Building On Top of Each Other
I once had a mathematics teacher who would say something that bugged me: what we’re doing is easy. I am barely being hyperbolic when I say that this teacher would say this for every single concept we learned. Therefore, I couldn’t help but think that surely not everything could be this easy.
The saying would take on an ironic meaning to me, because even for the most difficult concepts we would learn in the class, my teacher still saw it as easy. I would frequently turn to my friend and exclaim, “How can it all be easy?”
He had no answer for me, and I thought that there wouldn’t be one. However, after finishing the course and reflecting on my experience, I can start to see what my teacher was talking about. It’s not that the concepts are super easy to understand. Instead, it’s that the leap from what we previously knew to what we learned is not huge, so it shouldn’t be too difficult to understand the new material.
Here is a concrete example. When I was in secondary school, I learned about algebra and functions. At the end of secondary school, I learned about more complicated functions and curves (quadratic, hyperbolic, and ellipses), plus I was introduced to the notion of vectors.
In the beginning of CÉGEP, I learned how to manipulate vectors by addition, subtraction, dot products, and cross products. This was mainly seen through the lens of physics, but also through my mathematics courses.
Fast forward to my final mathematics course (Calculus III), and I began to learn how to put these ideas together. Suddenly, vectors weren’t static anymore. They were affected by parameters such as time or angles, meaning they became curves.
Now, imagine trying to show this to a student who doesn’t already have a good grasp on vectors or basic functions. It would be basically impossible. The leap from knowing nothing about vectors to working with vector-valued functions is too drastic. However, if one progresses from simple functions and vectors to these more advanced topics, it’s more of a transition than a leap. The topic that proceeds makes sense given the previous material that was seen.
For myself in that class, it meant that once I learned the polar coordinate system and graphing in three dimensions, everything we did was both in Cartesian and polar coordinates. Taking derivatives or integrals was done with both systems because it was the obvious next step to take.
I can now say that I understand what my teacher meant by the material being easy. It’s not that it would be easy to anyone, but that it should be easy for us, given our progression (and since I had the same teacher for all my mathematics classes, I know they had a good idea for the progression).
The implication of this statement though is that the fundamentals are so important. If one wants to make the next logical transition to a new concept, the previous concept must be understood. By rewinding the clock all the way back, one arrives at the absolute beginning of learning the first important concept. If this concept isn’t understood well, then it can have consequences down the line in terms of why a newer concept doesn’t make sense.
Let’s face it: school isn’t made for us to fail. It’s designed in a way such that a student can succeed. Therefore, the progression should be appropriate, and the responsibility is on the student to understand the fundamentals before moving on. Yes, this can be difficult when one is in a class that seems to be moving on despite you not understanding, but that’s when either the teacher or the student needs to step up and take a moment to review. If not, a student is just pushing their problems down the road.
If you feel like you’re making a huge jump in your learning that you don’t understand, it’s likely that the fundamentals you learned aren’t completely absorbed. Strengthen them, and those leaps will become baby steps forward.