# Suburban Lion's Blog

## 2013/04/09

### What can we learn from the Sci-Fi classroom?

Filed under: Education — ryan @ 15:15

We live in an interesting time where new technologies are radically reforming how humans interact with machines and with each other. The field of education is no exception. Tech savvy teachers are likely to be familiar with new educational paradigms like BYOD, MOOC, and flipped classrooms. There's more to this movement than just fancy buzzwords. Teachers are forging ahead into new territory by bringing technology into the classroom and looking for ways to make the most of it. As these technologies continue to improve, these hi-tech classrooms are starting to look like something out of a science fiction story.

Many scientific advances have had their roots in science fiction. It seems pertinent then to examine how science fiction authors have depicted the future of education as a source of inspiration for the hi-tech classroom. In this post, we'll take a look at some Sci-Fi classrooms and see what lessons we can learn from them.

# The sources

The following are a collection of Sci-Fi classrooms from sources that I'm familiar with. I'm sure this list is not exhaustive, so please feel free to contribute others in the comments. Also, please note that some of these source materials are intended for mature audiences. I'll try avoid any major spoilers for those who are unfamiliar with them.

## Starship Troopers

Based on a novel by Robert Heinlein, Starship Troopers takes place in a future where democracy has crumbled and replaced with a militarist establishment. Humanity is engaged in an interstellar war against alien species, and enlisting in the military is the most efficient path to "citizenship".

The beginning of the film depicts the protagonist, Rico, in a high school history class. Despite the futuristic setting, the classroom largely fits the "traditional lecture" paradigm. Rico's history teacher is a retired officer who is missing an arm. The course is arguably equal parts history and propaganda.

While there isn't much use of technology in the classroom, each desk seems to have a touch sensitive computer embedding in it.  For Rico, this seems to be more of a distraction from the class than a learning aid.

The high school experience ends with high stakes testing.  A low math score ultimately places Rico into the infantry while his close friends are placed in flight school and military intelligence.  For added pressure, students seem to check their test scores on a public computer terminal.

We also get a glimpse of a futuristic biology lab, in which students dissect alien lifeforms.

While the lectures, labs, and high stakes testing are all too familiar, the film does raise interesting questions about the purpose of education. The school is structured like a factory to produce potential soldiers. It's easy to see why this militaristic society would structure education in such a way that cultivates students that efficiently follow orders. I tend to view this as a cautionary tale of what education might become in the hands of a powerful military bureaucracy.

## WALL-E

WALL-E follows the adventures of a trash compacting robot in a world where Earth's natural resources have been extinguished by rampant consumerism. With the Earth no longer capable of supporting life, the remaining population leaves the planet in large intergalactic cruise ships. In the film, we observe a brief scene depicting a futuristic pre-school.

In this scene, a number of small children are watching a video with a computerized narrator describing the letters of the alphabet. No adults are present in the room and the instruction is fully provided by a robot. This picture of education fits with the overall premise of the film, in that humans have essentially automated themselves into irrelevance.

Like Starship Troopers, the educational system is designed to perpetuate the existing authority. Even in this pre-school setting, students are conditioned by messages such as "B is for Buy-N-Large, your very best friend". It's implied later in the film that the residents of this spaceship are taught very little about life on the planet that their ancestors fled. The system is programmed to keep the residents living happily on the ship, and inhibiting any curiosity about to the circumstances that put them there. The residents are fat, happy and ignorant, and the fully automated educational system is designed to keep them that way.

## Serenity

Based on the Firefly TV series, Serenity treats us to a brief glimpse of River Tam's childhood. The young River has been identified as intellectually gifted and is sent to "The Academy".

In this scene, River is taking part in a history course. It begins with the teacher narrating in front of a holographic projection that displays a visual depiction of the events. The lesson describes the "Unification War", in which a number of planets rebelled against the Alliance's expansion. The teacher explains that the Alliance engaged in this war to spread peace throughout the galaxy and asks why the rebels would resist. River seems to suspect that there's more to the story and points out that "people don't like to be meddled with".

While this looks like a typical lecture classroom, each of the students appears to have touch sensitive computer screen embedded in the desks. Students seem to be interacting with the computer during the lecture using a stylus, but also are engaged in note taking with a traditional pen and paper.

