. And in the rest of this post I want to show you a beautiful, visual way of understanding projectile motion I learned from two excellent physics teachers, Brian Frank and Frank Noschese. Here’s the link to Brian Frank’s post on this topic that inspired this work.

Average velocity—the most important velocity of all

I’ve often told my students that average velocity is the sort of “all-knowing” velocity. If you know your average velocity, you can easily find an answer to he question “how much longer will this trip take?” and many others. And the power of average velocity isn’t limited to long trips along the interstate to grandma’s house, it works for all situations—even the two-dimensional problems where we rarely use this concept.

Here’s an example. Start with a person who walks three different displacements—first to Annie’s house, then to Bill’s, then to Charlie’s, as shown in this vector illustration below.

The red vector represents the total displacement , and we know the fundamental definition of average velocity is:

.

Or rearranging,

Average velocity is the vector that when scaled by the time of your trip, gets you to your destination.

Let’s see that on a map. Say I want to go from Atlanta to New York. Wolfram Alpha quotes the direct distance as 741 miles. But of course, if you want to drive from Atlanta to New York, you have to take roads, and they don’t follow a straight line path between the cities.

3 different possible routes to NYC. All will have the same displacement, and therefore the same direction for average velocity. But since the route through Toledo is likely to take much longer, the average velocity for that trip should be much shorter.

Finding average velocity for accelerated motion

Here’s a pretty basic kinematics problem:

Suppose an object is moving at 6 m/s and accelerates uniformly to a velocity of 14 m/s over a period of 4 seconds. What is the average velocity for the entire trip?

Students almost instantly will calculate

.

But one can gain much more insight with a graph:

The area of the trapezoid and red box are the same. This is the idea of average velocity—it is the velocity that you could travel at for the duration of the trip, and arrive at the same location as the accelerated object.

But what do you do in two-dimensional problems, where you can’t draw a single velocity vs time graph? You can still do graphical vector constructions. The key idea is the definition of acceleration:

rearranging gives us

The change in velocity is just the acceleration vector, scaled by the time. And for constant acceleration, . In essence, the change in velocity vector, is a sort of clock in the problem that grows longer as time passes. And we can find the final velocity vector by just adding the change in velocity vector.

Let’s start with a simple example. Suppose a projectile is launched at 20 m/s at an angle of 45°. Find the final velocity after it has been in the air for 2.8 seconds.

In two seconds, the change in velocity will be:

and graphically, we can see:

A diagram of the initial velocity, change in velocity and final velocity.

To find the average velocity, we can use the formula that holds true from 1-d motion as a vector equation (assuming constant acceleration:

We just need to add the initial and final velocity vectors, and then half the length of the resultant to find the average velocity.

For constant acceleration, the average velocity is just the sum of the initial and final velocity vectors, scaled by a factor of 1/2.

One interesting result we see from this is that the average velocity vector reaches the midpoint of the change in velocity vector. This should make some sense, since the average velocity must be exactly “between” the initial and final velocity.

Putting this to use to solve projectile motion problems

We should now be able to put this idea to use to solve projectile motion problems. Remembering the key ideas about average velocity:

1. The average velocity points in the direction of the final position.
2. The average velocity touches the midpoint of the change in velocity vector.

Problem 1: A projectile is launched at 20m/s at 45° on horizontal ground. Find where it lands.

Here, we know the average velocity vector must be horizontal. For the average velocity to be horizontal, we know that the final velocity must point 45° below the horizontal.

Doing the graphical measurements, we find that

And we know from previous calculations the time required for velocity to change enough to produce a final velocity of 20m/s at 45° below the horizontal is 2.8s.

Graphical solution to the projectile problem 1

So the total distance traveled by the ball is:

But wait, there’s more. This method can solve much more complex problems as well.

Problem 2: Find the maximum height reached by the above projectile.

We know that when the projectile reaches its maximum height, will be horizontal, so we can find the time by measuring the length of the change in velocity vector, and dividing it by . Then we determine the average velocity, and scale the average velocity by the time. Graphically, here’s what it looks like:

Finding the maximum height reached by the projectile.

From our construction, the average velocity is:

and scaling this by the time tells us

In components:

This is in good agreement with the traditional solution, provided by Wolfram Alpha.

And there’s even more—we can solve the gnarliest of projectile motion problems—those that used to require the dreaded quadratic formula.

