The Forgotten Side of Math: Conceptual Analysis
Discussing the Difference between Computations and Concepts behind Mathematics
Mathematics had, for the longest time, been well-known to be a discipline of computations. From the beginning of our education, we were taught how the basic arithmetic operations worked as representations in the real world:
- let us say we have four apples in this basket, and we add another five more, now we have nine apples total
- how about we say you have ten oranges and I took away four, now we have six apples total
It is at this point in mathematics that we find the process the easiest. Even now, we still revert to that same old, same old principle of arithmetics. To some extent, we still indulge ourselves with the idea of fractions represented as a piece of pie partitioned into equal slices. However, what made the next stages difficult for us? How did we go from enjoying arithmetics to struggling in the depths of Algebra? Even more so, Calculus?
Well, at some point in our education, the conceptual background of the mathematical topics we were taught began to stop. Conceptual analysis, the tool that enables us to model the world in the language of math, had been lost. But, what is conceptual analysis? Why and how was it lost as we progressed further into higher levels of math?
Definition of Conceptual Analysis
Conceptual analysis is the partner to computational analysis. Instead of focusing on techniques or steps in solving a problem, it focuses on the reason behind why these techniques or steps are being used by utilizing the background and the connection of a particular topic to other topics. It showcases in which situation a mathematical topic is used and why.
At its most extreme, conceptual analysis allows the connection between pure mathematical concepts into applications in physics and engineering. It shows that mathematical topics, from different levels of education, are not mutually exclusive, that is to say, they are not isolated to just themselves.
Conceptual Analysis within Pure Mathematics: Algebra, Limits, and Calculus
Most teachers today settle for, do this, do that, in their discussions. Oh, the derivative of a constant is zero. Just copy the formulas on the board for determining a derivative of a function, then answer the seatwork. But, who cares? Why does it matter? How is this useful?
See, the people who regard Algebra to be useless in real life are the very people who were never given the conceptual background of what Algebra is. To put it simply, Algebra is the mathematics of the unknowns, where these unknown quantities are represented as letters, or in Algebraic language, as variables. Algebra provides different methods to solve for these unknown quantities using already known ones. That’s how the quadratic equation works: it determines the value of x for quadratic functions.
Algebra isn’t at all exclusive to its domain. Calculus can be seen as an extension of Algebra, with the introduction of limits. By its very basic principle, limits allow us to set an independent variable to approach a given numerical value, or abstractly, towards infinity. This is done to see how a function:
- behaves towards infinity, an abstract quantity we are not capable to numerically approximate
- becomes equal to, as it approaches a discontinuity in its graph
This would imply that limits, itself, is capable of using different Algebraic functions. By extension, limits would also work on Transencendal Functions, a group of functions outside the domain of Algebra, such as Trigonometry, Exponential, and Logarithmic Functions. Hence, a pre-requisite to mastering limits is to master Algebraic and Transcendental Functions.
How is the idea of limits connected to Calculus? Well, Calculus is defined as the mathematics of change in terms of ratios between the difference of two quantities in terms of their coordinates in the Cartesian plane. This ratio takes the form of a line in two-dimensional space: how much two points change in the y- and x-directions. This construction of change using lines provides a weakness in functions that curve. Calculus aims for these ratios to be as precise as possible by eliminating the amount of error yielded by the unfit approximation of a curve. It does this by taking these two quantities and making them approach each other until the two points converge into one. The keyword is approach. Does that sound familiar? Limits are used to set the difference between these two quantities to approach 0. This yields an exact measure, instead of an approximation. This form of Calculus is often referred to as either Calculus 1 or Differential Calculus. The usage of limits in Calculus 2 or Integral Calculus is different entirely.
To answer the question of why the derivative of a constant is 0, let us instantiate a function y = 5. This function generates a straight horizontal line that intercepts the y-axis at the point (0,5). This line does not rise and continues from negative infinity to positive infinity as (x,5). The mere idea that it does not rise would cause the difference in the y-direction to be 0, yielding the numerator of the ratio to also be 0. Hence, the derivative of a constant is 0.
Perhaps, the question of the other coordinate may arise. What happens if there is no change in the x-direction? Well, configurations such as these are not considered a function; a function cannot map its x-coordinate to more than one y-coordinate.
These explanations aren’t at all part of computational analysis since they tackle an entirely different part of mathematics. For a brief example, the Power Rule states that if an independent variable is raised to some exponent, that exponent is copied as a constant in the front, and that same exponent is subtracted by one. The idea behind why this makes sense is in conceptual analysis, by placing an algebraic function raised to an exponent to the definition of a derivative.
Conceptual Analysis between different disciplines: Calculus and Physics
Physics, at its most basic, is the study of motion through space and time. In this configuration, motion, or the movement of an object, can be expressed as a function in either Algebra or Transcendental. The measure of the change of these movements can be expressed as limits to nullify error and can be generalized further into Calculus.
How Computational Analysis began to eclipse its counterpart
In my perspective, computational analysis stopped being a concern in education because it seems to be a luxury. Sure, it explains why and how math topics work and how they relate to each other, but by itself, it is incapable of solving problems. It may help determine which math topic to apply to a given situation, but it cannot solve it.
There are also some mathematical topics whose conceptual counterparts are difficult to explain, either because it is vastly connected to a multitude of different background topics, or because it is extremely abstracted. It takes time for them to be discussed, and some curriculums or professors are incapable of sustaining that time demand nor are professors prepared to explain them clearly. Hence, most professors skip the ideology and go straight to teaching techniques to solve them.
To some degree, problems can be solved without the need for conceptual analysis, further implying its luxury status. Even more so, most professors are only concerned about the student’s education in that course and do not regard its importance as a pre-requisite to the student’s next courses, i.e., pre-calculus, basic calculus, differential calculus, integral calculus, multivariate calculus, and differential equations. Without sufficient understanding of why and how limits work in basic calculus will confuse the students as they go through differential and integral calculus.
Computational analysis is best used specifically for focusing on only one topic. Conceptual analysis is best used generally for a wider view of the different disciplines. For a student studying mathematics to truly master it, both counterparts are necessary.
