Light and Water: Post Eight

So far in my own inquiry into water and light, I've really been focused on understanding the role of viewing and illumination angles. Here are some ideas I'm more and more committed to:

• Shallow or glancing angles tend to reflect more off the surface of the water (like a mirror), This makes it so that being down by the lake you are more likely to see reflections, whereas being up high (looking down) on a lake you are less likely to see reflections. This relates to post two and post five
  
• A low sun behind you illuminates strongly the objects across the lake that are to be reflected, but the sun light only hits the lake with glancing blows (which mostly reflect away from you). The combination of strong illumination on the object and low levels back scatter from the sun make it easier to see the reflection.
  
• On the other hand, a high sun hits the water with more penetrating rays (not glancing), which refract into the water. In many cases, that light scatters off particles suspended in the water, which go off in all directions. In this case, the high amount of scattering creates noise that overwhelms any mirror reflections.
  

These last two bullets seem to concern post one and post seven

But there certainly is a role to be played by wind roughing the surface and the polarizing filter on my camera. So here are some photos that I think speak to this:

Here is an example I do think is likely wind, although glancing angles could still helping. This was also taken on Kidney Pond at Baxter State Park, looking toward Mt. OJI. I'm still doubtful that wind can fully account for this phenomena I posted about with the receding mirror boundary, but my mind is still open.


Here is one that I think has an effect from the polarizing lens (compare the real sky to the reflected sky). This photo was taken from the Stillwater River by the University of Maine.

I promise I'll get back to posting about teaching soon, but for now, I'm happily sharing my inquiry with you.

Light and Water: Post Seven

In this video, I show how the whiteboard can function as a mirror or as a sheet depending on where I shine the light. Does this have anything to do with any of other posts?

Light and Water: Post Six

This was picture was taken from Kidney Pond. Any guesses as to why I think this picture is interesting?

Light and Water: Post Five

In a series of posts, I have been sharing some of the fun and perplexing aspects of nature of light and water. In this post, I share some fairly contrived observations that seem to touch upon that perplexity in a new way.

For these photos, I took a blue marker and painted "a lake" on a small whiteboard. On my large whiteboard, I painted a blue face. Then I held the small white board up against big white board a a right angle (right by the face), and took some photos from various vantage points.

As you progress along the three photos, I change my viewing angle from glancing to more normal to the small whiteboard. What happens seems to be like what I showed you in post two, where the lake turned from a mirror to a sheet.

Some questions that I've been pondering over

1. How Can I explain the whiteboard phenomena
   
2. How might this be the same and/or different from water phenomena, especially post two?
3. Has this changed or refined my thinking about any of the previous posts?
4. In what ways is the whiteboard like a mirror and not like a mirror? And Why?

And for, fun, I added these photo as well.




And just to show you that this isn't a trick with markers



And lastly, a link to a movie

More natural and contrived observations are on the way.

Light and Water: Post Four

So far, I've been sharing photos that show bodies of water appearing as mirrors and appearing like sheets, depending on the conditions.

On twitter and in blog comments, I've heard about a range of ideas:

• wind roughing the surface of the water
  
• minerals, sediments, glacial flour in water influencing the color
  
• angle of the sun and the viewer changing what gets reflected to viewer, and
  
• those same angles changing whether the light mostly reflects off the water or refracts into the water.
• Update: The influence of a polarizing filter
  

In this post, I show three photos taken from Ship Harbor in Acadia National Park. In the first photo, the water acts like a window, letting us see into the bottom. With the second photo, I take few steps, turn my head, and the water changes character dramatically. How would you explain the difference? What, if anything, does this have to do with the ideas above and the phenomena shown in post one, post two, and post three.





In this third photo, the water shows its many different faces in a single shot.

Light and Water: Post Three

Here is post one. Here is post two.

In this post, I have a sequence of three pictures taken at Kidney Pond in Baxter State Park around the time the sun was setting. What, if anything, does this have to do with how we should be thinking about post one and post two?

Update: On twitter, the idea has been put forth by several that this is the result of the winds dying down as the sun goes down. The seeds of doubt for me is that it doesn't account for the receding nature of the boundary over time. I think it might have more to do with the portions of the lake receiving direct sun light. In other words the "image" of the trees is always there, but it's hidden in the noise of the scattered light. The image is "hidden" behind a curtain, and as we pull back the curtain, we see more and more of the image that was always there.

Light and Water: Post Two

In post one, I showed two photos of a glacial lake appearing quite different in the morning and the afternoon from the same location.

