Exploring the Pros and Cons of Variable-Based Physics Problem-Solving

[symbolipoint]

Once students reach Physics, no matter students' major fields, they are forced to (just experience where I attended) handle MOST of the setup and algebraic work using only variables, FOR ALL NUMBERS; and then to solve the whole problem or exercise still all in variables. Then, the last thing to do was to substitute the given and the known values to finish finding the value or values asked. This usually was the required way in the Physics Series of courses for STEM students and as I would guess, would continue that way for the Physics majors. Doing this way was never seen as anything bad in other science or engineering course, but was usually required for Physics.

Now I ask; Is this style or method good, or is it bad? I like it , most of the time. Is there any justifiable claim that this is not always good, or is bad? Keeping things as variables let's you see where all the numbers are going whether known, unknown, or constants. Keeping all in variables let's you find a formula for what you're solving in case you have any need to use that same formula for several of essentially the same problem. The manner also let's you write a computer program to give number results very quickly. The only reason for this way being bad, is that the person learning did not yet study and learn basic beginning algebra.

So, good or bad? Tell how!

 

[hutchphd]

The only reason for this way being bad, is that the person learning did not yet study and learn basic beginning algebra.

Absolutely. Operationally (in my experience) the problems arise because many folks who "know" Algebra were never actually taught how to use it. I think it should be a requirement for passing any substantive physics course to understand and use Algebraic analyses.

 

[kuruman]

Definitely good! You were taught well and you touched almost all the points why. One reason: once students get the answer in algebraic form, they can perform dimensional analysis to check it. It cannot find missing dimensionless factors such as $\pi$ but it can tell if one dropped a power of time $t$. Another reason: if a correct numerical answer is given, but the students' answers don't match it, it is much easier to backtrack the algebra when the equations are all written in symbolic form. A third reason: students get to recognize recurring motifs when they see them in symbolic form, e.g. $\sqrt{2gh}$, across a variety of problems. A fourth reason: intermediate numerical calculations breed cumulative roundoff errors which may result in a numerical answer that is sufficiently off in algorithm-graded homework to be marked incorrect when it isn't.

When I wrote questions for introductory physics tests, I did not assign numbers to the relevant quantities, just symbols. My arguments to the class for doing so were all of the above, plus "This exam is designed to test how well you can do physics, not how well you can push buttons on your calculator."

I also encouraged students to use spreadsheets for answering homework problems with numerical answers. The idea is to array all the numerical variables in input cells and then write the appropriate formula relating them in the output cell. This way, if they discovered an error in the input or output cells, they could just change the appropriate cell and the correct answer would pop out without having to recompute all the intermediate parameters. I can see no downside to the "variables-only" approach when one does physics where the goal is to understand the basic principles.

I am not an engineer, but I believe that with engineering the process of algebraic manipulation is different. Engineers take basic principles and design useful objects. That requires one to be able to translate basic principles to numbers in order to design something that actually works. This translation is often based on approximations of the idealized physics equations which means the use of numerical parameters and coefficients in addition to the variables in symbolic form.

 

[Vanadium 50]

I leave it entirely up to the student.

However, I also tell them that where the symbols go out and the numbers go in is where I stop giving partial credit.

 

[robphy]

In my experience, to many students,
getting the numerical answer to the textbook problem is the end goal,
especially since that's what's in the back of the book [for the odd problems].

And, it may have been that in high school,
substituting in the given numbers appropriately,
indicates[ reveals(?), etc...] the algebraic-unknowns left to solve for algebraically-with-numerical-coefficients.
..,.as if, chipping the crud away, leaving the gold-nugget whose numerical value is the last to be revealed.
Essentially, "Find x, in terms of the given numerical quantities".

Maybe these shouldn't be called "answers" in the back of the book... but "checkpoints".

I would think the real goal
is to determine the relationship between the physical quantities
in the given situation. How does x depend on a, on b, on c?

In the past, I have tried to suggest that the problem really shouldn't end
when getting the numerical answer in the back of the book.
...But to reflect... at least ask, how does the result change
if a is doubled? or halved?
Or a is very small or very large compared to b [a quantity with the same units as a]?
(Sometimes I would have a follow up part to get the result with a different a,
secretly trying to make the intermediate numerical steps tedious to repeat.)

On Friday (tomorrow), in my algebra-based class,
I'm going to try something...

I am going to do the ballistic pendulum problem on the board, following the textbook example,
but with a different numerical value for one of the givens.
I am going to solve it algebraically [as usual and as I prefer]
and plug in the values for my version, then for the textbook's version.

