Office Hours: Anticipating Student Thinking

Teaching Situation

A student comes into office hours and wants to know why the final velocity of a block sliding down a frictionless ramp of height h is not equal to v_{final =} v_initial + Sqrt(2gh)

Your Homework:

Describe two different lines of thinking that could have led a student to honestly arrive at this question.

Describe how your discussion with the students would likely need to differ depending on which line of thinking the student had taken.

* Note: I actually observed this situation–two times in fact.

Kinetic Energy and Temperature

A long-standing discussion among colleagues has concerned the understanding of temperature. Central to our discussions has been the heuristic, which is often taught, that temperature is a measure of kinetic energy or average kinetic energy (per something).

There are many touchstone examples we have discussed that have drawn our attention to critical features of not only the concepts and their connections, but what it means to understand them.

Included in these touchstone examples are

1. An ideal gas in an external gravitational field. How do we think about the kinetic energy and temperature varying with height? Theoretically, intuitively, and empirically.
   
2. The free expansion of a non-ideal gas. How do kinetic and inter-molecular potential energies change? How does temperature change? Why? How are these changes related?
3. A gaseous systems composed of boson and fermions in near classical regime. How do kinetic energies compare?

From these discussion, we have enriched our understanding of all the ways we might think about temperature, how it is related but distinct from average kinetic energy, what it has to do with heating and entropy, how the theoretical construct relates to the ways in which we attempt to access it through measurement, and importantly when it suffices to associate temperature with energy per particle or mode.

Ralph Baierlein in "Thermal Physics" writes the following in a section called "Temperature recapitulated" in a subsection called, "Temperature is deeper than average kinetic energy":

"The misconception that introduces this subsection is propagated with the best of intentions: to make temperature easier to understand. The root of the conceptual error lies in this: a belief that the purpose of absolute temperature is to tell us about a physical system's amount of energy. That is not the purpose of the temperature notion. Rather, temperature is intended to tell us about a system's hotness, its tendency to transfer energy (by heating). All physical systems are capable of heating or cooling others. The purpose of temperature is to rank the systems with respect to their ability to heat one another."

Light and Shadow: The best questions comes from students

Last year, I watched a lot of video of students working through worksheets on topic of light and shadow. Their questions led me to think about this problem:

The worksheets had students think about this situation: a mask with a circular hole, a screen, and a small (i.e., "point source") bulb are arranged like this


Students are supposed to think about "where" the light goes when you raise it. But one group went about thinking about what would happen to the shape and size of the spot of light as you move it from aligned with the hole to farther and farther up or down? Students questions are the best ones.

So, How does the size/shape of the bulb change as you move the bulb up and down? First give your intuitive guess... and then do some work to figure it out and convince someone else.

(I have three very different approaches to answering this. My most elegant came just last night as I was falling asleep.)

Facilitating Discussion with Peer Instruction

Here are two trappings I've seen recently:

#1: Validating the right answer the moment you hear it, thereby short-circuiting any dialogue about the problem.

Recently, I heard a college instructor say, "I agree with you," anytime a student said something correct, and "Are there any other ideas?" anytime a student said something incorrect. My sense was that the phrases "I agree with you," and "Are there any other ideas?" were honest attempts by the instructor NOT to say "That's correct" and "That's incorrect". I caught on to this pattern of talk quickly, and I imagined students would catch on within a few days or weeks.

#2: Requesting that a nearly correct student "say more" in a manner that implies, "Could you please restate what you said using the correct terminology?" rather than, "I'm really interested in your idea, please say more about that?"

This can be subtle, as it often has to do with tone of voice, body language, and subtle phrasings. When an instructor does this, however it really gums up the dialogue. Because students pick up on the fact that it's not about their ideas, this kind talk quickly descends into a game of "the teacher is thinking of a number between 0 and 100." Students either start opting out or just trying to guess what the instructor wants them to say.

The Big Picture

Facilitating discussions is hard work. In my mind, the number one priority is getting students to invite me in to their learning through their talk. Largely, this means that they are sharing their ideas and thinking, not trying to guess what's in my mind or avoid being wrong. When I see instructors falling into these kinds of traps, I see this priority being undermined.

Some people might think the solution to trap #1 is to have a better poker face–don't have any "tells" that give away the right answer. I think that approach is flawed because it's trying to stop the symptom, but not the cause. I don't need a poker face if I am not listening to student ideas primarily through a lens of correct and incorrect.