Problem 3: A projectile is launched at 20 m/s at a 45° angle from a 30 m cliff. Find the place where the projectile lands.

This problem is a bit more tricky—we don’t know the final velocity, the final location of the projectile, or the time of flight. Still, we can do it.

The basic idea of this approach still holds true:

1. The average velocity points in the direction of the final position.
2. The average velocity touches the midpoint of the change in velocity vector.

In this case, we have one other critical piece of information—the projectile lands on the ground, 30 m below its starting point. When we take the average velocity vector and we scale it by the time in the air, it must reach the ground. This constraint limits us to one solution.

It would be hard figure out the solution to this problem by hand—you’d essentially be thinking something like “When I multiply the average velocity vector (which is the vector between the starting point and midpoint of ) by the time (1/10 the length of ), I need to get to the ground. But this is a piece of cake for Geogebra. Here’s what it looks like:

Solution to the 3rd problem, generated in Geogebra.

Unfortunately, Wolfram Alpha can’t spit our a quick solution to this problem, so I have to do the algebra:

First find by analyzing the y-motion:

Solving this quadratic gives us

Now use the horizontal velocity to find the final position:

Boom.

If you want to play with my Geogebra construction to solve other projectile motion problems, here’s a link. Warning—I am by no means an expert in Geogebra, so my file is more of a hack. I’m sure a pro like Dan Cox could turn it into a wonderfully elegant physics homework solver that would slice and dice every projectile motion problem under the sun.

Should you teach this to students?

First, I just want to marvel in how awesome it is to be able to do fairly difficult projectile motion problems without ever needing components or the quadratic formula. I also love, love, love how much this method really builds on the critically important idea of average velocity.

I also value how much intuition this method gives into how the velocity vector changes through its flight, and I feel like it gives a greater sense of meaning to the quantities you’re dealing with—displacements, velocities, and changes in velocities—it reminds you of the vector nature of these objects. I think if I spent a bit more time with this, and paired with some sort of animation of the velocity vector as it moves through the air, and then put this initial/final vector analysis on top of the animation of the projectile moving through the air, it could help students develop a deeper understanding of the underlying physics of projectile motion, without having to constantly worry about which of the nasty kinematics equations to use.

At the same time, vectors are one of the most confusing topics for introductory physics students, and the tendency to drop vector notation and treat everything as a scalar is a strong one, so I worry there are a ton of places in this approach where students could easily go wrong, which makes me pretty hesitant to really try this out with my freshmen.

Still, the coolest thing about working on this for the past couple of days has been to discover an entirely new approach to something I thought I’d mastered years ago and felt I had nothing left to learn about. The biggest question I have is how do we help students (and some faculty) to see that there are moments like this waiting for them in whatever they may choose to study—if they are willing to seek them out.

Re-designing the symbology of teaching

A big thanks to Bo Adams for tweeting this studio360 story about hiring a graphic design team to redesign the typical symbols of teaching—the apple, one room schoolhouse, ruler, etc, which are a very tired and cliche symbology of both school and teaching.

The story is excellent—I forgot how much I love listening to design teams talk about the process of their work. Even more incredible are the visuals, released under a CC-non-commercial license.

I love design—the metaphor of connecting the dots is an outstanding one. I’ve fancied myself as a designer from way back in high school when I spent a weekend laying out a program for a conference using PageMaker on a Mac Quadra, and it pains me to see how widely used comic sans is in the teaching world. This is why it’s great to see a group of professional designers give some time assist the teaching world, and why I’m even more thrilled to see the increasing emphasis on design and design thinking in schools. Check out the work of Studio H and Emily Pilloton (check out her great TED talk) to see just a glimpse of some of the incredible work going on.

How interesting would it be to take this challenge and give it to students as well? Could we rethink all the tired symbols of students and learning as well—the piles of books, lockers, dunce caps and more?

Heat and the work done by friction

It’s been a while since I’ve written about a purely physics related topic, so I thought I would try to write up an interesting discussion I’ve been having with some s about colleagues about the work done by friction, heat, and how best to teach these concepts to introductory level students (9th graders for me).

