HOW WE TEACH
Mathematics is a challenging subject, and even those who enjoy it may find it too abstract at times.
Simply put, an abstraction is a summary of a concept. In math, this summary is often expressed using Greek symbols in the form of mathematical formulas. If you are the one who observed something special and summarizes it, the process can be quite intellectually satisfying. If you try to summarize something you observed but realized that others have observed more keenly or made better summaries than you, you can still learn a lot, and that is also quite satisfying. But if you are simply an audience member, listening to someone's summary about something you cannot relate to, then that kind of abstraction is what people tend to dread in math.
The predicament of our math education is that we are attempting to condense thousands of years of mathematical development into just 12 years of lessons. As a result, we have to make a lot of summaries. But the more we summarize, the more abstract math becomes, making it increasingly challenging for students.
In order to fit so many years of discoveries into only 12 years, the school syllabus has no room for many actual mathematical activities that took place during math's development. In the classroom, students mostly see a summary of these activities, along with some weak attempts to explain their context.
In this article, we showcase how we reintroduce these mathematical activities back into our curriculum through the use of mini-projects. We will use trigonometry as an example, a topic that many students find either challenging and abstract or simple but tedious.
Modelling A Cube
A generation ago, calculators began to find their way into the math curriculum. But we believe that soon the use of computers will become as common as calculators in the classroom.
Computers open up many options in terms of what we can do in class. For example, students can learn about trigonometry by modelling a cube on a spreadsheet.
People who use spreadsheets know how we can record our data and turn it into a line graph. For example:
Because the positions of the eight corners of a cube can be represented by numbers, when we enter these numbers onto a spreadsheet, we can “draw” the cube on the screen. Then, by rapidly recalculating these numbers (using trigonometry) and refreshing the screen, we can create the illusion that a cube is rotating. Here is what a student did:
Students nowadays are exposed to the mesmerizing computer animations on their screens. But there is such a disconnect between school work and the actual world that it wouldn't occur to them it is possible to understand and create their own computer animation using just what they have learned in school.
People think we have to learn a lot of math before we can start using it in real applications. On the surface, that make sense because for example, it is more convenient if you gather all your ingredients first before you start cooking a dish. Otherwise, imagine the hassle if you only realize you are out of some essential ingredients while you are half-way cooking.
However, our math education can feel like a never-ending process of gathering ingredients. Every time we meet one teacher, we learn a few ingredients from them. We do not know what to do with these ingredients except there is an unspoken trust that a next teacher will come and show us how to turn those ingredients into a dish. But when you meet the next teacher, more often than not, the same thing happens - they teach you a few more ingredients, and leave it to the hypothetical next teacher to show you how to cook the dish.
In the end, most students spent years and years gathering ingredients without knowing what they are cooking, Many would not stick around long enough to ever find out.
While having a good grasp of the basic ingredients helps a lot in learning the applications, the truth is, not every ingredient is needed for every dish. With some ingenuity, we can cook something simple with just the ingredients we currently have. We think it is better to do that with students that as soon as possible.
For example, any student whom you would expect to do well in O-levels would be capable of completing our cube-modelling activity. It is still a challenging task. Students will have to develop grit and learn to make wider connections than what is required for the exam questions. But all the math required are already tested in O-levels. Instead of spending all their school years practicing exam questions, some students will be better engaged if they spend part of their time on these challenging but instructive activities.
But before you attempt to model a cube, there are simpler activities to hone your skills . We show some "simpler" examples next.
Ancient Methods, Modern Technology
In one activity, we task our students to calculate the distance between any two locations on Earth, given their longitude and latitude. Here is what a student did: