Math Awareness Month Part 2: Infinite Series

Today's page for Math Awareness Month is about a recent video that caused some huge debate. I saw the video a month or two ago, and was very intrigued. I showed it to some of my friends, and we were arguing about the content for quite a while. It also spread rapidly around the math department at Andover, with some teachers bringing up in their classes.

Take a look at the page and try some of the exercises. You will find the outcomes very interesting and mind-boggling. The concept of infinity is difficult for any human being to grasp, making it tons of fun to think about.

http://www.mathaware.org/mam/2014/calendar/infinity.html

Comment below what you think of the video. Do you think it is accurate? What do you think the fallacies are? How could this be a part of string theory if it is mathematically flawed?

In math class last week, we were given the following problem:

I then did the math and determined that the limit would be -1/12. I then called my teacher over, and pointed to that answer. Recalling the video, I asked him if I could rewrite that -1/12 as 1+2+3+4+5+6+7+... as my final answer. Thankfully, he got the reference. In addition to being a funny anecdote, the fact that people got the joke shows how wide of an audience this information has reached and captivated, which is amazing to see.

Math Awareness Month Part 1: Magic Squares

I explained a bit in my last post that April is Math Awareness Month, as well as linked to the poster on www.mathaware.org. In honor of this occasion, I plan to make my posts this month relevant to the pages on the website and the mathematicians hosting them.

April 1st was a day on magic squares, and I am honored to have been the host of that page. There is a recent performance of me doing it, tutorials on how to make various magic squares, and different activities and questions that can further your magic square experience. Click here to see the page.

Math in the News: The Influences of Politics

Though mathematics is normally a pretty concrete subject, people's intuition for it is not. Probability and statistics in particular is a very difficult area for us to grasp, as you've seen with the Monty Hall Problem I talked about a few weeks ago.

Here is another example of mathematical aptitude being influenced by an outside source, but this time, it is not just a matter of lack of skill or desire to be correct. It is also influenced sometimes by political views, as Kevin Drum shows in this news article. Check it out!

http://m.motherjones.com/kevin-drum/2013/09/politics-destroys-math-ability

Math in the News: Teacher Salaries

One of the biggest issues in math education is teacher quality. I have discussed this in both of my TEDx talks about this topic.

We explained it the best we could in our Capstone Research Paper (link is at the top of the page), but I think this New York Times article that just came out describes it fantastically. So, I couldn't resist posting it here.

http://www.nytimes.com/2011/05/01/opinion/01eggers.html?_r=0

Enjoy!

Math in the News: Rota's Conjecture is Solved

One of the things that lots of people seem to be oblivious to is that mathematics is developing and innovating just as much as any other discipline, which I allude to in many of my presentations. There are many conjectures, or unsolved problems, out there that mathematicians are working on and trying to prove or solve.

Rota's Conjecture was a problem like this, in the branch of matroid theory. This is a diverse area of mathematics that isn't taught or mentioned in the American school system (another concept I allude to in my presentations). So when I read this article about Geoff Whittle solving the problem, I thought it would make for a great post. Here is the story:

http://phys.org/news/2013-08-rota-conjecture-year-old-math-problem.html

Math in the News: Giving Students More Independence

About a month ago, I was speaking at a TEDx conference that was themed around education. During these TEDx conferences, there are always live speakers as well as videos chosen by the organizers that fit well with the occasion. One of the videos we watched really got us thinking. Here it is:


This talk is not directly mathematical. It isn't meant to be a math education talk. But, these points apply to math education as well.
For example, the mathematician Euclid had nothing. He pretty much was starting from scratch, just like the kids described in the talk. So what did he do? Well, after creating five axioms (foundational ideas that can be concluded with basic logic), he started asking questions, and answering them with mathematical proofs. He would then think about other questions, and find ways to answer them using everything he had discovered. This was the basis of his book series called Elements.
Students should be able to approach math in a similar way. A teacher could lay out five axioms (or develop them with the students), and then back off. He or she might also provide terminology (line, triangle, square, circle, angle, bisect, trisect, etc.), but the students can discover the rest. By learning math this way, they will understand everything they are doing so much better because they created it. For more on that topic, check out my Capstone Research Paper (link is on the top of the page), and scroll to "Chronological Cognition."

Math in the News: Playing with Pentagonal Tilings

This last weekend, I attended MathFest 2013, which is held by the Mathematical Association of America in Hartford, Connecticut. I gave a talk on magic squares, and got to see many other fascinating speakers. Many of my future posts will be content I learned at this convention.

One of the speakers was Frank Morgan, who talked about tiling the plane with different polygons, primarily pentagons. During the presentation, he mentioned that he wrote a news article about it in the Huffington Post. Since the presentation was so interesting, I thought this article would be a perfect "Math in the News" post.

http://www.huffingtonpost.com/frank-morgan/bees-honeycomb-mathematics_b_1318258.html

Math in the News: The Twin Prime Conjecture

A common question people wonder is "What do mathematicians do?" People know that scientists make advancements in science and engineers make advancements in engineering, but what about math? Similarly, mathematicians make advancements in math.

This is often an odd concept to people, since math seems like it is all figured out. How could it be taught so definitively if there are still things to figure out? But, there are tons of conjectures out there that people are trying to prove.

One of them is called the Twin Prime Conjecture. You are probably aware that a prime number is a number that is only divisible by one and itself. There are an infinite number of prime numbers (click here to see a proof). Twin primes are two prime numbers that differ by two. For instance, 3 and 5 are twin primes because they are both prime and differ by two.

We know that there are an infinite number of primes, but are there an infinite number of twin primes? This is one of the things that has yet to be figured out, and the Twin Prime Conjecture is asking this question.

Recently, a big jump was made in trying to prove this conjecture. So, I will post the article from News Scientist that describes the latest advancements on it.

http://www.newscientist.com/article/dn23644-game-of-proofs-boosts-prime-pair-result-by-millions.html#.UcxiPkIWHFI

Why Has America Dropped to 32nd in Mathematics?

In school for the last four months, we have been doing a project where we find a contemporary issue in America or in the world, and figure out how to fix it through research, conducting an interview, and putting together a research paper, promotional piece, and presentation. My two friends and I decided to do ours on the issue of America having dropped to 32nd in mathematics in comparison to other countries in the world. Watch this video we made to see a general overview of the issue:

I also strongly encourage looking at the following website, which gives a fantastic visual overview of some of the facts and statistics of the issue of our mathematics drop. Click here to view it.

I'm not going to go into detail about the solutions we talked about, but I linked to our research paper at the top of the page. It is pretty long, but it goes through all of the things we could find to fix the problem. I think it is worth skimming.

Our group will be doing our presentation on Monday. I am also giving two TEDx talks over the summer about this issue. I will post links to them once I have done them.

Math in the News: The Pythagorean Theorem in Football

I mentioned last week that I wanted to focus this month on the famous Pythagorean Theorem. Since I usually reserve the second week for a "Math in the News" segment, I was looking for an example of the Pythagorean Theorem in the news.

Though I couldn't find a controversial article about it, I did find a fantastic video that showed its applications to football. And it was done by the National Science Foundation in their news section, so I think it counts. Click here to see the video.

My friends were just joking with me a few weeks ago about how in a sports game, players wouldn't actually be thinking about the math behind what they are doing. It's true that they probably aren't solving for the hypotenuse of a triangle in this dynamic moment. However, they actually do need to take an educated guess as to what angle and speed they should be running at. This does require number sense, which is definitely mathematics. The actual math might not be applied directly, but it is definitely present.

When I talked about connecting game theory with penalty kicks in soccer, this example actually uses the mathematics. Teams do hire statisticians to analyze the skills of the players on both sides, and will give their players advice on what to do. I'm sure coaches of football and other sports do this as well. Why else would they watch the opposing team's previous games to prepare themselves?

I had never thought the Pythagorean Theorem could be applied that way, and it is another reminder of the influence that mathematics has on our lives.

Math in the News: Facebook Math Problem

A few weeks ago, I suddenly opened up my Facebook page to see that I was tagged by several people in a post about this article. Once I saw it, I immediately wanted to discuss it on Cool Math Stuff.

