Tag Archives: teaching

What Math Really Is

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.

Science in the US

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.

[1]  http://news.yahoo.com/crisis-us-science-looming-physicists-warn-135405711.html

Proposed Changes to Secondary Math and Language Arts Curriculum

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)).

Language Arts:

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.

The Problem with the Public School System

A recent report on the Atlanta Public School System concluded 178 teachers at 41 schools were involved in a cheating scandal where “the educators, including 38 principals, were either directly involved in erasing wrong answers on a key standardized test or they knew–or should have known–what was going on”.1  Better scores on standardized testing would mean better funding for the district.  The scandal was brought to light after a 2009 mark in the district’s improvement along with erasing patterns on scantrons.  Back in January the district was notified that it had until September 30th to make changes or face losing their accreditation.2 Jennifer Oliver, vice president for communications for the Southern Association of Colleges and Schools, noted in an email interview with CNN at that time that “a loss of accreditation could mean that ‘graduating seniors may be limited in scholarships and college admissions if a program requires students to come from an accredited institution and/or uses accreditation of the high school as part of the criteria for evaluating students'”.

This is merely a side effect of the dogma of rewarding districts that perform well on examinations proctored by the districts themselves.  Although this in itself would be difficult to fix as too many proctors are needed.  Perhaps for smaller districts, adjacent districts could swap proctor personnel for test duration and exam submission, and larger districts could bring in an army of state proctors;  however, this seems like it may be impractical.  We could also ask the question about why students are not doing sufficiently well on the exams in the first place.  This problem seems to be twofold:  teacher quality and environmental circumstances.

Two possible reasons why teacher quality is poor are funding and pay requirements.  Better funding for salaries could attract better, qualified faculty, but alas, the poor performance on tests makes this a cyclic problem.  There is also the issue requiring higher payment for applicants with graduate degrees.  This makes no sense whatsoever.  If the requirements for a teaching position are constant, what difference does the degree make?  All that matters is if the applicant can teach.  Removing this requirement could allow the education industry (among others) to hire applicants with advanced degrees (who may be more qualified) at the same pay.  This would increase industry demand for Masters/PhD applicants (since they would have no additional hiring cost) and in turn increase consumer demand for such degrees.

There is also the environmental factor.  Neighborhood violence can certainly impede the success of education.  This seems difficult to address, particularly while gun laws remain a joke.  Who cares about future academic prospects when gun/drug hierarchies saturate a district and offer an immediate sense of autonomy and control–or a distracting threat for that matter?

[1]  http://edition.cnn.com/2011/US/07/06/georgia.school.cheating/

[2]  http://edition.cnn.com/2011/US/01/18/georgia.schools.probation/