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A couple of days ago, I came across this awesome post about showing your work blog on the great math blog, Without Geometry, Life is Pointless.

A quick summary is that some kids go crazy explaining every step of their work in words, and miss the point that writing the mathematics out algebraically allows you to communicate the same ideas, less ambiguously and more succinctly.

Avery has a great example from al khwarizmi’s 8th century math text. Read this, and you’ll see why books of arithmetic ran hundreds of pages in the middle ages, and algebra was topic reserved for the greatest math minds at university even in Newton’s day.

Al Khwarizmi’s text (translated)

“If some one say: “You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times.”

Modern notation

$\left(10-x\right)^2=81x$

This has got me thinking about my students, our correction process, and why I think things might be a little different in physics.

Here’s a problem from my last test, stripped of the context

Suppose that there are 2 downward forces acting on an a basket at rest, 20N and 2N, and a third, upward force is supplied by a spring scale. What force does the spring scale exert on the basket?”

Typical student work:

$20N + 2N = 22N$

Though the answer is right, I marked this wrong, and asked the students to correct it, where they have to:

1. Solve the problem again from scratch, explaining each step of their work with a sentence.
2. Explain the mistake in detail, looking for deep misconceptions that led to the mistake.
3. Explain a deeper insight in the problem. This may be solving a different, but similar problem, but most often should be explaining a feature of the problem that isn’t always evident.

Here’s an example of a correction:

1. Solution
Since the basket is at rest, we know the net force is zero:
$F_{\textrm{net}}=0$
There are 3 forces acting on the basket (the magnetic force, the gravitational force of the earth, and the tension force of the spring (upward)).
$\vec{F}_{g,E\rightarrow B}+\vec{F}_{m,m\rightarrow B}+\vec{F}_{t,s\rightarrow B}=0$
Solving for the spring force, we find
$\vec{F}_{t,s\rightarrow B}=-\left(\vec{F}_{g,E\rightarrow B}+\vec{F}_{m,m\rightarrow B}\right)$
Assuming down is the negative direction, and plugging in values we find
$\vec{F}_{t,s\rightarrow B}=-(-20 \textrm{N}+-2\textrm{N})=22\textrm{N}$

The force exerted by the spring is 22N, upward.

2. My original mistake was not starting with Newton’s first law, and instead simply adding up the forces.
3. My deeper understanding is that I can see if the net force known to be zero, I would not be able to solve for the tension force. So if the scale were bouncing up and down (a changing net force) it would be not be possible to determine the mass of an object in the basket.

Of course, rarely do corrections completed by students ever reach that level of explanation, but most of them do begin to see that equations aren’t ideas we simply pull out of thin air, plug values in and write down answers. They represent ideas and are only valid in certain conditions.

So in physics, I’d say showing your work is vital, and when a student is stuck, one of the best things to do is explain every step in English, rather than just searching for equations. But I do like Avery’s point that I want my students to see the value of mathematical notation for expressing ideas clearly and succinctly.

Maybe I’ll try to create some exercises where students have to explain what an equation is saying, or when it is valid…

1. October 19, 2010 5:57 pm

Maybe I’ll try to create some exercises where students have to explain what an equation is saying, or when it is valid…

See Equation Jeopardy from Eugenia Etikina and Alan Van Heuvelen:

More examples can be found in their Active Learning Guide:
http://www.pearsonhighered.com/educator/product/Active-Learning-Guide/9780805390780.page

There’s also an instructor’s editions with tips and explanations for some of the lab activities. The book is very modeling friendly!

• October 19, 2010 9:42 pm

This is neat. I’ve never really cared for playing jeopardy in class, mainly because I’ve thought the questions encouraged too much memorization. But this a very interesting take.

2. October 19, 2010 6:08 pm

One more thing I forgot:

An MIT course asked students to have a “commentary” column for their problem sets and exams.

Use the left hand side for your figures and mathematics. Use the right hand side to explain and COMMENT upon the figures and mathematics.

The kinds of things to be included on the right hand side would include the goal of the problem, definitions (in words) of your symbols, explanations of the equations you invoke, checks on the units of your answer, and checks on limiting and interesting cases, and occasionally, comments such as “how do I eliminate T?” or “why doesn’t my answer depend upon g?”

This is mostly just a matter of organizing what you would be doing anyway, but it encourages you to take a step back from what you’ve done, take a critical, perhaps even skeptical look, and see if it all hangs together.

From: http://web.mit.edu/8.01l/www/probset.shtml (includes example)

I tried this last year on quizzes (since I didn’t collect HW), and then I got lazy enforcing it. 😦 Anyway, I like using the right hand column because the words do not get in the way of the mathematics/physics. It seems like a comprimise between showing your work and letting the beauty of the physics stand out.

• October 19, 2010 6:12 pm

This format would be good for HW and classwork, too, so kids could remember what they did when looking over their work later. I know it takes more time, but I’d rather have them solve a fewer problems with a more rigorous analysis. This also means you couldn’t ask as many questions on a quiz. Other things for the commentary would be “which model applies and why” and “is your answer reasonable” and “what assumptions are you making,” etc., etc.

• October 19, 2010 9:45 pm

Yes, I’m with you on fewer problems in greater depth. And thanks to Kelly and Mark, most of my highest level concepts are judgment concepts on knowing when to apply the model.

• October 19, 2010 9:44 pm

This is cool, but I’m not sure I really love separating explanations and algebraic/graphical work. I really want kids to have to specifically address their calculations and explain them.