First, try to solve this math problem:

6 ÷ 2(1 + 2) = 

Even though it seems simple, there is lots of debate about the answer on social networking sites. Click here to view the article.

When I first saw the problem, I thought the answer was undoubtably 1. When I did the problem, I had multiplied by two while I was simplifying the parentheses. So, I assumed that people who got 9 as the answer were forgetting the order of operations.

Before I talk about the article, I was excited to see that this was a debate between people. Any time where the general public is talking about mathematics, especially in a argumentative way, is a huge accomplishment for society.

Reading the article, I was intrigued to find myself in the minority. I did not think of the parentheses as a substitute for the multiplication symbol, but a quick way to write two times the quantity one plus two.

What if the problem was written like this?

6 ÷ 2 • (1 + 2) =

If the problem were written like that, I would have gotten 9 as my answer. Since the 2 was separated from the parentheses, "Please Excuse My Dear Aunt Sally" would put the division first and then the multiplication.

However, without the dot there (more formally known as an interpunct), I do feel that the answer should be one. It seems like the most practical perspective on the problem.

Take an expression like 2a. It is implied that the 2 and the a are a single quantity. If you were using it in an actual problem, it is far more likely that you would be dividing, say 8, by the whole quantity 2a than 8 by 2, and then multiplying this by a.

Similarly, let's say you had 2(a + 3). Again, there would be no doubt that you should distribute the 2. This problem would then become 2a + 6, which seems like the correct thing.

If you ended up with 1 ÷ 2(a + 3), there would be no practical case where the one were to be divided by the two first.

Also, I think that the order of operations was put into place to make algebra and geometry more structured. For instance, say you had rhombus where the diagonals cross, or bisect, at a (3x³ + 9)° angle, and you had to find the measure of the sides which were of length 25 - 2x². As you probably know, a rhombus is a quadrilateral with all four sides equal (a square has all four sides and all four angles equal).

There is a theorem that states that the intersection of the diagonals of a rhombus is always equal to 90°. I think this is pretty cool on its own, to know this angle all the time. So, we can assume that 3x³ + 9 must equal 90°.

3x³ + 9 = 90
3x³ = 81
x³ = 27
x = 3

We must then plug this into the expression equal to the side measure to find our answer. First, let's look at the expression.

25 - 2x²

What are we being asked to do? Since two is a coefficient for x², we would square the x first, then multiply by 2, and then subtract that from 25. This would be agreed upon by all algebra teachers. What happens if we plug the 3 back in?

25 - 2(3)²

If there was no order of operations, we would probably do this problem from left to right. This would give us:


25 - 2(3)²

23(3)²

69²

4761

You can probably already tell that 4761 is way bigger than this side length would be intended to be. Even if you didn't make this estimate, this attempt at the simplification was nowhere close to the agreed order. By implementing the order of operations, there is no debate that the order should be PEMDAS. This clarifies lots of algebra when you are substituting terms into equations.

What if the expression were 108 ÷ 2x²? I still think most of us would agree that the 2x² is meant to be its own term, and would therefore be divided as a quantity into the 108.

But after substitution, we get the following:

108 ÷ 2(3)²

108 ÷ 2(9)

If we did the 108 ÷ 2 first, we would end up with the wrong answer. Even though the surface definition of PEMDAS would ask for this to be done first, it is not a practical approach to the problem.

However, I did see that the SAT or ACT would expect students to receive 9 as an answer. Since there is clearly not a correct perspective to have, I would encourage you to comment what you thought the answer to be and why. Just like with Pi vs Tau, this is a post that is actually a lot of fun to debate about.

The Museum of Mathematics

I have been trying to find a news story once a month that pertained to mathematics that I could discuss. Last month, I ended up using a website I found, but this time, I actually got a news story. On Sunday, I was told by multiple people that the Museum of Mathematics was on CBS, which was exciting for me. Click here to see the news story.

I have met Glenn Whitney on a few occasions, including at Gathering 4 Gardner in Atlanta (click here to see a post I did about that experience) and the World Science Festival Street Fair in New York City. Each time I see him there, they have set up their Math Midway, which features many fascinating ways to have fun with mathematics, including the tricycle with square wheels and a machine that solves quadratic equations by the crank of a lever.