Some people might think the solution to trap #2 is to not care about terminology. While that's possible, I don't think it's realistic. I think the solution is to distinguish the activities of "shopping for ideas" and "connecting with disciplinary formalism". The problem is in trying to do both at the same time and in the same way.

I certainly enjoy watching other people teach, because it gives me an opportunity to reflect on my own teaching. There are times when I "gum up the dialogue" and focus on correctness when I shouldn't. Sometimes by watching others, I understand better why we all fall into these traps and what impact it can have on students.

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.

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?

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.

The Setup

When I first started teaching in "somewhat-reform-minded" college physics courses, I remember learning that I wasn't supposed to give students the answers. I was also never supposed to "lecture" or tell students if they did something right (or wrong). The implicit philosophy was that somehow this was supposed to make me a better teacher and that it would make students learn more. "Students have to do it themselves" was the takeaway message.

Now, I understand that physics is not a spectator sport (students DO have to do it themselves with guidance, encouragement, and support), but I also know this

What we do TAs in preparing them for "reform" courses is a set-up:

• First, it is a setup to take away from students something that has for so long been valued (answers) without replacing it with something else that can be valued (equally or more than answer)
  
• Second, it is a setup to take away TAs' easiest and most natural way to praise students (for getting right answer) without replacing it with tangible ways to praise students in other ways
  
• Third, it is a setup that we fail to give TAs any reasonable tools for interacting with students in ways that tacitly change the value-system in the classroom

As I see it, we often blame students (for just wanting answers) and blame TAs (for not being good reform teachers). Often the blame game involves identifying the bad "beliefs" and "attitudes" that students or TAs have about learning. This, I tell you, is part of the setup. We put TAs and students in a fundamentally "bad" situation, and blame them for the "bad" things they do.

The way I see it– we have designed a systems in which classroom discourse with and over worksheets dominates the classroom environment. And here's the problem. Worksheets can only really do two things: make statements and ask questions. And those questions tacitly send the message that answers are what matter.

Think about all the things that worksheets can't do? Worksheets can't value questions, or creativity, or having a sense of curiosity or wonder, or individuality. They can't value patient problem solving. They can't smile or get excited with you. They can't value persistence through confusion.

Once again, this is part of the setup. We tell TAs not to value (or give) answers, but we pit these TAs against a system in which all the material structures around implicitly value answers. Questions on worksheets demand answers. But if TAs or students seek out answers, we blame them for not having sophisticated beliefs.

We have put TAs in the garden of "answers" and have told them not to let anyone bite.

So what would I do differently with TAs?

Man that's tough. But the answer has to involve thinking of TAs as capable of being good teachers and of students not inherently seeking answers. We have to think of simple skills that we can have TAs practice that will help them grow and that will help establish different classroom conditions. Conditions in which TAs and students aren't constantly having to avoid doing "bad" things.

One thing that I'd have all my TAs practice:

I'd have my TAs practice listening to what people say and writing it at the board in real time.

For many of my colleagues, the blackboard is sacred place where only correct science ideas can be written. Think about this: How comfortable do you feel writing WRONG ideas on the board. This is something I had to get over, but it's not hard with practice. I want TAs to practice writing what their students say (right or wrong) in such a public space. I want the blackboard to become ingrained a sacred place where student ideas get written.

Why do this?

• First, writing requires listening and interpreting. Paraphrasing what students say requires a different kind of listening than purely evaluating does.

• Second, writing gives TAs a more tangible job to focus on rather than what is mostly automatic (i.e, responding to the correctness of ideas). Telling TAs to write down student ideas is WAY better advice than telling them "don't give away answers", and is more easily carried out than more sophisticated "talk moves".

• Third, writing at the board means that TAs are not constantly facing their students for appreciable amounts of time as students express ideas. This may sound silly, but this takes away a lot of the pressure of TAs to respond. It naturally builds in wait time, which is something I want my TAs to practice. Writing at the board protects me from getting into I-R-E patterns of discourse. With the face hidden more of the time, my poker face becomes less important.

• Fourth, it is hard for new teachers to verbally value students' wrong ideas in authentic ways with out lots of practice. But writing ideas down concretely values those ideas by making them public and durable. This helps to build the culture of "not valuing only answers" in a way that TAs can do.

