A Study of the Development of the Paradigms in Physics Program

During the first week of class, doing a variety of active engagement strategies to set the tone for the rest of the term.

A lot of the thinking about Week 1 involved planning ways to engage the students in actively doing physics. Writing many “sticky notes” and posting them on the office wall provided a way for this faculty member to keep track of various ideas about what to say and do in class. During an interview at the end of Week 1, the faculty member reflected upon what had happened:

I looked at that chapter in (the textbook) about (the opening topic) and made a list of the pre-requite knowledge that people ought to have and I also thought about what kind of sense making do I want students to be able to do around these problems and what skills are needed for those activities... So on the first day of class – you can see on my board (points to white board on wall of the office) about a million post-its on it –these are all the little things that I wanted to make sure that I talk about before we really launch into the meat of the course material.

Day 1 had consisted primarily of providing information about the course and having the students work individually on the pre-assessment of their sense-making skills. Days 2 and 3 involved the students in a variety of active engagement strategies:

I knew...that these students might not have as much experience with active engagement so I tried to do a variety of active engagement strategies during this week to set the tone for the rest of the term. (During the second class), we did small white board questions and we did an interactive lecture and today we did a small group activity and we've had some whole class discussions where a lot of students were encouraged to contribute ideas to the discussion and those were all conscious choices on my part. I really wanted to make sure that there were a lot of student voices in this first week.

An important aspect of using such active engagement strategies is learning student names quickly. Being able to refer to speakers by name makes it easier to encourage students to speak up, to listen closely to one another, and to respond to what another student is saying.

Engaging students in an interactive lecture.

When asked what an interactive lecture is, this faculty member responded, “So when I get up to the board, I ask students a lot of questions.” The purpose of such questions is to find out what the students already understand and to hear how they are thinking about a topic. When working out a problem on the board, “I invite students to tell me what the next step is.” This prompts all the students to be thinking about how to solve the problem. Their responses open opportunities for discussing what the various options are and why one might choose to go one direction rather than another in solving the problem. This faculty member also warmly welcomes students' questions as useful contributions for all to consider in the midst of such collaborative thinking.

Asking small white-board questions.

Small white-boards are boards about \(30\times40\) cm (\(12\times16\) in) that students pick up, along with a white-board marker and eraser, as they enter the classroom. Small white board questions have many uses (See: (http://physics.oregonstate.edu/portfolioswiki/strategy:smallwhiteboard:start).

On the second day of class, for example, this faculty member used a small white-board activity both to engage every student in a review and to assess quickly the students' level of knowledge. After requesting “Write down something you know about Newton's second law,” this faculty member provided some coaching, “write large so everyone can see” and some encouragement to participate even if feeling bewildered, “Write a question mark if you don't know or if you're not sure what to write.”

While the students were responding, the faculty member roamed the room to see what students were writing and also to collect a variety of useful examples. Holding up one board so all could see the response, \(F=ma\text{,}\) the faculty member asked every student to make a judgment, “What do you think? Thumbs up? Thumbs down?” Holding up a second board, with \(\sum F=ma\) (with arrows over the \(F\) and \(a\)), the faculty member asked, “What is the difference?” This process engaged students in comparing the responses on the two boards, thereby reviewing the vector nature of quantities such as force and acceleration, ways vector quantities are represented in equations, and the difference between referring to a single force or to the vector sum of all the forces acting on an object.

Gathering a variety of responses on boards from around the room and holding them up in a central location helped to ensure that the students would not perceive the board being discussed as produced by a particular student. This eased any embarrassment for the author of a response being discussed while enhancing all of the students' attention on reviewing the nuances of talking and writing about forces.

Facilitating a whole group discussion in which many different students contribute their ideas.

Silence sometimes follows a question asked of a whole group of students, particularly those used to listening to a lecture rather than participating in a discussion. To avoid such a long silence, this faculty member asked small groups to discuss an issue among themselves before launching a whole group discussion. A pretest, for example, had asked each student to identify whether a series of algebraic expressions, such as \(m_1 m_2 a/(m_1+m_2)\) or \([(m_1-m_2)/(m_1+m_2)](a)\) have the dimensions of a force. The faculty member asked the members of each small group to talk about their responses, to see not only if they had the same answer but also if they had the same or different reasoning. ¹

After a few minutes, the faculty member began a whole group discussion with “I'd like to hear something about what you've been talking about in your group.” After someone shared one group's thinking, the faculty member asked the whole group, “What do you think about that?” Such a follow-up question prompted other students to listen and to think about what a student was saying. “What else came up in your discussions?” prompted a response from other groups, and “What else?” continued the sharing of student thinking. This was an initial foray into thinking about the similarities and differences between checking a result's units such as kilogram meters/second2 or checking its dimensions such as (mass)(acceleration) or (mass)(length)/(time)2.

