Based on today’s discussion about scientists’ notebooks, I drafted the following rubric for assessing students’ notebooks in class, following the guidelines of a lesson over at SGSI.

Personal Relevance: Does your notebook demonstrate that are you engaged and thinking rather than simply going through the motions of copying down notes and data?

Requirement	page #s – tell me where to look	Grade – I’ll do this
Evidence of your ideas and their progression: this should include things that you do understand, things you don’t understand, and show the process of your thinking.
Attempts to interpret, explain, or draw inferences from observations–not just observations and data.
Questions that arise as you conduct research and your attempts to answer them.
Evidence of personal expressions, personality, individual style, engagement or creativity
Sketches, drawings, or diagrams that convey thinking, capture observations, or depict what was done.

Public Relevance: Is your notebook useful and understandable for others familiar with your research. Can they follow the sequence of ideas, experiments, and data?

Requirement	page #s – tell me where to look	Grade – I’ll do this one
All days are dated and pages are numbered
There is some organization (such as labels or titles) among the “chaos”.
There are references to peers’ work or ideas when they influenced your own thinking.
There are details and accurate accounts of what you observed, such that your findings are well documented
There are detailed, accurate descriptions of procedures, making it possible to go back and repeat or “troubleshoot”

Today we discussed our rights and responsibilities as students and instructors.  In 2-3 weeks, we’ll revisit our list to both assess how we are doing and decide if any changes needs to be made.

Below is the full list of what they came up with, but my two favorite are rights they gave me.

“To change the curriculum and steer the class in productive directions”

“To reject unacceptable work”

Students Rights:

To get help–either from peers or the instructor
To share ideas and opinions and to feel comfortable doing so
To know and understand what’s expected of them
To have constructive criticism
To be treated fairly­–both in how we treat each other in class and with grading
To know of changes to class or class policies in a timely manner
To ask questions
To be yourself

Student Responsibilities:

To be respectful of others and their ideas
To be prepared and engaged­–to be in attendance, to have work completed, and to listen
To be come to class with an open-mind and willingness to learn
To ask questions when you don’t understand or are unsure
To participate and function well in groups
To come in with a good attitude
To contribute to a safe learning environment–both emotionally and physically

Teacher Rights:

To change the curriculum and steer the class in productive directions
To have the class’ attention and participation
To be respected
To expect that students will do good work
To reject unacceptable work

Teacher Responsibility:

To be available
To make class fun and engaging
To maintain a safe learning environment–both emotionally and physically
To help prepare students to be successful in ways they are expected to be
To teach ideas that are useful and/or meaningful
To encourage students
To give feedback in a timely manner
To treat students as adults and as individuals
To be on time and to not let class go over the scheduled time

In this post, I just want to motivate why a gradient gets complicated in a non-euclidean space, and in a following post I’ll work through how to take a gradient. The reason we care is because to find the force from potential energy, we need to take a gradient.

In a euclidean space with cartesian coordinates, the gradient can be taken this way

Grad(U) = (∂U/∂x) x-hat + (∂U/∂y) y-hat + (∂U/∂z) z-hat 

One way of interpreting the operation ∂/∂x is that  it “nudges” whatever the scalar function along in the direction of x, and then looks at how that things change. It also keeps track of how far the nudging has occured, and it takes a ratio.

This “nudging” is really important to get a handle on. See, in a euclidean geometry described by cartesian coordinates, a nudge in the x-direction is always the same nudge no matter what; and a nudge in the x-direction never ever causes a nudge in the y-direction or the z-direction. That’s actually what this metric

ds² = dx² + dy² + dz²

tells you. The 1 in front of dx²  tells you that the dot product dx.dx is a constant of unity. The fact that there is a (hidden) zero in front of dx.dy and dx.dz tells you that nudges in x are not influenced by or influence nudges in y or z directions.