The two photos below were both taken in the morning. The first one was shot at 7:13 AM right on Lake Louise, and the second one was shot at 7:34 AM as I was little ways up the trail to the Beehive. Once again, we see the glacial lake appearing as a mirror and then appearing as an emerald sheet.

The questions now are: Why does the Lake look different in these two photos? How does this explanation compare to your explanation for post one?

Light and Water: Post One

The two pictures below are from Banff National Park. Every morning, the glacial lakes began as mirrors, but by afternoon those same lakes would turn emerald. The question is why?

Five Things about Learning

I was asked by a friend recently to state five things I have learned about learning. Here is what I came up with in the moment.

(1) You can't give someone knowledge, you can only put provide them with opportunities to create new knowledge for themselves and with others. To teach is to provide such opportunities and to help others make the most of them.

(2) We learn in a lot of different ways–playing, talking, listening, observing, doing, reflecting, reading, writing, collaborating, practicing, tinkering, just to name a few.

(3) Emotional engagement and social interaction are not only integral to learning but inseparable from thinking and knowing.

(4) The nature of expertise and naivety are deceivingly complex, no matter how hard we try to reduce them to simpler forms.

(5) Learning and education cannot be reduced to a science. Not because it is art, but because notions of learning and education are framed by discussion of morality, philosophy, and culture.

The Many Roles of Content Knowledge for Teaching

Years ago, I was listening in on a group of college physics students who were working on some fairly standard torque-balancing problems. They had been given situations like the one below and they had to decide whether the situations were balanced or not.


The group had an interesting strategy that I call the “equal exchange” strategy. For example, students would take the “two blocks at the 1-notch” and replace it with “one block at the 2-notch”, because that was an equal exchange. For this situation, the strategy quickly reveals the answer, because each side now has “2 blocks and the 2-notch”, as shown below.


In working with the other graduate TAs and the professor running the prep session the week before, no one had used or even mentioned this strategy, neither as a strategy they would use or that students might use. All of us simply summed and compared the torques, by writing out 2*2 = 2*1+1*2. And we did the same for nearly every situation.

For me as a novice teacher, I was intrigued by what the students were doing. To me, it was thrilling to witness these students, all by themselves, inventing a novel way to solve the problem that I had never considered. Part of this thrill was that students were doing something different from me, but a large part of the thrill was wrapped up in me knowing that what they were doing was valid, despite being different.

Here’s something interesting to think about. For me, the physics knowledge I had to use to evaluate the validity of the students’ strategy was, in some ways, special to the task of my teaching, because it wasn’t the same physics knowledge I used to solve the problem. I (along with all the physics graduate TAs) summed the torques in order to compare the net torque numerically. The students’ strategy involved getting the situations to be visually comparable. The fact that I could see our strategies as being related and both valid is a kind of content knowledge that I needed to adequately assess what the students were doing.

Of course, some of the problems students had to work on were much harder than the situation above. So, the strategy to get all the blocks in one place can get a lot more complicated. Take for example, this situation:

In this situation, the number of moves not only goes up, but you have to do some more daunting proportional reasoning. As these students got to ever more complicated situations, the students were taking a lot longer than the other groups, and making more mistakes.

The question, for me as a teacher then, was, “At what point, if ever, should I step in to help them to discover other, perhaps more efficient, strategies?”

First, it’s helpful to reflect on some things. First, recognizing why their strategy was becoming increasingly difficult required that I have a particular mastery of the physics content and the physics reasoning. Recall that to solve the problem myself, I didn’t need to consider proportional reasoning or multi-step problem solving, because I just had to sum the torques. But now in this moment, in order to assess students’ progress moving forward, I had to be able to think about the physics concepts and problem-solving strategies in a particular way that was different than before. I had to be able to project the students’ problem-solving strategy into the future and into different problems in and make hypotheses about where it might lead them.

As a teacher, I could have chosen to engage students in developing their strategy, by helping them to be careful with proportional reasoning or with planning out more effective moves; or I could have chosen to nudge them toward my more efficient strategy. Given different goals and constraints, there is no right answer about what to do. But, for a me to make an informed decision, I had to be in the position of listening and making sense of what the students were doing. In order to be in that position, I had to have a unique mastery of the physics content and reasoning that, I’d argue, went well beyond being able to solve the problem myself.

Seeing other Connections

Looking back on this moment, other question for me as a teacher are these: “What does their strategy imply about what they are likely understanding well? What does this strategy imply about what students might not yet understand?”

To me, the students strategy shows me that they are likely making sense of Torque as Mass x Distance. They understand that idea well enough to know that there are variety of ways to get an equal torque by changing the mass and distance. In particular, most of their reasoning fell along the lines of, “if you triple the mass, you better third the distance.”