I visited MoMath on December 27th, which might have been a mistake on my part because it was packed with people. However, I was happy at the same time that a whole room of people were going to walk out seeing mathematics in a new light.

On my blog, my posts are mainly about pure mathematics. Pure mathematics analyzes math simply for the sake of analyzing it, while applied mathematics looks for its applications to science, engineering, finance, and society. I do have many posts labeled Practical, but even these are strictly mathematical. I never introduce science and engineering concepts (mainly because I don't know them).

MoMath gave me the opportunity to learn how math can be applied to other fields. I have ridden the square wheeled tricycle, which is a way to apply mathematics to transportation, in a fun and creative way. Using some geometry, you can find a path for any shaped wheels, which I found pretty cool.

If you are interested in pure mathematics, that is what this blog and many others cater to. If you are interested in applied mathematics, go to MoMath. It really is a fun way to look at math that you never would have done before.

Pi vs Tau: Pi's Rebuttal

As you may know, I have been strongly intrigued by the tau movement. This movement is promoting the idea that we should not be using pi as the circle constant, but rather 2π, which has been renamed with the greek letter tau.

This movement was started by Bob Palais of University of Utah when he wrote the article Pi is Wrong which was published in the Mathematics Intelligencer. Physicist Michael Hartl proceeded to write The Tau Manifesto, and founded Tau Day on June 28th (instead of Pi Day on March 14th) as well as the website www.tauday.com. This gave dozens of reasons in geometry, trigonometry, physics, and statistics why tau is more practical and natural than pi. On a side note, I currently have the world record for tau memorization at 2012 digits, but that's nowhere near the world record for pi memorization at 67890 digits.

If you have some more time, The Tau Manifesto is a fascinating read. I'd also recommend watching this video done by Vi Hart for an overview of the tau movement.

You can also click here to see all of the blog posts I have done giving details on why tau is better than pi.

Just two days before writing this, I found that someone was inspired by Hartl's Tau Manifesto to write The Pi Manifesto, which presents an interesting rebuttal for tau. This wasn't recent news in mathematics, but it was definitely news for me!

I found that all of the reasons for either side can actually be debated. I think this is the first post where there is actually a debate over what is the right answer.

An argument for tau is that the radius is what a circle is measured by. Dictionary.com defines circle as a closed plane curve consisting of all points at a given distance from a point within it called the center. Clearly, the radius is the main measurement here.

Pi supporters would then argue that the radius can only be found by taking the diameter and dividing it by two. When looking at a circle on paper, it is impossible to pinpoint the center and find the measurement out to the end.

Yet, constructing the circle requires knowing the radius. If you were to use a compass, you must put the point where you want the center to be and trace a line with a constant distance around it. This is the true way to form a circle. For practical uses, you would need circles to see what restaurants were within ten miles or something. In this case, ten miles is the radius, and you are constructing a circle with this measurement.

This example requires the choice between constructing the circle easier or deconstructing the circle easier. The construction requires the radius, while a circle already given uses the diameter.

An argument for pi is that the area formula, which is one of pi's most common uses, is messed up by this change.

A = πr^2  –>  A = 1/2τr^2

Yet, tau-ists argue that this is more natural. First, there are many other shapes that have 1/2 at the beginning of their area formula. Triangles, trapezoids, and n-sided polygons all use formulas with a 1/2 in it. Second, the area formula can be proven (I will in a future blog post), and this proof's last step is to multiply 2πr^2 and 1/2. It turns out to be more natural just to tack on the 1/2 rather than hiding the 2π with it.

This is again an argument where we need to choose between having a natural number or an efficient number. It is clear that tau is the constant that belongs, but we have an opportunity to simplify the equation.

Both manifestos list other reasons that take longer to explain, but are very interesting (I think a lot more interesting than the two above). These involve finding measurements of the unit circle, graphing trigonometric functions, and rewriting Euler's identity.

