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ШУхCOMPILATION: modeling with less math?
Date: Wed, 24 Mar 2004
From: Andrew Frank
Subject: Modeling and Mathematically Challenged Student
I teach general physics to juniors and seniors who are not very mathematically gifted. How does the Modeling approach deal with these students?
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Date: Thu, 25 Mar 2004 08
From: Matt Greenwolfe
All students at my school are required to take physics, and I teach the non-honors sections, so I have about the same teaching assignment you describe. Modeling has mostly been a good fit for this group, and I've only had to do a few things to make it more accessible. Since modeling, I've been able to create a Physics II course, which is a second year of physics that is not at the AP level. This is because the students have responded so enthusiastically to modeling in the first year course.
In the Constant Velocity and Constant Acceleration Models, I teach them to solve the problems using the slope and area of the v vs. t graph, and never develop the formal kinematic equations. I find they are much more likely to understand what is going on and apply the right mathematical relationship at the right time if they just use these two fundamental principles and don't have a list of equations to memorize.
In the net force models, I teach them to solve the problems by drawing the force vectors tail to head, to scale, and measuring their answers. We don't mention components. I have several large blackboard protractors for whiteboard use, and let them solve force problems with ruler and protractor on homework and tests, even up to the final exam. I never teach them to solve the problems with trig, but those students who are comfortable from their math classes start to figure out how to use trig anyway, because its "easier than drawing all those diagrams," and during white board sessions, they start to teach the other students how to do it. I've had years where all the students end up using trig by the end of the year, where almost all of the students are still using rulers and protractors, and years where its ended up with some solving it one way and some the other way. If they are uncomfortable enough with trig that it would become a barrier, I would rather that they understand the physics by drawing the diagrams. I've been so pleased with the conceptual understanding that the students get from this approach, that I've started teaching forces using the tail-to-head method even for my AP classes.
I've found that a more extensive Unit 1 is needed for this group, primarily to establish the idea of the different kinds of proportionalities and what they mean. They've studied proportionalities in their math class, but generally don't remember anything important about it. I do several simple experiments in unit 1 that give them more practice with this and don't use the pendulum. Throughout the year, I use ranking tasks with each experiment to get them to think about the proportionalities and how to combine them. This has worked really well.
I slow down, compared to an honors or advanced group. We take the whole year just to do modeling and CASTLE.
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Date: Thu, 25 Mar 2004
From: Hugh Ross
I teach physics to juniors with full range of mathematical abilities. I can't emphasize enough how modeling has opened many doors for those students with weak mathematical abilities. By the end of the course, most of those students who previously had no clue of how to represent a linear relationship mathematically are quite adept at developing these relationships with physical systems. Some of these students have previously failed the mathematics portion of our state's graduation exam and pass it with no difficulty after having taken physics.
I don't attempt to teach some applications, such as trigonometry with vectors or using the quadratic equation with projectiles, but these mathematical approaches are ancillary to the main MODELS in physics. I strongly believe that the modeling approach works with lower levels of students with minimal algebraic skills.
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Date: Thu, 25 Mar 2004
From: Clark Vangilder
Andrew's post is the $64,000 question. How do you model mathematics when mathematics is a modeling tool? There is no simple answer to this, as I think this post will demonstrate. It can be done, but it is very hard. I apologize in advance for posing more questions that I can answer...but I am just a man.
It's crucial that we understand that the mathematical model is a representation and coordination of geometric descriptions as understood by means of many conceptual systems. For instance, we say that the horizontal and vertical components of projectile motion are independent of one another in spite of the fact that a mathematical function where one variable is dependent on the other can be formed to describe that motion. The mathematical model is dependent but the physical one is not. Hmmm...
Consider the "velocity" formula, v = d/t. Can we really group or partition distance according to time? How do you group 6 miles in 2-hour chunks? I can group 6 miles into three 2-mile chunks because miles are the same unit, but I can't group different units in terms of one another without a new conceptual framework. We suppress the shift in conceptual systems and ignore what those shifts mean in the model. The mathematical model is often effective but it does not always map well to the physical process. What does the mathematics of velocity mean?
Some random thoughts...
1. Representation and meaning are not the same thing.
2. Concept and representation are not the same thing.
3. Model and meaning are not the same thing.
It is our duty to expose the boundaries between meaning, model, concept and representation. The Modeling Method is certainly rich enough to afford this expense, but the road is long, difficult and magnificent. The three basic modeling questions of "what do you see...what can you measure...and what can you change?" do not address "what do you mean?" when you write this equation or that equation. I am not critical of the modeling method; instead, I am immensely thankful for its life-changing influence on my own learning. Modeling is much much deeper than what we see in the workshops and the curriculum therein.
The basic question to ask is "what do you mean by that?," whatever "that" is. WARNING: make sure that you are ready for that question before you ask it...it's like pulling the cord on an inflatable life raft while driving a Yugo. One of the problems that we face in science education is that we produce a great blob of numbers and units when our students do not really understand "number and unit" in a purely mathematical sense.
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Date: Fri, 26 Mar 2004
From: Rob MacDuff
Andy Frank wrote: "I teach general physics to juniors and seniors who are not very mathematically gifted. How does the Modeling approach deal with these students?"
The answer to this question is the subject of an entire summer workshop (PHS 542: Integrated Math & Physics) and so it is hard to deal with on a list serve. The course takes the perspective that mathematical understanding is greatly enhanced by physics and physics understanding is greatly enhanced by mathematics.
As we move more and more into cognitive modeling instruction (stage II) where the focus shifts to an understanding of structure and the use of morphological reasoning (reasoning based on structure), the kids who can't understand mathematics are the smart ones because the way we do math doesn't make much sense (see Clark's post).
Just consider this: 2 times 3 equals 6, 2 times 3 inches equal 6 inches, 2 inches times 3 inches equals 6 inches squared. Now ask yourself, are all these operations which I have indicated by times; Are they the same? If you think about this for a second or so you should be experiencing the effect Clark talked about with the inflatable raft while driving a Yugo.
When presented correctly, the math can make perfect sense within the context of describing what is taking place in any phenomenon. I have been working with Clark on how to do modeling with heavy-duty-mathematically-challenged students, and here is something that they can now do easily. You might like to try it. Take three quarters (of a circle), and divide it by one third (of a circle); clearly show your work using diagrams alone. A correct response is not an answer but the explanation of the answer. You might like to try answering these questions as well: "what does your answer mean and why does it mean it?" Enjoy!
My point here is, why drop the math? use the science content as a means of making mathematics meaningful. If students are capable of drawing vectors, and drawing vectors tail to head, then they should be able to talk about what they are doing using both English and math at each step in the construction and why they are doing it.
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Date: Sat, 27 Mar 2004
From: Matt Watson
Matt Greenwolfe wrote: "I've found that a more extensive Unit 1 is needed for this group, primarily to establish the idea of the different kinds of proportionalities... This has worked really well."
In similar efforts, I began writing additional Unit 1 worksheets that would provide more structure and practice for curve fitting. I succeeded in providing more practice, but found that most students learned curve fitting by rote rather than embracing the proportionality or the experiment behind it. Limited success aside, I'd be willing to share.
COMPILATION: modeling with less math?
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