In my intro physics course, we have many table-size whiteboards for students to work on, but the room I’m in also has 10 large whiteboards on the walls. Today, I told students to stop using the portable whiteboards and to start doing problems on wall boards. This did several amazing things.

• It allowed me to easily see from across the room whether or not groups had simply gotten stuck or were making progress.
• It allowed all the students in a group to have access to the board without anyone having to look upside down or become boxed out.
• It provide students much more space to write and draw, which dramatically boosted organization and clarity of their work.
• It helped me not hover over students when I came around, lending more agency and authority in their direction.

I am seriously trying not to use the word “conceptual” anymore– not to describe the kind of understanding I want my student to have and not to describe the kinds of questions I often like to use to assess student understanding. It is not a very useful word. It’s not a useful word, in part, because it’s a word that my colleagues use to mean various thing that I don’t. When they hear the word “conceptual”, I think some hear something like, “the easy physics” or the “basic physics”, or “mathless physics”. For others, I think it means something like “deep understanding.” For others it’s just a word to describe those tough questions that physics education researchers have come up with to stump students. For others, it means a multiple choice question, where students don’t actually solve a problem.

In fact, I’ll say I don’t actually use the word very often. It turns out that my colleagues do. I mention some question I asked my students and they say, “Oh yeah, those conceptual questions.” They smile and nod. I wasn’t thinking it was a conceptual question. I was thinking of it as a question that illuminated some aspect of my students’ understanding, or a perplexing question that inspired my students to think, talk, and argue. I was thinking of it as a question that demanded that they pull together more than one tool from their toolbox, or to pull out tools that they rarely get a chance to use (and so are clumsy with). I wasn’t thinking of it as a conceptual question. I was thinking of it as a useful question for teaching and learning— useful for me to find out what my students know or don’t know, or useful for my students because it provoked meaningful engagement with disciplinary ideas and skills.

There are, of course, other alternatives to the word. Skemp, for example, uses “relational” understanding and contrasts that with “instrumental” understandings. And I think this gets at something closer. But I don’t think a new word is the issue. I’m not sure what the issue is exactly… For some, it seems to be a difference in what we think is important for students to know and be able to do. That’s an issue of values. For others, it’s a difference in what we take as evidence of understanding. That’s an issue of assessment. For others, it’s a difference in what understanding is. That’s an issue of epistemology. I don’t mean epistemology in the abstract philosophical notion. I mean one’s personal sense of what it means to understand something. Like when my student told me last week that they knew how to solve every problem but still didn’t feel like they understood anything. By the way, this student did ace the exam. Epistemology is a feeling you develop, a personal sense of what understanding feels like when you have it and what it feels like when you don’t.

Of course, our values concerning learning, the ways in which we assess, and our personal epistemologies are not separate… but neither are they build together coherently. Developing them and working to make them consistent is tough work. For me, myself, I suppose it’s a lot of what I am trying to work out with this blog

Yesterday, I asked the question, “Is every color in the rainbow?“:

Here are some places our discussion went:

Yes, the rainbow has all the colors.

No, there are obvious examples of colors not in the rainbow: brown, black, gray, periwinkle, etc..

The rainbow provides the “toolbox” to make all the colors, but it doesn’t have all the colors. Colors like brown, pink, white are not in the rainbow, but could be made by mixing rainbow colors. This leads to question, how many tools are in the toolbox

Idea #1: Some believe that the only true colors in the rainbow are  ROYGB(I)V, with colors like red-orange being blends of red and orange. ROYGBV are seen as the basic colors that you make all the other colors with. Most with this idea agree that red-violet is not possible in the rainbow because they dont’ overlap.

Idea #2 Others thought that the only true colors in the rainbow are RYB, with colors like orange being simple blends of red and yellow. It is generally recognized that purple is an issue with this idea. There are some very exciting ideas about how purple might be ending up in the rainbow, including that rainbows aren’t 2D but are twisted around themselves like a tube so that R&B do overlap to make purple. Another idea is that purple shows up because rainbows are often double rainbows, so that the stacked nature of rainbows allows purple arises when the bottom of one overlaps with the top of another rainbow. I like how topological these ideas are.