But if you remember back to my first post, we had this metric:

ds² = (1+α²) dx² + (1+β²) dy² + αβ dx dy

In this metric, we have to three things to consider when nudging:

(1) A nudge in the x-direction is not always the same nudge depending on where you are, unless α is a constant; and similarly for the y-direction.

(2) A nudge in the x-directions isn’t the same nudge as a nudge in the y-direction, unless α = β.

(3) A nudge in the x direction will co-occur with a nudge in the y-direction as along, unless both α and β are not equal to zero

Another way of thinking about this is like this, imagine you had the vector x-hat, which is a unit vector. Imagine you tried to drag x-hat in the x-direction. In a euclidean space, x-hat would still have a magnitude of 1, and x-hat would still be facing in the x-direction. However, in a non-euclidean geometry, two things can happen. One kind of that can happen is that x-hat can grow or shrink. That growing or shrinking is caused by terms like α² or β². Another kind of thing that can happen is that you’re x-hat vector, which you dragged in the x-direction, is all of sudden pointing a little bit in another direction. That’s what the term αβ does. It twists your unit vectors around.

This growing, shrinking, and twisting is what we’re going to have to keep track of in order to take a gradient. How we keep track of this is the subject for another post, but I hope you get the big picture–a nudge is not just a nudge but a nudge with a growth and a twist.

OK, so building off my last post, the question I want to work through is, “How do free particles move in spaces that are non-euclidean?”

Let’s use Andy’s suggestion of an object sliding around on a parabola, and we’ll project that parabola onto the x-axis, to see what motion is like on the 1D projected parabolic space. But let’s take away gravity, because we want to consider free particles that are subject to no ‘external’ forces.

We know of course that in the 2D world, the particle would move with constant speed from one side of the parabola to the other. The reason it would move with constant speed is because, without gravity, there is no way for the object to change its kinetic energy, or direction for that matter.

What I want you to imagine is you are looking down into the parabola from above, and you can only see what’s happening in the x-direction. It’s like you have no depth perception. To you, the ball would seem to speed up as the ball goes down the parabola and slow down as it goes back up the parabola. That’s because some of the speed is “hidden” in the y-direction, which you can’t see. Keep this in mind, because our mathematics has to jive with this intuition.

Getting the Projected Parabolic Metric

So let’s take the parabola y = 1/2 x²

As I said in my previous post we typically measure distances like this

ds² = dx² + dy²

But we can transform this 2D euclidean space to our 1D non-euclidean space by using the fact that

dy = x dx

Plugging that into ds² = dx² + dy² gives

ds² = (1+ x²) dx²

Recognize that the term “x” is just the value of the slope of parabola at the point x, so it’s completely consistent with what I’ve talked about in the first post.

Newton’s 1st Law in the Projected Parabola

Now, I’ve already established that the object moves with constant speed in the 2D euclidean world. So let’s set a value for that velocity and just call it “v”.

With a velocity v, the distance in the 2D geometry would be s = v t or ds = v dt, where s is measure along the arc length of the parabola.

This allows us to write

v² dt² = (1+ x²) dx²

which I’ll rearrange to be

(dx/dt)² = v² / (1+ x²)

or

dx/dt = v / Sqrt (1+x²)

Here is where we should check out intuition, because the value dx/dt is the velocity in the x-direction. I said that the ball appear faster near x =0, because there none of the speed is hidden in the y-direction. Plugging in x =0, we get dx/dt = v. That’s good, because that’s when none of the velocity is hidden. If we take the limit as x gets really large, the velocity goes to zero. And that also makes sense because the parabola gets so steep that you can’t see any of the velocity–it’s all hidden in the y-direction.

