October 28, 2017 Filed in: Articles

Do you shake your fist at the skies when you look over your student’s math work? Do you find yourself cursing the math teachers in your school? If you do, you are likely not alone. But are our grievances and grudges justified? As a physics teacher, it can feel very perplexing to watch one’s students struggle to use grade 10 mathematics in our grade 12 physics classes. Let’s give our math teachers a break for the moment and explore a surprising factor in our physics students’ struggle: the dialect of math that we speak in a physics classroom.

This doesn’t look anything like what we do in a physics class. In this mathematical representation, there are no explicit clues that hint at the expression’s physical meaning. It could be describing anything, and that’s actually the point from the math class perspective. The pure mathematical relationships are very clear.

Scientific calculations always have a physical meaning. Having units allows students to properly interpret a mathematical result, especially a novel result. For example, a student might be finding the slope of a position graph for the first time:

Carrying units through the calculation allows students to recognize that the slope represents an object’s speed or velocity. Training students to interpret results like this deepens their understanding. For example, the slope result means: “if the object continues to move with a constant velocity, then for every second that goes by, it would travel 12 cm forward.” Even the awkward seeming gob of units in the quantity (5 km/h)/s provides a deeper understanding of the acceleration concept because it can be carefully interpreted: “for every second that goes by, the velocity changes by +5 km/h). This is only possible when units are present.

Hmm? It looks so innocuous, but conceptually it behaves very differently from the elementary school example! We don’t end up with five groups of seven velocities, or seven groups of five-kilo masses. We end up with a totally different kind of thing. We end up with momentum: one of the few conserved quantities of our classical universe. Neither mass nor velocity have this most wonderful property. While the number part of the calculation obeys the elementary school model’s pattern, the physical meaning is shaped in a dramatic way by the units

Watch out! In this calculation, the units for time do not agree and will yield an erroneous result in metres if the numbers are blindly plugged into a calculator. The careful use of units helps us understand when conversions are necessary, and we can train students to spot this in advance of their math work.

This multiplication operation follows our elementary school model of multiplication very nicely. Reminding students of this helps them activate their number sense, which helps them decide whether the converted result seems right. We can measure this impressive distance using a small sized ruler (a metre stick) or using a very large sized ruler (a mile stick!). If we use the small ruler, how many rulers would we need to cover this distance? How about with the bigger ruler? So, if we convert from miles to metres, do we expect the number part to be bigger or smaller? This is a powerful sense-making skill that is worth developing.

1 “pants” in British English means underwear.

2 From a conversation with Eugenia Etkina, October 2016