Analyzing Aspects of Difficulty

Physics teachers have a lot to gain by analyzing what factors contribute to the difficulty of the course they teach. For students, the priorities of the course are dictated by these factors, whether or not these priorities are intended by the teacher.

Conceptual understanding of physics is an important facet of a student's physics education, but this is not the only thing we strive to teach in our classes. Aspects of student learning could be categorized as conceptual (e.g. applying Newton's 3rd Law to an analysis of forces), mathematical (solving a quadratic relationship to find the time aloft for a projectile), skill-based (plotting data with error bars), and knowledge-based (knowing that g on earth is 9.8 N/kg), and we could certainly identify other categories as well (problem solving, scientific reasoning). Any physics teacher makes choices to emphasize certain of these aspects over others, based on their own personal priorities and priorities dictated by others (through a standardized test or curriculum or through administrative or departmental influence). How can we use these distinctions to inform decisions about what priorities to set in our curriculum?

I believe that a physics course should be difficult. I also believe that we have a responsibility to our students to examine the "difficulty" of the course we teach through the lens of categories like I mentioned above. In other words, if a task or problem is difficult for our students, why do they find it difficult? Do the concepts being studied conflict with misconceptions? Are students being asked to use unfamiliar math techniques or analytical skills? Are they having trouble recalling or accessing necessary knowledge?

To illustrate a distinction between conceptual and mathematical difficulty, for example, consider the diagrams to the right, depicting two varieties of the "modified Atwood machine." As part of a discussion of Newton's 2nd Law in a high school physics course, students might be asked to, "Find the acceleration of the hanging block."

In my experience, either of these problems would present a challenge for a ninth grade student. It's a challenge to recognize that the acceleration of the two-block system is affected by both masses, though the force of gravity on the large and small masses affects this acceleration in different ways. However, the problem on an incline would present a significantly greater challenge to an average ninth grader. Why is this second problem more difficult? Does it require a require a more sophisticated conceptual understanding, or does it simply require more knowledge of math?

Well, I'd say it requires both*, but I'd also argue that the added difficulty of the second problem in this case is overwhelmingly due to the trigonometry involved in the solution. If we choose to include this problem in our course, it should be because we want to increase a student's familiarity with trigonometry, not because we assume that this trigonometry is inseparable from the physics concepts. Of course, a student's understanding of physics concepts is often connected to their facility with the relevant math (though the connection between math and physics is less direct than we sometimes assume), but we physics teachers have some degree of flexibility to include only the pieces of the "whole story" we deem to be age-appropriate. We don't shy away from discussing gravitational field in a typical algebra-based high school course simply because our students can't compute the line integral required to show that this field is conservative.

In making choices about how to teach and what to teach in our courses, there's a lot to be gained by analyzing how the activities we include in our course develop both skills and understanding. In a Physics First course, our task is twofold (at least!): we are teaching physics and introducing students to high school science. In determining how best to accomplish these parallel tasks, we can continuously ask ourselves what conceptual, mathematical, skill-based, and knowledge-based aspects of physics our course is prioritizing.

*To clarify, it seems to me that vector analysis of the forces on an object on an incline is mostly a conceptual (or perhaps skill-based) task. However, I'd argue that the added challenge of applying this analysis to a calculation of acceleration is mostly mathematical in nature.

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Programming as Problem Solving

Computer programming provides an ideal environment for young students to develop the cognitive skills required to solve a problem gradually through trial and error... and error, and error, and error.

At a New York City public school with a focus on science and math, I observed ninth grade students in a class called Engineering, devoted mostly to introductory computer programming. Students had spent much of the beginning of the year working in Alice, a simple-to-learn 3D programming environment, but had moved on to a text-based, open-source language called Python. Python is rapidly developing a huge following in both the programming and teaching worlds, in part because of its simplicity, readability, and diverse and dynamic library of resources.