Instead of being reluctant to risk talking to the whole group, the students were willingly sharing their thinking about this basic sense-making strategy. Through this whole group discussion, the faculty member primed the students to be receptive to the recommendation that they choose checking dimensions rather than checking units when evaluating their answers.

Doing a small group activity.

There are many types of small group activities such as working on solving a problem, interpreting a computer simulation or exploring physical phenomena. These may occur over several minutes or several days. (See examples at: http://physics.oregonstate.edu/portfolioswiki/strategy:smallgroup:start.)

This faculty member planned to engage small groups in solving a challenging but familiar problem, similar to those the students would have encountered already in the pre-requisite introductory physics course:

So we did an activity today that I thought was sort of a 211 problem (first term of introductory physics sequence, a prerequisite for this course), but maybe in the challenge section of a 211 problem, so that it wouldn't be so unfamiliar that the students would get really stuck but they would have resources for doing the kind of sense making that I wanted to have happen.

Each group member had a white-board pen and an eraser so that everyone had access to the group's large white board and could contribute to the thinking.

This was a complex problem whose solution and discussion would likely take more than one class session. The faculty member said pause occasionally and brought the whole group together several times to provide guidance briefly as the groups encountered expected issues. When the groups had generated a solution, or at least made reasonable progress, the faculty member planned to invite groups to present and discuss their solutions, as displayed on their large white boards, with the whole group. Although all the groups had worked on the same problem, their boards likely would differ in how the group members had visualized the problem. The focus of the discussion was to be on positive aspects, “What do you like about this approach?” It was important to be supportive and positive as students became accustomed to what for many may have been an unusual culture if they were used to sitting quietly during lectures.

Doing a kinesthetic activity.

Kinesthetic activities involve students using their own bodies to help them visualize a physical situation (See examples at: http://physics.oregonstate.edu/portfolioswiki/strategy:kinesthetic:start). This may involve getting all the students up out of their chairs to move or point in some way to model some aspect of a physical situation. Students also sometimes generated such motions on their own. During the small group problem-solving activity during the third day of class, for example, some students moved their hands in ways to model the problem situation, which involved an inclined plane on which a puck was sliding. Some also used a small white board, placing it at an angle on the table, or holding it up at different angles, as a way to model the steepness of the inclined plane.

Doing a compare and contrast activity.

During the interview reflecting upon the first week of class, this faculty member noted that all the small group presentations would be about exactly the same problem. A way to reduce such duplication in the future would be to design the problem-solving activity so that the small groups would work on slightly different versions of the same calculation. Then they would participate in a wrap-up discussion in which the groups would present their results and ponder what the different results meant. (See examples at http://physics.oregonstate.edu/portfolioswiki/strategy:contrast:start).

Focusing upon the reasoning part of a solution during small group problem-solving activities.

In solving problems, students sometimes get started on a path that turns out not to be useful. Experts do this too. Experts, however, typically monitor the usefulness of what they are doing and will change directions as needed. To help students learn to monitor their progress, this faculty member planned to ask three questions when talking with small groups, questions based upon the research literature (Schoenfeld, 1992). The first attempt to use these questions went well:

So another thing, today I tried out using Alan Schoenfelds' three meta-cognitive questions, and whenever I walked up to a group, for our first conversation, that's what I did; I went through those three questions and the students were pretty game for it; it didn't seem like it was a burden to have to answer those questions; some of them struggled with answering those questions and I think, I hope they saw that as a productive struggle.

Later in the term, this faculty member reflected upon using this questioning practice:

I'm really loving Alan's questions...I'm using them a lot...His questions are: what are you doing? why are you doing it? And what will you do with it once you have it? I feel like I ask those questions and the students can answer and I feel very satisfied and I know what to say to them next; or the students can't answer and now I know what to say to them next...

Asking these questions consistently when interacting with small group seems to have changed the students' perceptions of the faculty member's intent and their confidence in explaining their thinking:

I also feel that for me, because I've said I'm allowed to ask these questions at any time, students don't interpret subtext, in the way that if I come up and I say, “Tell me about this part of your board,” students say “Uh-oh, it's wrong! (The faculty member)'s singling that out because I've made a mistake!” By asking these questions, it's fair to ask these questions all the time, then I don't get that; I get students explaining to me with more confidence about what they were thinking...they don't interpret my asking a question to mean that they've done something wrong.

This faculty member later modified this metacognitive strategy to asking students just two questions: “What are you doing? How will it help you?” The hope was that the students eventually would begin asking themselves these questions to monitor their own progress, not only with colleagues in class but also when working together on the homework outside of class and when responding individually during exams.

