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On the origin of origins and confusion between math and science

September 6, 2011

The more I teach physics, the more I see subtle differences between Math and Physics that cause great difficulty for my students. This one is on the nature and meaning of the word origin in the two disciplines, and it was also inspired by a post by Brian Frank, where he made many of the same discoveries as I did.

The problem came up when, inspired by Kelly and others, I decided to push my student to think more deeply about position, and so I defined an origin in the room with a long strip of masking tape. Along the way, much conversation ensued about what meaning of the origin was, and why it wasn’t a dot.

My students are used to the origin being (0,0) on a y vs x graph, which forms the bread and butter of what they see in Algebra. They can also sort of get the origin as some sort of hypothetical “starting point” but when you muddy the two, and present them both with the idea that the origin can be a line, and things don’t have to start at the origin, things get really confusing fast.

I saw this when asking a bunch of students this question” If an object has a positive velocity, is it necessarily moving away from the origin?” It took a while for them to discuss with one another and come to the idea that if you start behind the origin (negative position), and have a positive velocity, you are getting closer to the origin, and so positive velocities don’t mean getting closer to the origin. Asking this question now will pay huge dividends when we get to the idea of acceleration, and everyone becomes convinced that negative acceleration means slowing down.

Now here’s the tricky part. I asked my students to draw a graph of this situation, and one student in particular drew this graph.

Here this student is trying to draw an almost vertical line. And so I asked him to draw it with the object moving a bit more slowly, and he seemed to get stuck, and puzzled. He couldn’t draw it with a less steep slope. Another student jumped in and drew something like this:

But the first student seemed confused, and almost seemed to say “but this object is further from the origin.” And then it hit me that this student was seeing the origin as the (0,0) intersection of the position and time axis, and not the horizontal line where x = 0.

This was a big insight for me, and probably the first time I’ve fully realized this in 13 years of teaching. It tells me that my students are somewhat unwieldy with the incredible math tools they have. They can graph things, and they can label their axes ‘y’ and ‘x’, and they all know the silly way mathematicians have deiced to number quadrants on the graph, but when it comes to really understanding what a graph is trying to tell you, especially about something as real and tangible as a buggy moving with constant velocity, they struggle much more you might think, especially when you take the time to let them really work on a problem, and don’t simply push them along the rails toward the “right” way of doing things.

I’m not sure what the solution is to this. Sure, I think it would help if students thought of the axes as “vertical” and “horizontal”, rather than ‘y’ and ‘x’, if they weren’t locked into the “independent variables go onto the horizontal axis” religion from middle school science, and if they could think about what it means for a point to be in the 3rd or 4th quadrant, instead of just knowing it is so, but I also know that there’s an appropriate time and place to introduce these ideas, and maybe in middle school when students are first learning graphing, it’s better to give them something more concrete like ‘x’ and ‘y’ instead of horizontal and vertical (is this more concrete?), or a rule about independent variables that must always be obeyed.

Anyway, it’s good food for thought, and another reminder that if you give students enough time to think, they’ll really surprise you with what they’re thinking.

3 Comments leave one →
  1. September 6, 2011 7:48 am

    Nice post. At Maine, several researchers have been investigating math / physics issues at the upper-level, including classical mechanics and statistical mechanics. So these issues don’t just go away. When did this happen in your class, this year?…

    In my class, I’m hoping that taking the time to talk through all the possibilities for “where a reasonable person might thinking the x=0 is?” helped. But the course has a fast paced. We had one day to talk about average speed and average velocity, and now we are on to acceleration. We havent’ enough time to sort through positive / negative indications of direction with velocity, and now we’ll be on the positive/ negative indications with acceleration. We’ll see how it goes in a few hours I suppose.

    This also reminds me that I should write a post about Jeremy Roschelle’s work in kinematics, where he found that the idea that “acceleration pulls on velocity” is very generative for learning, and that drawing acceleration vectors at the end of velocity vectors (rather than both with tail at same spot) leads to more “acceleration pulls on velocity” metaphors.

    • September 6, 2011 8:28 am

      Brian,
      Yes, these ideas came up in our constant velocity discussions this year. I know the pace of college courses is much faster than high school courses, but I think the one day you’re doing velocity the next day your doing acceleration (if you’re lucky) is the very reason so many students struggle early on in physics.

      I love the idea of acceleration pulls on velocity, and I’ve made a mental note to myself to add this to our computational modeling exercise for CAPM when I get to writing it.

  2. September 6, 2011 4:56 pm

    A lot of this is a really sad commentary on how little meaning is attached to concepts in math classes.

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