Many of the students in the class were working on a coding a guessing game in Python, an assignment written by Tim Wilson as part of a public library of Python-related materials.  In this game, the computer selects a random number between 1 and 100, and the user enters integers until they guess the number correctly. I've included below a few examples of observations I made while watching students make their way through this challenge:

In developing a method of comparing a user's guess to the computer's selection, one student had written a guess test consisting of the following code:

    if guess = computernum:
        print 'You got it!'

This student was busy running the program and trying out different numbers, but was getting increasingly frustrated that the computer just sat there silent whenever he entered an incorrect guess. 

Another student was trying to print the number that the computer had guessed, to make it easier to debug her guess-comparison code. Another student tried to help by suggesting that she include the code "print computerchoice" but any time she ran the program she got an error message:

Traceback (most recent call last):
     File "/Users/Documents/Python/guessinggame.py", line 10, in <module>
        print computerchoice
NameError: name 'computerchoice' is not defined

Another student had successfully completed the necessary code for making and checking a guess, and he was trying to expand his code to allow for multiple user guesses, in sequence. He copies the section of his code that checks the guess and pastes it below the first "guess-checking" section, but he is puzzled when he sees the computer's emphatically repetitive response:

What is your guess?56
56
ha ha you got it wrong
ha ha you got it wrong 

An introductory course in programming is, above all, a course in problem solving. The world of programming plays by consistent rules, and solving problems within this environment means learning those rules. Large tasks can be broken down into discrete, testable chunks, and incremental progress is rewarded with immediate feedback. Creativity and collaboration go hand-in-hand, and successful and unsuccessful approaches are identifiable by the program they produce. Any differences of opinion on what approaches might work are best resolved by simply trying things out - as recently quoted by our nation's youngest billionaire, "Code wins arguments."

We physics teachers like to paint the universe as a place that operates by a few simple relationships, but demonstrating the simplicity of these relationships can sometimes be a little complicated. When faced with a complex problem, developing a successful solution takes a organized approach based in the assumption that a system is doing what it's doing for a reason. Rather than simply saying, "it's not working..." a successful problem solver admits "It's working... It's just not doing what I want it to do. Why is it working this way and how can I change it?" Programming is an ideal venue for practicing this skill. As each of the issues I described above were resolved, students were rewarded with a more sophisticated understanding of the "unsuccessful" program they'd written, as well as the modification that brought the program closer to completing the task at hand.

Later in the course, students are exposed to a 3D graphics module called Visual, which allows for fluid 3D programming within Python, or as it's referred to when used with this module, VPython. All students at this school take physics in ninth grade as well as engineering, and one of the later assignments in the course is to use VPython to program a 3D simulation of a projectile moving in two dimensions. What better way to help students arrive at an understanding of the rules nature uses to determine the position of a projectile than to ask students to code these rules themselves?

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The No Relationship Relationship

Students often have a difficult time interpreting data from a graph of two variables when the relationship between these two variables doesn't exist. Teachers should approach these situations with care, and use them as an opportunity for direct discussion of what "no-relationship" means in a scientific investigation.

On a recent visit to a New York City public school, I saw ninth graders complete a classic pendulum lab. Students measured the period of a pendulum at various different angles of initial displacement, and then measured the period for various string lengths. Students then graphed their results and posted them on the walls of the room for everyone to see. Though not many groups looked around the room at their classmates' graphs, if they had they may have been surprised at the diversity of results! I've included a sampling of some student graphs showing their results concerning the relationship between period and amplitude.

These graphs illustrate how challenging it is for students to accept the no relationship relationship. (Before you take me to task about the amplitude-dependence of the period of a pendulum, keep in mind that for the maximum amplitude measured here, the actual period deviation from a small-angle approximated pendulum is less than 5%...) Young people who have been graphing results in science class since elementary school are so used to seeing a trend in the relationship between two variables that they go bonkers when the relationship between those variables doesn't jump out at them.

