Scalars are vectors?

I think when most of us introduce vectors, we mostly focus on the differences between scalars and vectors. I try to convince students that almost everything they know about numbers applies to vectors. I tell them this because there are so many good things they can bring over from numbers, and only a relatively few number of ideas that can go awry. Here are some good things you want your students to bring over from numbers to vectors

The orientation of symbols that represent numbers (or vectors) are important because they tell you about their meaning

For example, 6 and 9 are not the same symbol. Of course, students exploit this property of number symbols all the time to write funny words in calculators.

Symbols that represent numbers (or vectors) can put into new arrangements in order to make it easier to carryout certain procedures with them

For example, 71 + 82  is the same as 71


When we do this, the meaning of 71, 82, and + don’t change. It simply allows us see how place values compare so that one procedure for adding becomes easier to carry out.

Symbols that refer to “whole” can often be described in terms of “various parts”

987 =  9 x 10² + 8 x 10¹ + 7 x 10°

987 = 900 + 87

987 = 460*2 + 67

This is something students struggle with even with numbers–decomposition and re-composition. So it’s not a surprise it’s hard for students to understand vector components, and odd things like vector components with tilted axes.

Getting back to Vectors

While it’s true that all these properties are also important with vectors, instruction with vectors often makes it seem like these things are special, unique, and weird about vectors. They emphasize how you must carefully move vectors without rotating them. They make a big deal about how you move vectors to carryout procedures of adding them. They make a big deal about how you can write vectors in terms of components.

I’m not saying that students don’t have to learn some new ideas,  procedures, and to learn to distinguish scalars from vectors. I’m just saying let’s not pretend that vectors are all that different than scalars. The difference is subtle, somewhat like the difference between a square and a rectangle. Both squares and rectangles are quadrilaterals. They have a lot more in common than different.

Magnitude and Direction?

Another interesting thing about how we teach vectors comes from something I recently re-read in A. Arons’, “Teaching Introductory Physics”. Arons reminds us that we teach most students that vectors are “magnitude and direction”. Arons points out that vectors commute upon addition, which is something also true for numbers. But Arons reminds us that this is not true for all things with “magnitude and direction”. Everyone familiar with rotations knows that finite angular displacement have both magnitude and direction, but that they don’t generally commute upon successive displacement. Arons points out that “magnitude and direction” is not only a wrong definition for vector. Nor is it just an incomplete definition. He seems to suggest that it is misleading and possibly not generative for later learning. He proposes that it should be emphasized that anything that commutes is a vector. This, of course, make scalars a kind of vector, which is actually how we kind of think about scalars as zeroth order (or rank) tensors. Scalars are a special kind of vector (or tensor) that doesn’t transform under rotation. This makes the analogy about squares, rectangles, and quadrilaterals a little closer to the truth.


I’ve heard a fair amount of these kinds of statements throughout the year in my inquiry course

“I’m not good at science.”

“I don’t think scientifically.”

“I can’t think the way scientists do”

“I don’t like science”

“I’ve never liked science”

So last week, I asked my students to write a blogpost about why a child in their class might say something like, “I hate math. I’m not good at math, and I never will be”. Specifically, I asked them to discuss the experiences this child might have had or be having that would lead them to feel this way about math and themselves. After having wrote their blog posts, they are now reading about Carol Dweck, and the effects of praise and mindset, and I’ve asked them to go back and reread what they and others wrote.

There was a large variety of responses in the blog posts, but the most compelling one’s were from students who identified themselves as that student. Here are some of things they wrote about. Bolded areas not in original, but were added for emphasis:

“I went through early elementary doing well in math. It wasn’t until those darn multiplication tables in 3rd grade that I made my first ever C on my report card. Ever since that experience, I’ve had a bad feeling toward math.”

“I was too intimidated to ask for help because I thought my teachers and peers would think I was an idiot for asking something that maybe everyone else understood.”

“In my experience I had several other students in my classes in 3rd and 4th grade that I remember as being pretty advanced and fairly fast when it came to learning math and science but especially math. I would find myself frustrated at not being as fast as some others and that frustration only added difficulty to my ability to understand math. Plus I just had hard time comprehending some of the more difficult concepts and I didn’t want to embarrass myself in front of others.”

“For my brother, he hated math because he made B’s and C’s whereas I would make A’s and B’s and he felt like he didn’t understand it as easily as I did so he wouldn’t want to try.”