We later find out that the Academy is actually a front for a program to turn these students in super-soldiers through a series of cruel medical and psychological experiments. River's skepticism of the Alliance's meddling turns out to be quite prescient.

## Star Trek

In the 2009 film Star Trek, we find one of the more visually striking Sci-Fi classrooms. A young Spock is depicted in a school for Vulcans, a race known for their strong devotion to logical reasoning.  The scene shows students in semi-spherical pods, where the students interact with a projected display.  Several adults can be seen walking between these learning pods, but the main source of instruction appears to be with the computer.

The computer asks the students various questions, particularly involving science and mathematics, and the student responds verbally with the answers.

As the young Spock completes his interactive instruction, he is met by several other students who proceed to bully him about his human mother.

The young Spock fights back out of anger, a course of action which is looked down upon in a society that values emotional restraint.

## Star Trek: Voyager

Within the Star Trek universe, the TV series Star Trek: Voyager provides another perspective on education. The star-ship Voyager gets lost in space after travelling through a worm hole and spends many years heading back home. In this time, one of the crew-members gives birth on board the ship. The child, Naomi Wildman, grows up aboard the star-ship and is essentially home-schooled by the crew.

The details on Naomi's education are limited, but she takes a liking to the Borg crew-member Seven of Nine who serves as a mentor. On occasion, Seven will assign her various instructional materials or ask her to carry out small tasks on the ship.  Several other crewmembers serve as teachers as well.  Naomi works diligently on these tasks and aspires to become the Captain's Assistant.

We also know that Naoimi spends a good deal of time on board the ship's holodeck. It's implied that some of Naomi's education comes from interactive holographic children's tales like "The Adventures of Flotter". This colorful interactive fairy tale is designed to teach children deductive reasoning skills, and Naomi needs to solve several puzzle to help the storyline progress.

In a later episode, the crew picks up several Borg children and the educational offering on Voyager are expanded to meet their needs. This results in the First Annual Voyager Science Fair. Naomi presents a model of planet, while the Borg children present a clone potato, an ant colony and a gravimetric sensor.

## Accel World

Accel World is a manga and anime by Reki Kawahara set in a future where humans can interface with computers via a "Neuro-Linker". This hardware allows individuals to interact in a virtual environment using thoughts to control their personal computer. The opening episode of Accel World depicts what looks like a typical classroom: a teacher at the front of the class lectures while the students are busy taking notes. The difference is that the teacher isn't actually writing on the board, but rather the hand movements of the teacher are transcribed to a digital blackboard which the students can see through the neural-interfaced computer.

We also see that all of the students are busy using hand gestures to control these computers throughout the class.

Like Star Trek, the synchronous nature of the classroom causes problems for the protagonist. Haru is short fat kid with low self-esteem who is regularly bullied by students that are bigger than him.

To Haru, the virtual reality system provided by the Neuro-Linker is his only escape from the hostile environment of reality. When a classmate offers him a strange program that will "destroy his reality", he eagerly seizes the opportunity.

## Ready Player One

Ernest Cline's novel Ready Player One takes place largely within an interactive simulation called OASIS. In a world that has been ravaged by climate change and a crumbling democracy, the brick and mortar schools are a potentially hostile environment and the protagonist, Wade, volunteered to pilot an educational program in this virtual environment. Users access the OASIS using a combination of a virtual reality visor, haptic gloves (and other accessories) and voice control. Wade's school is one of many in OASIS and he describes some of the advantages of a virtual school:

There were hundreds of school campuses here on Ludus, spread out evenly across the planet’s surface. The schools were all identical, because the same construction code was copied and pasted into a different location whenever a new school was needed. And since the buildings were just pieces of software, their design wasn't limited by monetary constraints, or even by the laws of physics. So every school was a grand palace of learning, with polished marble hallways, cathedral-like classrooms, zero-g gymnasiums, and virtual libraries containing every (school board–approved) book ever written.

On my ﬁrst day at OPS #1873, I thought I’d died and gone to heaven. Now, instead of running a gauntlet of bullies and drug addicts on my walk to school each morning, I went straight to my hideout and stayed there all day. Best of all, in the OASIS, no one could tell that I was fat, that I had acne, or that I wore the same shabby clothes every week. Bullies couldn’t pelt me with spitballs, give me atomic wedgies, or pummel me by the bike rack after school. No one could even touch me. In here, I was safe.