Watching students carry out this pendulum lab was fascinating. Nearly every group doing the lab expressed doubt and dismay when they noticed they were getting approximately the same value of period for every amplitude. (At least one group decided to do something about it: if they didn't get a result that was sufficiently different from the last angle they measured, they went back and did the run again.) One member of the group that made the correct graph shown on the upper right apologized that they "did the experiment wrong." The assumptions students brought with them to this lab led to quite a few discussions with the teacher about why they were so sure that their measurements were wrong, but I didn't get the impression that these conversations resulted in much self-reflection. (We could find out, of course, by asking these same students to examine the relationship between period and mass...)

This period-amplitude relationship was just one piece of the entire pendulum lab, and students' graphs showing the relationship between string length and period were more successful than the period-amplitude graphs I've shown here. But, applications in torture aside, it seems like pendulum's most promising role in a Physics First course is in encouraging students to examine their own role as scientists. With a little bit of restructuring to this activity, the widely varying results shown in these graphs could provide the fuel for quite a sophisticated discussion about experimental design.

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Active Physics and Inquiry

An independent school in New York City provides an excellent example of a successful application of the Active Physics curriculum, but aspects of Modeling Instruction could have potential to make the course even more dynamic.

Active Physics is a project-based curriculum with a conceptual focus, designed to be used with ninth graders. Active Physics groups concepts by themes, such as "Communication," "Sports," or "Home," in an attempt to make the physics more relevant to students' daily life. The work done in each unit culminates in a "Chapter Challenge," where students must apply their knowledge to solve a real-life problem. One independent school in New York City has been using the Active Physics curriculum since 1994 as the foundation of a physics course for all ninth graders. When I visited this school, students were studying the efficiency of various methods of heating water, and were just about to begin the "Chapter Challenge" of selecting appliances to meet the basic needs of an average family, capable of being powered by wind-generator with an output of only 2400W.

When class began, students were seated in lab groups, discussing a question from their textbook: "Are high-efficiency appliance worth the added cost?" Students' responses reflected a common misconception - conflating the efficiency of an appliance with an assessment of its overall quality: "Well, yeah they're worth it... they're better." "They're more durable, work faster, and just work better in general." When students were asked to present the results of their discussion, only one group in three appreciated the more subtle implications of the concept of efficiency, stating, "A higher efficiency appliance will make up for its cost with less power used over time," but even this group was confused by the difference between the terms "power" and "energy." The stage was set for an inquiry-based activity to root out the would root out these misconceptions and lead students to a more sophisticated understanding of the concepts of energy, power, and efficiency.

The lab activity for the day consisted of heating up a beaker of water on a hotplate. The procedure steps outlined in their textbook were summarized on a projector: "Measure: 150mL of water, initial and final temp of water, measure time appliance is on (increase temperature by 20˚C)." After a brief discussion of how to use the equipment, students got to work carrying out these steps. They made a few potentially problematic procedural choices along the way (measuring water volume with a beaker rather than a graduated cylinder, plugging in the hotplate before starting their stopwatch, resting the thermometer against the bottom of the beaker, for example), but the teacher caught most of these and gave suggestions for improvements when he felt it was necessary to do so. In class discussion, students struggled with how to use the values they'd measured to make the required calculation of efficiency, but the teacher coached them through the process (partly by referring them to a similar activity done a few days earlier with immersion heaters):

Teacher: "Who remembers how we calculate the thermal energy gained by the water?"

Student: "Was that the thing that was 4.18...?"

Teacher: "Yes, we need the specific heat of water. Anything else?"

Over the course of the discussion, each group eventually arrived at calculations that basically agreed with one another, confirming an efficiency of about 10%.

While watching students carry out this activity and discuss the correct method for calculating efficiency, I tried to imagine what the same basic procedure would look like using a whiteboarding approach. Students might start the lab by brainstorming steps they'd take to to collect whatever data they felt were relevant to a calculation of efficiency, then writing these steps on a whiteboard and presenting them to the class for discussion. Once students had carried out these steps with their lab group, they could attempt a calculation of efficiency (again, on a whiteboard), and discuss as a class whether the calculations they'd made were relevant to the central question of the efficiency of appliances. Different groups could even use different methods of heating the water: an immersion heater, a hotplate, a microwave...

