Research School Network: Equations in Context Research School Associate Fahim Rahman explains how he has altered his teaching of equations to increase the use of context

Twitter is a wonderful resource for someone looking to engage in dialogues surrounding education. While scrolling one evening I stumbled across a thread surrounding formula triangles. As a Science teacher, I often find myself having to deal with these problematic pyramids, so I thought I’d take a peek into the controversy

Students often claim they provide some means relief when rearranging equations, and if they have received proper instruction on how to use them, they can show promise in rearranging the smaller equations. However, when learning new content, they often ask what the formula triangle rather than take responsibility for constructing it themselves, and bypass a skill needed in rearranging calculations.

In my search I realised that I had a much deeper issue with formula triangles, and it highlighted a problem within my own practice. When talking about problems teaching equations in his book Teaching Secondary Science Adam Boxer raises a list of issues with using formula triangles, but one resonated with me.

​“They betray a poor understanding … that equations have meaning, they [formula triangles] encourage a robotic and algorithmic approach to solving equations.”

Equations are not just a tool to help you attain you an answer, they are a way of demonstrating a fundamental relationship between things that may be completely different. This is something I have previously done poorly in my own practice. Often in physics, the imperative to have students be mathematically capable, often wrongly overrides the necessity in having them understand how variables are interconnected. Rearranging equations is not a suitable substitution for understanding why a faster, lighter object has more kinetic energy than a slower heavier one or how seatbelts causing you to be in a collision for longer is, counterintuitively, causing much less harm to you. In these reflections, the slight adjustments I’ve made in explicitly teaching formula rearrangement is applying qualitative logic, rather than employing a mathematical instruction.

In teaching speed to Year 7, I introduced them to the initial idea that speed is the time taken to cover a certain distance. That when Usain Bolt broke the world record in the 2008 Olympics, his 100m distance covered in the shortest time meant he was the fastest. The idea that 60mph in a car means that for every hour driven I am covering a distance of 60 miles. This was then used to provide a foundation for the equation and its appropriate units. By using this approach, they were able to link ideas to schemas they had already provided and use them to re-approach rearranging formula. Instead of now showing them how to rearrange, I ask: ​“If I’m travelling at 10m/​s for 1 second how far do I go?”, ​“If I travel at 10m/​s for 3 seconds how far do I go?”, ​“What did you have to do to arrive at that answer?”. This then enabled them to read the question to see it for what it was, a snapshot of reality that requires an explanation, rather than an equation question that was hidden behind contextual camouflage.

This worked well for my year 10s too. Specific heat capacity (SHC), the concept of the amount of energy a certain amount of material requires to change its temperature. This can be a difficult idea to grasp. Energy is an intangible concept and to discuss a material property that is related to it is often alien, all combined with an equation on top of that:

Thermal Energy = Mass x Specific Heat Capacity x Temperature Change

I used to teach this as definition, then introduce the equation then model a question on the board. Now I am taking steps to give context to what is being taught. If 1kg of a material has a SHC of 400J/kg°C, how much does it need to heat up by 1 degree? by two degrees? How much would I need if I wanted to heat it up by 2 degrees and I had 2kg of the material? In doing so not only are students gaining a much more concrete and reinforced explanation of the concept, but in then asking them: ​“What processes took place in your thinking to come to that answer?” or ​“What did you have to do to those quantities to arrive to that conclusion?”, they can see how the equation is formed or derived. By teaching formula triangles, we remove them from the vital context that provides. The process naturally evolves into: ​“It takes 400J to heat up 1kg of copper by 1 degrees. If I use 800J, how much does that copper heat up by?” By asking them what they did to arrive to that conclusion, and to go back and evaluate the mathematical steps behind that thinking. It becomes easier to develop the ability to think deeply about their own processes and practice the ability to rearrange and see it is not as daunting a task as it appears

How ironic that an indulgence in my dislike of formula triangles can lead me to evaluate my own questioning, explanation, metacognitive teaching. Perhaps there is some use for them?

Fahim Rahman is a science teacher at Durrington High School. He is also a Research school associate for Durrington Research School