101qs in Physics Class

Ugh, what a grey, rainy day we've had in New York City today...

I spent a lot of time indoors at my computer, and stumbled across Dan Meyer's 101qs site, which I hadn't seen before. The idea behind the site is for teachers to upload pictures and videos in the style of Dan's Act One prompts, where other teachers give feedback on what questions might come out of the picture. In a classroom, it would be the students suggesting questions that might be answered using the data. Meyer uses the photos and videos to introduce elements of drama and storytelling to make problem-solving relevant to students who might otherwise feel an aversion to it.

One video in particular stood out to me, maybe because it's more "physicsy" than many others I saw:

There's lots of information in this video, and it brought up all kinds of interesting possibilities for questions that could be answered: Is the acceleration of the train constant? If so, what is it? Does the train reach a constant speed before it leaves the station? If so, what is this speed? If not, how long might it take for the train to reach its top speed? What is the instantaneous speed of the train exactly 10 seconds into the video?

Of course, to solve these problems you need information that's not available in the video, but most of what you might need is freely available on the web. For example, the length of a SF Bay Area BART car is about 70 feet, and the train can reach a top speed of about 80mph. (In some similar videos, all the information needed to make estimates is included in the video itself... This video might work that way if an adult of "average" height was simply standing in the foreground.)

I've been thinking a lot lately about video-based data collection, especially as a potential solution for students missing essential lab days in a Modeling-based class. What I find exciting about Dan's approach, though, is the power of an open-ended question. Not only do students get practice applying physics and math, but they also get practice using creativity to exploit the ubiquity of useful data in the world around them. High school science has the power to change how students think - everything they need to continue to answer these questions is around them all the time, as long as they stay curious.

(btw, the answers I got were: yes: ~1.2m/s/s, no: ~30s, ~10m/s)

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Sharing Modeling Resources

A web-based hub for uploading and distributing Modeling curriculum materials is on the horizon. The potential significance of such a resource is huge, and a healthy conversation at this early stage can help ensure that it becomes both dynamic and user-friendly. 

In preparing materials for my own Modeling-based Physics First course, I've been looking through materials posted on a password protected portion of ASU's Modeling site. In particular, I've been spending a lot of time with two remarkable collections of materials developed and prepared by small groups of teachers in St. Louis (Debbie and Rex Rice, and Gabe de la Paz) and Pittsburgh (Shady Side Academy faculty, including Kathy Malone - recent recipient of the Albert Einstein Distinguished Educator Fellowship). Both collections have firm roots in the Modeling materials originally developed at ASU. However, in addition to revising these canonical materials, both collections also offer entirely original activities and even some major restructuring of the order of the conventional Modeling curriculum. In spending more time with these collections, I've been getting more experience with what I need in order to make efficient use of work done by another teacher, and it's been quite illuminating.

There are quite a few similarities between these two sets of materials. Both collections provide a large text file containing "Teacher Notes" for each unit in the curriculum. (The materials for one such unit are shown in the picture to the right.) These notes generally begin with suggestions on the Scope and Sequence and Instructional Goals of the unit as a whole, then go on to provide details about implementing the activities or worksheets contained within the unit package. Both collections include handouts or worksheets composed in Microsoft Word, and usually include one document file for each physical handout. (For example, a handout made to accompany a lab activity and a homework assignment directly related to that activity are usually included as separate files, sometimes in separate folders/directories.) The documents in both collections are formatted quite precisely, and I experienced frustrating formatting errors when opening some of these documents on my own computer. Most importantly, both collections reflect the great passion and expertise of the teachers who created them, as well as an enormous investment of time.

One key factor in the success of Modeling Instruction, I believe, is that it has been almost entirely teacher-driven. Workshops are run by teachers and interest in the method has spread largely through word-of-mouth. Curriculum resources are developed and shared by teachers as well, though this practice is somewhat cumbersome at the moment. However, like the ASU Modeling site, the online home of the American Modeling Teachers Association (AMTA) is already hosting sets of materials developed by high school teachers, such as the extensive work of Dr. Matt Greenwolfe. This site is going to be updated any day now to include a Modeling Curriculum Repository, in what will hopefully be a big step toward an online hub for Modeling-related resource sharing for teachers by teachers. Like any big crowd-sourcing project, there are some logistical hurdles, but I'll share here some of my thoughts and reactions to how this might be accomplished. 