Inviting small groups to report out their answers and to talk about looking for mistakes as well as building confidence in one's answers.

This faculty member's goal was to help the students develop strategies for evaluating results obtained by the lengthy problem-solving processes characteristic of the upper level courses. Early in the course, the faculty member reflected:

Today we did an activity where the students worked in small groups to produce a solution to a problem and I'm going to ask students to report out their answers on Monday and I'm planning on a lot of time for this discussion. It's going to be a little tough because the students have all solved the same problem, so my expectation is that students are going to see that they got the same answer and stop listening and so it will be a challenge to me to get them to focus on the reasoning part of the solution.

One possibility to reduce the duplication of multiple small group presentations was to ask each group to tell about one thing they did to evaluate their final answer. The intended outcome was to be a list of named strategies for making sense of their solution to the problem:

And I've also asked them if they've produced equations that they feel are answers, to talk about strategies that they can use to build confidence in the answers, to look for mistakes in their own answers, and so I'm hoping that during that discussion, I'll produce a list on the board of strategies, and that list will then be an anchor for all of our discussions about sense making...and I'm hoping the students will produce names for them (sense-making strategies).

This discussion went very well. Among the strategies identified and named by the students were “Dimensional checking, direction/sign of answer, reasonableness of answer, graphical analysis, limiting/special cases, compare with what you know, put answer back into start, proportionality, and assumptions/idealizations.” This list became the initial sense-making framework for use in class, on homework, and during the exams.

Listening closely to what students are saying in their small groups and during class discussions to learn more about how they are thinking.

During the interview about the second week of class, this faculty member was ecstatic, even though an activity had taken much longer than anticipated:

Wednesday was the best class I've ever taught in my entire career! (laughs) I could not have planned for it to have gone better! I started with a warm up question that I thought would take me 10-15 minutes...And the students, when they were talking with their peers, it became really apparent that this was a really challenging problem for them, more challenging than I had anticipated, and so we ended up spending a lot more class time on it than I had planned.

The faculty member had changed the process for presentations. Instead of having the small groups present their problem solution already written out on their large whiteboards, they instead re-did their solutions step by step on the wall-sized whiteboard at the front of the classroom:

For the whole class discussion, I invited students to come up to the board and do the solution and having the students write their solution on the front board made the presentations so much better than what had happened on Monday (when the small groups had simply presented their boards to the whole group). The students think much slower, they talked about each step, students in the audience actually asked questions, students in the audience contributed sense-making, and so some of things I had wanted to happen on Monday were happening on Wednesday

One of the presenters made mistakes that another student noticed; the ensuing discussion was respectful and supportive:

and beautifully a student who came up to the board made mistakes and those mistakes were caught by an audience member and we had a really nice discussion about them without embarrassing the people who were up in the front. So I felt like the students making mistakes and then not having a humiliating experience with that provided a really nice example for the whole class.

Another presenter explained an unexpected yet correct approach:

(Another student) didn't make a mistake in his solution, but he did it in way that is different from what I would do, so I was able to talk about the difference between those strategies, which was really nice.

Although this extended presentation and discussion took more time than planned, it provided an excellent model for the type of collaborative thinking that this faculty member wanted these students to develop.

Asking students what to do next when working a problem on the board.

In many traditional physics courses, students watch the instructor work through an example problem on the board. The instructor may provide commentary about each step in the solution but the students typically do not participate in developing what is written on the board nor contribute to what is said. During interactive lectures, however, this faculty member engaged students both in proposing and discussing next steps:

When I'm doing problem solving like this, I want the students to be contributing ideas...(I'm) asking them to tell me what to do next...(This) helps me to identify places where I think more explanation is needed; when someone contributes an idea that needs discussing, that obviously prompts me to discuss it in an authentic way.

This process included encouraging students to think about how to get started on a problem, welcoming student contributions even if incorrect, eliciting common stumbling blocks and discussing them explicitly, and elaborating on suggestions that seem to need more explanation.

Incorporating sense-making prompts on worksheets used in class and on homework.

Some of the problem-solving tasks were announced orally or simply written on the board in class. Others, however, were carefully designed and described on worksheets and in the homework. Early in the course, these included explicit sense-making prompts such as asking students to consider special cases, to plot a function and to physically interpret its shape, to compare an answer with an expectation based on prior knowledge or everyday experience, to analyze if and how the answer depends upon certain physical quantities, and to check that units and dimensions are the same on both sides of equations.

During the interview at the end of the fourth week of class, this faculty member reported on the students' progress in using such sense-making strategies. Checking dimensions was occurring frequently:

I'm hoping that they're becoming more efficient at sense making, that they practice a few different strategies and that they can employ a few different strategies. I see students doing a lot of dimensional checking because that's what I expected them to do, that's sort of low hanging fruit in terms of sense making.

