# The Math Hole

A general dump for mathematics students.

## Goals for the New Math Curriculum

I would like to take a minute and outline my goals for this summer curriculum project. Some are related to design, others to outreach. In short for anyone who is reading for the first time, I would like to redesign the middle/high school mathematics curriculum from the ground up using computers for computations.

- Create a curriculum plan that effectively integrates technology in a vital way. Computers are not a “passing fad” in my opinion, and I intend students to learn how to tell a computer what to do via a programming language in order to solve mathematical problems. I wish to focus more on the process of mathematical modeling and nurturing the logical thinking and problem solving skills that everyone claims mathematics is good for, but the current curriculum stifles in a devastating way.
- The new curriculum must cover all the current CORE mathematical education standards. I would rather have more freedom in my design, but if it’s ever to be realistically adopted I need to show that under the new curriculum the students can still learn the material currently required for standardized tests and, I suspect, improve their scores.
- Remove algorithmic hand computations unless there is a significant mathematical concept to be learned from doing them by hand. I want to place more emphasis on students being able to accurately model problems and develop a solid set of problem solving skills and utilize a computer to help them actually compute their answer.
- I would like to setup talks at various schools nearby to present my ideas and discuss the possibilities of adopting such a curriculum and address concerns that teachers may have. I would also like to outline the many benefits of switching to such a curriculum. For instance, computer science/computer literacy is a subject that is being pushed heavily towards high schools to address the increasing demand for computational [insert favorite noun here]-ists. In the world we live in, I believe it is necessary for students to have a solid set of problem solving skills that mathematics and computer science lend themselves to. One vital point I wish to make: I am NOT for pushing students to learn a programming language for the sake of learning a programming language.
*Computer Science and computer literacy is not equal to programming.*I believe it requires a motivational balance for the students to even consider learning a programming language. I intend to use programming primarily as a tool to do mathematics, and mathematics to motivate learning programming. I believe by integrating the two, the motivation for learning either becomes apparent. - If I’m speaking at area schools, I might as well seek out conferences nearby to present at. I would welcome suggestions for mathematical educational conferences, with an emphasis on technology or not. The word needs spreading.
- I want to get lots of input from local teachers and administrators. I want to get their opinions and concerns on such a curriculum and get an idea of how to address these concerns. I need to know what people are concerned about if I’m to make a curriculum that stands a chance.
- How to address the lack of technology in students homes, and how to account for this in the school setting. This also goes for addressing the lack of accessible technology in most schools.
- Finally, I want my curriculum to easily adapt to changes in technology. One major concern I’ve heard is that technology becomes outdates very quickly, and we can’t base a curriculum on a piece of technology that will be outdated in <5 years. I agree completely, and I have an idea of how to handle it. I need to keep this in mind when designing the curriculum and make it clear that it should be a living, breathing entity (although hopefully it won’t turn on us Cylon style!) that easily adapts to change in technology, among other things. I feel this is a lesson learned from the ISETL programming language someone posted in my comments previously.

I think that covers most of the major things I wish to do over the summer. I also plan on putting together a small book with a colleague, which if we manage to get it physically working will be a hit for sure. I know I would buy it! We’ll see about that last one though.

I would love to hear concerns or criticism about any of my ideas, as usual.

## “When will I ever use this stuff”

While thinking about Computer Based Math (CBM), which I have been doing a lot lately, I started to wonder if I could sum up my thoughts on how the curriculum should change into a single question or phrase. Here’s the one that’s been popping around in my head: “What math is going to be useful to students after graduation?”

There are a lot of things that immediately come to my mind. For instance, I believe that learning how to estimate and using mental math is an extremely useful skill to have. Unfortunately, it only seems to be stressed in grade school and rarely touched afterwards in favor of finding “exact” answers. When I’m shopping for food in the store, and I’m comparing prices between two different items, I’m not concerned ever with finding an exact answer. I round up or down for both the price and weight (or volume, quantity, whatever) and consider a rough calculation to determine what’s the better buy. If it’s really close, I just pick one. That’s the stereotypically example, but it’s something that (should) actually used on a daily basis. You could also consider someone wanting to compare statistical facts, trying to understand if they even make sense. You don’t need a calculator to do a quick and dirty calculation.