• Efficient subdivisions of documents can help clarify which materials are relevant to a given topic or activity. Both of the collections I've mentioned were made to be downloaded as a package. As a result, the "Teacher Notes" pertinent to each package are rather lengthy documents. The notes that are specifically related to one worksheet or activity can occasionally be difficult to locate, and isolating one activity from the unit as a whole can be slightly tricky. This makes sense given the overarching goal of Modeling Instruction - Modeling is a unified approach, not a collection of activities to be blended piecemeal into a traditional curriculum. But as more and more teachers convert to Modeling, I believe the need to isolate individual activities will become greater. I think there's a lot to be gained in breaking down these large unit-based collections of files into smaller chunks. Individual files in the unit can contain a few closely related handouts (a couple of related worksheets with teacher notes, for example, or perhaps a lab handout and with both pre-lab and post-lab supplements). I've posted an example of collection of resources I've prepared for an investigation into friction here. 

• Maintaining a hierarchy of organization on the web (rather than within a single .zip file for a unit) can provide both flexibility and manageability to the resource collection. AMTA President Mark Schober has told me that, in his opinion, the ideal resource-sharing mechanism would include access to both individual materials and recommended collections of materials comprising an entire unit. These collections would be curated by an individual or group for use with specific student populations, like playlists of songs curated by notable taste-makers or music communities. The choice of what to include in these playlists evolves over time, and the files contained within the playlists evolve in parallel. The inertia of large collections of curriculum materials can make them slow to change according to teachers' evolving needs and wisdom, and a system based on individual files is bound to be more nimble. 

• If teachers have access to multiple versions of similar materials, user ratings and recommendations can identify versions that are unique, valuable, and appropriate for use with different student populations. This is a model already in place in many communities, including our own community of science educators. For example, exemplary resources designed for use with PhET applets can be nominated for a Gold Star, which identifies "high quality inquiry-based activities that follow the PhET design guidelines and that teachers find useful." The challenge of such a system is to find a balance between maintaining democracy in contributions from individual teachers and keeping the collection of recommended materials concise and manageable for teachers using the site. 

• SBCD - Standards-Based Curriculum Development. Fellow physics blogger Kelly O'Shea just published a fantastic post about bundling established stablished standards for her course to facilitate communication with students about what will be covered on tests. This got me thinking that curriculum materials hosted on the AMTA site might be organized according to standards as well. That is, a worksheet on motion maps could be tagged (in a database and on the document itself) as relevant to one or more standards, such as O'Shea's CVPM1 - I can draw and interpret diagrams to represent the motion of an object moving with a constant velocity. Perhaps the entire Modeling curriculum could be broken down into agreed-upon standards, varying from "I can design an investigation into the relationship between acceleration and mass for a constant force" to "I can solve problems involving the separation of two slits and the distance between bright fringes in an interference pattern from a laser." Specific standards could likely be consistent between courses of widely varying ages and student populations - a more sophisticated course would simply include a wider variety of standards and different supporting materials. For example, the CVPM1 standard above would be a part of a Physics First course and a university course, but the materials used to support this standard might look quite different for such different courses. 

• To avoid formatting errors, documents can be distributed in robust, universally readable file formats. In producing my own curriculum materials, I've taken to saving a version of everything in PDF format, to ensure that I'll always be able to open a specific version to print it out for use in class. Both Greenwolfe and O'Shea have chosen to distribute their materials as PDFs as well. A disadvantage of this method, of course, is that the PDF format cannot be edited, but a text file or word processing document can be included as well to make updates more flexible. Alternatively, a universally available word processing application like GoogleDocs could be used to avoid such formatting inconsistencies. Andrew Stillman, an administrator of the online professional development site YouPD, has advocated such an approach.

• Creative Commons is key. Most Modeling documents I've come across have included at the bottom of the page a copyright tag like "© Modeling Workshop Project 2006," indicating that the work in the document stems directly from the original work done at ASU. Sometimes individual authors are identified and sometimes they aren't but this tag ensures that the owner of the intellectual property contained within the document is crystal clear. AMTA Executive Officer Dr. Colleen Megowen has told me that it is a priority for the AMTA to prevent the materials from being used for commercial purposes at any time in the future, but according to this page the copyright isn't explicitly protected against commercial use. It seems to me that without a watertight license, the entity that officially owns a given piece of work (whether it's the AMTA or an individual teacher who has designed their own materials) could at some point choose to restrict access to it, or aim to make a profit off its distribution. There's nothing wrong with teachers making money off of work they've done, but the alternative of an open source Modeling Curriculum Repository seems even more attractive. By attending a workshop, a teacher would be introduced to a wealth of free resources and a community of like-minded peers, both of which would aid them in their transition to Modeling Instruction. In order to protect this dream, a Creative Commons license can be used by any original author of material to prevent the work from being used commercially. Looking through descriptions of the various CC licenses paints some striking pictures of what the Modeling curriculum at large might look like!