I emailed a prominent advocate of Modeling Instruction to ask about crossover between Active Physics and Modeling Instruction, and she told me that "Active Physics and Modeling Instruction don't go well together." Modeling Instruction is about developing basic models for the most fundamental interactions in physics, whereas the projects in Active Physics tend to highlight more complex applications of these concepts: efficiency of electric appliances, acoustic properties of instruments (watch your ears...), how to build a DC motor or generator, etc. Both of these approaches have merit, and it seems to me that there's a lot to be gained in exposing students to aspects of both. That is, a whiteboarding approach might have avoided the more "cookie-cutter" aspects of this particular lab activity on heating water (and probably brought misconceptions to the forefront more effectively), and a project-based "Chapter Challenge" in a Modeling course might give students a better appreciation for how even the simplest models they develop can be applied to their daily life.

In my observations, I've noticed a trend among teachers of Physics First: in the absence of a single universally-accepted ninth grade physics curriculum, teachers tend to pick and choose aspects of various programs that appeal to them. This dynamism is healthy and exciting, but there is something particularly thrilling about the momentum that has been building around Modeling Instruction. A lot of aspects of Modeling just feel like the right way to teach physics: whiteboarding, student-designed experiments, modeling phenomena with multiple representations, and it's tremendous to see the Physics First movement marching forward hand-in-hand with Modeling Instruction. Still, we'd be wise to keep in mind the potential benefits of a diversity of approaches and try to maintain some of the freedom and flexibility that characterizes so many ninth grade physics classrooms.

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Colleague Response to a PF Inquiry on the Scarcity of PF Research

I get teased by some friends about it, but I'm a very proud member of a couple of physics teacher clubs. In one, around 20 or so individuals meet monthly to discuss experiences in physics teaching and share ideas. Recently, a teacher wrote to this group to inquire about teachers' experience with Physics First, and a friend in the group named Yoav Bergner wrote a response that very eloquently gets at the heart of some of the questions about its effectiveness. Yoav is a former educator, and is currently an education post-doc in the ReLATE program at MIT (as well as a builder of fine furniture and a fellow education blogger). He gave me permission to post his response here... and to include this photo of the phenomenally beautiful Ms. Pacman butcher block he made as wedding gift for a friend!

Now, this blog isn't usually a venue for major criticisms of Physics First (I'm such a fan of the movement after all, did you know?). Yoav's response contains a lot of different viewpoints and opinions on PF, and a few of these positions don't match my own personal opinion. But one major point that Yoav makes is hands-down irrefutable: there's just not enough research yet done to show that a PCB sequence is more effective, on average, than a traditional sequence. Anecdotal evidence shows plenty of success stories, and indeed plenty of failures as well, but hard numbers for or against are scarce. I'd add to that, though, that "more effective" can mean a lot of things: FCI gains, Lawson's reasoning test gains, increased enrollment in a higher-level physics course, increased student affect toward science, or even simply (as Yoav himself mentions below) the number of students who end up taking a physics class at least once in their lives. Teachers and researchers across the country have been responding to this need with research of their own since back in 2003 when Pasero's State of Physics-First Programs was published, and I know personally of a few programs and schools with data that are just around the corner.

Anyway I apologize for the interjection... Yoav's words from here on out (citations below):

The Physics First argument is often made unduly complex; there may not be much to it beyond the desire to increase the enrollment of high school students in physics. Neuschatz and McFarling attribute poor performance by the United States on the TIMSS to the limited reach of physics, arguing that, “our prime shortcoming is not the poor job that is done when physics is taught, but rather the fact that so few students take it, and that fewer still get beyond the basic introduction” (Neuschatz). Other defenses for inverting the traditional sequence, such as the claim that chemistry and subsequently biology build on concepts from physics, are countered by the traditionalist’s view that physics simply requires more mathematical sophistication. But math prerequisites have subsequently excluded many students from taking physics at all. Exposing “the largest number of people” to the “broadest range of physics topics” might be viewed as reason enough for teaching physics in ninth grade (Bessin).

