I am planning out a course that I’ll be teaching next semester for called, “Teaching Physics”. A major question I am wrestling with is, “What do I hope these students to walk away from this class knowing and being able to do?” Some thoughts that crossed my mind today.

Less than two years ago, Andrew Heckler opened a colloquium at the University of Maine with his take on the most compelling and important contribution of physics education research. From my best memory, he said something like this: “Our community is coming to the conclusion that it’s impossible to teach physics well without knowing how your students think and relate to physics content.”

Several decades ago, J. Minstrell wrote that it is a necessary part of teaching physics for “you and your students to know their initial conceptions before commencing a unit of study… Both the teacher and students must be aware of, and verbalize, students’ initial ideas.”

David Hammer wrote in the lates 90s: “A curriculum succeeds, not by guiding the flow of learning and instruction, but by helping to establish an arena of activity rich with opportunities for student and teacher discovery.”

There is something particularly interesting about Jim Minstrell’s recommendations for getting started in teaching physics. He doesn’t suggest you go out and start reading Physics Education Research articles. He doesn’t suggest you start using research-based curriculum materials. (Although I doubt he would recommend you not do these things). What he does suggest is this: Start by listening to your students’ ideas. Start by providing both your students and yourself with opportunities to learn about their ideas. Start by asking questions and constructing activities that engage those ideas. This is also reflected in David’s notion that the curriculum doesn’t succeed because it guides student thinking, but that it provides the teacher and students opportunities to find out about each others’ thinking.

At the end of the day, it is your students you need to come to know about. It is your students’ own mind that they need to know about. Yes, research materials can  help you create opportunities to learn about your students’ ideas. Yes, research articles can help you refine what it is you think you are listening for. But the work of learning about your students ideas is always now. …

What does this mean for my course? I’m not sure, but more and more I think I am committing to teaching this course in such a way that the students in my class experience student thinking– through examinations of actual student work, through observations and reflections of live clinical interviews I’ll conduct in class, through their participation in learning activities with students we invite to class, through interviews they will have to conduct, etc. Yes, we’ll get to reading some physics education research and examining research-based curriculum and teaching strategies, but not until we’ve spent time thinking about our own ideas about what students are thinking and doing, and why they might be thinking and doing those things. I’ll try to provide them with the opportunities to discover student thinking, and I’ll discover what ideas they have for thinking about student thinking. At some point, I’ll invite them to learn about how physics educators and  researchers think and how they have aimed to create curriculum and instruction based on what they think. I hope to do this partially by reading, but partially by inviting those people into our classroom.

I don’t see much reason why we can’t talk to researchers and educators over video. I don’t want the things we experience to be distant. Not distant student data or quotes. Not distant curriculum and curriculum developers. Not distant researchers.

1. Start by listening to your students. By listening carefully and trying to understand their explanations, their predictions, or even the motivation for their questions, you can gain insight into their present understanding.

2. Ask questions that are qualitative. Avoid questions that require the manipulation of formulae and/or technical words unless you specifically want to find out whether they can correctly pick and grind a formula or to find out what a particular word means to them. I believe questions that ask for a qualitative explanations or comparison can be used effectively to probe understanding of ideas.

3. Ask questions that are relevant to common situations. There is a tendency to try to think up some bizarre situation to “trap” students into displaying their “alternative” conceptions. This isn’t necessary. In fact, it appears that many students have more trouble describing or explaining a common situations, probably because it is so similar to situations for which the initial conceptions were developed.

4. Ask questions that require inferential thinking. Once I know my students clearly know the observations, I want to know how they structure the phenomena to give them meaning. I typically ask for a prediction, a generalization, or an explanation. “If you do this, what will happen? Explain why you think that will happen.”  “You’ve now made several observations of … what can you say in general about the situation? What do all the observations together tell you about the nature of…”   “Explain how… happens.”   “We see that … happens. How would you interpret that?”

5. Clarify the observation first. Prior to probing their organization of thought, you may want to ask for their observations. Frequently, I find their perceptions were different from mine. In other words, before you ask them to explain or interpret, you may want to find out whether they saw the phenomenon as you did.