Idea#3: There are an uncountable numbers of colors in the rainbow. We tend to only name a few colors we are familiar with, but we could point anywhere on the rainbow a see a new color, which we could name whatever we want. Our class has been using the name “Walch” as the example new name for a color.

Metallic named colors such as copper, bronze, gold, silver, chrome are not in the rainbow. There is the idea that gold just means “shiny” yellow and that silver means “shiny gray”.

Some colors leave people uncertain about. Pink, for example, might be in the rainbow, because we could consider it a shade of red, and red is in the rainbow. White might be in the rainbow, since some people believe that adding all light colors together makes white. Many state that it depends on what we mean by “color”, and others think it depends upon what we mean by “in” the rainbow.

The definition of color is certainly an issue. One proposed definition of color is this: “We know it when we see it”. Another definition was more like  this: “Any place on the rainbow you can point to and give a name is a color” Another is, “ROYGBIV define the true colors”… or “RYB define the true colors.”

In human pigment, white (or albino) is a lack of color. Thus white is a lack of color. On the other hand, projectors work by shining three colored lights, and those lights combined make white light. On the other hand, mixing a bunch of paint colors together gets your black or brown.

Prisms make rainbows by spreading light out into different colors. What happens when you collapse a rainbow back down? Does it turn white? Turn colorless? Do the rainbow colors change because they overlap more?

Other Questions that arose:

Why isn’t violet in between red and blue on the rainbow? It would be on the color wheel.

Is sunlight “white” or is it “colorless”?

When you mix all the colors, do you get white, black, or brown?

Are black and white colors?

What do we mean by tints, shades, hues?

How do rainbows work?

How do projectors, printers, computers, and paint work differently/similarly?

What causes color-blindness?

Are there colors that exist in the world that we’ve never experience before?

Why does light color seem to have different rules than pigment (paint color)?

More substantive concerns about the intro physics course:

Divorced Representations

Algebraic representations are the dominant medium students are taught to use for doing anything with kinematics. Students do learn a little about graphs, but they learn it separate from thinking and problem-solving. They simply learn that you can graph data and that an instructor could ask you to interpret a graph either on a test or as a game. They are never asked to use graphs to help organize their thinking or to be their abacus when solving problems. They are also asked to do a few exercises with strobe diagrams, but once again, they are isolated exercises divorced from organizing one’s thinking about phenomena and solving problems. They are just another kind of question an instructor could ask you. Divorcing kinematic representations from each other and also divorcing representations from problem-solving is a big concern here. But my central concern is that the representations are not taught as representations of phenomena. They are all just things that an instructor could ask you interpret. Here’s a graph-interpret it. Here’s a strobe diagram-interpret it. Divorcing representations from their authentic epistemological function is a serious problem in terms of teaching scientific literacy or problem-solving. This is on top of the fact that the algebra is also not taught as a representation, but merely a set of four equations one should learn how to pick, rearrange, and plug numbers into.

No Quantitative-Conceptual Understanding

I want students to have a conceptual understanding of kinematic relationships that allows them to think quantitatively.  One litmus test I  use to assess students’ understanding of acceleration is this one: “An object accelerates from rest at 10 m/s per second. How fast is it going in three seconds?” If a student goes to grab an equation, I know something is wrong. There are other litmus tests such as, “A car steadily speeds up from 50 mph to 70mph, what was it’s average speed?”  I’d like students to be able to say 60 mph quickly and confidently without grabbing an equation. If students don’t have ways of thinking about these questions conceptually and quantitatively, then the fact that they can plug numbers into equations should hold no weight in an assessment of their understanding of physics. I don’t care if they can do algebra problems pretending to be physics, if they don’t know any physics at all. Many of our students can’t do algebra problems pretending to be physics, and almost none of them know any physics. There are a couple reason for this:

• The curriculum spends no time developing the concept of constant velocity. Rather it just harps on the difference between average speed and average velocity.
• The curriculum spends no time helping students to understand the concept of instantaneous  velocity, and how it is different from constant and average velocity. Without instantaneous velocity, acceleration is bomb shell.
• The curriculum divorces the concept of average velocity from accelerated motion. All they get is a list of equations.

