Why branching out from central topics is a more effective learning system than jumping from horizontal up to the next horizontal
The traditional approach to teaching quantitative disciplines in our educational system is to cover a topic in it’s full breadth – on, more or less, a single plane of complexity without making any leaps to matters that build on those topics. This creates a dogmatic view that one should leave no stone unturned at a given level of difficulty before daring to make any leaps ahead. This creates disillusionment in many students by way of them growing impatient, waiting to learn some mathematics that can let them “tackle the world” in a more meaningful way and see the workings of the universe from a higher vantage point. The other reality of this is that whether learning a sport, weightlifting, a skilled trade, mathematics/physics, or programming, different people find different aspects initially difficult and others easy.
Giving a student as many tools as possible initially, allows them to keep moving and branching off onto things that they find intuitive instead of being stuck on that which comes with difficulty to them. Such an approach builds engagement that nurtures heuristic sense for the overall field that they are trying to master, through connection of dots – built on successes. It is then easier to circle back to a topic that seemed difficult and tackle it head on once much around it has been “colored in” or – plugged into jigsaw puzzle of the mind’s eye.
This of course does not replace the need for going in depth on fundamentals, but simply balances the vertical and horizontal pathways of a learning journey. The problem with many educational systems is not they go too in-depth on fundamentals without making vertical leaps to more complex topics, but that they spend year after year covering surface level material on basic topics. Students should focus on problems of varying levels of difficulty, challenging themselves as much as possible on the fundamentals, while making leaps to special/adjacent/complex topics to accelerate their journey to unconscious competence.
A tree’s branches and leaves grow simultaneously with it’s trunk and roots – a big trunk does not develop branches and leaves one day or one year as a binary event. A martial artist should learn strength, conditioning, and flexibility in tandem with their martial art. A novice programming student should start learning basic data structures and algorithms as soon as they cover variables, statements, loops, and objects. There’s no reason a calculus student doesn’t tackle a simple Ordinary Differential Equation, tied to a real world example, upon grasping derivatives and maybe having done some optimization problems. Likewise, a physics student can examine in parallel the forces of gravity, electricity, magnetism, and mechanical springs to draw similarities and differences between the mathematical underpinnings of the forces that exist in nature. It would be very beneficial to start interpreting coordinates as vectors and rotations as matrix transformations to get a head start on linear algebra when learning about kinematics/motion.
There are many students, including – in my experience – even in engineering schools, that for one reason or another do not see that every seemingly too abstract or overly trivial aspect of math is used in the real world to build great technologies or improve business processes.
Quantum mechanics uses linear algebra and differential equations to describe the subatomic world in a probabilistic manner. It is used to create semiconductor devices that make up the digital world now all around us. Trigonometry is used to express the electromagnetic waves which those devices use to communicate with each other and extract their phases and frequencies. Most of these mathematical operations boil down to fractions and series of algebraic manipulations. Thus we see how providing an example of going from abstract to functional to computation in a real world application can be a huge inspiration and provide a bird’s eye view to a disinterested student who everyday wonders as to why they are learning fractions and everyday the next year why they are staying up late over triangles. Of course one can provide a number of such qualitative overviews and simplify them as needed to the level of education of the students.
To give a quick recap of the benefits of branching out to more advanced/special topics as early as it makes logical sense:
Reduces intimidation of more advanced topics by showing a direct path from simpler to more complicated topics: showing how quickly they are within reach.
Creates motivation by revealing the bigger picture: showing where some abstract concept comes into play or how some operation fits into tackling interesting systems.
Keep in mind that one still has to delve deep into each topic on its own merit and explore the related abstract concepts and mathematical formalism for a true understanding. Branching out allows for a quicker learning path to grasp and retain all of the operations one will be using.