I want to use this space to think about a conversation I had with Mike Shatz, a professor at Georgia Tech, with whom I’m collaborating to bring computational modeling to my 9th grade physics classes.

Mike and one of his graduate students led a 2 week workshop for the Atlanta Public Schools this summer, and before that, they wanted to talk to me about my approach to teaching. I described the modeling process I follow in considerable detail, and shared with them some of the modeling materials for the first few modeling units. They thought that this approach might be a good way to structure part of their workshop. Important note: Mike and his student were not teaching an official modeling workshop. In fact, this was a workshop for physical science teachers, and they were using their interpretation of the modeling methods to give these teachers some understanding a more student centered instructional approach like modeling.

Mike and his grad student, Daniel Borrero worked through the following units, in this order” Constant Velocity, Constant Acceleration, Balanced Forces, Unbalanced forces. After this, Mike had a pretty interesting argument for me, and a few other observations.

Mike told me that he thinks Modeling teachs force decomposition too early. As part of the balanced forces unit, students are interoduced to how to add individual forces to compute the net force. Shortly thereafter, in the unbalanced forces model, students learn to solve problems like a block sliding down an inclined plane, and one method for solving these problems is to tilt your axes and so that the x-axis is parallel to the surface of the ramp, and then decompose the gravitational force into two components, one perpendicular to the ramp (which is counteracted by the normal force) and one parallel to the ramp (which causes the block to accelerate down the ramp).

Doing this decomposition is both conceptually and computationally challenging for students, especially for those with weak mathematical backgrounds, and Mike’s critique is all this takes away from the big idea that you’re trying to teach—the object accelerates in the direction of the net force, and students are getting all bogged down with components and trig functions, and might very well lose sight of this.

I think Mike’s alternative would be to spend all your focus on understanding that it is the net force that dictates how the motion of an object changes, doing a lot of lab work and computational modeling using VPython to understand this better, and not getting bogged down with trying to describe, decompose or calculate the individual forces that make up the net force until you’ve mastered this idea.

There are parts of this argument that I find very compelling—and in an upcoming post I’ll share a activity the Mike and Daniel did with their workshop, but with only a couple of weeks before the start of school, I don’t think I’m going to be able to fully revise my curriculum to take these ideas into account.

So what do you think? Would mechanics be easier to understand if we pushed off force decomposition and just focused on the net force?

17 Comments leave one →
1. August 1, 2011 10:15 am

I have stopped doing problems that require forces vector decomposition. No pushing objects into the ground at an angle, no objects on inclines. Not worth it for the majority of the population.

However, Matt Greenwolfe had his students do those types of problems with scaled vector diagrams, head to trail addition. No components. An interesting approach I’m too scared to try just yet.

• August 1, 2011 10:42 am

Frank, try the vector addition diagrams! They rock, and you get at all sorts of conceptual goodness when you draw them (like which force is going to be bigger when you have two forces at angles, etc… they get a much more gut feeling for those).

2. August 1, 2011 10:41 am

In Honors, we didn’t do vector components until UBFPM (two units after BFPM) last year. So in BFPM, we did those problems with force vector addition diagrams. I liked the way it worked last year. It felt like we had enough time with forces before breaking them into components.

In regular physics (aka Physics!), we’ve been doing only problems where the forces were unbalanced either horizontally or vertically, but never both. We never need to tilt our axes, though we do deal with components if there are forces at angles. By the end of the year, some of the stronger students in that class do start to think about a more general case, and I mention the idea of tilting axes to them, though we still don’t do problems like that. By that point, they’d rather use energy for it anyway.

• August 1, 2011 11:45 am

Since I try to follow all of your awesome ideas, this is what I do too, and I agree, VADs are incredibly powerful. I’m in total agreement that students shouldn’t use components when adding vectors until they beg for them. And I think part of Mike’s critique comes from just looking over the modeling mateirals and not having an in depth discussion about how these ideas are taught in practice.

Still, I see some value for the type of work he’s talking about doing—seeing an object move, and asking students to figure out the net force on that object based only on how the motion of the object is changing.