Component-focused Curriculum

After 1D kinematics, the curriculum goes to 2D kinematics. From 2D kinematics, we move to 2D forces. Both pretty much jump into finding components, again making the preferred tool algebra. We don’t spend any time in class talking about 1D forces, or talking about what forces do and why. Thus, students don’t understand the most basic idea of Newton’s laws, which is this: Pushing an object with 80N, is the same as pushing it with 100N when someone else is pushing back with 20N. Yeah, I said it, this is the most important concept for understanding forces. It’s not Newton’s 1st, 2nd, or 3rd law. It’s this idea (which I think Hestenes calls Newton’s 4th law): when multiple forces act on an object, wen can think of those multiple forces as having a combined effect that is a equivalent to effect of a single force which we call the net force. Because they haven’t been helped to make contact with this crucial idea in 1D, they are a total loss for 2D problems, except to follow mindless routines of finding components and writing down sum of forces. They have no idea what we are doing when we sum forces.

A student came to me before class yesterday and said, “I’m going to do well on this exam. I can solve all of these force problems, but I don’t have an understanding of what we are doing and why we are doing it.” All I could say was that I was glad that she could tell the difference between the feeling of understanding and the feeling that you can do what’s been asked of you. When a student can ace your exam and tell you straight-faced that they have no understanding of what they are doing and why, then there is a problem.

I have ten apples. I eat two apples. I now have 8 apples. I can write that story as 10 – 2 = 8

Let me retell that story. I had ten apples on the table and zero apples in my belly. Now, I have 8 apples on the table and 2 apples in my belly. I can write that story as 10 + 0 = 8 + 2

Let me retell that story. Today I ate two apples. I started with ten apples, but later I only had eight apples. I can tell that story 2 = 10 -8

If someone wanted, they could rewrite either of these expressions to be 10 = 8 – (-2). But just because you can do something, doesn’t mean you should.

For that reason, I don’t like this physics equation,

Ei = Ef – Wnc

It doesn’t tell a very good story.

It makes way more sense (to me) to write it as Ef = Ei + ΔE. This simply says the energy you have later is just whatever the energy you started with plus the change, or write it as Ei¹ + Ei² = Ef¹ +  Ef² This says that what you start with is what you end with, even if you rearrange things to be in different places.  I’d even be OK with ΔE = Ef-Ei = Wnc because then its clear that the work quantifies that amount of energy coming in or going out.

Any else think that writing KEi + PEi = KEf + PEf – Wnc is a bad story?

In my inquiry course, students take a group exam with their research team. In the group exam, students are presented with a novel situation that they have not encountered before, although it is one I am confident they have the ideas to make sense of over the course of 30-40 minutes. As a group, they have to collectively come to a prediction and explain why that prediction should happen using words and diagrams that are consistent with the ideas we have developed as a class. They then go make the observation.

If they get the prediction wrong, they have to do two things:

(1) Explain how they are making sense of what they did observe

(2) Attend to the flaw in the reasoning or diagram that led them to initial prediction

If they they get it right, they also have to do two things:

(1) Discuss a different answer that some other person could have thought would happen and why they would think this

(2) Discuss the flaw in that reasoning.

I don’t grade on correctness of initial prediction at all. I do grade on the clarity, consistency, and completeness of their explanations and diagrams. Any lack of consistency or completeness in the initial explanation, which is later addressed after the observation is fine by me. Of course that means they can’t just explain the right answer after the fact; they also have to go back to their original explanation to discuss the flaw in their original thinking–specifically address what inconsistency or incompleteness was present.

In the exam this week, the groups that predicted correctly seemed to have done a worse job than those groups who predicted wrong. By worse, I mean that their final explanations were less clear, more inconsistent, and less complete than the other groups. I’ve been pondering over why this might be the case. I think it’s for several reasons:

(1) When you get the prediction wrong, there is a much more authentic need to explain the discrepant observation. It is problematic that the observation turned out that way it did. This authenticity drives different engagement with the task.

(2) Groups who get it wrong spend a lot more time discussing. It takes time and effort to really put together a good explanation.

(3) Groups who get it wrong not only have to sort out the right explanation, but they need to sort through their wrong explanations. Sure, groups who get it right still have to create a fictional wrong idea and response, but it’s not the same as responding to your own wrong idea.

(4) For groups who get the prediction right, there is very little check on “getting it right for the wrong reason”. So, groups who get it right and observe have very little incentive to reconsider their thinking.