6. Listen (or read) carefully in a non-evaluative way to the answers given by your students. This is probably the most difficulty aspect. As teachers, we are prone to jump in and steer the students straight by telling them what to think. Students are prone to look to teachers for feedback as to whether they have the “right” answer. Fight this, if you want to know what they think. Be neutral in your comments about what students say. Help the students clarify their ideas, but do not evaluate those ideas yourself. Get them to evaluate their own or each other’s ideas. Students will be more willing to say what they believe if they are not graded on their specific answer early in the development of their ideas. There will be a time for grading later after ideas have been developed and used. When you are reading quiz or test results, rather than simply classifying answers as right or wrong, try classifying them as to the type of argument. What I find is that students often get the wrong answer for very good reasons, and they sometimes get the right answer for very weak answers.

Conducting these investigations in the classroom has changed the nature of my instruction. The focus is now on developing the understanding of ideas and applying ideas, ideas that are related to students’ own thinking. We are not matching through a textbook interpreting the ideas of some distant authority; we are building our own ideas.

An incomplete list of changes for next semester intro physics:

(a) Capitalize on the first day more. The first day was syllabus / take FCI day. Next semester, I plan on doing a two quick “mini-labs” and one problem-solving activity. I have 2.5 hours and I plan use it all to set the tone. … Also, a required HW after the first day is to find my office and to write their name on the whiteboard outside my office.

(b) Get away from current reading quiz system by doing a mini-SBG system. Instead of having a 5 multiple-choice quiz on reading, I am going to have 5 standards per week. I’ll explicitly assess those items (probably each twice) that week, but students can re-assess later weeks (details to be figured out). Each week I am supposed to give 2 reading quizzes each worth 5 points. In my new system 5 points will come from students showing mastery of those skills (at any time). 5 points will come from students filling out an online form where they have to write about “What was confusing from the reading?”, which must be completed before each class. This way I accomplish three things: I keep the incentive for students to read, I get feedback before class about what students think was confusing, and the assessments are more geared for diagnosing and learning.

For example, Week 1 might be

I can convert units

I can finding slope of a line given a graph

I distinguish distance, displacement, and position

I distinguish average velocity from average speed

I can interpret a position v. time graph

(c) Have warm-ups each day that make contact with content we initially learned earlier in the semester. Right now, the class runs like a freight-train. We need some reminders about what we were supposed to have learned along the previous stops. I want to keep these short.

(d) Use these quiz assessments and/or warm-ups to give more individualized feedback to students. I don’t find the computer exercises to be the most useful thing for students to do, but in the mean time they give students something to do while I write down individualized feedback on student work. In the future, I’ll hope to have better things for them to do during this time, but I think for now the computer exercises will have to do.

(e) Students will bring their clickers to my class. Right now students just bring them to lecture, which is one hour a week. Right now I am doing a lot of peer discussion-style stuff without clickers, so this is a no brainer.

(f) Manage lab activities with more scaffolding early on and organize lab time as to capitalize on dynamics between whole class and small groups. For example, I want do to the first lab activity together as a class. It’s a measure circumference and radius of different pipes to make a plot, to find slope to get an experimental measure of pi. Instead of just sending them off to do it, I want each group to measure one pipe (a different one) as accurately as they can, to find a value for pi based on their measurement alone, and to determine their percent error. At front of the class, there will be one giant plot, where each group will come up and add their data point. We’ll talk about the graph, and different ways we can get a better measure of pi by using everyone’s data— averaging everyone’s data and finding the slope. Each group will have to do the average method and the slope method to find pi and percent error, and use that to decide which one gives us a better estimate. As a class, we’ll report back what we found and learn and I’ll help students understand why the slope method works — it helps to get rid of any systematic error (i.e., look at the intercepts of the graph).