3. Brian Lamore permalink
August 1, 2011 11:12 am

Consider Hewitt: he deals with this topic strictly graphically. My 9th grade students (should) already know the parallelogram method of adding vectors. For your box on an incline they could combine the weight and normal force vectors and get the resultant directed down the ramp, which is the same as the component of weight directed parallel to the ramp. Same thing but no trig. The limitation is you can’t get at a quantitative force, but I think this is the point of your post: do they need it? By “too early” do you mean high school?

• August 1, 2011 11:48 am

Yep, see my above post. Graphical vector addition is the way to go in my opinion, and it should be sustained for as long as possible. I don’t particularly love the parallelogram method, simply becuase I think it’s a bit opaque for seeing how you are really adding vectors by drawing a parallelogram. Instead, we talk about the head to tail method, and constantly reinforce the idea by thinking of adding short displacements (5m east + 2m north). I also find that students have a lot of trouble with he word resultant. They tent to think that the final product of any vector operation is a resultant, and this is a big problem when you get to vector subtraction.

4. August 1, 2011 11:25 am

I leave off the inclined plane for my general physics class. Most years, the kids just get too bogged down with the math and do loose the big picture. They end up remembering procedures and steps rather than really understanding what is going on. I do have them work through the forces tutorial from Tutorials in Introductory Physics by Lillian McDermont et. al. The in class portion is wonderful, and they are all capable of doing it in groups with me helping through Socratic dialogue. The accompanying HW portion is very long, and does include a conceptual incline portion which I usually omit for that class. Most of the kids usually have a pretty good conceptual idea after they do this. So like Kelly, we do forces at an angle either pushing down or pulling up, and only have unbalanced forces in either the vertical or horizontal direction, but never both. These are juniors doing this, so I’m not sure how this would translate to freshmen. I always have a large percentage of the class (about 25% or more) who never really get it conceptually, and depend on procedures and (shudder) equations to get to the final answer. I am hoping to adress this better this year. (Keeping fingers crossed.)
How long have you been doing physics first? Do you like it?

• August 1, 2011 11:53 am

McDermott’s tutorials are great. I can remember taking a class with her at an AAPT workshop and our entire group couldn’t figure out the contact force pairs in the example of the magnet holding up the washer—such a great question. My freshmen can handle much of the material in Tutorials, but it helps a lot if I break it up into more bite sized chunks for them.

This is the start of my 3rd year teaching freshmen physics, and I love it. My school has been doing it this way for almost forever (since my wife graduated from there in 94), so it’s part of institutional memory. Our strongest students jump from honors physics to AP Chem as sophomores (taken with no previous exposure), and do the same thing again in AP bio as juniors. The one thing I don’t think it has really led to like I thought it would is explicitly pulling particular scientific themes like energy through the curriculum, and this is partially due to the pace and constraints of the sophomore and junior level courses.

5. August 1, 2011 5:36 pm

If you do projection with dot products instead of trig, the math is a lot more accessible. Of course, that means specifying angles with rise and run, rather than degrees, but lengths are easier to measure than angles anyway. I’d much rather have a ramp that rises 3cm over a run of 20cm, than one that is 8.5 degrees.

Still, coordinate transformation is hard for a lot of students, so delaying it until you really need it is probably a good idea.

As for the term “resultant vector”—I had to look up what it meant, and I was a math major. Why use such a complicated word for the sum of two vectors? Physics has enough vocabulary that there is no need to add more terms just to puff up the vocab list. Students know what addition is.

• August 1, 2011 11:56 pm

I’m with you on resultant vector—I use sum, and when describing forces, net force. It’s language used by my math department. I think dot products are probably a bit beyond my 9th graders.

