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.
As it is no surprise about the education in the US, science in particular has also been suffering. Last week in the April meeting of the American Physics Society, a group of physicists made precisely this claim. The primary data used in the basis of their argument was the decline in degrees in science. This data is certainly consistent with the argument that the US is falling behind in science. There is no debate that good performance in science predicates the choice of a degree in science. That is to say, the decline in degrees in science is mostly a consequence of poor performance in science earlier–rather than the cause of it. So the natural question is: why are kids less interested in science?
The everyday life of a kid is governed by three structures: school, parents, and other media they encounter (including television, internet, and social structure). It follows that one should be able to attribute the decline in scientific interest to one or more of these structures. Suppose we assume two things: (1) that there is a problem at some generation with respect to science advocacy in some of these components, and (2) that the extent to which kids assimilate to the scientific understanding of their mentors (i.e. parents and teachers) is strictly less than complete (that is, in an abstract isolated teacher-student situation, the student can only learn a proper subset of the teacher’s knowledge). These two assumptions ensure a gradual decline in scientific understanding over the course of future generations.
These assumptions make it no surprise that deficiencies in science can only get worse over time. This only leaves two more questions: how did the deficiency begin, and how can it be fixed? I might take a bold but not entirely unreasonable guess that much of it can be attributed to what I previously called curriculum that is not “readily applicable to everyday situations”. Who is in charge of deciding this curriculum? The answer to this is: precisely those who originally learned this curriculum. So again, it is no surprise why nothing changes. Many point to issues like parenting, television, and other media as the problem. But each of these are easily traceable back to the fact that members of each of these components (parents and media affiliates) had a similar education anyway. This further reinforces the claim that the curriculum itself might need to change.
Now if we change the curriculum, the question of teachers’ ability to implement it arises. This can be addressed with changes in college curriculum that prepare emerging teachers for a new curriculum in the primary and secondary levels. This seems like it might be the first promising step to address the issue in the long run. In an era where governmental budget deficits are high, the appetite for further investment in education (itself a long-term yet critical investment) also seems to be on the decline. It’s no surprise that a fair amount of politicians’ value placed on scientific investment puts it on the back burner in a budget crisis given that a vast majority of them have no background in science to begin with. This is yet another reason that a change in curriculum seems more promising in my mind than further monetary investments in education. And since the payoffs for this will not be immediate, it is all the more reason to begin a change in curriculum now rather than later. If a scientific curriculum tied closer to reality and pragmatic obstacles implants itself in those who will become the future generations, the hope is that this will in turn resolve the future problems associated to the parents, media, and politicians.
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.
With the termination of the military’s “Don’t Ask Don’t Tell” policy as of yesterday, I am reminded of the fundamental concept at play that delayed this resolution for so long. This fundamental concept is also at the core of many other issues: gender discrimination, racial discrimination, religious discrimination,…, X discrimination. This fundamental concept is at the heart of contemporary political partisan bickering, the wealth gap, and wars altogether. This fundamental concept is the scarcity of resources. When the set of resources available to a population in insufficient for that population, members of the population will inevitably compete for them. The ingroup bias has become the catalyst for nonuniform distribution of resources.
The ingroup bias follows from the ingroup instigation. The heuristic for obtaining resources is cooperative gameplay. We primitively wish to form alliances with people for the sole purpose of overpowering others (or groups of others) in order to ensure the acquisition of resources. This is the biologically intrinsic (and evolutionally reinforced) tendency which I call the ingroup instigation. Given the objective of the ingroup (to work together), the ingroup bias (the tendency to favor individuals in the ingroup and disfavor those in the outgroup) follows. If we assume this to be true, then the function determining which individuals will group is only governed by what best allows them to overpower other groups or individuals. By default this starts with proximity and special cases of it such as family (note also how fundamental physical forces operate). Then, once some groups gather many resources, it may be beneficial for them to team as well. This can be seen in political alliances, corporate mergers, residential segregation with respect to socioeconomic status (i.e. poor towns or rich towns),…,collisions of galaxies, etc.
It then follows that any minority or group of individuals who have less power become targets of the majority, simply because it is easy to take their resources. Whether it be a minority based on gender, race, sexual orientation, or religion, the characteristic of a group being a minority–not being in the ingroup of those with the quantitative or qualitative power–becomes sufficient for hindering their progress and in turn targeting their resources.
Hopefully one day we will realize that “potential knowledge” is a good with no scarcity that can in turn be distributed to everyone without limits. The amount of knowledge in a system will always be finite (but not constant), yet it will also always be an upper bound on usable resources in society (i.e. the amount of consumable resources in the system is dependent upon the amount of knowledge [on how to create such resources from raw resources] in the system at that time).