It all stared when one of my colleagues invited us to give feedback on a test he’d written for his 9th grade honors physics class. At the top of the test, he’d written a bunch of formulas for energy, including the following:

Energy is one of those topics can easily succumb to the formula zoo, and students can often focus too much on memorizing formulas and not enough on thinking about conservation of energy. I think to combat this, my colleague presents fairly complicated, multi-step energy problems that force students to think about many different types of energy, and energy losses, and in order to help them not be totally at sea, provides them with a number equations to use. Giving out equations is a practice I understand pedagogically, but don’t do myself, mainly because I try to work to minimize the number of equations we use, and worry that starting the test off with a list of equations would leave them thinking that physics is about the equations. But maybe the equation sheet helps defuse this by worry by giving them the equations, implicitly telling them they will need more than equations to do well on the test. I should think about this more.

The thing that really got me thinking was the last equation , which sent my physics antennae off. I knew there were lots of arguments why this isn’t fully right from a physics point of view, but I couldn’t remember them at the moment, and more importantly, I’m unsure how many of these arguments are really appropriate for a first year physics course.

So here’s the argument, reconstructed from some conversations with Mark Hammond, re-reading Matter and Interactions, and Bruce Sherwood’s papers, Work and Heat in the Presence of Sliding Friction, and Pseudowork and Real Work. If you’ve read these papers, or are familiar with this argument, you can probably skip this part, since it consists mostly of me trying to repeat Sherwood’s arguments for my own understanding.

Explicitly calculating the work done by the frictional force

Let’s start with a block, pushed across a table at constant speed, by a constant force . If the speed of the block is constant, the block must be experiencing a frictional force opposite its motion, and this force must be equal in size to in order for the net force to be zero.

Let’s look at this from an energy perspective, assuming the block is pushed a distance . We’ll start by considering a system consisting only of the block. The work done by the pushing force is . But since the kinetic energy of the system is unchanged, some other force must have done an equal amount of negative work on the block. And since the frictional force acting on the block is equal to , it makes sense to think the frictional force force of the table did of work. But there’s only one problem with this—if you feel the bottom of the block, it will be warmer; something must have increased the thermal energy of the block. So the table must have removed less energy than what you put into the block by pushing it, and .

We can view this in an LOL energy chart.

Energy LOL chart for a block pushed along a table at constant speed.

Revised energy LOL diagram to account for the thermal energy increase of the block. In this case, the work done by the frictional force can no longer be equal to the work done by the hand.

But how could it be that the work done by the frictional force is less ? We know the size of the frictional force must be equal to , so the only explanation is that is somehow smaller.

To understand this, Shrewood points us to the difference between the Work-Kinetic Energy theorem and the First Law of Thermodynamics. The Work-Kinetic Energy thoerem, usually derived in introductory courses by combining Newton’s Second Law, , and the constant acceleration equation, , tells us that:

For our object moving at a constant velocity () this equation tells us that the kinetic energy of the system won’t change. Note that this result is the same whether we choose a system consisting only of the block where the applied force would be equal to the frictional force), or of the block and the table, where the applied force would be equal and opposite to the external force of the floor on the table. In both cases, the net force on the system is zero.

The other law we should consider is the First Law of Thermodynamics, which tells us that the change in energy of the system is equal to the sum of energy transfers (work, heat, etc), across the boundary of the system. In both of our systems, we will assume that the only energy transfers are caused by work. In this case, the First Law of Thermodynamics can be written

For the case of the system with the table and the block (we will assume the contact table is stationary, and therefore the frictional force of the floor does no work), only the external force pushing the block is doing work. In this case, we can write:

But if our system is reduced to be simply the block, the frictional force of the table acting on the block is now doing negative work, . is a distance smaller than the total displacement of the center of mass , since we are allowing for the possibility for the the points of contact between the block and the table to not move as far as the entire block.

So in this case, the change First Law of Thermodynamics states:

Now consider the case where the the block and table are identical (suppose we have two identical blocks sliding past one another). In this case, there will be no heat transfer between the blocks, and the the change in thermal energy of the top block will be half the total change in thermal energy we calculated for the system of the two blocks ().

But when we apply this back to our analysis of the single block system using the first law of thermodynamics, we find a surprising result:

Knowing that , we can write:

Which is quite a surprising result. But this does allow us to fully account for the energy in the system. For a block sliding across an identical block, the external force does work equal to and half of that energy input is taken from the system as work done by the frictional force, while the other half is transformed into thermal energy of the upper block. This makes sense because that other half of energy that is taken by work would go into the lower block, and result in an equal increase in thermal energy for the lower block, which is what we should expect for identical blocks.

Sherwood takes this even further in his paper, where he considers the cases where the blocks are not identical, lubricated friction and more. But the basic takeaway that the work done by the frictional force acting on an object sliding across a surface is always less than .

