One thing that I struggled with a few years ago when first exposed to graphical problem-solving was all of the numbers. With my honors physics classes, I put a lot of emphasis on building their skills in exclusively symbolic solutions to problems - students are expected to do every solution with "no numbers until the end." That is, they have to first derive an expression giving the desired quantity in terms of nothing but given quantities; only then can they use any numbers. This approach has several big advantages:

- It's a pain to carry around units in the algebra (and necessary, because it's unthinkable to have "naked numbers"); symbolic algebra means that we don't have to
- We can re-evaluate for different values of the parameters easily
- We can check how our solution depends on the variables - should the Atwood's acceleration increase or decrease when this mass increases?
- We can
*learn*how our solution depends on the variables - maybe we didn't know that the mass would cancel out of the expression for the minimum speed in the Gravitron! - We can check how our solution behaves in special cases - should the acceleration go to
*g*if that mass goes to zero? - We can
*learn*how the solution behaves in special cases - hey, the position function of a falling ball with drag becomes linear in the large-*t*limit! - We can still check the units of the answer, without having to carry them through the algebra - if the units don't work, then it's not even worth plugging in the numbers
- This is a skill that ultimately is a part of how "big kids" do science - it's an great skill to have going into college science and math courses (the course that I took this summer had, out of about sixty HW and exam exercises, exactly three problems that involved numbers!)
- It helps to refine and develop student algebra skills (the more abstract end of the concrete-to-abstract progression of their skill development)

These advantages don't have to be lost in graphical problem-solving - students can still label the graph (naming every labeled quantity, whether it's known or not) and can still build slope and area relationships from the graph (involving those variables that they labeled). Solving these symbolically gets them the best of both worlds: graphical analysis and its lower barrier to entry and improved understanding, and symbolic algebra, with the skills listed above.

What does it look like? Here's a bit of student work, from a recent assessment on CVPM (constant velocity particle model) and the a whiteboard with the first (!) CAPM (constant acceleration particle model) problem that three students did this year.

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