But their strategy also hints that may not be having the opportunity to develop other important ideas. For example, they might not be learning that torques are summative (e.g., Net Torque = Sum of Individual Torques). If it’s important for students to learn this, a goal could be for me to make sure that this group is provided with an opportunity to learn that idea as well. It’s not just a matter of them learning a more efficient strategy, it’s about the opportunity to make contact with important physics ideas that they might not using their strategy alone.

The Big Picture

This example highlights for me that the role of content knowledge in teaching is wide and varied. The content knowledge I mention here is often referred to as specialized content knowledge. It's the content knowledge needed to evaluate a student solution that you may have never seen or thought about before. It's the content knowledge needed to project a problem-solving strategy into the future. It's the content knowledge needed to relate problem solving strategies with important conceptual knowledge. The reasons why this is content knowledge is that it need not have anything to do with students. An expert could have proposed these strategies in a journal of physics, and it could then be my job to evaluate the validity of that approach, or to see how that strategy would play out in a variety of situations, or to see what concepts are embedded within that approach. Some of that content knowledge is, in some ways, unique to teachers and teaching; because the range and variety of alternative solutions that teachers face are unique due to the fact that they are dealing with students. Thus, some of the content knowledge that teachers need to evaluate those solutions is unique to their tasks of teaching.

A big question for researchers is, "What kinds of content knowledge do teachers need for teaching? And where do teachers develop that knowledge?"

For me, I have developed a lot of that content knowledge by paying attention to students, by listening and reflecting on what they are doing. And I have further honed this knowledge by actively seeking out and reflecting on potential connections among what students are doing and the disciplinary knowledge and skills of physics. To be sure, I will continue to develop and refine this knowledge as I continue to teach in ways that allow me to listen and reflect on what students are doing. For this reason, how I arrange my classroom teaching in ways that allow me to listen to students is extremely important.

I hope this helps other to understand my concern of misconceptions listening, in that it provides less opportunities for teachers to develop the knowledge that furthers their teaching along.

I guess I do want to talk about misconceptions

In my last post, I used students' thinking about the seasons as an example to talk about the limited view that misconceptions brings to student thinking. In this post, I want to further that conversation with another famous example:

Passive and Active Forces

When students are asked to identify the forces acting on a book sitting on a table, some fraction of students will say it's only gravity pulling the book down. The table, they might argue, doesn't push up on the book. The table simply gets in the way–it blocks the book from falling.

One way of making sense of this is that students think of forces as pushes and pulls. In order for something to push or pull, then, it has to be active in some way. For example, you expect to see a person straining their muscles to push or pull. You expect to see a spring being compressed or stretched when it pushes or pulls. The table, students might think, can't actively push or pull, so it can't be exerting a force.

Here students seem to have the wrong idea, so we might tempted to think "forces as active" as a misconception, causing students to think that the table doesn't exert forces. From that view, we'd want to change the students' conception of force, so that they would understand why a table does exert a force. Here are four reasons not to think this.

Reasons #1 Not to think of this as a misconception

Wrong answers are not necessarily indicators of bad ideas. John Clement, for example, saw in students' notion of active force a quite productive idea. So instead of trying to change students' conception of force, John set out to help students to "see" the table as being alive and active. His instructional strategy aimed to help student see the table as a "springy" surface that just happens to be very stiff. In this case, it wasn't the conception of force that needed refining, it was the conception of table.

Reason #2 Not to think of this as a misconception

Ways of thinking you don't like now are often the ways of thinking you'll want back later. From an introductory physics perspective, we want students to think about surfaces exerting normal forces. So if they don't think of tables as exerting forces, that's a problem. But the truth is, later we'll want physics students to think of surfaces as merely constraints upon motion. So if students think of surfaces as blocking motion, that's (kind of) exactly how we want them thinking about it. So, what was once a misconception has suddenly become sophisticated.

Reason #3 Not to think of this as a misconception

Force is not a singular concept, and neither are students conceptions of it singular. Force is more of an explanatory framework. There are many bits and pieces that have to come together. The only reason it makes sense to think of the table as exerting a force on the book is once you've put all the pieces in place.

To put this more clearly, thinking of the table has exerting a force has a lot to do with having a commitment to the notion of equilibrium and a commitment to the notion of net force as a explanation for equilibrium. So, what seems like a question about force is really a question that about a person's level of commitment to a whole framework. Included in that framework is ideas that interconnect the ideas of force, net force, and equilibrium.