How this affects an example problem

Now consider the pretty classic physics problem that asks a student to calculate the work done by friction when a block is dragged across a level flood a distance, d. We already know that since the object is moving at constant velocity, the frictional force must be equal to the applied force, .

We have previously seen that the complete answer to this problem is where is some unknown distance less than the total displacement of the block, that depends on the material composition of the two surfaces that are interacting.

If we apply the first law of thermodynamics to the system consisting only of the block moving at constant velocity, we must also consider that the temperature of the block’s lower surface will be higher than the cold parts of the table it will encounter as it slides, and so there will also be a net transfer of heat () from the block to the table.

If we consider the system of the table only, we get

and if we combine these two systems we get:

And this makes sense—the work done by the external force is transformed into thermal energy which is distributed in both the block and the table.

Implications and questions for intro physics students

Obviously, it’s not wise to go into this level of analysis with a first year physics course, especially a 9th grade one. So the question is how much of this do you share with students? Is the equation a valid one for introductory students? It certainly allows them to come out with right answers on most problems we can devise at level.

My thought is that it is important that students are thinking carefully about systems when they are introduced to energy—to me, it is critically important that they understand conservation not from a “energy can’t be created or destroyed” or as , but that the total energy of a system can’t change unless there is a transfer of energy across the system boundary. Mastering this idea can pay huge dividends in later studies of chemistry and biology, and it’s one of the reasons I love energy LOL diagrams.

The old misconception zealot in me also doesn’t like the use of the word heat in this particular context—heat really should be the flow of energy between the system and surroundings due to a change in temperature, but I wonder if this distinction is too much for introductory students, and if introducing this distinction is counter-productive at this level.

And the idea that the work done by friction is equal to is useful for a ton of problems at the introductory level. If you decide to be strict, and not call the work done by friction, what is a better term that doesn’t open up the entire can of worms above?

Culture hacking: StoryCorps for seniors

Tonight, I went to a reading by Dave Isay, creater of the NPR StoryCorps project. The project is simple—find a person whose story you’d like to hear and invite them to a StoryCorps recording facility. There, you will interview that person with the help of a trained facilitator for 40 minutes. At the end of the session, you go home with a CD recording of the interview, while a second copy is archived with the Library of Congress. To-date, StoryCorps has recorded over 40,000 interviews, and is currently making a special effort to record the stories of teachers.

The reading consisted of Isay playing some of his personal favorite stories on the theme of love. The stories were tremendously moving; I think each brought me to tears. Along the way, Isay stressed the act of love that is listening. It’s an incredible gift to spend 40 minutes listening to someone’s story, and it reminds us of our shared humanity. I was reminded of this very thoughtful essay on Public Acts of Listening by Beth Friese for the edu180atl project, which shares a lot of the same motivations as StoryCorps.

As I was walking out of the program, I suddenly had a flash of an idea for a culture hack. What if you brought the StoryCorps idea to a school? I could think of so many ways that this could be an incredible project—start by collect the stories of the staff and share them on a blog. This could help to strengthen the connections of students with the adults around them. I also thought of how this might be used with students—collect the stories of the senior class. Underclassmen could select a senior to interview, and collect stories of things that senior is most proud of from his/her high school career, advice for new students, and so much more. Then the school could give the senior a recording of this interview as a keepsake, add it to the archives, and possibly even pass along some of the interviews to the next class of freshmen, which would help the seniors to create a greater legacy and pass along their wisdom to the underclassmen. The costs would be minimal—many schools likely have all the recording equipment they need to produce high quality recordings. I know I would love to have a recording of my 18 year-old self discussing my experiences in high school and hopes for the future, mostly just to see how clueless I was at that time.

So when will you tell your story?

Thoughts on “why a liberal arts education matters…”

Two days ago, the New York Times ran this essay by Vedeika Khemani, a graduate of Harvey Mudd College, on Why A Liberal Arts Education Matters.

I found the essay very compelling. Khemani argues that the “pragmatic approach taken by most Indian students to choose…to study whatever will land them a job…rather than intellectual exploration.” The one small point I want to quibble with is this quote:

Real-world problems rarely ever have textbook solutions. More than anything, the purpose of a college education is to learn how to think critically and what questions to ask.