The textbook gets an algebraic solution for part (a), then plugs in the numbers to answer part (a).
Then uses the numerical answer for (a) to use in the algebraic solution for part (b).
The textbook has the algebraic pieces there, but doesn't put them together
to express part (b) in terms of all of the given quantities algebraically.
The textbook is not encouraging or suggesting how the quantities are related.

It's easy to put the full solution together.

But here's the new idea:
Use Desmos ( https://www.desmos.com/calculator ) to

• first compute the result numerically
  (using the variable names to define numerical givens, turn them into sliders)
  and then construct an algebraic expression of the solution in terms of the sliders
  (which uses good typesetting so it looks like the printed equation
  [as opposed to a computer-code implementation in C ]).
  
  Thus, the sliders can be tuned to get a new numerical result.
  (The students use WebAssign... I wonder if someone will figure out that such a tool
  could work out all of their numerical answers if one student worked out the algebraic solution.)
• then---and this is the important part---delete the slider for the given quantity I had varied.
  Desmos now interprets this algebraic expression
  as a function of one independent variable [the deleted slider variable]
  and produces a graph, which could be used to explore scaling and limiting cases.
  
  One can restore the numerical value for the deleted given.
  Then, one can delete a different slider, and explore that relationship.
  
  The Desmos graph would be a [not-so-difficult-to-use?] crutch
  which could help encourage the goal
  of understanding the relationships between the [variables] parameters.
  [This crutch is an alternate approach towards analyzing the relationships
  without requiring the student have
  such mathematical skills at algebra and envisioning scaling and limiting behavior of functions.]

We'll see...

 

[symbolipoint]

A third reason: students get to recognize recurring motifs when they see them in symbolic form, e.g.

I find that reason to be very special (in a good way).

 

[Dr.D]

once students get the answer in algebraic form, they can perform dimensional analysis to check it. I

This is very important, and for this reason, I have always required my mechanical engineering students to obtain the symbolic answer, provided it could be obtained without the use of numerical methods.

 

[hutchphd]

The Desmos graph would be a [not-so-difficult-to-use?] crutch
which could help encourage the goal
of understanding the relationships between the [variables] parameters.
[This crutch is an alternate approach towards analyzing the relationships
without requiring the student have
such mathematical skills at algebra and envisioning scaling and limiting behavior of functions.]

While I understand your motivation I am not at all sure that this is a desired outcome. The ability to do such analysis without an on-line crutch is, IMHO, a primary goal of studying physics in a liberal arts (or engineering) education.
I realize this is fond and illusory goal, but I really dislike shortcuts in this arena. It is this skill that has allowed me to routinely outperform engineers with far better specific training. The reward for this skill is spectacular.

 

[robphy]

While I understand your motivation I am not at all sure that this is a desired outcome. The ability to do such analysis without an on-line crutch is, IMHO, a primary goal of studying physics in a liberal arts (or engineering) education.
I realize this is fond and illusory goal, but I really dislike shortcuts in this arena. It is this skill that has allowed me to routinely outperform engineers with far better specific training. The reward for this skill is spectacular.

While I would love it if my students had the same or better
intuitions and problem solving strategies than I do,
I would prefer to hook them or empower them somehow, even non-traditionally,
with tools to supplement analytic thinking like this
rather than to not have them do any analysis at all.

The existence of this thread already admits that
many students lack adequate algebra and mathematical skills.
But is pure traditional algebraic reasoning the only way to reason?
It seems to me there are other approaches... geometric and graphical approaches, computational approaches, simulations... In other words, traditional algebra and calculus are barriers to many.

Yes, I frown at the widespread use of TI-83s and new physics majors using WolframAlpha or integral-calculator.com instead of learning traditional methods of doing integrals. (But I admit I can't use a slide rule or do long division.)

But maybe this generation can take advantage of computational tools and computational thinking
and (maybe?) learn to do numerical computations to tackle problems that analytic methods can't.

 

[symbolipoint]

The existence of this thread already admits that
many students lack adequate algebra and mathematical skills.
But is pure traditional algebraic reasoning the only way to reason?
It seems to me there are other approaches... geometric and graphical approaches, computational approaches, simulations... In other words, traditional algebra and calculus are barriers to many.

The first bold part - Yes in fact drawing figures, diagrams, pictures is often an academically (and otherwise) necessary part of problem solving analysis in order to write the necessary equations/inequalities, even if they are to be done all in variables.

The second bold part - not for me; instead, traditional algebra (and some other features of undergraduate lower level required Math courses) was a ticket to being able to stay in the educational program; and also carried on to some extent after graduating and getting into some employment positions.