(5) Even if you get the prediction correct for fairly correct reasons, it may just seem obvious to you why the answer is what it is, and you may construct a poor explanation, just because you don’t feel like there’s much to explain. Maybe students have the right explanation in mind, but they don’t put time and effort into carefully constructing that argument in words and diagrams.

Anyway, it’s an interesting situation. I will say that no groups did poorly, but by far the best final explanations came from groups who were articulate and clear about their wrong reasoning to start with.

I’ve been having lots of conversations with our director of computational sciences about computational thinking. Among many things, we have been talking about, “What are the beginnings of computational thinking and how do we foster those beginnings?” I’ve come to see that the beginnings of computational thinking involve thinking about arithmetic and algebra as strongly interconnected and thinking about computation as involving creativity and insight.

Here are few examples that we discussed this week.

How would you calculate 21 x 19? Of course, there are many ways to do it. You could add 19 twenty-one times, or add 21 nineteen times. You could do 21 * 20 and then subtract 21. One interesting way we can think about 21 x 19 is as (20 + 1) (20-1). The reasons this is interesting is because it takes form (x+1)(x-1) = x² -1, which is then just 20² – 1 = 399.  Of course, we can generalize this to any distance from known squares, so that 18 x 22 = (20-2)(20+2) = 20²-2² = 396.  With this method, even products like 63*57 and 112 * 88 are a cake walk.

Another question we talked about was 26 x 27, which is of the form (x+1)(x+2) = x² + 3x +2, which gives us 25² + 3(25) +2 = 702. Less compelling, but still an interesting and different way of thinking about it than the standard algorithm.

The point of all of this isn’t just to figure out how to multiply numbers quickly. Rather the point is to (i) begin thinking about how to break down complex calculations into collections of much simpler ones, (ii) to begin to recognize how classes of similar problems might all be solvable by a common algorithm, (iii) to come recognize that there are often many different algorithms that can be used to carryout the same calculation and (iv) to begin to make contact with the idea that efficiency of an algorithm can depend greatly on what kind of problem you have and its structure.

I know there are lots of physics teachers and physics education researchers concerned with computation thinking. What do you guys think?

I’ve been thinking a lot about the sequence of odd numbers and its relationship to accelerated motion.

1, 3, 5, 7, 9, 11, 13…

In a previous post, I came to this sequence by discussing a special kind of motion that I defined as always covering 3x the distance in the second half of a trip than in the first. You can check that the odd number sequence meets this criteria.

Moving up the sequence, each number merely represents the Δx covered in successive intervals of time. Because they are equal time intervals, they also represent the average velocity over successive intervals. The sequence shows that the average velocity is increasing by 2 chunks each second. This of course means, that the ball has a constant acceleration of “+2”

You might think that we’d have to craft a whole new sequence for different accelerated motions. But you don’t. If we want a sequence with twice as much acceleration, we just multiply each number in the sequence by two. If you want to cut the acceleration in half, just half every number in the sequence.

You can also use this sequence to describe motions not starting from rest. For example, if you want something that starts with some initial velocity and speeds up, you just start somewhere else in the sequence besides the beginning. If you want something slowing down, you just move backwards in the sequence.

I’m not saying this is anything new. I’m just saying it’s interesting to think that sequence 1, 3, 5, 7, 9… is a representation for constant acceleration, and indeed any motion involving constant acceleration.

Does this way of thinking help us to solve problems?

Let’s say you want to answer the question, what is the acceleration of a ball that rolls down a 16ft ramp in 4 seconds?

All you do is try to come up with the sequence of 4 numbers that gets you to 16 ft. This one is easy: 1 *(1 + 3 +5 +7 ) = 16 ft.  Since the series steps in increments of 2, the speed is increasing by +2 ft each second. As a result, the acceleration is 2 ft/s each second.

I personally thought it would be hard to come up with the sequence for 200 cm in 10 seconds, but it’s not. The answer is

2 * (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19) = 2 + 6 + 10 + 14 + 18 + 22 + 26 + 30 + 34 + 38. This clearly has an accelerated of 4 ft/s each second. It’s easy to check my sequence adds to 200 by adding in groups of 40, which is the insight you need to make producing these sequences so easily. Getting good at this takes some practice, and it is still terribly difficult for many situations. But the idea is simple: The numbers you add up must be the odd number sequence or a multiple of it. 

