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.
It’s no controversy that education appears to be under par in comparison to what we might hope. I claim this has not only to do with home/environmental conditions for individual students and the ever omnipresent media (and lately, social networking media as well), but also with the curriculum in the classes they take–particularly math and language arts courses. What seems to be the case is that the math courses focus on topics that, while enhancing the mental discipline of students, fail to teach concepts that are readily applicable to everyday situations. Language arts courses emphasize composition (and in particular, stress the ability to make effective arguments) under the premise that students are already well-versed in the grammatical aspects of language (an assumption so far from the truth that the very idea of expecting them to write cogent argumentative essays is ludicrous).
I’d recommend keeping the middle school curriculum essentially the same (introducing key concepts of elementary algebra and geometry), but I’d vehemently oppose the idea of continuing this sort of curriculum for another four years. In 9th grade, students should be introduced to key concepts from set theory (set notations and operations, functions, equivalence relations, bijections and counting principles, and concepts from elementary number theory, including divisibility, prime numbers, and the fundamental theorem of arithmetic). With this collection of concepts in their inventory, they can spend 10th grade covering concepts from logic and proof writing–and applying them to concepts they learned the previous year (as well as everyday phenomena).
11th grade could then be devoted to a more serious algebra course–covering concepts of operations more thoroughly than the 9th grade course, identities, concepts like associativity and commutativity, matrices, inverses, basic definitions and examples of groups, rings, and fields, polynomials, and comparisons between integers, rationals, reals, and complex numbers, and the fundamental theorem of algebra. If students only took these courses and earned a C in them, I’d still argue they were better prepared for life than earning an A in the traditional courses.
An additional 12th grade class covering geometry more seriously could entail concepts of lengths, triangles, trigonometry, polygons, area of polygons, polytopes and hypervolume, axiomatic Euclidean, spherical, and hyperbolic geometry and basic properties of spherical and hyperbolic triangles, approximation of areas/volumes and method of exhaustion (leading up to a freshman course in real analysis (to replace calculus)).
I’d recommend middle school curriculum to focus a little more on grammar and sentence structure. A 9th grade course could solidify this with a rigorous treatment of the core concepts of language: parts of speech, morphology, syntax, and grammar. Concepts like word families, clauses (independent, dependent, subject, and predicate), transitive and intransitive verbs, and sentence diagramming should be emphasized.
A 10th grade course emphasizing argumentative writing would appropriately accompany the 10th grade math curriculum in logic and proof writing. I’d recommend abstaining from requiring students to read nontechnical works until the third year of language arts. This way, rather than getting distracted by a fiction or nonfiction story as a platform for an argument, emphasis is instead placed on the concept of argumentative writing itself. Smaller readings should be used in this course.
An 11th grade course could then more effectively do in one year what four years of language arts typically try to do. This course could focus on work-based and research-based extended essay writing.
Curriculum comparable to the AP English Literature course could make an appropriate optional fourth year class.
I thought I’d attempt to come up with a consistent convention of classifying mathematical statements. To do so we will simply think of the collection of statements as a partially ordered set where is interpreted as . And for each statement/node/point there will be two -valued functions that represent the number of incoming and outgoing edges (i.e. immediate antecedents and consequences). A point will be called a corollary if it has one incoming edge. A point will be called a theorem if its set of successors is a “large” subset of the poset (even if it is a corollary). A point will be called a lemma if at least one of its immediate successors is a theorem. A point will be called an axiom if it has zero incoming edges. And a point will be called a proposition if it is none of the above.