When I arrived in my World History classroom, several students were already seated at their desks. Their avatars all sat motionless, with their eyes closed. This was a signal that they were “engaged,” meaning they were currently on phone calls, browsing the Web, or logged into chat rooms. It was poor OASIS etiquette to try to talk to an engaged avatar.They usually just ignored you, and you’d get an automated message telling you to piss off.

I took a seat at my desk and tapped the Engage icon at the edge of my display. My own avatar’s eyes slid shut, but I could still see my surroundings. I tapped another icon, and a large two-dimensional Web browser window appeared, suspended in space directly in front of me. Windows like this one were visible to only my avatar, so no one could read over my shoulder (unless I selected the option to allow it).

School in OASIS bears a marked a resemblance to the traditional classroom. Students attend classes synchronously in the virtual environment. A teacher leads the class in something like a lecture format, but this lecture can be supplemented by virtual materials that would be impossible in a traditional class. For example, students can take a virtual tour of the human body on board a microscopic submarine in Biology. Students have opportunities to interact with each other as well and a "mute user" option makes it possible to avoid virtual bullying.

I think it's also worth mention that this type of virtual school is quite possible with today's technology. For example, virtual schools already exist in Second Life.

# Common Themes

While this is a small sample of Sci-Fi classrooms, I think there are some patterns here that are worth noting.

## Education is an institution of great power

First and foremost, these Sci-Fi stories depict education as a very powerful force in human development. This is something of a double-edged sword. Education can be used as a tool to enlighten individuals or it can be used to preserve a system of authority.

There's an old maxim stating that "knowledge is power". Sci-Fi takes this maxim one step further by showing us worlds where restricting access to knowledge can make individuals powerless. As a society, we need to take steps to ensure that information is open and available to avoid falling into a trap of historical revisionism.

## Lectures are here to stay

Most of these Sci-Fi classrooms seem to fit the template of the "traditional lecture". A teacher stands in front of the class and delivers information to the students. Perhaps part of the reason for this is that the "lecture" is so pervasive in our culture that we wouldn't recognize these as classrooms if they were structured otherwise. Our familiarity with the format means that we quickly recognize the scene as a "school". This format allows the Sci-Fi plot to make its point and move on in a short amount of time.

Despite the dated classroom format, the technology in the classroom opens up new methods of presenting information. The chalkboards have gone electronic and the Power Point slides have been replaced by holographic projections. The interesting part is that despite all this technology, these Sci-Fi classrooms often still have a human teacher. Computers might be used to disseminate information, but learning is guided by a "real teacher". This suggests that the role model that teachers provides is an important part of the educational process. The teacher is trusted to make an appropriate use of the technology to engage students in the practice of critical thinking.

## School is a social experience

Another related trend in these Sci-Fi classrooms, is that the students gather in the same place at the same time. This suggests that students interacting with each other is an important part of the school experience.

This is particularly curious in the case of Star Trek and makes for an interesting comparison with present day hybrid courses. In a typical hybrid course, students have both asynchronous "homework" time (typically done over the Internet) and synchronous "classroom" time. What's interesting to me about the Vulcan classroom is that it a synchronous environment with primarily asynchronous instruction. During the instructional time, students interact primarily with the computer rather than each other. It might be comparable with an online class that you were required to take in person.

It may be that the concurrency of having students gathered in a single location is to support the development of social skills, but in the case of several protagonists this has the negative side effect of bullying. Ready Player One solves this bullying issue by transitioning to a fully virtual classroom. Students can simply block electronic messages from others if they choose.

# Conclusion

While we obviously don't want to base educational practices on anecdotes from science fiction, I think that these Sci-Fi stories mirror the struggles of today's teachers.  On one hand, teachers want to adhere to practices that have worked in the past.  One the other hand, teachers can see the value in exposing students to new technology.   What we see in these Sci-Fi classrooms is the result of both of these influences.  They look like present day classrooms, only the tools have been replaced with the latest hi-tech gadgets.  In effect, the Sci-Fi classroom is generally just a superficial make-over of our existing cultural expectations of school.

