## HOW WE TEACH

###### ACCELERATED LEARNING

We believe in accelerated learning. But acceleration involves more than simply covering advanced topics ahead of schedule. When teaching students an advanced topic, we should consider their age and cognitive development.

For example, the notes below show our student, Charles, using calculus to prove Snell’s Law. He was a few months before 15 years old and did this after having about 5 lessons of calculus.

At first glance, his work does not look like the calculus taught in school or textbooks. But this is actually the original form of calculus. Isaac Newton, the person who invented it, called it “Method of Fluxions” and we adapted it to suit our teaching objectives.

To us, this is the better way to introduce calculus because it is less abstract. It helps a beginner to quickly appreciate the simple ideas behind calculus and almost immediately use it to investigate the cause and effect in nature, which is what calculus was invented to do.

From First Principles

The calculus taught in school today consists of many “rules.” However, it is easy to lose sight of what these rules are really about.

Essentially, each rule is just a way to skip a few steps. If you are willing to write down every step like Charles, you can actually do calculus from first principles without relying on these rules. Below, we have boxed out some steps and labeled them with their associated rules:

In calculus, rules are useful for streamlining explanations. However, jumping straight into them when teaching beginners often obscures the true nature of calculus and shrouds it in mystery. There is significant educational value in showing beginners that calculus is essentially like the algebra they already know, but applied in ways they may not have imagined.

Once students understand this, we can then explain how they may replace the steps in the boxes by quoting the respective rules. For example, they could write “by the Chain Rule” or “by the Power Rule” and skip those steps.

In fact, this hands-on application approach is our preferred method for teaching the rules in math.

Cause and Effect

Using calculus, Isaac Newton investigated phenomena like gravity, demonstrating how mathematics allows us to predict events such as the passing of comets. By calculating how a comet’s trajectory changes when influenced by the gravitational pull of stars and planets it encounters, Newton and his contemporaries realized that several historical comet sightings were actually observations of the same comet returning to Earth over time.

Our students are not yet at that level of accomplishment. Therefore, we assign them simpler tasks. For example, Charles' task above was to investigate how light refracts and how we can predict the angle of its refraction. For beginners, completing such a task is an outstanding achievement.

In a related task, another student, XT, of the same age, used calculus to show that light reflection is a special case of light refraction. Here are her notes:

These tasks provide clear insight into why humans conceived of something like calculus.

It was the invention of calculus that provided a method to quantify and explore the relationships between cause and effect in nature. If, as we often say, science is all about understanding cause and effect, then it is calculus that enables us to do science at all.

Following the school syllabus, it would take several years of working through contrived problems before one may gain such insights.

Singapore method: Concrete to Abstract

The way we teach beginners a less abstract form of calculus aligns with the Singapore method of teaching mathematics.

For example, primary students in Singapore learn "model-drawing" to solve word problems. However, because many parents (and teachers not trained by NIE) are unfamiliar with model-drawing, it often leads to confusion.

One common confusion is the misconception that model-drawing is an alternative to algebra—and a poor one at that. However, saying that algebra is better than model-drawing is like saying that a dog is better than a puppy.

Simply put, model-drawing is a simplified and more concrete version of algebra. It is designed to help primary students quickly engage in problem-solving, thereby accelerating the development of their problem-solving skills. When students are old enough, the plan is still to help them transition into the algebra that adults use.

Since we also have students who study in international schools (where they don't teach model-drawing), we can share our observations. In terms of developing problem-solving skills, we observe that international schools are at least two years behind local primary schools. We believe this is because, without a temporary tool like model-drawing, international schools have to delay teaching problem-solving skills until their students are old enough to grasp algebra.

Caveats of "Concrete to Abstract"

The caveat of teaching one method first and transitioning to another later is that it creates a lot of chaos and confusion.

Timing the right moment for this transition is challenging because every student reaches the level of readiness at different rates.

While an acceleration technique like model-drawing benefits all students, it disproportionately helps the faster learners, increasing the variance in abilities within the cohort. Unfortunately, exam setters respond by creating more difficult problems, and eventually, some of these problems become so difficult that they make model-drawing impractical. Even if a student can solve those problems using model-drawing, they must be highly skilled and likely ready to learn algebra. The proliferation of such problems in exams creates an urgent need for transition, which makes the classroom more chaotic for teachers.

Based on our observations, the Ministry of Education addresses this chaos by providing various levels of pull-out programs for primary students, such as GEP, E2K, subsidized Olympiad training, and Math Master-classes. Additionally, there is heavy investment in training teachers, and government policies support an ecosystem of private tuition and enrichment classes to offer something for everyone.

"Concrete to Abstract" is a Continuum

To help students transition out of model-drawing, some enrichment classes choose to jump directly from model-drawing to algebra. However, centers that follow the recommendations of the Singapore method usually prefer to provide a more gradual and supportive transition by exploring an intermediate step between model-drawing and algebra.

For example, the BCA method, which is taught in many centers nowadays, involves using a subset of algebraic techniques that are more abstract than model-drawing but less complex than full algebra. While everyone teaches a slight variation, more challenging algebraic concepts involving fractions or negative numbers are usually excluded.

Consists of only a subset of the algebra that adults use. Harder algebraic techniques like fractions and negative numbers are excluded.

Applying "Concrete to Abstract" on Advanced Topics

While model-drawing can accelerate the development of problem-solving skills, we have shown that implementing it on a national scale requires an enormous effort that spans both the public and private sectors to manage its challenges.

Perhaps this is why we are not aware of any attempts, other than our own, to apply this acceleration principle to teach more advanced topics like calculus and beyond. After primary school, acceleration typically only involves allowing students to fast-track the curriculum.

Our Accelerated Program

When designing an accelerated program, it’s important to consider its objectives. Is it to let students complete the same syllabus as others but in a shorter time? Or is it to free up more time for activities that enhance the quality of their learning?

Ideally, we prefer the latter. Seeing the big picture, understanding concepts from first principles, and having a clear insight into why something like calculus was invented—these are all elements that add significant value to our students’ education.

Just as model-drawing helps to jumpstart a primary school student’s development of problem-solving skills, applying the same acceleration principles to more advanced topics can jumpstart a secondary school student’s maturity in mathematical thinking, better preparing them for university.

However, depending on the situation, we may need some stepping stones to reach this ideal.

Hence, our approach is to first help students become confident with their schoolwork by fast-tracking their syllabus. Once they are ahead, we supplement their learning with enrichment activities like what Charles and XT did here.