All of the things I have posted on my blog are really cool aspects of mathematics. And they do fit into mathematics curriculums perfectly. For instance, while going over prime numbers in fifth grade, you can teach why they go on forever. It isn't too difficult of a concept, it just requires a couple seconds to explain what a factorial is.
But there are lots of changes that can be made in mathematics education that will enhance not only the fun, but the application as well. In my opinion, there are 3 P's to cool math concepts, that a good chunk of my posts fall under. They are proofs, patterns, and practicality.
I have done numerous posts on proofs and patterns, but very little on practicality. Last week's post was a practicality one; it studied a type of problem that you run into in daily life. True, you hopefully won't run into that situation when you are being held captive by the cops, but certain situations in sports and economics will stick you with that type of choice to make, where the dominant strategy isn't always the best.
Practicality is the center of school education. School is preparing you for the outside world, getting you ready for when you have to pay your taxes, finding a good discount at the mall, or even playing a round of poker. And school puts you through the sequence of Algebra, Geometry, Trigonometry, Calculus, and if you continue forward, you might run into Statistics, Linear Algebra, Computer Science, and Discrete Mathematics.
On the topic of mathematical areas, the survey link at the top of the page is for some data for my presentation coming up in New Delhi, India. If you can take a moment to fill that out, it would be great. And it pertains to branches of mathematics.
Back to what I was saying, is this Algebra/Calculus sequence really what we need to succeed in life? Andrew Hacker, a political scientist at Queens College, published an article taking his stand on the topic. The article is a little long, but it is truly worth the time. Click here to read it.
Vi Hart, a "Mathemusician" who makes some really amazing videos for Khan Academy, recently made a video showing algebra and how it is significant, as well as really amazing. This video is also really cool, whether you read this article or not, but the inspiration for it is clear. Click here to see the video.
Coincidentally, I got to meet her at Gathering 4 Gardner this past March, and I found out about this through some other people I met at Gathering 4 Gardner.
I also have a strong opinion on this case of whether Algebra is necessary to teach in school. Though I don't really have a qualified background like Andrew Hacker and Vi Hart, I do speak for a student who just completed the Algebra curriculum and is watching my peers go through it.
Hacker brings up a very good point that Algebra is not a good model for real life situations. When playing football with friends, you are not calculating the distance of the throw by finding how long the parabola representing the arc made by the ball after the quarterback releases it is before it crosses the x-axis while determining how many yards per second the ball is traveling so you can plug that into d = rt and by solving for t, determine how long it will take for the ball to get to the ground, and then plugging the x-intercept, which took approximating the radicals in the quadratic formula when realizing that the equation could not be factored evenly, in for d to find the rate, which you must then switch your speed to in order to be in the right spot at the right time so you can catch the ball without having to dive or stutter. That is completely crazy. What a good player does is approximates where the ball should land, and tries to judge what speed he must take to get to that spot, understanding that he may have to stutter for a second or lunge forward to make the play.
This is absolutely correct. But also, consider the fact that subconsciously, you are following those exact steps, just without the numbers. You must consider the speed the ball is going, when it will be at a reasonable height off the ground, and how far you have to go in order to receive it. Of course, you are not actually taking data points to construct a parabolic figure and using some distance formula I've never heard of to get a number that will be divided by however many milliseconds it took to get from point A to point B to find the precise velocity of the ball. But we do need to have somewhat of an understanding of this information.
Hacker would rather have every student's big focus be quantitative reasoning: mathematics that can be applied to real-life issues. This also makes sense, as you need a grasp on how to keep your numerical aspects of life under control.
Yet, people already seem to be doing fine. We seem to have a well enough grasp of quantitative reasoning to keep America's middle class strong. Yes, the economy has gone under due to people buying things that they couldn't afford. But even if quantitative reasoning was the class of study, there would still be people that didn't pay well enough attention and not using their money wisely. And just like how people would struggle through algebra, they would struggle through quantitative reasoning.
If people already have enough quantitative reasoning skills to survive in the economy, then we don't have to teach it in school. However, we can get a little more advanced in quantitative reasoning, which will require a little algebra.
Again, I want to allude to the fact that I have no qualifications or experience in teaching mathematics. I do want to propose what I think could potentially work well from seeing news, my peers, and the courses I have taken in math and science.
I do believe that a basic understanding of algebra is somewhat necessary. Aside from the fact that quantitative reasoning will begin to require algebra, there is also a more personal reason. In late middle school to early high school, you are not yet sure of what your career will be. Through a study of your most important algebraic concepts incorporating the things Vi Hart showed us in her video, and the things I post on my blog on a weekly basis which neither are taught in the current curriculums, students might want to continue to study mathematics further. If we are adding some of these patterns and proofs into the classroom, then we are adding a chance of inspiring students to pursue a mathematical job, which requires a calculus sequence more than a quantitative reasoning sequence.
