The Power of Per

 

How many times have you put an equation on the board and seen the lights go out around a classroom? Seeing 32 students simultaneously pressing the “disengage with lesson” switch. There are probably many reasons why students either struggle with the mathematical side of physics or find it to be the area with which they are reluctant to engage. This isn’t about the problems, but rather about an approach that initially bypasses equations, and builds mathematical and physical confidence in students without them realising they are being engaged! Rather than focusing on equations the trick is to focus on units and to unleash the powerful physics lurking inside. Most of all we want to unlock that little “/” sign, and recognise the “power of per”.

As much as we may like to think that maths is the language of physics, that is not a useful sentiment when teaching most students in schools. Personally I don’t even agree with that (even as a mathematical physicist), in our contexts English is the language and the maths provides a useful shorthand for describing the relationship between quantities, and so we should focus on the underlying language to build understanding rather than resorting to equations to be simply descriptive. Equations mean different things to different audiences, to the experienced physicist they may well contain the insight into the physical universe but to most GCSE students they are purely mechanical devices for getting an answer.

I’m going to use a couple of examples that have worked for me. Let’s start with weight and mass; rather than worry too much about W=mg, my approach is to focus on “g” and more specifically the units in which g is measured. In terms of teaching I begin with stating that g=9.8 N/kg; I immediately articulate the units “Newtons per kilogram” and the draw attention to the word “per”, asking questions such as “when did you last use the word per in everyday life?”, “what does the word per mean?”, and “is there anything we could replace “per” with?”. Let’s be honest, “per” is not in the everyday vocabulary of most students, but “every” is. Going back to the units I then like to flesh them out with what the units measure and so I go from “Newtons per kilogram” to “Newtons of force for every kilogram of mass”. Now the physics is revealed, 9.8 N/kg simply means 9.8 Newtons of force for every kilogram of mass on which the force of gravity acts. So if the mass in question is 5kg the lesson and students know the meaning of g, then they use reasoning to determine W rather than just substituting values into an equation. I found that most students would then produce the equation for themselves, indicating real understanding as opposed to rote learning.

A more complex example is acceleration. Gaining a genuine understanding of acceleration, as opposed to the output of an equation, is our goal, and again the units can both deepen this understanding and help bypass the need for an equation. I must admit that I find the form of the units “m/s2” to be unhelpful; I know what a square metre looks like but have no idea about a square second. I much prefer the form of the units on which I was brought up, m/s/s. To start extracting meaning from the units we have to use some punctuation, rather than just saying “metres per second per second” we need to pause, take a breath and insert a comma and say “metres per second” …pause… “per second”. Swapping in every for the second “per” we get “metres per second” …pause… “every second”. Hopefully students recognise the first “metres per second” as a measure of velocity, and so we now just add in the idea that in acceleration velocity changes and so the units become “change in metres per second (or velocity), every second”, and thus the physics is revealed. This can then form the basis for calculating acceleration as well; given an initial and final velocity, and a time over which acceleration takes place, it is quite straight forward to see how the units allow us to perform the calculation without even using an equation.

For example let the initial velocity be 1 m/s, the final velocity be 7 m/s, and the time taken to accelerate be 3 s. In 3 s the velocity increases by 6 m/s; assuming the acceleration to be constant, we can then suggest that if a change of velocity of 6 m/s happens over 3 s, then in every second a change of 2 m/s takes place. Here’s our answer, a change of 2 m/s every second, is 2 m/s/s. Not an equation in sight!

Equations are barriers to learning for many students. Approaching these types of problems through the units gives one way in which students can develop understanding of physics and experience success before the need to introduce the formal equation.

The next part can be read in https://markwhalleyeducation.blogspot.com/2023/05/vips-very-important-pers-of-power-of.html (VIPs: Very Important Pers), in which I look at constants a little more and also give a list of the key VIPs.

Thanks to @missneutrino for the splendid graphic.