(g) Instead of checking labs in notebooks,  students will do (most of) their labs on their whiteboards. Data on board, graphs on board, calculations on the board. They have to run me through what they did, and I check them off.  Some labs, however, are supposed to be graded more closely. For these, I’m not sure what I’ll do, but I think I’ll do a group check out plus individual check out on paper, which will focus on procedures students need to know how to do, such as fractional uncertainty, weakest link rule of error propagation, how to linearize data, and how to interpret slope. I could maybe make this more SBG as well.

(h) Continue having students work problems on large wall whiteboards. This is going well. I want to use a combination of a whole-class discussion and group-group conferencing. My new job next semester is to have clearer goals in mind for what students are supposed to have learned from doing that problem, and to make sure those issues arise during discussion or conferencing.

(i) Give students clear time frames to work within, and use a timer to monitor time. I’m OK using more time when needed, but I want to explicitly make that decision, not just forget about time. This is mostly about using class time efficiently. I have 2.5 hours with them. We can get a lot done, but I have to be vigilant about squeezing the most out of the time we do have.

(j) Differentiation and balancing individual vs. group work. After students finish their whiteboard problems, I want to give them the option of doing another problem together or doing their second problem “flying solo”. For “solo” problems, I’m gonna steal from Frank and have answer key in the back of room, where students can give themselves feedback on the problem. While those students are flying solo, I can attend to those students or groups who are still struggling.

(k) Lastly, I want to make sure I have a one-on-one conversation with every student after the first exam. Not sure, how I will do this, but it’s important. I’m also going to email absent students with a note saying something like, “Hey, we all noticed you weren’t in class today. Hope to see you on Thursday!” I already started doing this with my most frequent absentee students, and it has dramatically improved their attendance and likelihood to come talk to me about physics.

This is a paragraph from MTSU’s MTeach Website

Are you interested in teaching Physics and Mathematics?  Are you a Physics or Mathematics major with a concentration in Education, and are you interested in certification in both areas?  The Noyce Physics-Mathematics Teacher Scholarship offers up to $10,000 a year for a maximum of two years.  In exchange, the recipient will agree to teach for two years in a high-need Tennessee school for each year of scholarship awarded.

I don’t have data to back this, but almost all of the MTeach students that I know (those focusing in physics) are struggling to do well in their classes. Some of those students are failing my physics class. Others have just let me know personally that they are scraping by in their intermediate / upper-level math and physics couses. These students are surprised and concerned, because they’ve typically always done well.

I think one big reason they are struggling is because they are taking on more math and physics class than they can handle. Taking 2 physics courses and 2 math courses and 1 or 2 education courses sounds like a nightmare. (Aside: I never took more than 4 classes per semester while in college, and I often took just 3 classes). I think the reason they are taking more math and physics classes than they can handle is because it involves $10,000. I’m not positive, but I think the scholarship is now $15,000 dollars to do both math and physics. It’s also possible that students are being strongly advised to do this, and independent of the money, they feel like this is what they are supposed to do. Whatever the reason, students are doing it, and seemingly struggling as a result.

Surely, an environment in which students are scraping by on a bloated academic schedule in order to get a scholarship can’t be the best environment for future teachers. Right? I mean, I’d rather invest $15,000 for a good physics or math teacher than $15,000 for merely an accredited teacher in both areas. There’s a big difference between taking classes and immersing yourself in the content of a class. I worry that that $15,000 is forcing students to take to many classes, for which they superficially learn a lot of things instead of deeply learn a few things.

Let’s face it. These scholarships are really important to many students. One student in my intro physics class works 30-40 hours per week on top of being a full-time student. If this student passes my physics class, they are eligible to apply for the scholarship. If not, they have to wait another year.

Throwing money at problems, especially teaching, can have unintended effects. Overall, the MTeach program is in its infancy; there’s so many program logistics that will get tweaked and changed. Someone over there should look at the hard data to see how out math/physics students are faring. If they aren’t faring well (like I suspect), we’re going to have to figure out why, and ask the question, “Is our scholarship policy incentivizing failure?”