Thinking about functional behavior and considering special cases also were occurring:

I'm pleased to see that more students are thinking about functional behavior, which is harder, but I am seeing students do it. Today I was pleased that students started talking about what happens if the thrust is zero, what happens if the air drag is zero? What do we expect then? I don't think students are very good at that (considering special cases), but even I'm sometimes not good at that; so I think it's just important to practice that.

Other less sophisticated strategies also were occurring:

I also mentioned that some of the sense making strategies that they have are not very sophisticated but sensible, like have two different people independently do the same problem and see if they get the same answer, or look up the answer using Mathematica or an integral table or something, those are not sophisticated but anything to build confidence in your answer; I hope the students are experiencing that.

The emphasis on explicit discussion and use of sense making seemed to be working. As this faculty member and assistants gained experience in designing and using sense-making prompts, a problem-solving format evolved for students to use: first in starting a problem by anticipating what the solution should be like, next doing the problem, and then evaluating the result: “anticipate, do, evaluate.”

Grading homework with feedback on the sense-making aspects of a problem.

Fortunately, the grader of the homework had understood the importance of paying attention to and providing feedback on the students' explicit articulation of their sense-making strategies in writing up solutions to the problems. During an interview midway in the course, this faculty member commented upon the sense making evident on the students' responses on an exam:

I graded the exam and I saw a lot of sense making in the question on the exam that I was happy with and I think that is both to do with the fact that we're expecting it in class and on the homework and that I told them, I instructed them that if they used sense making on the exam, if they wrote down something that they knew to be wrong and they could explain to me why they knew it was wrong then I would give credit for that so I saw a lot of discussion in that way.

This exam policy mirrored the emphasis in class and on the homework of monitoring one's answers for mistakes as well for reasons to believe that one's answer was correct.

During office hours, engaging students in working problems together; being a resource for them as they help one another, rather than doing all the talking.

In addition to holding regular office hours, this faculty member chose to observe the “open door” policy for the upper-level courses, that students could stop in to ask for help whenever the office door was open. This provided access for individuals as well as groups of students as needed.

One-on-one conversations during office hours can allay concerns about students who may otherwise seem at risk:

Actually I was pleasantly surprised at one of the students who was saying really good things and doing well (during office hours). In class I was really worried about him...but in office hours I saw that he was not struggling as much as I had thought; in class it was just the nature of his questions and the contributions that I thought he was making in his small groups; it was clear that he was not following very well, and when I talked with him about the homework on Thursday, I got a completely different impression; I think he's just rusty.

Office hours also were enjoyable for this faculty member when groups of students worked together productively with only coaching as needed:

I had tons of students in office hours yesterday, which was a pleasant change. In office hours, we're in a mode where students are working together and I was there as a resource. I got to observe students solving problems and I was really pleased to see that some of the students were really using sense-making strategies in the way that I had hoped that they would “in the wild” so that was confirming.

For students who needed more extensive help, this faculty member asked questions rather than telling answers:

Yesterday I was pleasantly surprised at the students who were at my office hours... there was one group of students who were in the low score cluster on the second exam and they were asking good questions and...I asked them sort of Socratic questions to move them along the homework solution and they were answering them; they were giving very sensible answers; so I was just really pleased with that, they were having trouble putting it all together but when I asked them specific questions they were doing ok.

As the topics of the course became more advanced, more students chose to come:

I'm seeing the same students I've been seeing all term and then some new students who are realizing that they are not as comfortable with the course material as they thought they were.

The office hours provided additional opportunities for practicing solving lengthy problems symbolically with help available as needed.

Designing examinations to assess sense-making skills explicitly as well as content knowledge and mathematical fluency.

Grading physics examinations that require problem-solving typically involves assessing the answer and if wrong, perhaps giving partial credit for some aspects of the problem-solving process. Although sense making presumably occurs as strong students work through an examination, explicit requirements for sense making signal to all of the students the kind of thinking they should be doing not only to guide and monitor their progress but also to evaluate their answer.

This faculty member asked students to make explicit the sense-making strategies they were using for that final step of thinking about why they have confidence that their answer is correct or why their answer may not be correct if they happen to recognize this at the end of working on the problem:

I think class is going reasonably well...There was lots of evidence of sense making on the exam; of course I asked specifically for it, but it looked like the students were doing it and doing it somewhat successfully. A lot of instances where students got a wrong answer and during the sense making say, wait, I recognize that something went wrong, not sure what it is yet, but something is wrong

For students to think about and write such explicit sense-making comments takes time, however, which impacts how many exam questions one can ask.