Reason #4 Not to think of this as a misconception

Concepts and language are not the same. If you don't mention anything about force, and ask, "What's holding the book up?", every child and student will say the table is holding the book up. So even though they don't think that the word "force" should describe what the table does, students DO think that the table is interacting with the book. From this perspective, students have the right idea, they just don't think the word force should apply.

The Big Idea

I think my point is this. It's fine to think about misconceptions, as long as it doesn't stop you from doing the kinds of thing I just did.

What did I do?

• First, with the help of John Clement, I spent time thinking about the potential productivity of students' prior knowledge and how I might capitalize on that for classroom learning.

• Second, I tried to see connections between what students know now, and what kinds of knowing I might expect them to know soon (and also down the line).
  

• Third, I thought about all of the ideas that students would need to have in place, and resisted the temptation to think of the problem as simple application of isolated knowledge

• Fourth, I spent time thinking about what the ideas students have, and resisted the temptation to evaluate students' ideas based on vocabulary alone.
  

I said I didn't want to talk about misconceptions

I have written a few posts now on the subject of misconceptions:

• Here I discussed why I often avoid talking about misconceptions
• Here I discussed how I think we often misconceive misconceptions
• Here I discussed a kind of classroom listening I call "misconceptions listening"

Let me start this post by saying, I see that there are some potentially good things that come about by focusing on misconceptions:

1. It can draw attention to the relevance of prior knowledge for learning
2. It can draw attention to the importance of conceptual knowledge in learning
3. It can draw attention to the shortcoming of traditional instruction to promote conceptual understanding and take into account prior knowledge.

But I also think there is a lot of bad baggage that comes along with that focus:

1. It's promotes a deficit-view of learner, focused on what's wrong with students (e.g., students aren't just blank slates, they're worse than blank slates)
2. It's anti-constructivist in its focus. It fails to drawing attention to the ideas that students do have that instructors should look for and try to build on.
3. It often conflates student discourse, students' prior experiences, and students' conceptions. How we talk, think, and experience are connected, but they aren't the same.

These lists aren't exhaustive. I think there are more goods things, but also a lot more bad things.

Anyway, I thought I would spend another post rambling on about misconceptions:

First example: Misconceptions about the Seasons

A commonly described misconception is the one where students say that the earth is closer to the sun in the summer than it is in the winter. One way of explaining why students might say this is that people have an generalized notion of "closer means stronger", arising from having had many experiences of being closer or farther from a variety of sources-- heat from a fire, sound from a speaker, smell from rotten food. We even get closer to people to feel their love.

If an instructor thinks that students have a "season's misconceptions", than the goal of instruction might be to rid the student of their misconception, or overcome it, or elicit it and confront it. However, if an instructor instead thinks the students' conception is the idea that "closer means stronger", than that instructor probably wouldn't think of trying to eradicate that idea. That idea seems like a good conceptual basis for understanding lots of ideas– the 1/r^2 fall off of fundamental forces, why radiation and sound intensity fall off, diffusion, etc.

Compared to what college students typically say to explain the seasons, I'd really rather have someone say, "The sun would be closer in the summer, because that would explain why it's warmer–you are closer to the hot sun!" At least that idea makes sense, and is an explanation. Lots of college students say it's "because of the tilt". But they say that simply because it's the answer they've been told.

I have found that I can get a lot of traction by asking students this question instead: "Why in Maine is the sun out for 16 hours in the summer, but only 8 hours in the winter?" Students will bring up lots of ideas. They will have lots of false starts, and attempts to explain. They will notice puzzles and inconsistencies in what they are saying. New questions will arise. But almost no one brings up a model in which the earth is closer in the summer, and if they do, they recognize that this doesn't explain the difference in day light hours.

It's interesting to think about this effect. When the thing to be explained is "why warmer", a common explanation from people is "it must be closer". But when that prompt is "explain why sun is out for more time", students bring up different ideas. To me, this is important, because science is the process by which we take into account more and more aspects of a phenomenon (or range of phenomena) and try to build a more globally coherent explanation. From this perspective, students have lots of 'localized' explanations.

For that reason, a focus on misconceptions misses the point. If my students are only trying to explain "why warmer", then "closer" is a pretty good explanation, even if wrong. Eventually, of course, students are going to get around to trying to explain "why warmer" and "why changing day light hours" and "why the hemisphere are in different seasons" and why "sun rises in different locations throughout year". Explaining all of that that sounds much more difficult, to me and to my students. I'm going to need to be patient for that happen.

As I see it, my job as a teacher isn't to correct misconceptions. Rather a big part of my job is to help students make contact with important aspects of the phenomena, and for me to press upon the coherence of their explanations with respect to the evidence, arguments, and tools that they currently have at their disposal.