We seem to make a big deal in intro mechanics that x- and y- motions are “independent”. There certainly are in a limited number of cases (constant forces, linear drag forces, and linear restoring forces, for example) in which the force dynamics described among cartesian coordinates are separate. But this condition isn’t true in most situations (gravitational forces, electric forces, magnetic forces, non-linear drag, non-linear restoring forces). Still, in many situations, we can still find a coordinate system in which the forces are separable. For example, all the non-linear restoring forces (gravity, electric, rubberbands) are separable in (r,θ) coordinates, but the magnetic and non-linear drag forces I don’t think are separable, at least not generally so.

I could be wrong, but I think all (or most) force (fields) that can be described by the gradient of a scalar function will have at least one coordinate choice for which the force dynamics separate. This requires some further inspection, but it has to do with curl being zero. Linear drag forces are, of course, non-conservative; but they “nicely” separate due to a cancellation of factors that would otherwise couple them. This nice cancellation doesn’t occur with v² dependent drag forces, so non-linear drag forces are not separable.

I’m not sure, butI think when we teach this notion of “independent motions”, we are confusing independence with orthogonality, or perhaps separability with orthogonality. X and y cartesian coordinates are certainly orthogonal, but motions described using x and y coordinates are only independent (or separable) under circumstances in which they aren’t coupled.

Anybody know the history of teaching “horizontal and vertical forces” are independent? My suspicion is that it is an over-generalization from projectile motion.

Every student who wants to apply to the college of education at MTSU has to be interviewed by a bunch of different people. I’ve been asked by several students to do this. I don’t really know the purpose of the interviews, but every student explains it like this: “You just ask me about why I want to be a teacher and then you fill out a form.”

I’ve decided that “Why do you want to be a teacher?” is the stupidest question ever. In fact, I can’t hardly imagine a worse question. So I have been thinking about what questions I am going to ask students instead. Here’s my rough draft of possibilities:

• What’s your most meaningful learning experience (// in school // out of school)? Tell me about what you learned and how you learned it. What made that learning experience particularly meaningful?
• What role did other people play in helping to make that learning experience so meaningful? Are there other factors that helped?
• Of all the skills and knowledge that a teacher needs to engage students in such meaningful learning (like the one you described), which skills do you believe are most important  // believe take the deliberate practice to develop?
• What do you think great teachers do throughout their careers to develop these skills? What do you see yourself  doing over your career to ensure that you continually work to develop this set of “hard-to-develop” skill set?
• In the short term, what goals have you set for yourself? Why are these goals important to you? What specifically are you doing to work toward those goals? How will you know if you are or are not making progress toward those goals?
• Tell me about something that you understand well now, but really struggled to understand early on.
• Tell me about your biggest fear / worry in becoming a teacher. Why does this worry you?
• Outside of teaching and school, tell me about three things that you value most about the yourself and the life you lead. Why do you values these things? How did you come to value these things? What impact does this have on others around you?

I’m hoping you all will weigh in and help me figure out what’s best to ask.

Issues that are on my mind today about the intro physics course I teach:

(1) Students don’t get enough individual feedback from me, partially because almost everything in class is done in groups, but their high stakes assessments are done individually. This is not good for students (especially those that are struggling). Plus, for me, it makes assessing and diagnosing individual students difficult. I need to get students doing more individual work for which they can get feedback, even if that’s not part of the model for the class.

(2) What little individual feedback they do get from me (mostly on lab reports) is for things that are borderline irrelevant / on the margins of importance. This makes me spend my time “grading” student ability to write their lap report in the correct format rather than “assessing” their understanding of disciplinary ideas and skills. That, and it gets students really focused on the wrong things—looking for ways to make sure they dont’ lose points in the future rather than learn.

(3) The overall grading system has so much “stuff” with so little substance– reading quizzes, project presentations, project reports, discussion quizzes, clicker participation, checked labs, and graded labs. There is nothing in those parts of the grading system that point them to what is important to learn. Rather, it seems to point them toward what bases are important to have covered to get a good grade or be positioned to get a good grade (if one performs well on tests). Students who struggle to understand (and do poorly on tests) can’t spend time trying to learn and understand because there are too many hula-hoops to jump through in order to guarantee enough fluff points that a good test grade will even matter.