Of these Sci-Fi classrooms, the one that gives me the most hope for the future is Voyager.  Naomi's education is a balanced mixed of numerous sources.  She spends a lot of time learning from holographic children's programs, but also electronic textbooks, human role models, and hands-on science experiments.   The important point to be made here, is that the new technology adds to the educational experience rather than just replacing existing tools.   The presence of pen and paper note-taking in the Serenity classroom is another telling indicator that good teaching doesn't necessarily require advanced technology.

As far as the educational technology goes, some of these Sci-Fi devices might not be too far off.   The difficulty is going to be integrating them into the classroom effectively.  It needs to be used in a way that contributes to the learning experience or it runs the risk of becoming a distraction.  In some ways, this is already a issue.  How much lost classroom time do teachers owe to fiddling with projectors, doc cams, and Power Point?  Perhaps the focus of educational technology developers should be on streamlining the existing tools into a seamless classroom experience.  Creating voice controlled lecture tools might provide a nice stepping stone between the classrooms of today and the ones we see in science fiction.   Any technology that frees up a teacher's time to focus on students' learning is likely to be a welcome addition to the classroom.

One thing's for certain, it sure is an exciting time to be a teacher!

## 2012/05/13

### 5 Recent Mathematical Breakthroughs That Could Be Taught in Elementary School (but aren't)

Filed under: Education,Math — Tags: — ryan @ 13:02

In a previous blog post, I made the claim that much of the math curriculum is ordered based on historical precedent rather than conceptual dependencies. Some parts of the math curriculum we have in place is based on the order of discovery (not always, but mostly) and while other parts are taught out of pure habit: This is how I was taught, so this is how I'm going to teach. I don't think this needs to be the case. In fact, I think that this is actually a detriment to students. If we want to produce a generation of mathematicians and scientists who are going to solve the difficult problems of today, then we need to address some of the recent advances in those fields to prepare them. Students should not have to "wait until college" to hear about "Topology" or "Quantum Mechanics". We need to start developing the vocabulary for these subjects much earlier in the curriculum so that students are not intimidated by them in later years.

To this end, I'd like to propose 5 mathematical breakthroughs that are both relatively recent (compared to most of the K-12 curriculum) while also being accessible to elementary school students. Like any "Top 5", this list is highly subjective and I'm sure other educators might have differing opinions on what topics are suitable for elementary school, but my goal here is just to stimulate discussion on "what we could be teaching" in place of the present day curriculum.

# #1. Graph Theory (c. 1736)

The roots of Graph Theory go back to Leonard Euler's Seven Bridges of Königsberg in 1736. The question was whether or not you could find a path that would take you over each of the bridges exactly once.

Euler's key observation here was that the exact shapes and path didn't matter, but only how the different land masses were connected by the bridges. This problem could be simplified to a graph, where the land masses are the vertices and the bridges are the edges.

This a great example of the importance of abstraction in mathematics, and was the starting point for the field of Topology. The basic ideas and terminology of graph theory can be made easily accessible to younger students though construction sets like K'Nex or Tinkertoys. As students get older, these concepts can be connected to map coloring and students will be well on their way to some beautiful 20th century mathematics.

# #2. Boolean Algebra (c. 1854)

The term "algebra" has developed a bad reputation in recent years. It is often referred to as a "gatekeeper" course, which determines which students go on to higher level mathematics courses and which ones do not. However, what we call "algebra" in middle/high school is actually just a subset of a much larger subject. "Algebra I" tends focuses on algebra as it appeared in al-Khwārizmī's Compendious Book on Calculation by Completion and Balancing (circa 820AD). Consequently, algebra doesn't show up in the math curriculum until students have learned how to add, subtract, multiply and divide. It doesn't need to be this way.

In 1854, George Boole published An Investigation of the Laws of Thought, creating the branch of mathematics that bears his name. Rather than performing algebra on numbers, Boole used the values "TRUE" and "FALSE", and the basic logical operators of "AND", "OR", and "NOT". These concepts provided the foundation for circuit design and eventually lead to the development of computers. These ideas can even be demonstrated with a variety of construction toys.

The vocabulary of Boolean Algebra can and should be developed early in elementary school. Kindergartners should be able to understand basic logic operations in the context of statements like "grab a stuffed animal or a coloring book and crayons". As students get older, they should practice representing these statements symbolically and eventually how to manipulate them according to a set of rules (axioms). If we develop the core ideas of algebra with Boolean values, than perhaps it won't be as difficult when these ideas are extended to real numbers.