Now that I've stepped off my soapbox, let me clarify that this is not intended to be a set of recommendations for how to structure the AMTA site or the files contained within it. Rather, this is just an early collection of personal thoughts on a topic that I find quite exciting. As the work of assembling both the site and the materials goes forward, I hope that this conversation evolves. What are YOUR thoughts?

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Intervention in Modeling

Concept-related intervention by teachers to correct or redirect student thinking can interfere with processes of peer-instruction and inquiry, but without intervention into the complex social dynamics of a high school classroom, the trust and courage required for these processes to be effective can be slow to develop.

As I've visited various ninth grade physics classes, I'm often faced with a question that teachers who employ inquiry-based instruction face every day: When to intervene in student thought-processes that are headed down the wrong track? For an outside observer like me, a policy of little to no intervention is almost always best, as it's crucial to the observation process to tread very lightly on the environment a teacher has created. But for the teacher who has committed to an inquiry approach, this question gets wrapped up in all sorts of conflicting impulses. Just how helpful is concept-related teacher intervention during, say, the small group discussion phase of a whiteboarding activity?

Anecdotally, my observations have suggested that the short answer is, "not very." In situations when students will be presenting group work to the entire class, pointed Socratic questioning seems most efficiently used when the entire class can benefit from witnessing and participating in another group's thought process. Rerouting this group's thinking prematurely denies every other student in the room the opportunity to think about why that particular line of reasoning doesn't hold up. Teachers might limit a group-by-group Q&A to "one question per group," but in practice this gives students an excuse to sit around doodling cartoons on their whiteboards while they wait for that one question to be answered. I've talked with teachers who like to plant correct ideas throughout the room in the group phase of a whiteboarding process in the hopes that this understanding will grow throughout the class as the whiteboards are presented. However, this takes for granted that such "idea planting" is effective in the first place. Surely these conceptual seeds can be more effectively sowed through a short hands-on activity or a more targeted "auxiliary" whiteboarding problem than by teacher-driven explanations.

It's essential, however, to draw a distinction between concept-related intervention and social intervention into the dynamic between students that can make peer-instruction succeed or fail. In one class I observed, a teacher intervened to delegate responsibility when two members of a group didn't seem to be contributing to a lab activity: "Why don't you help "M" work on the algebraic representation and you help "E" with the motion map?" These students made an attempt to obey these instructions, but "M" and "E" clearly didn't want any help from them, and they eventually gave up and resumed their previous unproductive behavior. I got the impression that the students were used to having their contributions shot down, probably in quite a few more environments than this one physics class. It's unrealistic to expect ninth graders to navigate the sometimes vicious hierarchies of academic or social capability on their own, yet we often ask them to do so. An inquiry-based physics class can provide a more level playing field for these types of interactions than a locker room, but in order to generate trust and courage in students, a teacher has to act as a constantly vigilant referee.

Colleen Megowan's PhD dissertation out of ASU describes four paradigms of the roles teacher play in four modeling-based courses she observed: teacher as scout leader, teacher as stern but kindly parent, teacher as coach, and teacher as general contractor. Here is an excerpt from her description of a ninth grade physics class (illustrating the stern but kindly parent paradigm):

[Students] appeared to feel comfortable saying what they thought to each other and to the teacher, even to the extent of challenging the teacher’s assertions (about physics) if it conflicted with their own commonsense concepts. There was no evidence that they were afraid of ‘looking stupid’ to one another or to the teacher. They behaved as though knowledge resided in their peers as well as their teacher... However, there was very little effort invested by students who took the lead in whiteboard preparation in making sure that their disengaged group-mates could make sense of the whiteboarded information. The teacher often put these disengaged students on the spot by directing questions to them in the whole-group discussion, and when this happened, their more engaged groupmates often rescued them with whispered cues and gestures.       (Megowan, 82-84)*

The classroom environment described here is a direct product of the teacher's "stern but kindly" interventions that have directed class discussions, whiteboarding, and hands-on work since the first day of school. As the latter half of the citation reveals, there are certainly aspects of the peer-instruction process that might still be improved upon, and the teacher's behavior suggests a very gradual, deliberate intervention intended to do exactly this.

Most of all, it is clear that the students in this class are operating within an environment of mutual trust. Over a few months in this classroom, they have gained the courage to examine their own thinking, and to learn from mistakes they and other students have made. It's the challenge of each individual teacher to determine when their interventions enrich this process for students and when they detract from it, but resources for teachers (in the form of Modeling Instruction workshops, or support material for an activity or worksheet) can provide some assistance in meeting this challenge.