First, I couldn’t agree with this statement more. But, I think one of the best ways to learn this is by studying in fields like physics, math and computer science—fields that to me, embody the liberal arts. I think the flaw is our education system, which presents many of these subjects, especially at the introductory level, simply as long exercises in finding the solution that’s printed in the back of the textbook. In to many of these classes, students never ask questions of their own, and never take on problems with that require approximation, simulation and don’t readily yield closed form solutions. Properly taught these fields are all developing questions that probe into the unknown, and and then developing a systematic approach to finding the answer to those questions.

So how do we encourage people students to embrace the liberal arts further? One critical ingredient I see is reminding students of the vital role of the sciences in a true liberal arts education.

Growth mindset benefits carry into old age

Here’s a very interesting article that hit the NYT Sunday magazine a little while ago, and I just got around to reading in Instapaper.

A Sharper Mind, Middle Age and Beyond

A few quotes:

One essential element of mental fitness has already been identified. “Education seems to be an elixir that can bring us a healthy body and mind throughout adulthood and even a longer life,” says Margie E. Lachman, a psychologist at Brandeis University who specializes in aging. For those in midlife and beyond, a college degree appears to slow the brain’s aging process by up to a decade, adding a new twist to the cost-benefit analysis of higher education — for young students as well as those thinking about returning to school.
…
Still, when Dr. Lachman and Dr. Tun reviewed the results, they were surprised to discover that into middle age and beyond, people could make up for educational disadvantages encountered earlier in life. Everyone in the study who regularly did more to challenge their brains — reading, writing, attending lectures or completing word puzzles — did better on fluid intelligence tests than their counterparts who did less.
…
And those with the fewest years of schooling showed the largest benefits. Middle-age subjects who had left school early but began working on keeping their minds sharp had substantially better memory and faster calculating skills than those who did not. They responded as well as people up to 10 years younger. In fact, their scores were comparable to college graduates.
…
When young adults think about college, they think about career opportunities and possibly the social benefits. What they don’t realize is college education has long-term benefits well beyond first job and social contacts.” The same could be said for continuing education.

This is a great long read, and it again points me to the idea that the fixed/growth mindset framework carries far beyond limited “do better in school” confines I’d placed it in. Our abilities, are far more malleable that was once believed, or we are often told.

So if it’s true that believing one’s abilities are not fixed, and that practice and hard work yield such tremendous improvements—in cognitive ability, in willpower, in reducing racial prejudice and now in reducing mental decline in old age—I once again wonder why these lessons get so little attention in many schools, and too often, fixed mindset thinking is is reinforced with achievement oriented praise, rigid tracking and ability grouping.

More on AP from Harvard’s Eric Mazur

In case you aren’t reading it, Education Guru Grant Wiggins has started blogging in a big way. His posts on transfer and his six part series on the student voice in education, are outstanding.

Today, Grant wrote a post writing up a recent trip he made to Harvard to visit Eric Mazur, father of Peer Instruction, who told Grant this:

Mazur also noted in our conversation that his years of experience on the Physics AP design committee made him less than enthusiastic about AP’s. He has data showing that student who got 5s on the Physics AP do worse than other Harvard Physics students who did not take the AP’s – a sobering thought.

I’m not surprised, and I’ve also heard similar stories about AP physics having no bearing in physics performance in other colleges. My guess is that the majority of those 5′s come from AP physics B, a inch-deep, light-speed marathon through all of introductory physics: mechanics, fluids, thermodynamics, waves, light and optics, electricity and magnetism, and modern physics. In fact, this course covers more material (at a much shallower depth) than your standard first year physics course in college. Also, while AP B has developed more of a conceptual focus, students still tend to see it as mostly a sea of equations. So it doesn’t surprise me that a lot of students coming out of it, even with 5′s struggle in Mazur’s physics class which really stresses a deep conceptual understanding. I could even imagine that students coming into Harvard with 5 on the AP physics might have a bit of overconfidence (they are Harvard students after all), and work less hard in the course or and so be less likely to seek out help when they start to struggle. Another interesting takeaway from this tidbit is that there are students at Harvard who didn’t take AP physics, and if Mazur is able to make statistically significant comparisons, there must be a lot of them. While this should be obvious, it’s a common misconception some students have that they need to take “every AP” their school offers in order to have a shot admission at a school like Harvard. Not true.

As Grant Wiggins suggests, the fact that earning a 5 on AP physics shows some negative correlation with performance in a college physics course among Harvard students should be very sobering indeed. It’s one more bit of information that tells me the value of taking an AP course that is so focused on content and earning a particular score on the AP test is meaningless, and possibly even harmful to your understanding and future progress in the subject.