RTOP: Solid grasp of subject matter?

Here is another item from the RTOP:

The teacher had a solid grasp of the subject matter content inherent in the lesson.

At first glance this seems pretty silly for a classroom observation. But digging deeper, and reading the clarifying paragraphs gives this statement new and wonderful meaning:

"This indicates that a teacher could sense the potential significance of ideas as they occurred in the lesson, even when articulated vaguely by students. A solid grasp would be indicated by an eagerness to pursue student’s thoughts even if seemingly unrelated at the moment. The grade-level at which the lesson was directed should be taken into consideration when evaluating this item."

This isn't about having subject knowledge. It's about knowing a subject matter well enough that a teacher can see fragments of disciplinary knowledge in all the things that students say and do in the classroom. A scientist could easily score low on this, despite having mastered the content, if that content mastery didn't allow them to listen and "see" the beginnings of knowledge in classroom discourse.

An Example from Physics

In the physics classroom, an example of this came up this semester. A lecturer was presenting on the topic of constructive and deconstructive interference, and was discussing lasers as an example of constructive inteference A student asked a question at some point about this being like polarized light. The lecturer was thrown off by this question, and went off on an explanation for why the two had nothing to do with each other. While it is true that polarizers and lasers are different phenomena, arising from different mechanisms, there is a lot of conceptual overlap between the two. In particular, with both situations there are waves, and the phenomena involves thinking about the degree to which waves are or are not aligned with each other. In both cases, those alignments can be described with an angular measure. The difference is that one involves an alignment of phase relations and the other involves an alignment of oscillating planes.

While I'm positive that the instructor understood both concepts fairly well, his understanding didn't help him to see meaningful connections between both content areas and the students' question. For that reason, I might score this RTOP item low.

Anybody have any good examples?

Teacher as Listener: What are you listening for?

One of the items on the RTOP is, "The metaphor 'teacher as listener' was very characteristic of this classroom."

Overall, I think it's a good thing for a teacher to be a listener. Over time, however, my views of what this the metaphor means have changed. I am more interested in how a teacher listens than if they listen.

One kind of listener I see I would describe as a "misconceptions listener". The misconception listener has several characteristics:

• They almost exclusively listen to students' ideas through a lens of correct and incorrect (rather than listening for the possible productive beginnings of ideas, or whether or not a student's idea involves appeals to evidence, or to consistency, or whether or not a students' reasoning is plausible, mechanistic, compelling, particularly lucid, etc.)

• They are often aware of lots of misconceptions and difficulties. They often, but not always, utilize classroom strategies that aim to elicit and confront them.
  

• They have a difficult time letting incorrect ideas become the focus of discussion (unless it's to discuss why the incorrect ideas are wrong). They subconsciously fear that an instructor's engagement with (or silence about) wrong ideas is tacit endorsement for those ideas being correct.
  

• When students are off the mark, they use Socratic questioning strategies to guide students back toward saying the right things (and almost never use dialogic questioning strategies to help everyone, including the teacher, get to know their ideas better).
  

• They often have developed a good poker face to use with students. Because of this, they too often engage with students in a way that involves a significant degree of deceit.
  

Let me state that there is nothing wrong with (1) listening for and having concern about the disciplinary knowledge that may or may not be evident in student ideas, (2) being aware of common difficulties and building instruction around them, (3) using questioning strategies that aim to nudge students along, (4) deciding at times not to discuss with an entire class a confusing idea, and (5) having a good poker face.

My concern is with instructors whose whole range of listening behaviors falls narrowly within the confines of "misconceptions listening". Misconceptions listeners never really listen to their students' ideas on their own terms, because they are always on the look out for what's wrong with students' ideas. Because misconceptions listeners never really listen to their students' ideas, they are unlikely to grow as a teacher. My concern is that misconceptions listening is not a generative practice. In my experience, it seems to be a dead end for many instructors.

Inverse Problems:

I feel that we give students too many problems in which they have reduce the complexity of the world down to a single number. We describe some complex situation involving an object moving, and students should find the average velocity–a single number to describe some important feature. Or, perhaps, we give students some combination of resistors in a circuit, and students should find the equivalent resistance–a single number that describes something important. Here are some forces acting on an object, find the acceleration.

How often do we ask students to come up with 4 different situations in which the average velocity of a trip would work out to 55 mph?

How often do we ask students to come up with 4 different ways to get an equivalent resistance of 10 Ohms?

How often do we ask students to describe three different situations in which an object would roll down a ramp with an acceleration of 3 m/s per second?