OK,  what we might want to do is write Newton’s 1st laws for a 1D parabolic geometry. So what we want is to rewrite this expression so that it looks like

d²x/dt² + Correction[x, dx/dt] = 0

And we can do this by taking a time derivative  of our equation for dx/dt and rearranging.

d²x/dt² – v * d/dt [ 1/ Sqrt (1+x(t)²)] = 0

d²x/dt² + v * x / (1+x²)^3/2 *dx/dt = 0

Now this is pretty ugly, but it puts it in a form like Newton’s 1st law. Our corrections term is the messy part, because it is non-linear function of x and also depends on dx/dt. It might be tempting to think of this whole term as  velocity and position dependent “force”, but we don’t have to. I want to stress that this is just how particles move in our 1D projected parabolic space without forces. I’ll show you later how we can add forces.

The truth is, it would have been prettier to not take that derivative, and cast this in term of energy. One way of doing this is to multiply our expression for (dx/dt)² by 1/2 m, to get

1/2 m (dx/dt)² = KE / (1+ x²)

where KE is just a constant.

Adding Back in Gravity (the easy way)

To add back in gravity, we just have to realize that KE is no longer a constant in the 2D Euclidean space. But since the total energy in 2D euclidean space is still constant, we can write

KE = Total Energy – PE = TE – mgy = TE – mg (1/2 x²)

1/2 m (dx/dt)² = (TE – mgy) / (1+ x²)

1/2 m (dx/dt)² = (TE – 1/2 mg x²) / (1+ x²)

where TE is just a constant expressing the total energy in the 2D Euclidean space.

I’m pretty sure this works. I suppose we’ll see.

Adding Back in Gravity (the hard way)

This will be the subject of another post, but basically we’ll start with our Modified Newton’s 1st Law expression, and we’ll have to figure out how to take a gradient in a non-euclidean space.

As an aside,  I’d like to point out that SO far, we haven’t had to use calculus of variations or Langrangians.

I’m a big fan of thinking about wrong ideas. One wrong idea I’ve been spending a lot of time thinking about for over a year is one that I came across while working through a piece of curriculum intended for physics majors to learn more about this relationship

F = –Grad(U)

The materials had students think about a rock on a mountain, and how the force on the rock would be related to the slope of the mountain at that point. The  materials went on to have students think about a topographic map, where the height of the mountain could be described everywhere with some function z = f(x,y) and the potential energy by U = mgz. They had students eventually build up to considering this expression

– mg * [(∂f/∂x) x-hat + (∂f/∂y) y-hat ] ,

which would appear to be the a “projected gradient” of some sort.

In this post, I want to explain why this expression is (1) not a gradient and (2) not a force.

So, let’s imagine an object moving on some randomly shaped “frictionless” surface.

And let’s say that we can describe this surface as before, with z = f(x,y). This surface could be a ramp, or a bowl, or a complicated ice sculpture. It doesn’t matter, but I want to consider the case that the object can’t fall off or fly off. Let’s also imagine that gravity is involved and that the gravitational potential energy of the object is

U= mgz, as was assumed in the rock on the hill case.

So by substitution we can write

U = mg f(x,y), as was done in the curriculum.

Now if there was no surface involved (i.e., the object was just free-falling in gravity), then we could certainly relate  the force due to gravity  to the potential energy in the following way

 Fg = -∂U/∂z z-hat= -mg z-hat.

However, that’s not going to be true with our surface constraining the motion. Given this, however, I do agree it’s tempting to consider the expression

– mg * [(∂f/∂x) x-hat + (∂f/∂y) y-hat ],

which involves taking some the partial derivatives of U with respect to x and y (instead of z). The question is, is this a force or perhaps a projection of a force? It’s certainly tempting to think that this is, because it looks like the gradient of the potential, and it looks to be happening in the x-y plane.

It turns out this isn’t a force, because taking the partial derivatives in this way does not constitute a gradient. The reason it’s not a proper gradient is interesting (to me) and has to due with the fact that taking partial derivatives like this only makes a gradient in the special case of a euclidean geometry. To do so, I will have to convince you the transformation of U(z) to U(x,y) is very different than what it at first seems.  So, we’ll have to back track a bit.

See, in a 3-dimensional euclidean space, we measure distance by the following metric:

ds² = dx² + dy² + dz²

But what we have is different. We have a 2D surface embedded in a 3d space. Given that  z = f(x,y), we can write the differential dz = (∂f/∂x)  dx + (∂f/∂y) dy and substitute that into our 3D metric to get the following metric for the 2D surface:

ds² = dx² + dy² + [(∂f/∂x)  dx + (∂f/∂y) dy]²

This can be simplified and rewritten as

ds² = (1+α²) dx² + (1+β²) dy² + αβ dx dy

where I’ve just named ∂f/∂x  as α and ∂f/∂y as β.

OK, so now have this 2D non-euclidean surface described by the above way of measuring distances. It is weird, because you have these correction factors such as α² and β² that augment the distance traveled depending on the slope of the surface where you are measuring. We also have this weird cross term that involves dx dy. These terms are what make space non-euclidean. It’s like your ruler grows and shrinks and even twists as you move around the surface. Or, if you will, those corrections exist because we are only looking at x and y and the correction factors add back in the distances in the z-direction.

So the following questions arise:

1. How do free particles move on such surfaces?
2. Can we calculate forces by taking the gradient of the function U = f(x,y)

OK. OK. Let’s step back again and clarify why these are the two important questions:

In a Euclidean geometry, free particles can be describe by the follow equation

m d²r/ dt² = 0 → This is just Newton’s first law.

and many types of forces can be calculated  using by F = – Grad (U)

This allows us to write Newton’s 2nd law as,  m d²r/ dt² =  –Grad (U)

All of this works for Euclidean geometries and for forces described by conservative vector fields–but not all conservative forces, just forces described by conservative vector fields. There’s a subtle difference, because Normal forces are conservative (i.e. they do no work), but they cannot be described by a conservative vector field.

OK. So, what we’d like to do is write a similar equation for particles on our surface. The problems we’ll end up running into are these:

1. Free particles on our 2D surface can’t be described by d²r/ dt² = 0, and
2. We’re going to have to be smart about how we take our gradient.

I want to take up these two obstacles in two other posts. So, next I’ll talk about free particle motion in non-euclidean spaces. And then I’ll talk about gradients and external forces in non-euclidean spaces. Hopefully, I’ll get to a fourth post where I talk about why classical mechanics and general relativity are the same thing. And last, I’d like to talk about the mistakes students made in this curriculum, especially the ones that got me thinking about these issues so deeply. Wish me luck.

I’m a big fan of either using 10 m/s/s or 22 mph per second for g, and giving students reasonable numbers when they are doing exercises that involve projectile motion. The real world won’t be so kind when they take data, so I want to start them off where they have a fighting change of reasoning proportionally on the basis of concepts. I’m trying to fight the “iterative plug-n-chug” approach, by doing things we can think through.

But, in one of my classes, I’ll need to be using 9.8 m/s/s, because that’s what they’ll have on their exam. And I won’t have reasonable numbers always, because the problems I’m supposed to provide example are not picked by me. So, I have decided to do a lot of  “figure out a number you know is too high”, “figure out a number that is too low”, and “make a good guess”, before doing any plugging into equations.

Take for example, this kind of question I am supposed to work through as an example:

A ball is thrown vertically upward with a speed of  7.2 m/s. On the way down the ball hits a flag pole that’s 1.2 meter above from where it was thrown.

It’s a goalless problem, so I’ll have to decide something to solve for. Let’s go with, “How long will the ball take to get to its highest point?” … Here we go. Well, we know that the ball loses 9.8 m/s every second, but it only needs to lose 7.2 m/s. That must mean that the time to reach its highest point is less than 1 second. How much less? Well, it’s certainly greater than half of a second, because half a second would be  time to lose 4.9 m/s.  So, a number that is too high is 1 second, and a number that is too low is 0.5 s. I would guess it’s probably pretty close to 3/4 of second, because 7.2 seems pretty close to in between 4.9 and 9.8.

