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.

Images of Teaching

I was reading a recent book chapter by Russ, Sherin, and Sherin. In the chapter they discuss, among other things, four images of expertise in (mathematics) teaching:

Teacher as diagnostician

"Examining the mathematical thinking of students, looking for symptoms, and diagnosing their underlying causes"

Teacher as conductor

"Directing and shaping the classroom discourse... to orchestrate whole-class discussions in ways that advance the mathematical learning of the whole class"

Teacher as architect

"Selecting and implementing curriculum materials...choosing tasks to use with student as well as deciding how those tasks should be carried out"

Teacher as river guide

"To be flexible in the moment...responding quickly and effectively...responsive to the context, to students, and to what occurs in the moment."

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I like this breakdown, both for thinking about research on teaching and how we conceptualize teaching, but also for thinking about my own teaching. How do I imagine myself as a teacher? In which of these images do I feel competent? In which do I feel more novice?

I will say that my weakest area is as architect- especially thinking about the design of a whole course. I haven't had a lot of experience designing courses, but I think I am also weakest here because I am a decent enough in the other three areas that I get away with not being a good architect. In this sense, the willingness and ability to improvise is both an asset and a liability. I have become more aware this year of my needs to mature as an architect, and have taken some steps to reflect on and enact some novice architect moves. One professor here at UMaine has been helpful in nudging me in this direction.

I'd say conducting is where I am strongest. Of course, I still need a lot of improvement, but I feel pretty comfortable and confident navigating whole-class discussion. This is a lot based on my experience in teaching as a summer kindergarten "teacher" in an Americorps program, as a tutorial TA and instructor at UMD, and even more recently as an instructor and PD facilitator at UMaine. But I also had the opportunity to watch a lot of good conductors. In particular, during my second year at Maryland, I watched David Hammer lecture everyday. A few times I even got to substitute teach for him, and try it out. Having role models helped a lot, but it helped even more that these role models were often the same persons coming in to observe me and give me feedback. Now that I think about it, I also got a lot of good feedback from my mentor teacher in the Americorps program. Mrs. Patterson was always nudging us to make sure that children had opportunities to learn from mistakes, and to not let them feel bad OR to let those mistakes just pass by.

As a diagnostician, it shouldn't be surprising that I feel adequate in some ways and less adequate in others. This is because I feel more confident as a conductor than as architect. I am good at diagnosing when I am conducting, but I need to improve my skills at designing more and better opportunities for diagnosing.

I think the river guide image is the hardest for me to assess. I feel like I am almost always playing river guide, largely because I fail to play architect well enough, but that doesn't mean I am good at being the river guide. I suspect my expertise is patchy. There are certain rivers I feel very competent reacting to rapids and obstacles; but there are other rivers where I would be lost and tumble over ungraciously. Being the river guide means you both know the terrain have all the skills down, but that you can perceive and react quickly and appropriately. It'll be interesting after my first year at MTSU to revisit this list.

Anyway, where do you all feel more or less competent as a teacher?

Eleanor Duckworth is a Poet

I read this over and over, and then over and over. It is the first Paragraph from "Teaching as Research" by Eleanor Duckworth in "The having of wonderful ideas"

"What I love to do is to teach teachers. I love to stir up their thoughts about how they learn; about how on earth anyone can help anyone else learn; and about what it means to know something. I love to help them feel that any aspect of human endeavor is accessible to them and that they can make it accessible to any person they teach. I love to try to find ways into a subject that will catch everybody's interest; to find out what people think about things and to find ways to get them talking about what they think; to shake up things they thought they once knew; to get people wrapped up in figuring something out together without needing anything from me; to help build their fascination with what everybody else thinks, and with the light that other people's thinking might shed on their own. I love to see the most productive of questions be born out of laughter, and the most frustrating of brick walls give way to an idea that has been there all along."

A glimpse into the nature of the feedback I give to students

I went through the first four weeks of feedback I gave to students this semester, and this is what I found.

Impressions? Feedback on my feedback? What does yours look and feel like?