# #3. Set Theory (c. 1874)

Set Theory has its origins in the work of Georg Cantor in the 1870s. In 1874, Cantor published a ground breaking work in which he proved that there is more than one type of infinity -- the famous "diagonal proof". At the heart of this proof was the idea of thinking of all real numbers as a set and trying to create a one-to-one correspondence with real numbers. This idea of mathematicians working with sets (as opposed to just "numbers") developed momentum in the late 1800s and early 1900s. Through the work of a number of brilliant mathematicians and logicians (including Dedekind, Russell, Hilbert, Peano, Zermelo, and Fraenkel), Cantor's Set Theory was refined and expanded into what we know call ZFC or Zermelo-Fraenkel Set Theory with the Axiom of Choice. ZFC was a critical development because it formalized mathematics into an axiomatic system. This has some suprising consequences such as Gödel's Incompleteness Theorem.

Elementary students probably don't need to adhere to the level of rigor that ZFC was striving for, but what is important is that they learn the language associated with it. This includes words and phrases like "union" ("or"), "intersection" ("and"), "for every", "there exists", "is a member of", "complement" ("not"), and "cardinality" ("size" or "number"), which can be introduced informally at first then gradually formalized over the years. This should be a cooperative effort between Math and English teachers, developing student ability to understand logical statements about sets such as "All basset hounds are dogs. All dogs are mammals. Therefore, all basset hounds are mammals." Relationships can be demonstrated using visual aids such as Venn diagrams. Games such as Set! can further reinforce these concepts.

# #4. Computation Theory (c. 1936)

Computation Theory developed from the work of Alan Turing in the mid 1930s. The invention of what we now call the Turing Machine, was another key step in the development of the computer. Around the same time, Alzono Church was developing a system of function definitions called lambda calculus while Stephen Kleene and J.B Rosser developed a similar formal system of functions based on recursion. These efforts culminated in the Church-Turing Thesis which states that "everything algorithmically computable is computable by a Turing machine." Computation Theory concerns itself with the study of what we can and cannot compute with an algorithm.

This idea of an algorithm, a series of steps to accomplish some task, can easily be adapted for elementary school instruction. Seymour Papert has been leading this field with technologies like LOGO, which aims to make computer programming accessible to children. Another creative way of approaching this is the daddy-bot. These algorithms don't need be done in any specific programming language. There's much to be learned from describing procedures in plain English. The important part is learning the core concepts of how computers work. In a society pervaded by computers, you can either choose to program or be programmed.

# #5. Chaos Theory (c. 1977)

Last, but not least, is Chaos Theory -- a field of mathematics that developed independently in several disciplines over the 1900s. The phrase "Chaos Theory" didn't appear in the late 1970s, but a variety of phenomena displaying chaotic behavior were observed as early as the 1880s. The idea behind Chaos Theory is that certain dynamic systems are highly sensitive to initial conditions. Drop a shot of half-half into a cup of coffee and the resulting pattern is different every time. The mathematical definition is a little more technical than that, but the core idea is relatively accessible. Chaos has even found several notable references in pop culture.

The other core idea behind chaos theory is topological mixing. This could be easily demonstrated with some Play-Doh (or putty) of two or more colors. Start by combining them into a ball. Squash it flat then fold it over. Repeat it several times and observe the results.

The importance of Chaos Theory is that it demonstrates that even a completely deterministic procedure can produce results that appear random due to slight variations in the starting conditions. This can even be taken one step further by looking at procedures that generate seeming random behavior independently of the starting conditions. We live in an age where people need to work with massive amounts of data. The idea that a simple set of rules can produce extremely complex results provides us with tools for succinctly describing that data.

# Conclusion

One of the trends in this list is that these results are easy to understand conceptually but difficult to prove formally. Modern mathematicians seem to have a tendency towards formalism, which is something of a "mixed blessing". On one hand, it has provided mathematics with a firm standard of rigor that has withstood the test of time. On the other hand, the language makes some relatively simple concepts difficult to communicate to younger students. I think part of the reason for this is that the present curriculum doesn't emphasize the rules of logic and set theory that provide the foundation for modern mathematics. In the past, mathematics was driven more by intuitionism, but the math curriculum doesn't seem provide adequate opportunities for students to develop this either! It might be argued things like "new math" or "Singapore math" are helping to develop intuitionism, but we're still not preparing students for the mathematical formalism that they'll be forced to deal with in "Algebra I" and beyond. Logic and set theory seem like a natural way to develop this familiarity with axiomatic systems.

