Occasionally, there may be some variation in this linear mathematics course pathway, e.g. a high school student may have the option of taking a statistics course somewhere in the mix, which is wonderful, but unfortunately I've often seen this statistics course referred to (by students and staff) as the alternative to a calculus course for those who "aren't as good at math", which, though it may end up being somehow to some degree true about the course because of the way it's taught and/or the way it's been perceived over time (e.g. it's become a self-fulfilling prophecy for Statistics to be the "easier" mathematics course), it is a huge insult to "real" statistics. But, I digress... And perhaps I'll be qualified to write a more in-depth piece specifically on statistics and statistics education a year from now after I've piloted my school's first statistics program.
In actuality, mathematics as a field is not a rigid linear sequence of courses; rather, its structure ought to be considered more like that of science or even history. In science, there are fields like biology, physics, chemistry, etc. We understand that they are distinct sub-fields, yet we also understand that they have shared natural principles, and although perhaps we must begin by studying one at a time, we can thereafter consider how one affects the other in nature, producing disciplines of study like biochemistry or biophysics, and there is even the study of string theory that seeks the elegant union of all sciences. In history, we understand the distinction of US history, European history, and Japanese history, but we also understand that they do at times interact, for how can the story of the US be told without telling the story of the land from where its founders came, or how can Japanese history be complete without mention of its military exchanges with the US? The histories are at those junctions intertwined, two sides of the same story.
The structure of mathematics must be understood the same way: a set of distinct yet interrelated sub-fields. Geometry is not "higher than" algebra, trigonometry is not "higher than" geometry, and calculus certainly is not superior over all the other high school mathematics courses, and certainly not the end-all of mathematics as a whole field (though it is so elegant and really beautiful). And don't even get me started about the so-called "Pre-Calculus" course... What the heck is that anyway? (I taught it for three years and I can't tell you.) I mean, what is it really? What is the essence of it? And does its content have any true relation with its name? (I think not, but perhaps more on that another time.)
Actually, some even consider mathematics as a whole to be a sub-field of science, at the same organizational rank as biology, physics, and chemistry. Biology is the study of life, physics is the study of matter and its motion, chemistry is the study of matter and its reactions with other substances, and mathematics is the study of patterns. And again, everything is interrelated, intertwined; understanding in one area grants enlightenment in the others.
Doesn't this wider perspective of mathematics make it so much more curious and fascinating? College math majors will understand this far-reaching, diverse view of the field of mathematics as they have to take a variety of math courses in their programs, many of which are not prerequisites to others (e.g. discrete mathematics, linear algebra, real analysis, game theory; but calculus is a prerequisite to many of them, demonstrating what I said before about calculus certainly not being the superior, ultimate epitome of mathematics). But of course, these individuals are not the ones who need to be convinced of the beauty of mathematics since they must have already glimpsed it to commit thousands of dollars to higher education in advancing their study of mathematics. Most others, however, come to this realization of mathematics too late, or not ever at all, their traumatic past experiences in the high school mathematics classes they've taken having made them callused to the very thought of every being intellectually stimulated and amazed by anything with the word "math" in it. How students view mathematics in school will 90% of the time be how they view mathematics beyond school. So it is my hope that secondary mathematics education can shift in ways that will steer students away from the rigid, linear perspective of mathematics as a field to a perspective of the truer, broader picture of mathematics as many distinct but wonderfully interrelated sub-fields before those students grow into adults that will perpetuate the former perspective.
I think one way to make this shift is to throw in as many forks as possible into the secondary mathematics course sequence, leading in to as many different diverse types of mathematics courses as possible. Here's one diagram for a diversification of mathematics pathways that I sketched over a year ago:
For starters at my school, less than 10 years old and still very much in its formative stages, I'll be teaching the very first statistics course of this school. I'm very excited to lead this first step at my school towards broadening students' perspectives of the wide, vast, beautiful world of mathematics.