Who are we kidding? Apparently not my LAs

Here's an example of the kind of worksheets students are doing in college physics courses around the country and also here at my university. I have no doubt that students working through these worksheets (especially in groups) is better than having them silently watch a graduate TA work out problems at the blackboard. But who are we kidding with this stuff?


Apparently not my LAs:

On the heels of a reading about the difference being "univocal" discourse and "dialogic" discourse, this is what one LA had to say about the class she teaches in that uses these worksheets:



I love how this LA feels that this worksheet is a trap for both the teacher– "The teacher is limited to this because of worksheets..."– and for the students, "You must realize the exact answer it is asking you for"

This LA also realizes that what appears to be open questions about science are really closed questions, "Questions could have more than one answer... however, you realize that it was asking for a particular answer."

And how this LA connects the worksheet structure to how students behave: Students are asking questions like,"What are they looking for here?"

This should disturb us–not about our students, but about our worksheets. What are we doing to them? These students aren't asking about the physics, they are asking about what they are supposed to put down on a piece of paper. Seriously, don't get me started on a conversation about who is "they", anyway.

But here's the question:

If my LAs aren't duped by worksheets after reading just one paper about the nature of discourse in math and science classrooms, why are so many of my colleagues?

Why I am doing more science with my LAs

OK. So here's the thing. This semester, I am teaching a one-credit "science teaching and learning" seminar for undergraduate learning assistants (LAs).

I have ten undergraduate science students for a little over an hour per week. Each of them helps to facilitate some classroom learning activities in a course they did well in previously. Four of them teach in introductory chemistry in what is called "Peer Led Team Learning". Six of them teach in introductory physics using "Tutorial in Introductory Physics"

My class is supposed to help them be successful in their classroom teaching experience by giving them tools for questioning, listening, and promoting productive group work. A secondary goal is to expose them to variety of issues in teaching and learning with the hopes that they will take an interest in teaching, and decide to become certified to teach in secondary science.

When I first started teaching this class, I was closely following the model of the Colorado Course. They are doing great things over there, and they have over a hundred LAs per semester distributed across many departments and colleges. I visited last October for their LA workshop and got to experience firsthand what they are doing.

For me, their course structure and guide was a great starting place, until I realized something:

My LAs don't necessarily value conceptual understanding, inquiry, or deep-engagement in science.

Sure, they have moments where they do. Somewhere we all realize the importance of deep learning and the deep shallowness of school to engage most with deep learning. But those moments are tempered by the realities of their (mostly bad) college science courses, where they are required to memorize a million things and perform well on high-stakes tests. My LAs value doing well in school, and I don't blame them. They want to go to med school. They want to get good engineering jobs. In the worlds they live in right now, GPA is currency.

So where does my course fit in? C'mon. Let's face it. Do I really think my students are going to come around to valuing inquiry and conceptual understanding by reading some papers about teaching and learning? Do I really think my students are going to value inquiry and conceptual understanding just because they are teaching in classrooms where students work in groups? The courses they teach in are "band-aids" at best. There is lots of pseudo-teaching going on. Students sit in groups and are dragged by the teeth through some reasoning that a worksheet demands of them.

I don't want the papers we read or "pseudo-reform" teaching to be the thing we hang our hats on. Sure, I want us to draw on various papers to inform our discussion and to challenge our own thinking. Sure, I want them to reflect on and critique the teaching they are engaged with.

So what did I do? Most importantly, I have committed the class to doing more science and doing that science together. If we a read paper about problem solving, I engage them in problem solving. If we read a paper about formative assessment, I run a science lesson in which I model formative assessment. If we read about conceptual vs. algorithmic problems, you bet we're going to be doing both of those and discussing the difference.

After we've done some science (and have a shared common experience around doing some science), then we can talk about teaching and learning through the lens of the paper we've read.

I am certainly still struggling to do a good job with this course, but at least I think I've nudged in the right direction. One of my persistent concerns is that I have made too many "one shot wonder lessons". They are getting to experience some good one day lessons that are perhaps fun, engaging, and involve the beginnings of deep learning with content; but teaching is also about the coherence of those lessons over time and sustained engagement. In what ways am I misrepresenting the profession? Could I do any differently with a little over an hour each week? I think I could if I made sure I was teaching the same science topic each week. But that would require some serious planning, and not just mid-semester adjustments.