It sounds like you... Two things that stand out about it are...
You seem to be making an important distinction about...Why do you think...?
Thank you for... It really helps me to understand...
It's great that you are thinking about...
You have brought up a lot of...
I love how you have earnestly you have tried to...
What I like about what you've done here is...
I also like how you...this shows me that you are...
You wrote...Do you think...?
I appreciate how much...This helps me to concretely understand...
I'm wondering how you...
It sounds like you are trying to...
Thank for you being honest about...
Keep up with..., and I hope to hear more about...
It sounds like one your strategies is....
How do you think this is different...?
It is nice to see you having an emotional and intellectual connection to...
It sounds like you are trying to think of ways to...
I like how thoughtful you were about... I wonder...
You mentioned something... Was it...?
You mention this... By that do you mean...?
You have written very thoughtfully about... it sounds like...
You seem very aware of...
I agree it can be hard to know...
I appreciate how you’ve...
I hear what you are saying about... do you think...?
I like how you...This is a great thing to keep doing because...
Others have also expressed your concern, which seems to be... Is that right?
I can see how your were trying to...
I really like your idea about... This shows you are really thinking about...
I especially like how you made connections to...
It seems like two things: ... and... Do I have that right?
It is clear that you thought deeply about this, especially your idea about...
I like how you have written a rich description of... In explaining the ideas, you’ve included...and...
You mention... What experiences have you had with...?
I like how you are looking for connections here... This is something expert learners do all the time...
It sounds like you... And you are concerned that...
This indeed sounds like a complex problem....
I like your goal to...I have also...Let me know how it goes
It's clear that you tried to think why... But you also included how... which helped me to understand that...
In your reflection, you highlight...How are those things going...?
You have done a nice job relating...This shows me that...
You seemed very concerned about... the empathy you express here is...
I think you did a nice job of initially describing...I'm wondering what else you might include to help me understand...
You pointed out that...Why do you think...?
It's very clear that... What do you think...?
You wrote clearly and thoughtfully about...
I hear you saying that you'd like... and also that...I can certainly relate to...One suggestion is that...
The last thing you write here about... is an interesting problem. What made you think about that?
I hear what you are saying about...
It sounds like you are saying...

A powerful question

A powerful question I have learned to ask students is, "What do you notice?"

One of the reasons why this question is so powerful is because students can't be wrong. Only a student can know what they notice.

I first picked up this question in an education talk about poetry. The speaker asked us "What do you notice about the poem?" NOT "What do you think the poem means?" Collectively, we noticed rhymes. We noticed rhythms. We noticed themes. We noticed alliteration. We noticed contrasting words. We noticed metaphor. We noticed allusion. We noticed a shift in mood across stanzas. We noticed repetition. We noticed imagery...

Out of our noticing, meaning of the poem emerged.

And so, too, I have learned to ask this question in science. Present students with phenomena and ask what they notice... not to predict what will happen. Not ask them to explain. Not solve for a number. Simply ask them to notice and ponder and let questions and meaning emerge.

Pointing to Learner's Excitement with Science

This semester I learned how important it can be to point out to people when they are having fun learning and doing science.

Learning, learning science, especially science class isn't always going to be fun. Sometimes it's going to be frustrating, either for students or for teachers or both, for a variety of reasons. Inquiry, especially, comes in fits and starts. Conversations, debates, presentations are probably going to go on, at times, too long, causing tedium or frustration. There are days that people just seem less excited to talk and to share, and to "dive right in" to science. Hopefully those days aren't too often, but they do happen. Even in a given day, you can hit some slumps or even walls, from which the class may or may not rebound from.

It even varies with students and classes over the years. Some students will be excited about science often. Some students will be excited about science rarely. Some years, there are classes that just click together with their science learning better than others. But with any class and any set of students, there are moments where a whole class becomes excited together about some science thing–a conversation, a puzzle, a phenomena, an experiment, a question, whatever. In these moments, even those students who are rarely excited get wrapped up in the excitement.

I am convinced that pointing out these good times pays big dividends, simply by saying, "I want you to remember this feeling you are having, the feeling you are having talking about science and learning together."

Now, I don't think you should point it out too early, like while the excitement is still building–that might disrupt the "flow" and the "excitement." But I do think it's worth pointing out before the excitement has died down completely, while the feeling of excitement is still palatable. Pointing it out, I think, helps buffer the bad times, because people remember that good times. I think it also helps to build students' identities around science and science learning, because you are pointing to their own emotional capacity to feel good and to get excited about doing and learning science.

Next year, I am going to more deliberately do this. We'll see how it goes.

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?

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.

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.

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.