(4) I have growing concerns about differences in performance outcomes among different populations I teach. Without going into details, there are certain combinations of race and gender are faring well and certain combinations that are faring not-so-well. This is probably typical of college physics, but it’s still weighs heavily on my mind.

What we know about stacking transparencies…

Cyan + Magenta = Blue

Cyan + Yellow = Green

Magenta + Yellow = Red

One Theory to Explain this pattern is the following

Cyan filters block out all RED (leaving primarily blue and green) — blue/green looking cyan jives with their intuition

Magenta filters block out all Green (leaving blue and red) — blue/red looking magenta jives with their intuition

Yellow filters block out all Blue (leaving Green and Red) … leaving question why does green/red look yellow? Not-intuitive

This theory is somewhat uncommitted about what happens with oranges, yellows, and violet;

The theory goes:

Cyan (blocks red) + Magenta (blocks Green) = Blue Left Over

Yellow (blocks blue) + Magenta (blocks green) =  Red Left Over

Cyan (blocks red) + Yellow (blocks blue) = Green Left Over

Another Theory Goes like this

Cyan enhances blue and green maybe even yellow and dims other

Magenta enhances blue, red, and violet (and may even yellow/orange) and dims others

Yellow enhances yellow and dims others

The theory goes like this

Cyan + Yellow gives Blue, Green,Yellow enhancements , making an overall Green Appearance … this jives with their intuition

Magenta + Yellow gives Blue, Red, Violet, and Yellow, Orange enhancements … Not sure why this come off just Red?

Cyan + Magenta = Blue (enhanced twice), Green, Red, Violet, Yellow, Orange … why does this come off as just blue?

Both theories have gaps and puzzles. However, Caroline invented a theory, which seems to fill in a lot of gaps for both theories.

Caroline’s Extended Theory of Opposition

Blue / Orange are Opposites (…we do find that stacking orange and blue gives a dark appearance)

Violet / Yellow are Opposites (…we didn’t have violet transparencies… but yellow and purple gave a dark red)

Red and Green are Opposites (…we do find that stacking red and green gives dark appearance)

The Theory Goes like this

Since Magenta contains both Blue and Red, it probably blocks out both oranges and greens!

Since Cyan contains both Blue and Green, it probably block out both red and oranges!

Yellow blocks out Violets. Since it must also oppose blue, does this mean that yellow contains orange? 

How Caroline’s Opposing Theory Fills in Theoretical Gaps

Caroline’s Theory fills in a lot of holes in theory #1, by detailing what happens to Orange, Yellow, and Violet…

It says that Cyan (blocks Red and Orange), leaving not just Blue and Green, but Yellow and Violet

It says that Magenta (blocks oranges and greens), leaving not just Red and Blue, but also Yellow and Violet

It says that Yellow (blocks violet and blue), leaving Red, Orange, Green, and Yellow

Caroline’s Theory also fills in holes of theory #2, because it can explain complicated mess of why Blue and Red end up

Blue + Red + Violet + Yellow + Orange = Red . This is because Blue/Orange and Yellow/Violet are opposing, leaving only red unopposed

Blue (x2) + Green + Red + Violet + Yellow + Orange = Blue. This is because Red/Green Oppose, Yellow/Violet oppose, and One Blue is opposed by an orange, leaving only one blue unopposed.

In my inquiry class, we are sorting through a surplus of observations about what happens when we stack various colored transparencies. I decided to narrow the focus down from all transparencies to just CYM, so that we can try to pin down some things in a certain arena without having to worry about everything all at once.

We have (at least) 3 ideas about how colored transparencies work. Here’s is what I think those theories are:

Theory #1: Transparencies are Color Activating (or Enhancing)

White light has all the rainbow colors available (ROYGBIV), but white light by itself is mostly colorless. It is colorless because none of those colors are activated. When white light hits a yellow transparency and then goes through it, the yellow component of white light becomes activated (or now visible). All the different colors of light get through, but only the yellow light is activated. Transparencies work by activating some portion of the rainbow colors but not all of them.