Observers might also note that all five of these proposed topics are related in some form or another to computer science. Computers have been a real game-changer in the field of mathematics. Proofs that were computationally impossible 500 years ago can be derived a in minutes with the assistance of computers. It's also changed the role of humans in mathematics, from being the computer to solving problems using computers. We need to be preparing students for the jobs computers can't do, and my hope is that modernizing the mathematics curriculum can help accomplish this.

Do you have anything to add to this list? Have you tried any of these topics with elementary students? I'd love to hear about your experiences in the comments below.

## 2012/05/03

### Pre-Calc Post-Calc

Filed under: Education,Math — ryan @ 14:00

Gary Davis (@republicofmath) wrote an article that caught my attention called What's up with pre-calculus?. In it, he presents a number of different perspectives on why Pre-Calc classes have low success rates and do not adequately prepare students for Calculus.

My perspective on pre-calculus is probably far from the typical student, but often times the study of "fringe cases" like myself can provide useful information on a problem. The reason why my experience with Pre-Calc was so atypical, is because I didn't take it. After taking Algebra I, I had started down a path towards game programming. By the end of the following year, where I had taken Geometry, this little hobby of mine hit a road block. I had come to the realization that in order to implement the kind of physics that I wanted in my game I would need to take Calculus. I petitioned my counselor to let me skip Algebra II and Pre-Calc to go straight into AP Calculus. They were skeptical at first, but eventually conceded to my determination and allowed me to follow the path I had chosen.

Skipping from Geometry to Calculus meant that there were a lot of things that I needed to learn that first month that many of my peers had already covered. I had never even heard the word "logarithm" before, had no idea what e was, and had only a cursory understanding of trigonometry. These were the topics I had missed by skipping Pre-Calc, and I was fully aware of that, so I "hit the books" and learned what I needed to know about them. By the end of that first month I had caught up to the rest of the class and by end of the semester I would be helping other students with those very same topics.

I think the most obvious difference between myself and the "typical Calculus student" was the level of motivation. Many of the students in Calculus were there because "it would look good on a college application". I was there because I wanted to be there. A common problem throughout math education is the "When am I ever going to use this?" attitude. I already knew where I was going to use the math I was learning. I had an unfinished game at home that needed a physics system, and every new piece of information I learned in Calculus made me one step closer to that goal. If you had ever wondered why a 4th order Runge-Kutta method is better than Euler's method, try writing a platformer.

The second difference was a little more subtle, but there were some conceptual differences in how I thought about exponential, logarithmic, and trigonometric functions. The constant "e" wasn't just some magic number that the textbook pulled out of thin air, it was the the unique number with the property that $\frac{de^x}{dx} = e^x$ and $\int e^x dx = e^x$. When it came to sine and cosine, I would think of them like a circle while my other classmates would picture a right triangle. They would hear the word "tangent" and think "opposite over adjacent", but I thought of it more like a derivative. Sure, I had to learn the same "pre-calc" material as they did, but the context of this material was radically different.

A couple years ago I suggested that Pre-Calc should be abolished. The trouble with Pre-Calculus (at least in the U.S.) is that the course needs to cover a very diverse array of questions which includes exponential, logarithmic and trigonometric functions. I would argue that these concepts are not essential to understanding the basic foundations of Calculus. The math curriculum introduces the concept of "slope" in Algebra I, which is essentially the "derivative" of a line. There's no reason why we should be sheltering students from language of Calculus. The concepts of "rate of change" and "accumulation" can and should be connected with the words "derivative" and "integral", long before students set foot in the course we presently call Calculus. As students become more comfortable with these concepts as they relate to lines, parabolas and polynomials, then gradually step up the level of complexity. When students start to encounter things like surfaces of revolution, then they'll actually have a reason to learn trigonometry. Instead of trigonometry being the arbitrary set of identities and equations that it might appear to be in pre-calc, students might actually learn to appreciate it as a set of tools for solving problems.