Nuclear weapons : Cold War :: APs : college admissions

I wrote recently that students seemed locked in an AP arms race, and them more I think about it it’s a very good analogy. Some very competitive high school students feel like they are locked in some life or death struggle with nameless competitors for precious few places at the “good” colleges (I plan to blog more on what a bogus notion it is that there are “good” colleges). And this plays out much like the drama of the Cold War. Just like the US during the height of the Cold War, we didn’t understand the struggle we were in, or the enemy we were competing with, yet we still felt the need to stockpile ever more powerful and useless nuclear weapons in the hopes of deterrence. Meanwhile, the Soviet Union was doing exactly the same thing, but this pursuit of military might was also hollowing out their economy and leading to an economic and political collapse. I suppose it’s over-stating things to compare AP’s to useless nuclear weapons and burned out students to the the Soviet Union’s economic collapse brought about by a singular focus on military might (I’m also aware that it’s abusing history as well). But the there may be a lesson from history for helping to lower the head of the college admissions frenzy—many times in the history of the Cold War, tensions were lowered by changing the rules of the game and open communication between the two superpowers. Could we do the same today, by changing the rules of the college admissions process and encouraging more open communication? We could encourage students to forsake the AP courses they aren’t interested in for things that do interest them, encourage gap years, and help students to develop a life of meaning, something that more often than not, requires cooperation and teamwork with ones’ peers rather than dogged competition. It seems to me that might be just the recipe for avoiding the Mutually Assured Destruction of the AP arms race.

Following up on 1 hour of learning with my students

Today I tried to follow up on the 1 hour of learning idea with my honors physics classes and weave it into a discussion of helping them make decisions about their schedules for next year, and even more importantly, a plan for how they’ll make decisions to lead meaningful lives in high school and beyond.

Establishing meaningful rituals

I started by handing out the Renaissance Man article along with the selection of suggested learning experiences recommended by Jeremy Gleick. After that, we discussed the article, and I was surprised by how much students enjoyed the article, how likable they found Mr. Gleick, and how relatable they found his accomplishment, while still recognizing what a tremendous accomplishment it is to devote an hour each day to learning. I think I found this surprising because previously, some of my students have found some of the tales of students accomplishing extraordinary things (like this student who attended a UN conference on climate change) un-relatable and a bit of a turn off.

In our discussions, students could clearly see how interesting Jeremy Gleick was, and when I asked, “would you want to eat lunch with this guy?”, many responded with an enthusiastic yes. When I asked students if they could see themselves doing this, most said they could, if… and then we listed a ton of things keeping them from doing this. If they gave up another hour of sleep, if they didn’t have so much homework, if they didn’t have so many extracurricular involvements, etc.

My second class said what was interesting was that Jeremy had created a ritual of learning. Students could see some of their own rituals in their lives, be it a sport, debate or an instrument. We also talked about how difficult it is to maintain a ritual when you don’t enjoy it. I really like this meaningful ritual notion, and am going to try to incorporate it more into my language.

How students’ schedules play into rituals and a life of meaning

This was a perfect segue in a topic students are currently thinking about a lot—their schedules for next year. One of the most pressing questions for my students is whether or not they will enroll in AP Chemistry next year as sophomores. Students who take AP Chemistry take it as their first chemistry course, and the pace of the course is very fast.

In order to help students make this decision, I asked a couple of former students who had struggled with the decision to take the course last year to write up their experience and advice about making the decision in a couple paragraphs that I might share anonymously with students. The students wrote beautiful reflections that do a great job of characterizing the difference between AP and honors chemistry, and the questions one should consider when making the decision.

I tried to help the students see that the experience they’ve gotten in my course, and the experience they are likely to get if they choose to enroll in AP are simply different. One isn’t harder than the other, no more than sprinters are better athletes than distance runners. Our classes have different goals, but it can be a good thing to be stretched in different ways.

We then discussed why one should choose to take the AP Chemistry course, and I tried to present a few reasons—you’re interested in a possible career in the sciences, you feel like you might benefit from a very fast paced course that is going to push you to keep up with the pace of the course, and stretch you to learn more material than you thought possible. But I also said one thing that I think students should not be using as a factor to drive their decision—whether or not it looks good for college.