What’s the actual answer? About .73 seconds.

OK. Now on to another question. “How high will it go?” Well, the average velocity of the ball during the trip will be right in the middle of its starting velocity (7.2 m/s) and its ending velocity (0 m/s). That makes its average velocity 3.6 m/s–I can divide by two in my head. Given that I’ve concluded in the first part a travel time of 3/4 of a second, this must mean it must have traveled less than 3.6 m, because it would have taken an entire second to travel 3.6 m, and it didn’t have that much time. My reasoning is that it should go 3/4 of the way to 3.6m in 3/4 of a second. This is actually pretty easy to reason about, because 36 = 9 * 4. So, I’d say it would have gone about  2.7 meters during that time.

OK, so what’s the actual answer? 2.6 meters

OK, so how much time to hit the flag? Well, it took 3/4 of a second to go up. It would have taken another 3/4 of a second to go down, but it didn’t quite go all the way down because it hit the flag. So, How far did it get? Well, it got a little over half way down (2.7-1.2 = 1.5). Let’s just call it half for good measure. Now it might be tempting to think it took half the time to go half way down as the whole way down. That would be true if it was moving at constant velocity. But it’s moving slower earlier and faster later, because it takes time to speed up. Given that, we know that it must take MORE than half the time to cover half the distance, because that’s the slow part of the motion. By that alone we can say that it must have taken more than 3/4 + 3/8 = 9/8 of a second. So we have a lower bound of 1.1 s and an upper bound of 1.5 seconds. Let’s just guess that’s it took 0.5 second, because that’s more than half of .75 seconds.

That would mean the whole trip should be about 1.25 s = 0.75s on the way + 0.5s on the way down seconds.

The real answer? 1.2 seconds.

OK. So, some of this is a bit much. But, I didn’t have to use equations.

 

Yesterday I met a new physics teacher in our area. He has no background in teaching, beyond the two weeks he has been in class already. He has a science degree and worked in the public sector. He is teaching sections of physics and chemistry on an emergency certification. He is looking for help, suggestions, ideas and advice–whatever he can get.

I am going out to visit his physics class next week.

What do you think is the most important thing to focus on with a new teacher like this, especially with no training? I have never had to work with a teacher with absolutely no training. I can only imagine that there are lots of places where he’ll need support; but right now it’ll have to be like triage, deciding where help is needed most. You can certainly tell he is a bit frazzled, although not much more than you would expect for anyone teaching their first year. From his perspective, what he needs is ideas for what to do in the classroom–activities, lessons, things to get student engaged.

Anyone out there been in his position: What kinds of mentoring and help would you have found most helpful? What kinds of mentoring support would have been frustrating or unhelpful?

 

 

Besides my inquiry course for elementary teachers, I am also teaching is a section of non-calculus-based physics. The class is highly structured, so that I am told exactly what to do each day. The basic setup is this:

(1) Students come in and take a 2-3 minute reading quiz, which consists of five MC questions that I write (so that each section has different quiz). They are supposed to have read “lecture” materials online, and quizzes are intended to have required reading but not full understanding.

(2) Students get into a group (8 groups of 4 I think) and work through a series of conceptual questions on a computer (10-15 minutes). This isn’t graded, but the questions give immediate feedback.

(3) Students then come back to listen and watch as an instructor works through a sample problem (20-30 minutes). The example problem is a goalless problem. It’s also probable that the instructor will discuss the computer exercises. I am told exactly what example problem to go over.

(4) Students then go off to work in groups on different but related goalless problems, structuring what they do, in part, on the sample problem that was just worked out.

(5) Students discuss their problem solutions with the whole class. They work on their problems in whiteboards, so that is also how discussions are structured.