OK, so maybe we should make sure students know how to estimate stuff on the fly. What else should they know? Is it really in our best interests to teach students how to solve linear systems of equations by hand, or how to factor/distribute quadratic or cubic equations? How about conic sections? What about teaching them how to model problems using mathematics in their lives? How about teaching them how statistics work so they learn to question the statistics we are bombarded with on a daily basis instead of just believe them. The answer seems obvious to me.

In college a student should take the appropriate level of mathematics for their future career paths although ideally we should impart some basic knowledge that is useful in a more real way than we like to pretend the current math we teach is. If a student is going to be an engineer, certainly the mathematics required of them should be quite a lot including some treatment of calculus and differential equations while a mathematician needs a considerable larger amount of rigor. How many students are currently forced to take courses up through College Algebra, most never remembering or even having learned how to apply anything they’ve learned? So many students fail college algebra repeatedly only to finally pass by the skin of their teeth and they leave school probably never looking back at math ever again. I feel that this is a massive failure on the part of schools, and it’s currently being accepted as a perfectly normal situation. And for what? What possible benefit to society does this give?

It’s an interesting question to consider. I have no right answers but I’m very interested to hear other people’s opinions about what topics they feel would be useful for students. I’m thinking mainly about the high school math curriculum and what students should know when they leave high school. College math curriculum is a whole other ballgame, that I may try to tackle in the future.

## Post Math Circle Presentation

I presented to the math circle today, and it went great. I feel the level of my presentation was perfect for the kids. I spent a grand total of 3 minutes talking to everyone, then I just let them work together for the rest of the time. I ended up editing my presentation that I posted before slightly, but nothing worth really noting.

The kids were between grades 4 and 7. Despite their age, I was absolutely astonished at how mathematically developed some of these kids are already. Their thought processes rival mine when it comes to approaching a problem and a lot of the questions they ask on their own are questions I have asked myself when first thinking about these problems.

Overall, I had a blast and I think (hope) the kids had fun too. I plan to present again in the fall, this time on Hilbert’s Hotel Infinity! I’m basing it on a book called “The Cat in Numberland” which I purchased a while back. It’s a book written for children in late grade school/early middle school and it handles the concept of infinity very well and succinctly while not directly talking about “infinity”. I highly recommend it for anyone with children around this age who might be curious about the concept of infinity (and you get notions of countable sets for free!).

Overall, today was a very good day.

## How to Win $540,000,000

With all the talk about the record high jackpot, and some thoughts from a friend, I decided to play with Mathematica to see if I could “buy” a winning ticket. Here’s a quick Mathematica file I wrote to generate some numbers. I decided out of laziness to ignore the powerball, and adding in the powerball would’ve been even worse for the odds.

T = Lottery = Table[RandomSample[Table[i, {i, 1, 56}], 5], {1000000}];

T1 = Sort[T];

This code randomly picks 5 numbers between 1 and 56 without repetition and stores them in the list T. T1 is the list sorted numerically for easier matching of the winning numbers. I decided to try “buying” 1,000,000 tickets at first, and I was not surprised at the result. MemberQ is a function that detects the presence of an element in T1, and returns “True” if the element is in T1 and “False” if it is not.

MemberQ[T1, {2, 4, 23, 38, 46}]

>>False

Hm. Well that was a waste of money. Surely 10,000,000 tickets will be a better bet, right?

>>False

Damn. I guess I could just feasibly buy all (56 x 55 x 54 x 53 x 52) / 5! =3,819,816 possible combinations in my fake scenerio instead of relying on luck, although with 10,000,000 tickets I would think the odds would be a bit better! It’s worth noting that the actual megamillions lottery combinations includes the powerball, which I didn’t bother to do in my fake game so there are actually (56 x 55 x 54 x 53 x 52 x 46) / 5! = 175,711,536 combinations for the real deal.

I certainly got my $5 worth of fun this time around. Did you?

