There is an almost ubiquitous misconception about what mathematics really is, and it’s a misconception that genuinely beckons a correction. I would take a guess that if the average person was asked “What is mathematics?”, they would respond with something along the lines of “well, it’s a bunch of rules that help you find certain numbers”. While this may have been a correct answer long ago, it is far from correct today.
Fortunately, what it actually is can be summarized very succinctly. Mathematics is simply the process of making assumptions and proving what follows. Hence, we all do math on a daily basis—either when talking to one another, or when thinking to oneself: “given what I know, I think that…”. This is math.
This is also why math courses through calculus are terrible—as they are absurdly misleading. Current curriculum is libelous to the discipline of mathematics and its participants, and action needs to be taken to address this.
At this point, high school language arts and composition courses may teach more math than actual math courses. Fundamental to mathematics is logic and its application in the context of sets. Logic was essentially nonexistent when I was in high school. Yet math and language arts classes implicitly assume that students have a solid understanding of it when they are asked to make arguments. Granted, as we are logical entities cognitively, we are trivially masters of logic. But in terms of conveying it communicably, improved training is necessary. I feel the overlooking of this necessity is a grave miscalculation that has hindered scientific thinking (an ability from which every citizen of the world can drastically benefit) for far longer than it should have. This needs to change.
There are two huge problems with the way math is taught at the secondary and lower-division level:
- It isn’t taught as what it is: language, logic, axioms, and consequences that form the foundation of knowledge and reason.
- The intense emphasis on algebraic operations of real numbers makes colloquial usage of the term “mathematics” an unrealized drastic overstatement.
Note that some mathematicians and logicians may refer to mathematics as a particular type of set theory and all of the theorems that follow; but I am more generally equating math to a formal language and all theorems that follow for any choice of a set of axioms and a logic system.
The key point not made (sometimes not even made in a mathematician’s career of coursework) is that math boils down to assumptions and consequences. Philosophy (or what I call archaic mathematics [don’t be offended]) evolved into math once philosophers started making well-defined assumptions and proved what followed. Prior to this, philosophers had assumed in fundamental yet immeasurable/unworkable objects like morality, God, existence, and truth. Their mathematical descendants have respectively transformed these objects into utilitarianism, universe, arbitration, and valence on propositions—thereby allowing consistent consequences. Now, centuries in the making, the ubiquity of the fundamentality (yet undefinability) of the former four objects expresses the collision course of mathematics: if it crashes—if pseudoscience saturates convention, then logic will no longer be a mechanism capable of transporting people.
So, how do we solve the aforementioned problems? One option is with a top-down approach. If professors can solidify the more accurate interpretation of mathematics to students, then those students who go on to teach at the secondary and post-secondary levels can induct the process. One may also attribute some of the blame to state legislation. 48 of the 50 states (the excluded being Texas and Alaska) have boarded the Common Core State Standards Initiative—mostly led by the National Governors Association and the Council of Chief State School Officers and consisting of “parents, teachers, school administrators and experts from across the country together with state leaders”.1 While I see usefulness in standard curriculum of mathematical operations, word problems, trig, and functions, I don’t see the need to spend 4+ years on it (assuming it’s taught efficiently). I’m also not surprised that most students don’t see the use in it considering how far removed it is. I think students could more easily learn concepts from logic and set theory. Such concepts are foundational to math, and hence knowledge, and are correspondingly more applicable to everyday phenomena. Moreover, a solid understanding of them can accelerate the students’ ability to learn the current standards. Even if legislation permitted, it would also take a qualified teacher with a strong background in set theory and logic to make the connections. This brings us to a funding issue, in particular that of salary incentives for qualified graduates.
Recall in the 1960s, after the launch of the Sputnik satellite by the Soviet Union, the US along with other European countries began a “New Math” campaign in which foundational math topics like logic and set theory were emphasized at an earlier age. The program broke down within a decade due to pragmatic issues: parents not being able to help their kids (an illegitimate argument, since after all, it is a reform and reforms mean new things) and ineptitude of teachers unfamiliar with the concepts (a legitimate argument that could be tackled with incentives for qualified graduates, which could also mitigate the first issue). The only other legitimate criticism I can find concerns the trade off of spending time on computational abilities versus investigative abilities. While I find the first to be important, I claim high school and college graduates could benefit more from the latter. I propose that primary education in math remain unchanged, and that we introduce logic, set theory, etc. at the high school level with problem sets that are roughly 50% computational and 50% proof based.