*Megowan's dissertation is a fascinating read! It's available from the ASU "Resources" site linked here, near the bottom of the "Doctoral Dissertations and Masters Degree Theses" section.

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Khan Academy II: Discussions and "Khanversations"

"Khan Academy" style instructional YouTube videos could be more effective for introductory physics if they used a discussion model rather than a lecture model.

I had a fine time last week at the WNET Channel 13 Celebration of Teaching and Learning (which consisted of about 30% substance, 20% patting teaching on the back for doing "such an amazing job," and 50% advertising), and I wanted to follow up on the post I wrote about Khan Academy.

Over the course of the day, I saw Sal Khan (the Silicon Valley superstar shown in the camera-phone screens to the left) give his standard talk, and then follow it up with an hour-long question and answer session. In general, I came away convinced that Khan's heart is in the right place, and that Khan Academy strives to be far more than a YouTube channel. The goal of Khan Academy, he said numerous times, is to off-load a number of tasks traditionally done by teachers in order to free up the teacher's time to do more valuable things. During the Q&A, I got a chance to ask Khan essentially the questions that I posed in the last post: What is the role of an explanatory video when we know that clear and concise explanations can be counterproductive to student learning? His answer was basically that students should have access to whatever resources that might be helpful to them, and they're taking seriously their responsibility to measure the effectiveness of the videos to identify which ones aren't working. Here's a quote from his response:

When I think about my own learning, there are some times when I learned something through the experiential, where finally when I had to write a program when I was doing some computer graphics, trigonometry finally kicked in... But for some things, you know, especially when I was doing higher level math, it really sometimes was a friend in a coffee shop giving me a clear and concise explanation. And I was just like, "Wow, that really hit the spot. That was really much better than what was in the book, and that got me through my stumbling block."

I agree with what Khan is saying here, but this response reveals a slightly simplistic view of how learning works. I can't deny that clear and concise explanations from friends or teachers have gotten me through some tricky spots as well. However, I'd also suggest that hearing those explanations in clear and concise terms sometimes didn't actually help me as much as other approaches might have. Precisely because I was hand-fed exactly what I needed to fill in the gaps in my understanding at that moment, a few days or weeks later, those gaps sometimes returned.

When I think about what Khan Academy videos might look like if they were truly out to correct student misconceptions about, say Newton's Third Law, I imagine something more like the "dispute between students" prompts you find in Lillian McDermott's Physics By Inquiry books (see my previous post on this topic). In the Khan Academy model, picture a "Khanversation" between two voices, in which both individuals make arguments supported by diagrams to support a claim their view is consistent with observations in the natural world. This approach would provide opportunities to bring common misconceptions out into the open and model effective argumentation for students as they practice these concepts and skills in their classroom.

In a 2010 review paper in Science, Stanford School of Ed Professor Jonathan Osborne calls attention to a great irony in many science classes - traditional science teaching fails to develop the skills of argumentation and debate that are at the heart of the way science actually operates. Not only do student-centered teaching methods help to develop these essential skills, they also facilitate learning of science concepts far more effectively. Osborne writes: "Learning is often the product of the difference between the intuitive or old models we hold and new ideas we encounter. Through a cognitive process of comparison and contrast, supported by dialogue, the individual then develops new understanding. Consequently, learning requires opportunities for students to advance claims, to justify the ideas they hold, and to be challenged." We should be teaching our students first and foremost how to navigate their way through this process, as this is a skill that will be far more relevant to them than any science concept. (excepting, of course, Newton's Third Law...)

One of the most productive aspects of whiteboarding is that students are expected to formulate a verbal argument to support their answer, and present this argument to the teacher and their peers. Not only does this give a teacher instant access to their students' reasoning, but the students themselves are constantly exposed to effective and ineffective arguments. What role might other methods play in this process? I have tried to use handouts to structure and spur dialogue between students, but I've never gone so far as to upload such a dialogue to YouTube. At first glance, however, this possibility seems intriguing.

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What's to Learn from Khan Academy?

The video lectures on Khan Academy don't address the complexities of how people actually learn. What might these videos look like if they did?

I was lucky enough to secure a free ticket to the Channel 13 Celebration of Teaching and Learning this Friday in Manhattan. Sal Khan is giving a talk about Khan Academy, the series of YouTube tutorials that have been touted as a revolution in education. Here's an example of Sal Khan layin' down some knowledge about Newton's Third Law:

There's a healthy discussion in the physics teaching blogosphere about why these videos aren't the revolution to education that 60 Minutes might lead you to believe. Physics teacher Frank Noschese makes a very strong argument on his blog in this post and others (there is also a nice set of links to other blogs at the bottom of this page).