## Math Circle Presentation – Guarding Art Galleries

Coming up soon, I’m presenting at the local math circle hosted at a neighboring university. I decided to focus on triangulations and some properties about them and round it off with some art gallery problems. I decided to make a quick update and post my file online for anyone else running math circles who might want some idea’s for discrete geometry. This math circle is aimed at middle school students, just for an age reference. Without futher ado, here it is. If you want to directly download it, here is the direct link. Let me know if anyone has comments. I’ll give a follow up post about how it went and whether it was too difficult for them or spot on.

## A New Math Curriculum

I’ve put a lot of thought into the future of math education recently, especially after thinking about computer based math education(CBME). I’ve been specifically thinking about the curriculum as a whole, and I can’t help but feel that the topics being taught in the “standard” sequence of algebra, geometry, trigonometry, and calculus are all wrong. I’ll try to outline what I mean by “wrong” and why I think this way and hopefully spur discussions.

The idea behind CBME is that the world is changing very quickly, mathematics included. With the invention of computers, we are able to tackle problems that would have been computational nightmares in seconds. This has spurred every single scientific field forward as we figure out how to better use computers to solve problems. The problem is that we’re training students to handle computationally difficult problems by hand, forcing them to master the algorithmic techniques of “factoring a quadratic/cubic expression” and “solving systems of linear equations” for example all by hand. What we *should* be doing, is teaching the students how to model problems using mathematics, and teaching them how to use a computer to solve these problems and then interpreting these answers.

What I think should be done is heavily incorporate computers into the education. There are a number of traditional responses against doing this, most of which don’t make much sense when you actually think about them (I might go through them in the future, but I will instead refer the curious reader to www.computerbasedmath.org). What I propose, is that we teach students how to program with computers alongside teaching them mathematics. It’s been said by some others that Composition:English::Programming:Mathematics, and I feel that’s a fair analogy. We need to know how to tell the computers what we want them to do, and to do that we need to program them.

What I simply propose is we go through the current curriculum, and determine what is better suited for students to know how to do by hand and what they should learn how to get a computer to do. I propose that in teaching students mathematics, we teach them a programming language to tell the computer these algorithms we previously required them to do endlessly by hand. Imagine all the instructional time that will be freed up by not focusing on solving things by hand, and imagine what we could do with it. We could instead spend the time doing something infinitely more useful, learning how to *model* problems using mathematics. The infamous “word problems” are things students hate doing more than anything else, and yet they are the problems most likely to be encountered outside of the classroom. When I student taught, I spent so little time on world problems that looking back I’m a bit ashamed. I feel I adequately prepared by students according to the current curriculum, but I feel I failed at teaching them anything they can use outside of a math classroom, which to me means that I’ve failed as a teacher.

I no longer believe the standard curriculum is correct and I’m trying to brainstorm which pieces of it are necessary in the “new curriculum” and which can be replaced by computers. The fact is in order to tell the computer what to do, you *have* to know what you are doing, otherwise you ask the computer the wrong thing and it will give you (surprisingly!) the wrong answer. I feel we can utilize programming to reinforce conceptual understanding of problems, which is the opposite of what most people would initially think when they think of “using computers” to solve math problems. It’s not a magic box, it relies entirely on what you give it.

Surely, teaching students mental math and estimation are very useful skills outside of the classroom as well some some basic algebraic techniques, but how far do we take it? What sorts of topics are useful in this new curriculum I’m proposing? To be honest, I’m really not sure. I’d like to think about this over the summer and get a rough idea of what’s beneficial to keep and what’s useful to use computers for. Some things stand out on the chopping block, yet others are in more of a grey area. What we have to keep in mind is that we’re preparing students for the world. The world doesn’t have highly polished problems that don’t make any sense to students for the sake of getting an integer solution, do why the heck are we teaching them to tackle those problems?

There are a tremendous number of things stopping this currently: training teachers to teach the new curriculum, standardized testing (which should be thrown out anyways), angry parents, angry politicians, angry teachers, people who dislike any major change. The real point that I want to make is that our math education in the US stinks. It could be argued that it stinks everywhere and as computers continue to play a bigger role in society, the longer we take to change our curriculum and methods the faster the US will fall even further behind the rest of the world. Not that I’m only for the US improvement mind you, I want math education to change worldwide because I hold no such loyalties, but it seems the only thing to lite a fire under the US population is any threat to our “position” in the world. It’s all silly politics, but the issue is nothing but silly.