While there is not widespread agreement that Physics First is a good idea, there is a general consensus that if it is to be adopted at all, a ninth grade physics course should be conceptual and should not focus on mathematical problem solving. This was hinted at in Lederman’s Physics First advocacy writings, in which it is recommended that mechanics, for example, be de-emphasized (Lederman, 1998). The idea is explicit in the AAPT report, “the emphasis in a physics-first sequence should be on conceptual understanding rather than mathematical manipulation,” but the rationale is perhaps best articulated by Hobson and also Hewitt. Hobson’s point of view is that a first physics course should address societal needs for conceptual understanding of modern issues, that mathematical problem-solving is appropriate for a second course. A first course should expose students to exciting topics: quantum physics, cosmology, global warming, pseudoscience, nuclear weapons (Hobson, 2005; Hobson, 2006). He cautions that “many physics teachers resist teaching conceptual physics for all. If we insist on teaching a math-based course first, we will continue turning away both science and non-science students in droves, and it will be essentially impossible to institute physics first” (Hobson, 2003).

Hewitt is a well-known proponent of conceptual physics via his classic textbook, which has been in print since 1971 and is used in the vast majority of physics courses aimed at non scientists. In 1982, he opined, “I think most teachers feel that the conceptual and traditional can be taught simultaneously. I think not, and will exaggerate to make my point. I suggest that teaching conceptual and traditional physics together is akin to teaching children to dance during the stage of life when they'd normally be learning to walk” (Hewitt). Recently, Goodman and Etkina have hinted at a successful “mathematically rigorous physics first course,” but their claims need to be scrutinized in the context of the following: at the school under study, the physics class is designed to mimic the content and structure of the AP Physics B test as much as possible; the full course material is spread out over two years of physics (second year optional); the only assessment is performance on the AP Physics B; and only 20% of graduating seniors end up taking the test (Goodman).

The sad reality is that there are almost no real data showing measurable success or failure in the Physics First effort. Dozens of articles and letters published in The Physics Teacher make claims with barely a shred of support (for example, Dreon, Ewald, Korsunsky, Taylor); meanwhile the San Diego school system experienced a rejection of the new curriculum by some affluent districts, which were subsequently granted academic independence to return to traditional sequences (Tomsho). There is scant evidence that decades of physics education research are being applied in Physics First programs––high school teachers are not by and large documenting their progress. There is scant evidence that decades of physics education research are being applied in Physics First programs––high school teachers are not by and large documenting their progress. Pasero’s 2001-2003 ARISE report on the state of Physics First programs acknowledges the following points (quotes due to Pasero, enumeration mine):

1. “Teaching a math-free physics course [is] very difficult,” Pasero observes, noting further that schools typically select one of the following strategies to deal with math-related challenges: (a) make algebra prerequisite, (b) offer two tracks, or (c) coordinate physics and algebra courses. “Negative comments [from students] were almost exclusively reserved for times when they felt the math was overwhelming.”

2. “Students’ favorite part of physics class was labs.”

3. “This may be the most significant finding of this study: Physics-first schools are not quantitatively documenting the degree of their success.”

American Association of Physics Teachers (AAPT 2006). “Physics First: An Informational Guide for Teachers, School Administrators, Parents, Scientists, and the Public”.

American Institute of Physics. Teaching Physics First

Bessin, B. (2007, March). “Why Physics First?” Guest Editorial, The Physics Teacher 45, 134.

Dreon, O. (2006, Nov). “A Study of Physics First Curricula in Pennsylvania”, The Physics Teacher 44, 521-523.