I think there is a lot of value to these inverse questions, including, but not limited to the fact that

- They require and value creativity
- There are many, many ways for students to be right.
- They seem to require that students engage with the concept (not the equation)
- Procedural aspects are used to check their solutions (not arrive at them)
- Attempts that don't work out can be revealing to students, but on their own terms
- A comparison of solutions across students can highlight important features of the concept

I'm wondering if this should only be used as a learning tool, or if such questions are also viable candidates for assessment.

Instantaneous Speed: Jokers Wild

Last semester in help study, students were solving a problem to find out how fast an object would be moving right before it hit the ground, if it were dropped from 1 meter.

One student thought that it would be moving at 10 m/s, because 10 m/s/s is the acceleration due to gravity, and it had fallen 1 meter to get that speed.

A second student responded that it would take a whole second to reach that speed (not a whole meter). He added that it should take 10 meters to reach a speed of 10 m/s.

The Difficulty

I like both of these students' thinking, because it helps remind me of something: There are many conceptual pieces to acceleration that make it difficult to understand. One of those pieces is the concept of instantaneous velocity, and how it is different and related to constant and average velocity.

See, the first student is thinking that, in freefall, 10m/s of speed accumulates in 1m of distance. It doesn't. It accumulates over 1 second of time (no matter how much distance). The second student seems to get the accumulation of speed over time idea (at least in words), but is thinking of the 10 m/s accumulated as if it were the constant or average speed. While the ball will end with a instantaneous speed of 10m/s, it's average speed over the entire trip down will be 5 m/s, meaning it only covers 5 meters not 10.

Instantaneous Speed Taught:

Lot's of people teach instantaneous speed as a limiting process. You can do this with calculus, or with graphs (slope of tangent line). You can do this by zooming in on your calculator until it looks linear. You can do this by looking at really short segments of ticker tape. Whatever.

I think all of that is fine, but I think there's a better way:

I want to think about instantaneous speed as something like this: The speed you would measure if you could somehow keep the object from gaining or losing speed (long enough) to measure.

You don't NEED a limiting process to do this. All you need is a ramp.


If I want to measure the instantaneous speed of a ball somewhere along the ramp, I just stop the ball from speeding up or slowing down beyond that point. By making the ball go flat, I can measure it's constant speed, and that will be a good measure of how fast it was going right before it went flat.

If I want to profile the speed of the ball in many places along the ramp, I just repeat the procedure for various locations along the ramp. This has lots of extensions: you can change it up by starting it with some initial speed. You can ask students to predict how the final speed will compare to speed half way down. You can ask them to compare the d/t on the ramp to d/t on the flat part. Yah, yah, yah.

The Big Picture

I think there is real value to this. First of all, it is completely consistent with the concept of instantaneous speed. Instead of finding the slope of the tangent line on a graph; we are creating a tangent line with our measurement, by making the ball travel with constant speed–that's what a tangent line is. We determine the slope of that tangent line by actually measuring the distance and time of the ball along the flat path, instead of calculating a slope on a graph.

Second thing: This allows students to approach the idea of instantaneous speed without a limiting process. Now don't get me wrong. I'm going to teach my students to think about instantaneous speed as speed at an instant, and as a limiting process, and as a slope of a tangent line; but I'm going to do that when I need to. When will I need to? When I can't repeat an experiment over and over again. See with the ramp, I can send the ball down as many times as I want and "interrupt" the ball's ramp motion wherever I want. But what if I only have one shot to let the object move? Or what if I am watching a movie on you tube? Or how would you interrupt a ball falling vertically? All you get to do is observe, not experiment. I those cases, I'm going t have to work with my students to find a new method of measuring instantaneous speed.

Third, the "interrupt" method of measuring speed, especially with the ramp, has nice ways of engaging with the relationship among instantaneous speed (anywhere along the ramp), average speed (along the entire ramp), and the constant speed along any flat part. Building them as a set of relationships rather than separate concepts is necessary for students to understand the question this post started with.

I've only taught instantaneous speed this way once, but I thought it went quite well for giving students a way to think about the concept. I'm certainly not saying it's magic bullet, but it's certainly an approach worth consideration.

Kinematics: Acceleration is the Jack of Spades

One question I like to ask introductory physics students is this one:

Starting from rest, an object accelerates at 4 meters per second squared. What will be its speed in 3 seconds?

When I ask this question, I am looking for students to not look for an equation. I just want them to say "12 m/s, duh!" or, "Well, it would be going at 4 m/s after on second, and 8 m/s after two seconds, and 12 m/s after 3 seconds" or, "Well, every second that goes by, it gains 4 m/s of speed, so in three seconds it will gain 12 m/s."