Because math is such a definitive discipline, it is rare for Cool Math Stuff posts to have comments. I strongly encourage you to comment on this post. After reading both manifestos, watching several YouTube videos, and seeing lots of press coverage, it is difficult to take a side. Comment what you think about the different arguments (I think the three I listed earlier are the closest arguments to call), and if you are a Tau-ist or not. This is one of the few chances where you can get into a debate about mathematics!

A Dumbing Down of the Riemann Hypothesis

Today is my first post on math in the news. I recently came across this article on the Riemann Hypothesis, which I had planned to talk about in India, but didn't get a chance to. Let me give a brief background and then I will share the article.

Back in 2000, Clay Mathematics Institute of Providence, Rhode Island announced the Millennium Prizes, which consisted of seven problems that had been stumping mathematicians for a long time. They set aside a million US dollars for any person who solved one of the problems.

I find it interesting just on its own that you can become wealthy as a mathematician. Other than the Nobel Economics Prize, math has its own way of getting a million dollars.

One of the more popular of these problems is the Riemann Hypothesis, which I wanted to talk about today. I will try to explain here what the Riemann Hypothesis is (it is a difficult concept, but online sources complicate it drastically), and then show the article.

First off, you might remember the number i, which is the square root of -1. This is not a normal variable that can just replace anything you want; it is always the square root of -1. You may have heard in math class the term "real number." A number that has just an i in it are imaginary numbers, like 2i or -5i.

A little over a year ago, I did a post about the complex plane. This takes our horizontal number line from first grade and makes it our x-axis. It then takes these imaginary numbers and makes those the intervals of the y-axis.

A point on the x-axis is a real number, represented with the letter a. A point on the y-axis is an imaginary number, represented with the term bi. A point that is just floating around somewhere not on one of these lines is a complex number. You can write it with the expression a + bi, with a being the number it lines up with on the x-axis and bi being the number it lines up with on the y-axis.

Say you had to take the equation y = x^3 - 2 and start plugging in values for x (replacing the x with a number and then figuring out what it equals). Most people would start plugging in real numbers like 0, 1, 2, 3, -1, -2, -3, and so on. However, this Riemann Hypothesis requires us to open up our minds a little bit. Rather than just plugging real numbers into equations, we have to plug complex numbers into equations.

The Riemann Zeta Function is the equation we are plugging these numbers into (a little side-note: the Riemann Zeta Function has a 2π in it for any tauists). For the Riemann Hypothesis, it is concerned to find when this equation equals zero. This is called the zero of the equation.

The real numbers that are zeros of this equation are called the trivial zeros. They are equal to -2, -4, -6, -8, and so on. The complex numbers that are zeros of this equation are called the non-trivial zeros. As far as we know, these non-trivial zeros all have different b values in our a + bi, but the a value always seems to be 1/2.

The Riemann Hypothesis is simply asking the question is there a non-trivial zero of the Riemann Zeta Function whose a value is not equal to 1/2. Imagine winning a million dollars after submitting a hundred page paper that answers just a yes or no question.

It seems like proving either side would be extremely difficult. This article gives a nice explanation of how they are going about proving the yes side of it.

http://www.rdmag.com/news/2012/11/supercomputing-solve-superproblem-mathematics

The article brings up a very good way to do it. If you find just one time where the a value is not 1/2 and it is complex, then the statement is proven. So, Yuri Matiyasevich decided to turn the supercomputers on and start cranking out values. They have not found any values without the 1/2, but they also cannot mathematically prove that it always is the 1/2, and that is where the dispute lies.

Let me finish by saying what I find cool about the Riemann Hypothesis. So what if some weird function might have a consistency with its x-intercepts? But there is a very practical and interesting connection.

You may notice how with the prime numbers, there really isn't any relation between them. I mean, the Fibonacci numbers are the sum of the two before it, the powers of two are one more than the sum of all the ones before it, the triangulars are the sum of the natural numbers, the squares are the sum of the odd natural numbers, but the primes have no relation like that. I have always wondered why that is, or if there was one.

If the Riemann Hypothesis gets solved, it will shine a light on the distribution of prime numbers. We will be able to see if they do have a pattern or if there is no pattern. Some might find the technological, algebraic, or financial parts of this problem interesting, but I think the practical aspect is really cool.

The Best Approach to Math Education

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.