Ewald, G., Hickman, J., Hickman P, and Myers, F. (2005, May). “Physics First: The Right-Side- Up Science Sequence,” The Physics Teacher 43, 319-320.

Goodman, R. and Etkina, E. (2008, April). “Squaring the Circle: A Mathematically Rigorous Physics First,” The Physics Teacher, 46, 222-227.

Hewitt, P. (1983). “Millikan Lecture 1982: The missing essential––a conceptual understanding of physics,” Am. J. Phys. 51 (4), 305-311.

Hobson, A. (2003, Dec). Letter to the Editor, The Physics Teacher 41, 508-509.

Hobson, A. (2005, Nov). Letter to the Editor, The Physics Teacher 43, 485.

Hobson, A. (2006). “Millikan Lecture 2006: Physics for All,” Am. J. Phys. 74 (12), 1048-1054.

Korsunsky, B. and Agar, O. (2008), “Physics First? Survey First,” The Physics Teacher 46, 15-18.

Lederman, L. (1998). “ARISE: American Renaissance in Science Education,” Fermilab-TM- 2051.

Lederman, L. (2005), “Physics First?” Guest Editorial, The Physics Teacher 43, 6-7.

Neuschatz, M. and McFarling, M. (1999). “Maintaining Momentum: High School Physics for a New Millennium,” AIP Report R-42.

Pasero, S. (2003). “The State of Physics-First Programs”, Revised March 2003, Fermilab-Pub-01/206.

Sheppard, K. and Robbins, D. (2002) "Physics was once first and was once for all" The Physics Teacher  41, 420.

Taylor, J. et al. (2005). “Curriculum Reform and Professional Development in San Diego City Schools,” The Physics Teacher 43, 102-106.

Tomsho, R. (2006, April 13). “Textbook Battle: Top High Schools Fight New Science as Overly Simplistic”, The Wall Street Journal.

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"Conservation of Momentum" in a Transition to Physics First

Two teachers at the same school facilitate essentially the same lab activity, but with different levels of commitment to the process of inquiry.

In a time of transition from a traditional BCP sequence to a Physics First sequence, schools or districts are often faced with a distinct challenge of limited personnel. A ninth grade physics class is fundamentally different from both a ninth grade biology class or a physics class directed toward juniors or seniors, and very few teachers have direct experience teaching Physics First. In the switch to ninth grade Active Physics in the San Diego public schools, for example, the transition was driven mostly from the top down, with grossly inadequate professional development to prepare teachers for the task they were undertaking. Many biology teachers were simply ordered to become physics teachers, and physics teachers who had no experience with younger students were dropped into a project-based curriculum with little quantitative emphasis. The program was doomed to fail from the beginning, and San Diego switched back to BCP five years later.

Much of the time, however, the remnants of a switch to Physics First can be more more subtle. I visited a public school near Philadelphia that offers a Physics First track as an option for high-achieving students. The majority of ninth graders take environmental science. During my visit, I saw observed two sections of physics for ninth graders, both based in Modeling Instruction but taught by different teachers. The same basic lab activity was carried out in each class - both labs involved the use of a track with two carts with photogates mounted to measure the speed of a cart rolling toward either end of the track - but there were important differences between the classes.

At the beginning of one class, the teacher wrote a simple prompt on the board: "Objective: graph ∆Pbluecar vs. ∆Predcar" and showed students examples of possible interactions between the carts (bouncing, sticking, etc.). Students set to work putting together the apparatus, making measurements, calculating values, and eventually, plotting these values on a set of axes.

In the other class, the teacher gave students a worksheet on which to record the results of three specific collisions (bouncing with both carts moving toward each other, bouncing with one cart still, and sticking with one cart still). After all students had completed the required calculations for each collision, he asked students to tell him the results of their collisions and wrote each group's result on the board. He then told students:, "Something needs to be true about those initial and final momentums. If you didn't get this, there will be a lot of things wrong when I correct this lab... This is the goal for this unit." The teacher then wrote "pi = pf" on the board, and explained the details of conservation of momentum to his students while they sat in their seats.
 