Many students will say they have no clue without an equation and then give up, or they will go looking for an equation and then plug into the equation vf = at + vo. To me, either giving up or plugging into that equation tells me something important about that students' understanding of acceleration–they have no idea what accelerations means, at least not in a way that allows them to tell a story with it.

For the students who do succeed with or without an equation, I ask them how far it went during that time. To me, I think about this by saying that, on average, it must have been going with a speed of 6 m/s (half way between 0 and 12). Since that average speed happened over 3s, it must have gone 18m.

Nearly every student I encounter goes to the equation x = 1/2 a t^2. And while there's nothing wrong with that, I know these students are victims of the "Big 4", because I am fairly certain about two things:

99% of students who goes to an equation won't be able to solve both of these problems:

(a) In 4 seconds, a particle is accelerated from rest (by a constant force) to a speed of 20 m/s. How far has it gone?

(b) A bowling ball is hit impulsively with a rubber mallet, causing it to roll across the floor at increasingly faster speeds. If the ball is whacked with the hammer once per second, such that it speeds up by an addtional 10 m/s with each hit, how far will it have gone in the first 3 seconds after the initial hit.

The answers are:

(a) Over the 4 seconds, it's average speed was 10 m/s, meaning it went 40 m in those 4 seconds.

(b) In the 1st second, it travels 10m. In the 2nd second, it travels 20m. In the 3rd second, it travels 30m. In total, it travels 10m+20m+30m = 60m

Students Answers

In the first problem, students will likely say that the distance is 20 m/s * 4s =80m

In the second problem, students will likely say that x = 1/2 at^2 = 1/2 (10)*3^2 = 45m

How do you make sure your students understand acceleration?

I'm not sure. But I think the answer is to try to ask them questions that require them to make both sense of and tell stories that involve acceleration.

Kinematics: Is Average Velocity Queen?

In college physics, students often learn a slew of kinematic equations:

I'm a big fan of only teaching this one:


There are four reasons I like just using this, both personally and for teaching. It's all-purpose; conundrum-focused; tool-and-representation-infused; and intuition-refining.

First: The equation works all the time. There are no caveats to remember. As I've gotten older, this is the first of my two "go-to" ideas in kinematics. I will say that I almost never use the 2nd or 4th equation from above to solve any real or textbook problem.

Second: You get to engage students with the conundrums of figuring out what the average velocity is for a variety of situations: unchanging velocity, steadily changing velocity, piece-wise changing, non-linear changing, etc.

Third: You can explore various techniques like weighted averages or riemann-sums using data from tables, graphs, and other representations. It naturally builds in ideas of approximation.

Fourth: With these tools, you get to explore the boundaries of intuition (e.g., looking at this graph, what would you guess the average velocity to be? Or the famous example of determining the average speed when traveling 45 mph one way and 60 mph the other way on a round trip)

This is me guessing what the average velocity should be for a few situations:

At the end of the day, for instruction, you are likely to still have to fold in ideas about instantaneous velocity, changing velocity, and acceleration. It's something I want to talk about, but I'm going to leave it for another post.

Equivalent Resistor Twister

Recently, I've been having a lot of discussions with colleagues about equivalence resistance, and the difference between teaching the concept as a mathematical tool for thinking (and problem-solving) vs. teaching the parallel and series equations in order to calculate resistances. So far, not a single colleague has demonstrated much facility with mathematical thinking about equivalent resistance situations without resorting to the equations. Once I show them how, however, they are off to the races, and often pretty excited about their new mathematical tool.

Here are two problems that are fun if you think about the concept and simply miserable if you approach it using guess-and-check-with-equation.

Problem 1
Using only 10-Ohm resistors, come up with 4 different ways of hooking them up to a single (ideal) battery such that the equivalent resistance across the battery is 10 Ohms.

Problem 2
Using only 3-Ohm and 2-Ohm resistors, come up with 4 ways of making a 1-Ohm equivalent resistor circuit element.

Curriculum Evaluation Task Force: Nearing the End

Since November, I have been working with a diverse group of math and science teachers to evaluate and put forth a recommendation for a new middle school physical (and earth) science curriculum.