How this theory explains and accounts for other phenomena:  This theory explains why we see green when we stack Yellow and Cyan transparencies in the following way: After enhancing the yellow light, all the unactivated light continues on where it reaches the cyan filter. The cyan filter activates both Blue and Green. So now, both green, blue, and yellow are activated. Green is obviously present from being activated by by Teal, but the blue and yellow activated light blend to make more green. Thus, yellow, blue, and green give the overall appearance of green. One group says this theory can also explain other CYM combinations, by noting that magenta must enhance much more red more than blue.

Theory #2: Transparencies are Color Changing

Transparencies work by changing the color of light. For example, white light is white. But white light upon a yellow transparency causes the white light to undergo changes which make it yellow. Of course, different colored lights upon the yellow filter will change the color in different ways. For example, the magenta light on yellow the transparency changes the light to be green light. Different colors have different rules.

I’m not sure if hidden in this idea is a “blending” or “averaging” idea–meaning that maybe we could come up with some rules. But it seems like the idea is “colors mix in weird waya” and you just have to go observe every combination to learn each rule. I think what I can appreciate about this idea is that it doesn’t assume that there should be simple rules: Why should we expect a yellow filter to have the same behavior when different colored lights shine through them? Can’t the change it induces be particular to the kind of light we shine on it?

One limitations of this theory (so far) is that it has a difficult time making predictions because it sort of says, “Go out in the world and find out what all the rules, because the rules are different for all the different combinations”. It’s feels almost anti-theoretical.

Theory #3: Transparencies are Color Filters

This idea also focuses on white light having all the rainbow colors.  Transparencies work by blocking some colors and letting other colors through. For example, many groups decided that we know both Magenta and Cyan must let a good amount of blue light through, because Magenta stacked on Cyan make blue. So a lot of blue light must be getting through. Most groups thought that Magenta must let mostly red and blue (and maybe purple, orange and yellow), and that teal must let through mostly blue and green (and maybe purple and yellow). They explain that blue light results from stacking Cyan and Magenta because it’s the only color that doesn’t get filtered out. I’m interested to see what these groups say about yellow as a filter.

The Big Picture: Assessing student theories? Assessing disciplinary knowledge?

Most of the time, we assess student ideas in terms of correctness. Most of you know which of these theories is more closely aligned with current scientific understandings. However, without correctness, how do we assess the “goodness” of a theory or of an explanation. I think the only refuge is to think about how the scientific community assesses scientific theory, because in science we don’t have the answer key to tell us what scientific theories are correct. Surely there are some criteria for judging some theories to be better than others. Surely we must have ways of assessing scientific explanations in light of the current ideas and evidence we have.

I’d say that right now, I think the “activating color theory” is the best theory… It has the most specificity (details which filters enhance which light), coherence (tells a consistent story for each situation), and has explanatory power (explains all CYM combos).  #2 is almost simply a statement of observations at this point. #3 has some good beginnings, but at this point, it still needs some fleshing out. Yellow in particular is going to represent some serious hurdles. We’ll see where this leads us, and I’m happy to keep all of theories along, pressing upon them in different ways that I think will nudge us all along together.

Of course, to be fair, I also have reason to believe that not everyone has well-distinguished theories. I think many hold some combination of #1, #2, and #3 and they certainly aren’t well articulated or committed theoretical frameworks. So Where do we go now? They have a homework to write about their theory for transparencies (how they work in general for CYM in particular), and how their ideas explain stacked CY, YM, and CM.

Andrew Heckler, a physics education researcher at the Ohio State University, wrote an interesting paper about the consequences of prompting students to draw free-body diagrams. Heckler found that asking novice college students to draw free-body diagrams as part of physics problems has some interesting (negative) consequences for their problem-solving. It’s worth reading yourself, but here are some notes.