OK. Where to next? One way of moving beyond writing down a million sequences for every new problem is to benefit from the fact that a sequence of odd numbers is always a square. For example,

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

…

In general if you are summing to the Nth odd number, then Σ (2n -1) = N² … this is of course the same as saying that accelerated motion goes like t².

Aside: You can prove this equality by showing that difference between two successive squares is always an odd number… n² – (n-1)² = n² – (n²-2n +1) = 2n-1

This tells us that our original sequence of odd numbers, which we said shows an acceleration of “+2”, goes like Σ(Δx) = n² , where n is the final place in the sequence. In other words, if you go for 10 chunks of time, you get to 100 chunks of distance.

This allows you to some really nice estimations: Let’s say you are trying to figure out what acceleration is for a ball covering 372 ft in 19 seconds (starting from rest).

Let’s use our rule of squares: 19² = 361ft, which is the distance that an object accelerating at 2 ft/s/s would get after 19 seconds. Given that we landed at 372,  the acceleration must be a little big bigger than 2ft/s/s? How much bigger? Well, 372/361 = 1.03… or three percent further than you’d expect. Thus the acceleration should also be 3% greater than expected, or  2.06 ft/s/s

If you want to write the sequence down, it’s just 1.03 * (1+3+ … + 35 + 37) = 1.03 (19)² = (2.06 /2) (19)²

Why worry about this?

I don’t think you have to. But I do think that reasoning from well-understood special cases is an important skill. The a =2 case is really compelling and interesting to think about because

The successive distances are simply the sequence of odd numbers

The successive positions are simply the sequence of square numbers

Understanding how this special case relates to other cases is also interesting

(1) Different magnitudes of acceleration are merely multiples of the odd number sequence

(2) Starting with some initial velocity is merely starting mid-sequence

(3) Slowing down is running the sequence backwards

I know I have a different way of thinking about why accelerated motion goes like time squared, and I have some news tools make it really easy to estimate accelerations, positions, etc

I’ve been blogging over the past week about how some students in my inquiry course are unhappy about my capstone-for-an-A grading policy. Here are what some students had to say in the anonymous feedback I asked for:

“I know there are some people who aren’t as invested in science as I am and are now settling to make a B simply because they’re not passionate about discovering something new, or don’t enjoy science. I feel that, in a way, it’s like punishing those who don’t like science, even if they put forth their best effort”

“I’m worried that I can’t get an A in this class, and I’m an A student. It irks me that if I try my best, I’ll still just get a B”

“I find the grading policy system extremely frustrating, because why try? You won’t get an A unless you do a paper… I’m so frustrated, I am willing to just settle for the B.”

“I’m worried that I can’t get in A in this course without sacrificing A’s in the my five other courses”

“I don’t understand how you can say a 100% is the same as 83%. 100% should be an A no matter what“

Emphasis is added throughout.

Mindset is a really interesting issue here. The effects of a schooled culture is a interesting issue. The real and perceived importance of grades is an interesting issue.

Thoughts?

I’m feel fairly capable fostering “thinking about phenomena” and “understanding of concepts”. I feel I can motivate it. I feel like I can engage students in it. I feel like I can even break down that such thinking and concepts into bite-size parts for student consumption. I feel like I can structure sequences of activities and questions that help students grapple with their own thinking and understanding of concepts. When I’ve done all that–when I’ve motivated tasks and students understand concepts–I feel fine teaching students procedures for solving problems and helping them to become proficient and careful in working through problems.

But I am lousy at teaching procedures for procedures sake. Granted, I don’t want to teach mindless procedures. But the truth is I have to. I am teaching a physics class where students need to become proficient at procedures that make no sense. For this purpose-of learning how to teach procedures without concepts–I need to think more like Khan Academy. I need to think more carefully how to motivate procedures, how to break down procedures into consumable parts, and how to sequence student contact with aspects of those procedures. Lastly, I need to foster up in my gut a sense that I care that students learn this (even if I don’t feel I should). I can become better at teaching procedures. It is a new goal of mine. Even if I never have to teach mindless procedures again, it will help me teach mindful procedures as well.