After you have completed your math fundamentals from second to sixth or seventh grade, you can then take your traditional Pre-Algebra or Introduction to Algebra course. Following that can be a Fundamentals of Algebra course, which would involve the basic concepts you cover in algebra:
- a moderately rigorous study of linear equations (just a little less detail than the current curriculum)
- a less rigorous study of quadratics (solving for x and some simple transformations and interpretations)
- an overview of the other four parent functions (radicals, cubics, rationals, and absolute value)
- a very brief overview of trigonometric functions (just a basic idea, no involved studying)
- some basic number theory concepts (types of numbers, sequences)
- a review of probability from previous years
- an incorporation of details that will spark interest in the minds of the students
Rather than covering two full years of algebra, you can put the most important parts into one. You can cut out that month of factoring quadratic equations, the week of conversions between slope-intercept and point-slope forms, and the daily twenty minutes spent reviewing number eighteen from the homework because it required too many steps. Though algebra isn't so necessary throughout regular life, you need a basic understanding for most fields of science, engineering, and technology, as well as for pursuit in math itself. By erasing most of algebra from your mathematics curriculum, it becomes impossible to teach high school physics and chemistry classes while kids are still working through their quantitative reasoning course.
After students have a grasp of the idea of algebra, they could then enter a Quantitative Reasoning course, but not the same exact type as Hacker proposed. This type of course might have:
- some finance concepts that will be critical for life outside of school
- a continuation of the probability concepts from the introductory algebra course
- an overview of game theory with a focus of real-life models and situations
- an overview of economics with a similar focus
- an overview of mathematical logic with a similar focus
- an overview of inductive and deductive reasoning with a similar focus
This type of course will prepare you not only for the situations directly involving numbers, but making rational decisions in whatever field you are in, and starting to think critically, which is a popular goal in the people I know and admire.
After this, a student may be a freshman, sophomore, or junior in high school. By then, they will have an idea if they want to go into a STEM (Science, Technology, Engineering, Mathematics) field, or if they would rather pursue other interests. If they would like, they could follow a calculus type of sequence like so:
Integrated Mathematics A (some more detailed review of algebra, and a thorough geometry course)
Precalculus (some more trigonometry, and a thorough precalculus course)
Calculus (a thorough calculus course)
This would prepare them for a job involving lots of higher level mathematical thinking. Students who do not plan to follow a mathematical career can do a more practical mathematical study, such as the following:
Integrated Mathematics B (some review of algebra, an overview of geometry/trigonometry/precalculus)
Probability and Statistics (a thorough statistics course)
Discrete Mathematics (a course that teaches combinatorics and number theory)
This series of courses would be much more relatable to daily life. The STEM students already learned the fundamentals necessary in Quantitative Reasoning after their first algebra course, but the students with other interests can become more advanced in the practical areas of mathematics. By the way, integrated mathematics is a course that combines Algebra and Geometry concepts that I saw as a one year version of Algebra II/Geometry at Phillips Exeter Academy, which is one of the six high schools I am looking at. It seems like a good idea because while we still use the traditional calculus sequence, at least you are getting through it quicker. This integrated mathematics course is important to keep, because the algebraic and geometric models found in areas like statistics and discrete mathematics must be interpretable. However, it would be much less detailed than the course that I named Integrated Mathematics A, since this thorough understanding of Algebra II and Geometry is not necessary for the jobs that these students would want.
Then, they would get their thorough statistics course, which would prepare them for analyzing or collecting data, as well as teach some more advanced probability calculations. If they get through that, they would be ready to take a Discrete Mathematics course, which is also mathematics that is applicable to real-life situations.
I was not surprised when I saw the reaction of the mathematics community, to immediately try to defend Algebra's case. But we did not jump to defend pi's case when Bob Palais and Michael Hartl presented tau. Yes, algebra has so many cool things and does have some practicality. But just like with pi, math education has to change.
Hackler wanted to take algebra out of your required courses. I disagree with this as well. Without a basic algebra course, students cannot understand lots of the proofs and patterns people like myself talk about, they can't find out if they would like to move forward in a STEM profession, and they can't do as much in their Quantitative Reasoning sequence. However, instead of spending three years on Algebra and Geometry, spend two on it and use that third year to teach Quantitative Reasoning and let the students choose where they want to go.
Yes, I have no qualification or background in teaching mathematics to students. This course selection may not be in the correct order, have the correct names, or even be the correct courses. But as a student watching this whole process happen, I can tell that some students would value from something like this STEM-preperatory sequence while others would have much more of a benefit from something like the other sequence. And everybody gets the best of both worlds.