One problem involved asking students to simply identify the forces exerted on a basketball rolling across a ‘frictionless’ floor. (correction: the problem states that friction is small enough to be ignored)* Students who were asked to draw and then identify were more likely to identify a “force of motion” than students who were not asked to draw a diagram. Why might this be so? I’d speculate that it’s simply compelling to draw an arrow in the direction of motion as part of a sketch. Then, upon seeing what one has drawn, one is persuaded into thinking that it must be a force. My argument is that it isn’t so much that students have a force of motion misconception, but that there is a dynamic between what one draws, what ones sees, and how one responds. Drawing an arrow in the direction of motion is part of the dynamic by which students engage in thinking that there must be a force in that direction.

Another problem from the study involved students having to figure out the minimum mass needed to get a box initially moving where in the problem the box is being pulled on by both sides with known but different forces and there is friction. Once again, some students were just asked to solve the problem, and other students were first asked to draw a FBD and then solve the problem. With this problem (as with the others), students were more successful in solving the problem when they weren’t asked to draw FBD.

Many of the students who were successful used intuitive approaches that were not taught. One of these approaches Heckler calls the two-step method, in which students first simply subtract the two pulling forces, and then set them equal to the friction force. Some students even went so far to draw 2 different diagrams, one with only the pulling forces opposing each other. And then a new one with the combined pulling forces opposing the friction. In contrast, students are taught to draw 1 FBD that shows all the forces, and then they are taught to write out a complete ΣF statement. The students’ intuitive approach has several benefits. First, it has a divide and conquer strategy–if you can’t figure everything out, start with what you know and work from there. Second, it allows you to figure out the direction of the friction force along the way, instead of having to guess and then adjust at the end if you find you’ve gotten a friction with a negative sign. Third, since the strategy makes sense to the students, they have ways of spotting errors and correcting mistakes along the way. When students take the expert approach, they are more likely to make mistakes and less likely to correct mistakes.

Overall, Heckler found that students did not typically see the FBD as a way to help organize the problem or to check for consistency. Rather, FBDs were more of just something an instructor was asking you to do. In fact, many successful students would draw a wrong FBD, and then proceed to ignore it, so that they could solve the problem correctly using an intuitive approach. And many students who drew incomplete or wrong FBDs often still solved the problem correctly using an intuitive approach. Still, overall, students who weren’t prompted to draw diagrams did better than students who were.

Intuitive Approaches in Energy in my Classroom

Speaking, of intuitive approaches. Last week, I showed students how to draw energy pie charts instead of starting with equations for energy conservation. This led students to use some intuitive approaches that were successful, but quite different than the formal approaches. In one problem, a roller coaster started a height of 85 cm and then goes around a loop with a radius of 17 cm. Students were asked to find the speed at the top of the loop. The formal approach would have students write

PEi + KEi = PEf + KEf

mgH + 0 = mg2R + 1/2 mv²

but several student groups noted that 34 cm was 40% of 85 cm, which meant than the potential energy on the loop was 40% of the original , leaving 60% of the initial energy for Kinetic.

They then wrote this equation

KEf = .6 PEi

1/2 mv² = .6 mgH

I let students go down this path, knowing that this approach might not be easy to implement all the time. Instead  of steering them away from it in the moment, I let them continue. In order to make sure they had an opportunity to make contact with the formal approach they would be expected to use on the exam, I then had them explain their solution to another group, and that other group share their approach, which was more closely aligned with the formal approach.

The Big Picture

I’m pretty convinced that students have a wealth of problem-solving strategies and reasoning skills that go untapped when we teach formal methods to soon. It leaves these formal approaches disconnected from the the good things students have to bring to the table. Of course, I know that students’ intuitive approaches will need to be formalized at some point, and that many intuitive approaches will run into problems later. But I feel that teaching students to use formal approaches without helping them anchor it to their own sensibilities and ideas is much more problematic. I’d rather help them to refine and objectify their own approaches, and introduce formality as authentic need arises.

* Note that many physicists will initially have a problem with this. They’ll say, “Rolling on a frictionless floor? That’s impossible. This question is flawed!” Remind them that students often believe that force is required to maintain motion, and that this is a misconception. Then ask them if they have a similar misconception that rolling (or spinning) objects must be maintained by a torque.