I think this issue of Pre-Calc is really a symptom of a larger problem. The mathematics curriculum seems to be ordered historically rather than conceptually. I've heard Pre-Calc described as a bridge to Calculus. This makes sense when you consider the historical development of Calculus, but not when considering the best interest of students in today's society. Leibniz and Newton didn't have computers. Who needs bridges when you can fly?

## 2011/08/26

### Mathematics as a Foreign Language: a Tale of Two Classrooms

Filed under: Education,Math — ryan @ 23:59

Last Thursday's #mathchat topic was "Is the spirit of mathematical thinking being swamped by a focus on technique?". One of the things that caught my eye during this discussion was a comment by David Wees suggesting that we teach math more like programming. I've proposed something similar to this before, but as the conversation continued into the details of learning how to program I started to think of the process like learning a foreign language. While I quickly came to realize that there were differing views on how foreign languages should be taught, I think there might be something to this idea. The human brain has built-in hardware to assist in learning language. Can math education take advantage of it?

Mathematics has its something of its own written language. A "conventional mathematical notation" has emerged through a variety of social influences. Some of those notations "just make sense" in the context, while others are adopted for purely historical reasons. As an undergraduate, college mathematics was like learning a foreign language for me. I had no idea what "$\forall n \in \mathbb{R}$" meant. Aside from "n", those symbols were not used once in any of my previous courses! It was culture shock. I eventually adjusted, but I now understand why mathematical notation can have such an intimidating effect on people.

What follows are my experiences with learning two foreign languages and how I think the difference between the two methodologies relates to the "math wars". I had 2 years of Spanish in high school and 3 semesters of Russian in college. I'm going to refer to the teachers as Mrs. T and Mrs. R respectively, for reasons that I think will be obvious later.

Mrs. T's Spanish class was held in a portable classroom at the edge of the high school. The classroom held about 30 students and the air conditioning barely kept out the 100-120 degree desert heat. I must give Mrs. T some credit for being able to do her job under such conditions. The classes often started with practice reciting words and phrases, followed by worksheets in groups and ending in a quiz. "Capitones, vengan aqui", she would say while slamming her hand down on the table in front of her, indicating that the students in the front row of the class were to carry everyone's work up to her. Everyday she would do the same routine, and everyday I wished that table would snap in half. We had done so many 10 point worksheets that at the end of the semester I came to the mathematical conclusion that the 100 point Final was only 2% of my grade. Being the little smart-ass that I was, I pointed out that I could skip the Final and still get an A. I don't think she liked that very much, because she threatened to fail me if I didn't take it. Aye que pena!

Mrs. R's class was much smaller, with only about 8 students. It was more like a conference room than a classroom. There was a U-shaped table that opened towards the white board, so Mrs. R could walk up to each person and engage in conversation. There was some rote memorization at first, while we learned the alphabet and basic grammar, but after the first few weeks of class Mrs. R started refusing to speak English in class. Class started with everyone saying hello and talking about his/her day -- in Russian. We role-played different situations -- in Russian. If I needed to know a word, I had to ask about it -- in Russian -- and someone would explain it to me -- in Russian. We watched Russian films and listened to Russian rock music. It didn't feel like a class, but rather like 9 friends with similar interests hanging out for an hour each day.

In both of the classes I learned much about the respective languages, but what really stuck with me in each case was the culture. I might not remember enough of the vocabulary to consider myself fluent in either language, but I'll still find myself singing along with Santana or Mashina Vremeni.

In the "Math Wars", the Traditionalists follow something similar to Mrs. T's method while the Reformers want math to look more like Mrs. R's class. Both methods "work", if test scores are all you care about, but there's a very subtle difference between them. In Spanish class, I always felt like I was always translating to and from English in order to communicate. In Russian class, I felt like I was articulating ideas directly in Russian. There's something beautiful about just immersing yourself in a different language until you learn it. I learned how to program in C by installing GNU/Linux and reading other peoples' source code. Sure I read a few books on the matter, but it was immersing myself in "C culture" that really solidified my understanding.

