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.

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?

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.