Many physics students struggle with algebra. As physics teachers we are constantly complaining to each other that the maths department hasn’t done their job because our students can not rearrange equations!

I was working with a student recently and he asked for help on rearranging an equation

I have noticed that many students who have trouble will be aiming to have their unknown variable ( in this case) on the left hand side of their finished rearranged equation.

So I tried to diagnose to this student to see if this was the case.

Teacher: “Would you be ok if when we finished rearranging this equation, the stays on the right hand side?”

Student: “I dont know what you mean…”

T: “Ok, lets look at a different problem. Are these two equations the same?”

S: “No, they are different”

T: “Really?! Can you explain how you know that?”

S: “If I rearrange the bottom one () I get ”

There are two separate problems here. The first is that the student made a mistake in trying to rearrange the equation in their head, and ended up with a rogue negative.

The second is that this student needed to rearrange the equation in first place to compare the two equations. I look at these two equations and see they are the same. This student didn’t. Why? I think the answer lies in the metaphor they have attached to the “=”.

As human beings, we understand many abstract concepts by attaching meaning through metaphor or analogy. You may take for granted that you understand what “=” means. Its only when you meet someone that has a different understanding than your own that you may reflect on your own understanding.

The metaphor I attach to “=” is a set of** balance scales. **

To me “=” means that one side of the equation is balanced with the other side. Both sides may look different, but the “=” sign tells me they are weighted the same.

This student (and many others) don’t attach this metaphor to the “=” sign. The metaphor/meaning my student attaches to “=” is *“ and the answer is“*. I think this is perpetuated/started by their use of a calculator. There, students type in a series of steps for the calculator to evaluate. Then the student pushes “=”, and the calculator says

*“and the answer is”*.

This is the reason my student couldn’t see the equivalence between and . They have a fundamentally different understanding of the “=” sign. And this is a very difficult belief to try and dislodge. It isn’t just a “fact” I can easily replace by explaining to them their error. This will require a number of activities to try and start to remedy the situation. To be honest, I am not sure where to start…

Last note: When discussing this with the other physics teacher at our school he mused that this would be the reason kids struggle so much with the proportion sign. Whereas seems a small step from = to us experts, it seems to give students huge trouble. But now understanding that they have a different understanding of = than us, it seems so obvious why they struggle here.