(6) Students go work on a lab. Most of the labs are not graded. They are just checked. I think 2 or 3 labs are graded more closely, but students are told which ones will be ahead of time.

Everything happening in any given day of class is around one concept. Class is 2.5 hours long and is twice a week. Often for the labs, they will collect data one day and analyze it the following day. There is also a 1.5 hour lecture, where all the sections meet. The lecturer runs some Peer Instructions with clickers, but also discusses the data students should have collected. They discuss issues in graphing data, linearizing data, etc, in order to prepare them for the next day where they do analysis in the lab.

As part of the lab class, they also have to carry out an independent investigation, write a paper, and give a presentation. I still need to learn more about what this is all about.

I’ll try to keep everyone updated on the how the flipped class goes for me.

Back when I was blogging using blogger, I posted a series about light and water.

First, I shared with you my intrigue over how a lake looked like a mirror in the morning and a sheet in the afternoon. To follow up on my intrique, I shared with you another pair of photos, in which another lake looked like a mirror and a lake from different vantage points (but at the same time of day). Some while later, I shared with you a rather contrived observation–a whiteboard can act like a mirror or a sheet in the same way a lake does, depending on your vantage point and how light shines on it.

This lent some support to the idea that angles are really important. One reason they are important is because shallow rays don’t penetrate  deeply, so it’s like you are seeing the surface. When the sun is low and you are low, you have a high intensity beam glancing off the surface, showing you the mirror effect off the relatively lat surface. When the sun is high, you still get some low glancing reflections, but the mirror image is overwhelmed by the scattering of rays that penetrate deeper into the lake. When you are high, glancing rays never have a change to get to you, so all you see is the sheet effect.

Today, I’m struck by how similar looking at lakes is to what physicists do when they investigate matter. Physicists shine light at all kinds of matter and look what comes out. They vary the intensity. They vary the frequencies. They vary the angles. What comes out tells physicists about the structure of the matter that light was interacting with. In that sense, looking at lakes, especially from different angles from different perspectives with different intensities, is the naturalist’s scattering experiments.

One of the things I don’t like about most of the modeling curriculum I have experienced is the tendency to jump into taking data. In the summer workshops I attended at ASU and the several mini-workshops I have attended elsewhere, I was pretty much always asked to go take some measurements of things.

(1) Go measure some distance and time data

(2) Go measure some force and distance data

(3) Go measure some current.

(4) Go measure some frequency and length data.

And then we graph the data and look for relationships among variables. Next, we’ll often try to make a best fit curve to model the situation and interpret the meaning of the parameters in the equation.

The problem is that I always felt like I was just measuring things and looking for relationships without purpose. There were no puzzles we had identified as worthy of my epistemic curiosity. We had identified no perplexing questions that made me wonder, “what kind of data should I take to help answer that question?” What relationships am I expecting and why? How will I know if the measurements I’ve taken are good enough to either support or refute one idea or another? None of that was going on–not for me at least. And that was the thing that was puzzling to me.

Now don’t get me wrong–empirical data is important in science. But it alone is not science, at least not to me. I have been more apprenticed into starting science with something perplexing and letting that perplexing situation be the source of ideas, arguments, and explanations that need to be sorted out. In my mind, there’s now a reason to take data–that data will be EVIDENCE to support for claims. To me, the the subtleties that entangle and distinguish data and evidence are crucial for understanding the nature of science–both to do science and to be scientifically literate.

Leslie Atkins is fond of this quote that supports this view:

Observation and experiment are not the bedrock on which science is built, but rather they are the handmaidens to the rational activity of generating arguments in support of knowledge claims. (Driver, Newton, & Osborne, 2000, p.297)

Now, I’m certainly not saying that scientists never just muck around with data, or discover interesting things by looking for at relationships, or plotting data, or trying to make sense of mathematical equations. So what am I saying? I’m not exactly sure. Maybe, I’m asking someone to explain to me what I’m not understanding about the modeling curriculum.