Saying I hated math covered up for answers I may have gotten wrong. “Well, I hate math so it was expected that I missed those.” This may be the way students hide that they do not understand. They may feel embarrassed or not as smart as their peers for getting the wrong answers.

I can definitely relate to this student because I was this student at one time, and sometimes still think of myself in this way now…. A students begins getting lower grades in math than desired, and might even start to compare his or herself to the rest of his or her classmates

I hated math too… maybe the student is comparing their ability to someone else’s and they are upset because they do not think they are as good as their friend.

My 6th grade algebra teacher told my mom “Jane Doe is a good girl and I never have any trouble out of her but she is just not good in math and I don’t have time to work extra with her”.  I remember her saying this to my mom with me standing right beside her.  I knew I didn’t catch on to math at the same speed or in the same way as some of my peers but up to that point I had not thought that I was really bad at math.  From that day on I thought I stunk at math…my teacher said so…..this was just my lot.

What are you noticing? Here are some of the things I notice

  • I see the impact that our current grading systems have on students’ feelings of self-worth
  • I see how children use shifts in (math) identity as a mechanisms for maintaining self-worth.
  • I see how school reinforces a view in which your worth as human being can be mapped to your place in a linear hierarchy
  • I see how school reinforces the view that intelligence and smarts are fixed attributes
  • I see how one of the primary activities of school children is to avoid looking stupid and to maintain one’s standing in the hiearchy

Model-based Thinking…

Today I watched group of people with none to little to some to a good bit of background in education research try to write a education proposal. Many of those people were scientists, and here are some thoughts from the day.

When a scientists thinks of a physical system, they think of its parts and how those parts are organized in that system. They think of the relevant properties of those parts and the system, and how those properties might be measured and what roles they play. They think of how those parts interact with each other and the rules for how those interactions play out (which are often dictated by properties). They think about the surroundings of that system, and how interactions with that system either introduce new parts, change  properties of certain parts, or change how those parts are organized. They think about how those changes act together to bring about functional changes to the system as a whole. All of this might be understood as physicists “modeling” that system in order to understand how it works. They think about systems in terms of models. They develop models, deploy models, think about the implications of models, reflect on the parts of model that aren’t flesh out. They use model to ask questions. They use models to make predictions. They use models to inform them about designing experiments and what might be measured. All of this modeling is informed by theory.

Despite the fact that scientists do all of this with physical systems, they struggle to play this same game when thinking about educational systems. They don’t know what the relevant entities are. They don’t know how to draw boundaries between systems and surroundings. They don’t know about the properties of the entities or know what properties are relevant. They don’t know how those entities arrange themselves and how those properties and arrangements lead to different functional states of the system. They don’t know how those entities interact with each or respond to external interactions. But even worse is that they also don’t even know that they don’t know these things. Mostly, they don’t even think that should be thinking in terms of models (or theories), so they don’t even try to think in terms of models. Rather, they just go on instinct. They go on things they feel, things they’ve heard, things they’ve experienced. What they do is no informed at all by theory. In many ways, they are novices in all the ways that new physics students are.

Aside: If you’ve ever entertained the thought that general “critical thinking” skills is something that exists, this is evidence that there is no such thing. People who can think critically in one context do not typically spontaneously think critically in other contexts. Critical thinking requires rich knowledge base from which to think.

The good thing about this group of people is this: They have some really good ideas. They are learning. They are growing. They are well-intentioned. They have some good experiences to draw from. Part of the problem is that they are missing lots of knowledge. Part of the problem is that they don’t know the game they should be trying to play.

I feel the same way about my physics students. They have some really good ideas. They are learning. They are growing. They are well-intentioned. They have experiences in the world to draw from. Part of the problem is that they are missing lots of knowledge. Part of the problem is that they don’t know the game they should be trying to play.

The best I can do with my students and these proposal writers is to enter into a dialogue that nudges their thinking along–provide them with some impetus to learn some new knowledge or rethink the game they are trying to play.

Whiteboard Change

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.
Ultimately, all of this helped me more quickly and accurately assess student progress and also provided them with a more effective collaborative space. Students really liked it. I really liked it. I think I’ll keep it, for now at least.
Aside: I’m becoming sold on energy pie charts.

Brian rambles…

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

Is every color in the rainbow

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)?

Back to Articulating What’s Missing

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.

Equations and Stories

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?

The benefit of not knowing

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.

The Beginnings of Computational Thinking

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?

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