The AP arms race

Long ago, taking an AP course truly made students stand out. It was seen equivalent to doing college work in high school. Strong performance on an AP showed interest in the subject, and colleges would often see this as a sign that those students were especially well prepared to go on to major in that discipline. But that is before the AP arms race led every student thinking he or she must take as many APs as possible just to have a chance of admission at college. Colleges have seen more and more students skip introductory classes with AP credit and struggle in subsequent courses, making them far less likely to grant credit for AP. They also see students who have 5′s in a subject and then don’t go on to do anything in the subject in college. Overall, APs become a greatly diluted metric of accomplishment, which only amplifies the arms race. It used to be taking 5 APs in a college career was unheard of. Now, some students think they need 8 or more.

Of course, this is ludicrous, and it’s a distortion of the advice you hear at many college info sessions about wanting to see applicants take the most “demanding schedule possible.” What colleges want to see is that students will take advantage of the many opportunities each college offers, and a good indicator of this is how well a student takes advantage of the opportunities available to him/her in high school. But opportunities is one and the same to AP courses—it’s only our desire to come up with shallow and easy metrics that ever to admissions officers, teachers, or students making this false equivalence.

I doubt Jeremy Gleik took every AP course at his Los Angeles private school. I say this, because I think if he did, he would never have found the time develop his one hour of learning ritual. And though it should be obvious, I think almost any college admissions officer would find his love of learning, which probably oozed from his application and his discussion of his 1 hour of learning ritual. The didn’t need to waste their time counting how many AP classes he took.

The risk of doing very time consuming things just to look good for college

This is the danger that students risk when they do things just to look good for college—they very easily can find themselves in over their heads in courses and activities they have little interest in. This will inevitably reduce their motivation to do well in these classes and activities, and that could have a negative impact on the college process. Even more importantly, if students fill up your time with demanding classes and activities they don’t enjoy, they may not have the time to find that ritual they will enjoy—the one that will add meaning to their lives.

My goal in having this conversation isn’t to get students to not sign up for AP chem. As I said before, there are many great reasons to take the course. But I do want students to think carefully about why they are taking the courses they choose, and how those choices fit into the bigger picture of learning to live a happy and meaningful life.

After these discussions I always wonder if they have any effect. But today, I got this email:

Hey Mr. Burk!
I just wanted to tell you that I really enjoyed class today and the article we read. I’m really glad we talked about the stuff for next year, and I thought some of the stuff you said will be really helpful in decision making. Just letting you know.

POMs-The currency of momentum

We’re studing momentum in my honors physics class, and I’m thinking back to all the fun we had last year designing a completely new unit for momentum, the Parcel of Momentum (POM).

I’ve spent the past few days trying to follow Kelly O’Shea’s great introduction to the Momentum Transfer Model (MTM) paradigm lab, but we had considerably more trouble than her class did. Ultimately, we got to the point where we were looking at a data table that looked something like this:

And we began to study this data looking for a pattern. Eventually, some students started to see that in some cases, the changes in velocity were exactly opposite one another, and these cases happened to be when the masses were equal. Later, they realized that in cases where the changes in velocity were not the same, the more massive cart had a smaller change in velocity than the less massive cart. And from there, students were able produce a nice graph showing that , that the carts seemed to be swapping something during collisions.

One student even said that it was like two people swapping money, and so this got me thinking, and I designed this with a little bit of free time today:

View this document on Scribd

So now I have a momentum currency for my class, and for a few minutes, I thought this would be cool for some sort of demo/activity in class. I’d give everyone a set amount of money, and then we’d simulate collisions where they would “transfer” momentum currency to one another. Finally, we’d tally all the individual transfers and see that momentum is conserved.

But then I got to thinking about this a bit more, and I’m not so sure this is a wise way to go. There are lots of ways that my momentum currency may confuse things more than I want. First, momentum is a vector, without creating some sort of perpendicular currency, I’m not sure that they will get that having more momentum doesn’t mean you can move in any direction you want. Second, objects end up with negative momentum, meaning they’re traveling in the negative direction. However, I have no way to track negative momentum, and so it would seem that this currency idea might mistakenly make students start to treat it like a scalar quantity, rather than a vector. Finally, I use the money metaphor a lot when dealing with energy, so I’m not sure we need a currency for both momentum and energy, and think that could get confusing fast.

So now I’m tempted to keep my POMs for myself, and not doing anything with them. But I’m curious if you have any suggestions about the usefulness of this activity or how to improve it.

So now