I'm bringing the science discussion from the Favorite Student Emails thread to its own thread because I am interested in discussing what levels of approximation are appropriate under what circumstances.
To get us started, I will quote two of Astrofraa's posts and give my responses to clarify why I wander around teaching people definitions that I know are wrong. I will also give parenthetical comments about the phenomena themselves using the examples I use with my students to help the non-scientists among us follow the conversation and maybe learn a little something along the way.
Interesting. This is getting very off-topic, but I would be interested to know your thoughts on Arons's argument in "Teaching Introductory Physics" for introducing the operational definition of mass experientially by actually pushing on masses and seeing how momentum and acceleration change? It seems to me that would be simple, experience-based *and* have the benefit of actually being correct and not needing modifications later on should they learn about more complicated ideas (or read science fiction :-)). But I haven't taught intro physics for non-majors in a long time, so I fully admit I may be naive in these matters.
I try to get my students to avoid the word "amount", precisely because it's not clear if they mean mass, volume, surface area, or some other, even less obvious quantity, so I definitely feel your pain. Who was it that said we can't have both accuracy and clarity, because all our models are by definition approximate, and therefore inherently wrong at some level? So I don't disagree with your principles at all; I just question (sincerely, not rhetorically) whether "amount of stuff" is *too* wrong to be useful.
But, I add again, I haven't tried this with non-majors, so I can't assert from my own experience that it works. And yes, this is equivalent to conjugate's brick example, I believe. The important thing is that they *experience* it, and that the definition is *operational* (which to a physicist means the definition is a procedure that leads to a measurement). If the definition is operational (push on it and see how much the velocity changes), then when you get to the weirdness, the definition doesn't need to change; you just automatically see how there are counter-intuitive implications of the same definition. I still don't quite see how the operational definition is less simple than the "amount of stuff" definition.
The thing that you aren't grasping is how many confused ideas the beginning non-major student already has that I need to address.
The operational definition of mass is initially confusing to the students because force, inertia, momentum, kinetic energy, mass, weight, volume, density, acceleration, velocity, speed, and distance are basically interchangeable concepts to the majority of my students. Seriously, I have discussions and grade reading quizzes that repeatedly confuse these notions even
after my students have read the introductory chapter on motion. Consequently, I have to start untangling the notions they already have before we do a hand-ons demonstration. Otherwise, what the students "learn" from their experience is not at all what we wish them to learn because I will have students happily inform me that the reasons you lurch forward when your car abruptly stops are
- due to Newton's third law where the action is the car stopping and the reaction is you going forward
- the force of inertia moves your body in the opposite direction of the acceleration
To emphasize the point of students not learning from their experiences what we expect them to learn from the hands-on activities, you would not believe the difficulties that my students have with the first lab in which they are supposed to form a hypothesis about the effect of aging on the mass of a penny, use the balance to ascertain the mass of ten pennies each, combine that data with their groups for a total of forty data points, plot the results, decide whether their hypothesis was supported, and then draw a conclusion based on their graph. Nearly everyone can use a balance (although I had a notable exception just this week where the group had the idea of a balance worked, but somehow reported that all of their pennies after 1985 had the same mass of 2.4 g and all of the pennies before 1985 had the same mass of 3.0 g and were incensed that I marked their data collection as poor), but they are not clear on how to determine if a hypothesis is supported, how to read a graph (despite this being our second graphing exercise), and how to draw a conclusion from the data.
Consequently, as a first step, I have to make sure that we do something that is very much within the students' previous experiences to get them started on the right track instead of using their mixed-up ideas from non-technical uses of the terminology in daily life and skimming the text. Starting from the idea that mass is stuff, volume is the space the stuff takes up, and weight is a force applied to the stuff works quite well. After a lengthy discussion with solid examples of why equal mass is not the same as equal volume, but equal mass will always yield equal weight (by application of F=ma to reinforce that force, mass, and acceleration are separate things because of their relation and their differing units) and bringing in the idea of density as the ratio of mass to volume, then we are ready to tackle the idea of net force.