Where do I begin? How can a graduate student such as myself push for such a large change that even if started today would probably take 5-10 years to implement? Does the world have that sort of time? I certainly hope so.

## Wolfram’s Education Portal – Computer Based Education

I just got an e-mail announcing that Wolfram’s Education Portal is in beta. It’s a neat step towards providing support for computer based math education. They currently have what amounts to as an Algebra and Calculus textbook with interactive examples using Mathematica. You don’t actually need Mathematica to use it, you only need to download their free Wolfram CDF player, which lets you view things made in Mathematica without having a license.

I would be interested to see if anyone tries to implement some aspect of this into their classroom, and how effectively it works for them.

This all comes back to Computer-Based Math Education which I posted about a while back. The basic idea is to teach students how to model a mathematical problem, use a computer to solve the problem, then focus on interpret the results. There should be more emphasis on modeling mathematical problems and interpreting the results versus actually doing computations by hand. Most of the algorithmic techniques we teach students tend to break down at anything applicable to the real world. The site above has some interesting articles outlining how you could implement such a system, if you had the infrastructure to support it. Luckily, a lot of schools are moving towards laptops for their students (such as the one I student taught at) and this could give students easy access to these program. Wolfram|Alpha is a beautiful website that allows students to search for more information than just solving problems. Check it out if you haven’t, because it’s an amazing piece of work and best of all…it’s free.

I’ll finish with a final thought. A lot of opponents to computer-based math education seem to think that teaching students to utilize a computer to assist in solving problems will somehow “dumb down” math education and allow students to get answers with no thought to the concept. Anyone who has programming experience or works with computers in any aspect knows this is generally not the case since as the old saying goes, “garbage in, garbage out”. So we instead “dumb down” the math problems we expect students to work on by hand and then wonder why they can’t even begin to fathom incorporating mathematical thinking in their lives?

There is a definite problem with math education in the United States and I think I see a potential solution or at the very least, something to help the problem.

## Project Euler – Addiction at it’s Finest!

Recently, I was introduced and subsequently hooked to the website Project Euler. The idea is simple, there are a large number of problems which require both mathematical problem solving skills and a small knowledge of programming. I see it as an opportunity to learn a programming language you’ve always been wanting to try out (like Java, C++, or Python), but I’m using it to learn a program I’ve been needing to get a better grasp on for some time now, Mathematica.

I’ve been using Mathematica for all my programs which is probably borderline cheating for some but only for a handful, mainly because Mathematica can crunch huge numbers lickity split and has the ability to quickly produce or check divisors of numbers.

A small group of friends and I have been spending all our free time trying to solve more of these problems using Mathematica. They quickly become challenging and force you to either consider the deeper mathematics lying beneath each problem to avoid brute-force solving the problem (which actually works in most early cases), and/or learning new programming tricks to implement your mathematical ideas.

I’ve always had a little CS person inside me crying for attention, and this is a neat way to feed that demon while learning new mathematics at the same time. I only wish it hasn’t consumed all of my free time.

I have to admit though, I’m learning to be extremely precise with what I’m trying to describe to get Mathematica to interpret mathematically because even the smallest error will throw my program to hell.

I’m having a blast!

## Post-2012 Joint Math Meetings

My time at the joint math meetings in Boston is at an end today. I wanted to share some interesting things I learned from the short course, and some neat pictures. I might expand upon each topic more in depth in the future to both give myself a reason to go over the material more thoroughly and provide a nice place online to brief someone looking to learn more about this field.

This short course on Discrete and Computational Geometry (DCG), taught by Joseph O’Rourke (who is probably my favorite researchers in this field) and Satyan Devadoss, has filled in a lot of gaps in my knowledge about the basic building blocks of computational geometry. In essence, in order to really research in this area you need to know about polygons, triangulations, convex hulls, and voronoi diagrams. I will briefly discuss a few of these items, namely the ones I have cool pictures for.