6. Matt Greenwolfe permalink
August 2, 2011 8:59 am

I started out teaching head-to-tail vector addition when I had our non-honors physics sections, figuring that at some point problems would reach a level of complexity that would call for an axis system and components, at which point I would introduce them. My AP students, in their second year with me, convinced me otherwise. We came to one of those symbols-only banked turn problems with friction, and I tried to tell them we had to use components now. They were so comfortable with graphical vector addition – and with thinking for themselves due to modeling – that they wouldn’t have any of it. They solved it with head-to-tail vector addition diagrams in two different ways, and the solution this way was quite elegant. They neatly avoided the typical traps that students fall into in this problem with very little guidance needed by me.

I’ll second Kelly. You get so much out of the tail-to-head vector addition diagrams. They have an explicit representation of net force and it’s direction. They are helpful even when there aren’t forces at angles, for example in a vertical motion with an upward or downward force in addition to gravity. You can see right from the diagram which forces should add or subtract. You no longer have to carry around negative signs like an accountant. You put them in because they conceptually make sense. There are so many fewer steps and ideas to convey to the students when teaching vector addition. They do have to think about the direction of each force, the direction of the net force, where the angles are in the diagram – the things with physical content that you would want them to think about.

You can answer all sorts of conceptual questions with graphical vector addition diagrams – quickly and easily – when they would take much work sorting back through several math steps with components. For example, what if you change the angle or amount of one of the forces, how do the other forces, net force, acceleration change as a result? Just manipulate the diagram and observe.

When we move on to fields in AP, the graphical vector addition diagrams are much more useful.

There are some types of conceptual questions that are better addressed with the idea of components. So I do mention them briefly at appropriate times. But all of the problem-solving – conceptual, quantitative or symbols only is generally done with the tail-to-head vector addition diagrams.

Rather than saying that we move on to components too fast, I doubt we really need components at all. I think the negative side of teaching components is that it boils things down into a rote procedure. The vector addition diagram continues to support conceptual thinking.

David Hestenes’ Geometric Algebra is a synthetic treatment of vectors that allows for algebraic manipulation of them as whole objects, with no need to break them into components. He has often spoken of the “coordinate virus” that infects our math and science education. I think he means by this, the reduction to a rote math procedure divorced from the geometrical visualization. When you have the geometric visualization, you can think creatively rather than follow a rote procedure.

7. August 2, 2011 11:47 am

Was at a workshop a few years ago, and the leader asked, “Why do we need to teach those problems (incline plane, etc)? Do they need it to understand physics, or is it to prepare engineers for their coursework? If it’s just for engineers, let them teach it and we’ll stick to physics!”

• August 2, 2011 11:38 pm

Right on. I really don’t see much of a need for the type of problem I drew in this post. I was just using to make a point for Mike’s argument.

8. August 2, 2011 4:45 pm

I ran across the work of David Tall at Warwick University in Britain a few years ago (and now I can’t for the life of me find the precise reference) that suggested student understanding of vectors diminished when components were introduced. He advocated holding off on components as long as possible. As I have followed this advice increasingly over the years, I’ve found the longer we hold off on components, the more students continue to solve problems graphically, even after learning components, and these are often the strongest students. Now I truly see the problem with “componentitis” or “coordinate virus.”

Before we started holding off on components, I had one student who backtracked on his own and started doing everything he could without components (perhaps because he was John Burk’s advisee?). During a group oral exam in a class in which this kid was the youngest student (and not yet in multivariable calculus, unlike the others), I tossed him a hard angular momentum problem. One of his buddies said “Whoa, T isn’t in multi… take it easy on him, Mr. Hammond.” In response the student cited a geometric interpretation of the cross product, invoked a little symmetry and announced the answer, to the astonishment of his older classmates. “We’d still be writing components!” whispered one on them.

You have to explain this to others, though. Some (parents, other physics teachers) seem to look down on a graphical solution, as if we are “dumbing down” the subject. I think the exact opposite is the case, however. The students are keeping their brains in the game when the actually use the vectors instead of reducing the problem to an algorithm.

9. August 4, 2011 8:11 am

It is hard to reconcile the “hold off on components” viewpoint with “computational modeling”, as the most common representation of a vector in the computer is as components. Of course, with a language like Python, you can have vectors as a type, and just add them, without having to remember each time what adding vectors means in detail.