Both of these classes served as the introduction to the same Modeling-centered unit on conservation of momentum, yet it seems to me that only one of them held true to the priorities of inquiry-based instruction. It's relevant that the first class was taught by a teacher whose Modeling Instruction training came early in his physics teaching career, while the second was taught by a seasoned physics teacher with many years of experience teaching AP. I learned in talking to this second teacher that the science faculty were a major force behind the decision to choose Modeling Instruction as a curriculum for their Physics First classes, but (on the basis of this one brief observation) the Modeling training they both received impacted the younger teacher more deeply than the older teacher.

At the end of my visit, I spoke with another teacher at the school who is teaching a "Pre-Chem" course for lower-track students, consisting of three units of Modeling chemistry combined with a less-quantitative introduction to five units of Modeling physics. (This teacher is also a Modeling Instruction summer workshop leader.) We talked about the complexities of transforming one's own teaching style, and the role that various teachers have played in the broader transition to Modeling at this school. It's clear that a successful transition to Physics First requires strong administrative support, but administrative support is no help at all if the teachers themselves are not committed to the classes they are teaching. Perhaps the successes of the program at this school are due in part to the flexibility of everyone involved to adapt to a wide range of teaching styles and expectations through this transitional period.

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Creativity and Introductory Physics

Open-ended, "creative" assignments can be an effective way of making science relevant to students, but there are more direct ways to exercise the creativity needed to excel in the fields of science and engineering.

"Creativity and Introductory Physics" is the title of an essay in the January, 2012 issue of The Physics Teacher on the importance of giving physics students opportunities for creative, divergent thinking (page 42 if you've got it handy).  In the essay, the author, a professor at Union University in Jackson, TN, illustrates how he tries to incorporate creative assignments into his courses, in topics ranging from magnetism to wave-particle duality. The teacher offers an example of a creative assignment where students are given an option to "create an original story that is critically dependent upon the concepts of Einstein's special theory of relativity," among other options.

For some students, assignments like these can bring seemingly irrelevant and abstract ideas into focus. In a general interest course about the weird world of modern physics offered at my alma mater, the final project for the course was quite similar in nature to this one, and I knew many humanities folks who expressed relief that they could receive "quantitative proficiency" credit for writing a poem about the impossibility of faster-than-light travel! But such assignments can sometimes bear only superficial relevance to the physics being studied, and I believe there are applications for creative thinking that more directly exercise the creativity that is essential for being a good scientist. One example is the student-designed paradigm labs that form the introduction to most units in a Modeling Instruction course. Another is the requirement of "Design Criterion" labs in the IB Physics Internal Assessment, where students must develop an investigation into the relationship between two variables of their choosing. (One student of mine investigated the relationship between the mass of iron filings mixed into a consistent sample of play-doh and the resistance of the dough. Another looked at the relationship between the time a spaghetti strand was left soaking in cold water and the applied force from a spring scale required to break that strand... One written example of such a project can be found here.)

In this short excerpt from a talk given by Sir Ken Robinson on divergent thinking and creativity, we learn that an individual score on some quantitative evaluation of divergent thinking generally declines drastically from age 3 to age 25:

I'd love it if Robinson gave us a few more details about this test (he refers to a book called "Breakpoint and Beyond" - I'll let you know if I end up picking it up), but this little result tells us exactly what we should be fighting in our physics classes - as our students get told how things work, they learn to see fewer possibilities in the world around them, to look outside themselves for ideas. I can't think of a better argument for teaching physics in ninth grade! Ninth graders are enthusiastic and creative thinkers, potentially unburdened (in some environments) of the pressures of getting the answer right as quickly as possible so that they can move on to the next question. What better time than early high school to reinforce to our students that science is something that is done with your own two hands?  What better tools to accomplish this than tangible instruments like the meter stick, the stopwatch, and the spring scale? (and the force plate, and the motion detector, and the video camera...)

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