First, it’s worth mentioning where DCG fits into the world. It is a field with a large amount of crossover between mathematics and computer science. It combines discrete geometry from mathematics and the study of algorithms from computer science, sometimes leaning more to one side or the other depending on the researcher.

We generally consider a problem involving point sets, or a set of points in , generally in or . I want to consider the *convex hull* of this point set S, which is the intersection of all convex regions containing all points of S. For , you can imagine stretching a rubber band around a collection of points and then letting it snap tight around every point. This is the convex hull of a region.

A *triangulation* of a planar point set S is a subdivision of the plane into a maximal set of noncrossing edges whose vertex set is S. It isn’t too difficult to see that triangulations are not unique, and there usually exists many triangulations for any given point set with >3 vertices. We wish to, in a sense, maximize each triangle in our set by maximizing the smallest angle in any triangle over all triangulations. This type of triangulation is called the *Delauney Triangulation, *and is immensely useful for reconstructing terrains, or modeling a 3D object, like the characters from a Pixar movie.

Research in these areas is awesome in the fact that it usually has direct applications to industry problems. It is also fun because it is a relatively new field, and most of the questions are still unanswered making it a good time to jump into this field.

A voronoi diagram is another of the ‘basics’ that you should have a good grasp on to make sense of computational geometry. A voronoi diagram of a point set S is a partitioning of the plane such that every partition contains points closest to S. The edges of each partition are equidistant from two or more points of S.

Without much work, you can compare the voronoi diagram of a point set and the Delauney triangulation of the same point set and realize that they are dual to one another.

There is a lot of interplay between these structures and their algorithms, and they really are the basic structures you see throughout DCG over and over. As I said, I might go through and give each one a more thorough treatment if I feel up to it, or if there is enough interest. I’m currently working on dusting off my programming skills to program a few algorithms to compute each of these into Mathematica, so maybe I’ll post those once I finish them. I know Mathematica can do them with it’s internal algorithms, I just want to get back into the swing of coding programs so I can actually get my hands dirty with some of these problems.

Overall, the Joint Meetings were a blast, and I wish I could’ve stayed longer to check out some more of the talks.

Finally, I’m glad I got to meet Joseph O’Rourke finally, because it turns out he’s downright amazing. Not only in his breadth of knowledge of a subject he practically pioneered, but he is one of the most easily approachable people I’ve ever met. Even after only having a few short conversations with him, he went out of his way to stop me in the middle of a ridiculously large crowd of moving people just to say “Hi!”, which says a lot about his character to me. I hope someday I get the opportunity to collaborate with Joe and get to know him better because if there’s a mathematician I’d really like to get to know at this moment, it would be him.

For now, it’s back to the board to finish writing up a research paper for our group for submission to CCCG.

## Joint Mathematics Meetings and MAA Short Course

As the holidays tick away, I’m preparing for my first trip to a major mathematics conference, the Joint Mathematics Meetings (JMM) hosted this year in Boston, MA. My reason for attending, and the thing I’m most excited about, is the MAA short course I’m attending. This year it’s ‘Discrete and Computational Geometry’, which is one of my main research focuses right now. My current research revolves around it, and I feel that Discrete Geometry is going to be one of my main fields of interest in the future.

It’s hosted by Joseph O’Rourke and Satyan Devadoss, both big names in the field. I’m especially excited to introduce myself and potentially discuss the research we’re currently working on. It’s a bit of a stretch for a graduate student, but I’ll be damned if I’m being funded to go to the JMM and sit back passively. This short course will quickly bring me up to speed on the main topics of interest in Discrete Geometry, and hopefully give me some open problems to bring back to my research group. They are also providing a complimentary copy of their textbook by the same name, which is an amazing gift.

Now I just need to start sorting through the *250+ pages of abstracts* and decide what I’m going to attend. There are several graph theory and combinatorics mini-symposiums I have my eye on, but I need to keep my eyes out for mathematical education seminars that look interesting.

All in all, it should hopefully be a productive trip. I hope to establish contacts while there and even better, find some people to research with! If anyone reads this and is attending either the meetings or the DCG short course, leave a comment so we can begin the introductions!