Math is a difficult subject. Even people who are keen to learn math can find it too abstract sometimes. But abstraction itself is not really the problem.
In simple terms, an abstraction is just the summary of a concept. If you are the one who observed something and made the summary, then that kind of abstraction is fine to you. Some people may even like it because it is intellectually satisfying. But if you are just listening to another person’s summary about something you cannot relate to, then that kind of abstraction is not fun at all.
The problem with math lessons is, we are trying to condense literally a few thousand years of math development into 12 years of mandatory education. Inevitably, we have to make a lot of summaries. But the more we summarise, the more abstract math becomes and so, the more difficult it is for our students.
In trying to fit a few thousand years into 12 years, the school syllabus has left out many math activities that actually took place while math was being developed. At our centre, our challenge is how to weave back some of these activities into our lessons. Because this is not common at the time of writing, we enjoy the creativity this brings to our work.
Trigonometry is a difficult topic for many students. In this article, we show how we use hands-on mini projects to teach students the abstract concepts in trigonometry.
Modelling A Cube
A generation ago, calculators began to find its way into the math curriculum. But we believe soon the usage of computers will become as common as calculators.
To help students relate better to trigonometry, we let them model a cube on a spreadsheet. People who use spreadsheets know we can record our data and represent it as a line graph.
Since a cube has 8 corners, and we can use numbers to represent the positions of these corners, we can enter these numbers into the spreadsheet and “draw” the cube on the screen. Then, we rapidly recalculate these numbers (using trigonometry) and refresh the screen. If our math is right, we would create the illusion that a cube is rotating. Here is what a student did:
Even though the common way to implement 3-dimensional rotation is through a “rotation matrix”, we design our mini projects to rely only on what our students already know — in this case, O-level trigonometry. In fact, having done 3-D rotation without "rotation matrix" allows us to show students later on why matrices are useful.
Nowadays, children are mesmerized by the graphics on their screens and monitors. But most of them do not realise what they learn in school already gives them the ability to understand how computer animation works. We can exploit this missed opportunity.
There are other activities that O-level students can do (or should do before they attempt to model a cube). For example, many people like a good origin story, as evidenced by the popularity of superhero origin stories in films.
In math, there are also fascinating origin stories, such as the ones you see in the National Geographic’s television series “Cosmos”.
These stories, when told well, can make you care about what you are learning. When you start caring, you become more curious to understand deeper. But most of the time, the stories do not include the actual math involved in the events. So, the ball is in our court now as classroom teachers to design activities that fill in these gaps.
In a series of such activities, we used the theme of how our ancestors managed to figure out the size of the world. We curated it specifically to teach the concepts of trigonometry, but also set the stage for many higher mathematics that emerge naturally from their quest.
For example, in one activity, we let students create a spreadsheet which calculates the distance between two locations on Earth, given their longitudes and latitudes.
If you have ever wondered why we use “radians” in trigonometry and not just stick to “degrees”, this activity can help a lot. In short, you will realise the calculations become simpler with radians. But we know this explanation can mean little if you have not done the activity yourself. So, we will not expound further. But rest assured, your teachers did not create “radians” just so they can have something to test you in the exams.
Descendants of Astronomers
We are descendants of astronomers. Long ago, our ancestors had to survive by learning how to read the stars in the night sky to predict seasonal events and adapt their lives accordingly. In another activity, our students retrace the footsteps of the ancient astronomers to build a sine table from scratch.
Math is not just a set of unrelated techniques compartmentalized into topics. As we continue on this quest to figure out how big is the world, other concepts like functions naturally emerge.
Most of the time, people use mnemonic devices like “soh-cah-toa” to teach trigonometry.
Without the opportunity to experience actual math activities like the ones we show here, we understand there is little choice but to adopt such memory devices.
The modern school syllabus has summarised trigonometry too much and, as a result, made it too abstract. Hence, students have to rely on mnemonics to help them remember the abstract facts that mean little to them.
But this is like building a makeshift raft by haphazardly tying a few logs together. You can only hope it lasts long enough in the water and not fall apart before you reach the next island.