I always use the example of the car traveling along the road at constant speed and ask what forces are applied to the car. The students from their own experiences can tell me the forces are due to the car engine, friction, air resistance, and gravity. This is the point at which I can then explain that forces have both magnitude and direction so that we consider the sum of forces acting in each direction separately and explain about the support force to counteract the weight of the car by applying Newton's third law so that the sum of forces in the vertical direction is zero. I can then explain why shutting off the engine keeps the car in motion for awhile (inertia due to the mass, see, weight and mass have very different effects on the car), but the friction and air resistance slow the car down until it stops (now we're at equilibrium and the sum of the forces is the car in all directions are zero).
With that idea of inertia, I can then use an operational definition of mass and inertia by discussing how mass affects the objects responses to forces (it's hard to get a massive object started moving and it's hard to get it stopped once it's going). At this point, I do an outer space example like Conjugate suggested on the other thread where we shake boxes of various objects to ascertain which box of the same volume contains steel ball bearings, pillows, and helium balloons. We can't weigh the objects because weight only applies to the force in a given direction due to gravity, but we can still use mass through inertia. Students are often confused about gravity itself, so this is a nice entry into both a discussion about how inertia (mass) still works everywhere and how gravity is a force between two masses, not just a static field on Earth. For the lesson, we wander around the universe delivering our unmarked boxes so that we have to determine what's inside by determining mass through inertia and then calculating weight for a planetary delivery to figure out what equipment we have to take with us to move the boxes once we land because a big box of just feathers on Jupiter still requires a trolley.
After all of those explanations and examples, my students are usually then firmly on board with the idea that mass relates to inertia, weight is a force with a direction, and volume is occupied space. Those are the ideas that appear on the test, not the early let's-all-get-on-the-same-page-about-mass-and-weight-not-being-the-same-thing quick and dirty approximation.
With the idea of inertia firmly in place, then I can address momentum and use the example of a kid on a skateboard speeding along the sideway next to a parked car. Clearly, the car is more massive than the kid/skateboard combo, but because momentum is mass*velocity, the car with a velocity of zero has no momentum while the kid/skateboard combo has much more.
To illustrate the ideas of momentum, velocity, acceleration, and distance, we use carts on tracks and motion detectors attached to computers so that the students get the chance to try different things and see the associated graphs. Doing this allows the students to focus on the physics, not the niggling details. If I started with students on chairs or carts on tracks to illustrate inertia, I will get responses dealing with force, friction, momentum, velocity, and acceleration, not mass. But by having mass and inertia already in place before we do the demonstration, I can easily disentangle the notions of momentum (mass*velocity), velocity (speed in a given direction like miles per hour east), and acceleration (change in velocity that also has a given direction like miles per hour per second east or 10 m/s^2 downward).
With those ideas then in place, kinetic energy is a snap because clearly mass*velocity*velocity is something very different from mass, inertia, velocity, acceleration, and momentum.
So, Astroafraa, while I certainly agree that having people learn about mass and inertia from an operational definition is the way to go, I have to set the ground work for that learning by starting from the inaccurate notion that mass is matter in order to meet the students where they are instead of directly jumping to the operational definition. Consequently, when a random person asks me the short version of the difference between mass and weight, I always give the mass is matter definition because it resonates with people in thirty seconds to get the difference instead of having to do a thirty minute spiel disentangling the difference between inertia, momentum, kinetic energy, force, and velocity, which the operational definition of mass tends to lead to random people off of the street to conclude.
At the end of the unit on motion that starts with the mass as matter definition, students can apply all three of Newton's laws and define mass as a measure of inertia, but I don't disabuse them of the idea of mass also being a measure of matter contained in an object. I'm too busy trying to get them to accept that while the velocity at the top of a ball's trajectory is zero m/s in the vertical direction, the acceleration is still 10 m/s^2 (because of the gravity) in the downward direction. The inaccurate definition of mass being matter existing side-by-side with the definition of mass being a measure of inertia is much less important to me at that point.