Our group consists of the following:

• A Grade 6 Math Teacher
• A High School Math Teacher
• A Grades 7/8 Life and Physical Science Teacher
• A College Physics Professor
• A High School Environmental Science Teacher
• A Grades 7/8 Earth and Physical Science Teacher
• A Grade 8 Math and Physical Science Teacher
• A Grades 7/8 Math and Physical Science Teacher
• A Grades 11/12 Physics and Chemistry Teacher
• A Grades 6-8 Earth, Life, and Physical Science Teacher
• A Grade 6 Math and Science Teacher
• A Grade 6 (Elementary School) Teacher for All Subjects
• A High School Earth, Physics, Physical Science Teacher
• 4 Physics Education Research Graduate Students

As you can see from the list, our group spanned grades 6-12, plus college. As a group we taught life science, earth science, physical science, physics, chemistry, and math. Our least experienced teacher had been teaching for a few years, while our most experienced teachers had been teaching for 3 decades.

We are now coming to the end of our work–last Saturday we met as a large group for the last time. On average, teachers put in about 100 hours of work each. They were paid for their time, and we always provided breakfast and lunch during our long Saturday meetings. In the coming weeks, we will be finalizing reports on our evaluation and recommendation.

For me, I'd say the most difficult part of the process has been to manage the tension between (1) pushing forward so they could accomplish the work within a timeline, and (2) providing time and space for norms, voice, and collective ownership to develop. My strategy was to almost always focus on (2), but to drive hard and with clear purpose on (1) whenever the group needed me to do so, and then to back off again to let the rewards of cultivating (2) pay off.

Although the road was rocky, I think we have accomplished much, largely in terms of planting the seeds for a strong professional learning community. The teachers feel they have learned a lot through this process, and they didn't want it to end. Throughout the day on Saturday, teachers were discussing how the were going to stay involved and not let the experience, knowledge, and excitement they've developed go to waste. A few teachers were talking about how they had already joined a curriculum selection team for math in their school. Other teachers are moving on to our 9th grade curriculum selection process. Others will be helping to plan our summer professional development academy and/or piloting in the fall. A mother-daughter pair on the team expressed to me how valuable this project has been them to spend time with each other on matters of their profession. I think this sentiment has been true for everyone. It has given just enough structure, time, and sense of purpose to allow teachers to come together on matters of their profession. I can only wish them good luck moving forward, as I won't be here to see it happen.

Impossible Trapezoid

In my science teaching seminar, we just read a paper on differentiated instruction, and I assigned for homework a writing assignment to discuss how differentiated instruction connects to what we have been learning about formative assessment (something we have been learning about all semester).

For class I was hoping to give to the learning assistants (LAs) a few different teaching scenarios. One scenario was going to involve some fake data from a "quiz" where math students had to do two different but related problems. In the scenario, 1/3 of your students bomb one question on the quiz, 1/3 of your students bomb the other question, and 1/3 of your students ace both of them.

I was going to have to the LAs look over the quizzes to decide what skills which students were struggling with, and have the come up with a plan for their class the following day that would further all students along in their understanding (hence differentiating the instruction based on formative assessment).

For one of the quiz problems, I was going to use this problem, taken from Dan Meyer's sample geometry tests. That is until I started to try to solve the problem my self. Can you figure out what's wrong with this problem?

The Getting Better 2 x 2

What have I gotten better at this year?

Giving concise & specific feedback to students. Especially when multiple things need attention, I have gotten better at holding back and just pointing out one thing that students need to work on and how they can do it.

I have gotten better at explicitly building in coherence across lessons, both through what I do in class and in what I ask students to do for assignments. I am still struggling to do this, but I am getting better.


What do I still need to figure out or work on?

Building more systematic coherence among (1) learning goals, (2) opportunities for students to learn them, and (3) assessment of learning goals. There are a lot of things I don't assess that I should be. I often shift priorities too much throughout the year, especially in classes that do not have high external accountability.

Getting to know my students better. I just recently learned something about one of my students that I should have known earlier- It has been his dream since 8th grade to be a science teacher.

One Situation: Multiple Force Models

I like this picture. And one reason I like it is because it reminds me that there is no correct way to a model a situation using a free-body diagram. Here are four free-body diagrams.







While the models becomes more sophisticated; I don't think any of these models is correct or incorrect. Depending on its purpose, sometimes the simplest model is the best one. All of this makes me wonder what dis-service we do to students when we over-emphasize the correctness of a free-body diagram and under-emphasize the fact that we are building and refining models. As we come to notice and become concerned about more nuanced features of a situation, we may (or may not) choose to attend to those features.

Snow Column Buckling

We got a really wet snow storm yesterday, allowing the snow to stick to the side of the wood on my balcony. In the morning, as it started to get warm, several of the columns started to buckle like this.


I thought it was cool enough to share. Doesn't the big kink look unstable? Oh yeah. I took this picture, turned away for a second; by the time I looked back, the column had collapsed.


Here's another baby one:





What physics do you see here? What does this picture make you wonder about?