Khan Academy lectures seem to me to be a new type of textbook for a sort of curriculum that has been around for ages. The problem is, we've seen that this curriculum just isn't effective. The idea that YouTube lectures can be useful to students isn't flawed in itself, but video resources for more effective pedagogical approaches just aren't posted on Khan Academy. Rather than bashing Khan, let's think about what types of videos might be used as part of more effective curriculum, like Modeling Instruction.

Modeling isn't about lecturing, of course. It doesn't matter whether the lectures take place in a classroom or on YouTube, lecturing just doesn't work. So, what video resources would be effective in a Modeling course? Much of the most valuable student experiences in a Modeling course can't be replaced by videos - hands on lab work, interpreting unique data, discussions with other students, presenting a whiteboarded solution to the class. Somewhere in the midst of all this I imagine there's room for, say, example problems worked out using language and representations specific to a Modeling course, but how would you prevent such concise explanations from interfering with a student's natural struggle to build their own understanding? Perhaps, as Derek Muller suggests in this video, students might benefit from watching a conversation between students as they gradually work toward a correct understanding of a concept or problem.

For me, the takeaway from Khan Academy is simply how easy it is for individuals to make simple instructional videos that are available to a very wide audience. There's still a ways to go in thinking about how such videos might supplement progressive pedagogy, but the method is there for the taking.

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Whiteboard Everything

 Whiteboards should be central to any inquiry-based approach. Whether students are asked to present their solution to a problem, their interpretation of lab results, or anything else that requires them to think independently, whiteboards are an ideal tool for this process.

The more I observe, it seems, the more opportunities I see for effective applications of whiteboarding. Whiteboarding is simply a highly effective way of getting students' thoughts out of their heads and into the classroom, where they can be critiqued and discussed. There are other ways of doing this, of course (simply asking students to raise their hands is one such way, "clicker" response systems are another, higher-tech approach), but whiteboards have unique flexibility and versatility. When a group of students works together to prepare a whiteboard for presentation, the peer discussion that goes into this activity is the first step toward correcting individual students' misconceptions. Plus, whiteboards are extremely economical!

On a recent visit to a school teaching Physics First, students in a rather large class were asked to present their results from a lab experiment. Each student was given one or two poster-sized pieces of sticky paper, and wrote out a summary of each section of a conventional lab report (hypothesis, procedure, results, analysis, etc.), which they then stuck to the wall for the class to see. This was a challenging task, and some groups took much more time to complete it than other groups. When everyone was done, the teacher then asked each group to present their posters, in turn. The room was rather restless, and during each group's five-minute presentation students in other groups found it hard to sit still. To save time, the teacher asked latter groups to skip the parts of their report that were essentially similar to things other groups had already spoken about. When the period ended, most of the groups in the room hadn't had a chance to present their posters at all.

Imagine the same activity done with whiteboards. The whiteboard is too small to record all the information in every section of the lab report with a big, bold dry erase pen, so the activity would have to be broken down into pieces. For each section of the report, a few groups would present what they'd recorded on their whiteboards and other groups would look on. Each of these presentations would be less than a minute long, and even the most restless students would find it easier to pay attention to their peers for this short time. The teacher would have the opportunity to focus closely on the aspects of the activity that differed most from group to group (in this case, the data and analysis), and could spend more time discussing with students the significance of these differences. Since each group's presentations were only a minute long, every group in the room could be assured an opportunity to present at least once, and the threat of being called on to present again would encourage all students in the room to stay alert.

On another visit, I witnessed a very successful application of whiteboarding that faltered a bit when many students in the room had made a similar mistake in the free body diagrams they'd recorded (the whiteboard pictured here doesn't show this mistake...). What followed was a lengthy, lecture-style instruction about how to correct this mistake in the students' diagrams. In talking with the teacher of this class later in the day, we agreed that one might instead make up a new free body diagram problem on the spot designed to hone in on this common mistake, or perhaps even refer to a database of problems specifically designed for this purpose. Students would be asked to set their original whiteboards aside and solve the new problem for class discussion on a new whiteboard. Rather than lecturing about how to correct the common mistake, the correct solution would arise out of this group discussion.

My Modeling workshop leader, this past summer, mentioned that whiteboarding is so entrenched in his current teaching that he wouldn't know how to teach any other way. If he was asked to teach a European history class, he would teach it using whiteboarding. I'm starting to see what he means as I begin to appreciate the power of creative whiteboarding in physics class!

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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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