For students to really learn math, they need to be immersed in the "culture of mathematical thinking". I might not agree with the term "spirit", but mathematicians seem to display a common pattern of asking very entertaining "what if?"s and seeking out the answers. You can find beautiful math in something as simple as drawing doodles in class. There's more mathematical thinking going on when two kids make up a game during recess than there is in a thousand worksheets. Our body of mathematical knowledge is formed through communication and peer-review. It's is such a shame to see math classes run like a dictatorship, built around memorizing a list of "techniques". Sure, mathematics is an essential skill in finance, data, and engineering, but lets not underestimate the importance of "asking questions" in our focus on "problem solving".

Proceeding with the question "what if we teach math like a foreign language?", what might we do differently?

Mrs. T might argue that repetition seems to work, and there's a substancial amount of evidence it does (at least in the short term). Math class already has its fair share of repetitious worksheets, but what if we shift the focus of the repetition to learning the "alphabet and grammar" of mathematics earlier like Mrs. R's class? We could start with "set theory" and "logic" then work up from a firm foundation. The benefits could be substantial.

Mrs. R might also argue that students need to be immersed in the culture of math. Students should learn about the history of math and be exposed to "mathematical pop culture". Let's laugh together at XKCD or collectively gasp in bewilderment at the arXiv. It's moments like those that make us human. Lets embrace them.

Embrace the "culture of math".

Of course, it would probably be a lot easier to do such a thing with a student-teacher ratio of 8:1. One can only dream...

## 2011/03/28

### VMATYC 25th Annual Conference: Day 1

Filed under: Education,Math — ryan @ 14:55

Last weekend I attended the 25th Annual Conference of The Virginia Mathematical Association of Two Year Colleges (VMATYC), Virginia's chapter of the American Mathematical Association of Two Year Colleges (AMATYC). This was the first educational conference I have been to since I started teaching developmental math two and half years ago, so it was a very exciting event for me. What follows is my account of the seminars I attended at the VMATYC and what I learned from the experience. I've tried my best to summarize the events I attended from my notes, but please contact me if there are any inaccuracies.

I missed the early sessions on Friday due to class, but made it in time for the seminar I was most interested in: The Developmental Math Redesign Team (DMRT) Progress Report.

# DMRT Progress Report

Virginia's Community College System (VCCS) has been in the process of “redesigning” the developmental math program for about two years now, and is now in the process of implementing some major changes to the way developmental math is handled at the community college level. The report was presented by Dr. Susan Wood, Dr. Donna Jovanovich, and Jane Serbousek.

Dr. Susan Wood began the discussion with a broad overview of the DMRT program. The DMRT began in 2009 with the publication of The Turning Point: Developmental Education in Virginia's Community Colleges, which highlighted some of the problems facing developmental math students. This document set forward the goal for the developmental education redesign, which is specifically targeted at increasing the number of students that go on to complete degree programs. The Turning Point also initiated the Developmental Mathematics Redesign Team. The following year, the DMRT published The Critical Point: Redesign Developmental Mathematics Education in Virginia's Community College System, which outlines the proposed changes to the developmental education program. Next, a curriculum committee began work on a new developmental mathematics curriculum, which is available here. These changes are slated for implementation in Fall 2011. Dr. Wood also made the point that these changes fit into a larger framework of the student experience, a cycle of “Placement/Diagnostic Instruments --> Content --> Structure --> Instructional Delivery --> Professional Development --> Student Support Services Assessment --> Placement/Diagnostic Instruments”.

Next, Jane Serbousek followed with more detail about the proposed DMRT changes. The content of the developmental math courses has been revised to better reflect what is needed to be successful in college. The content has also been reorganized from three five-unit courses, to a series of nine one-unit “modules”. The modules are competency based, and are intended to use a grading system of ABCF instead of SRU (Satisfactory, Reenroll, Unsatisfactory) which is currently employed. She noted that the question of “what constitutes mastery?”, is a difficult one. The intention of this modular framework is that students should only take the modules that are needed, as determined by the placement test, and work to improve their mastery of that topic before moving forward. This also allows for greater differentiation between students. For example, Liberal Arts students would have different developmental math requirements than students in STEM programs.

Part three of the presentation was led by Dr. Donna Jovanovich and discussed the goals of developmental math redesign. The three goals of the DMRT are (1) to reduce the need for developmental education, (2) reduce time to complete developmental education, and (3) to increase number of developmental education students graduating or transferring. Each of these goals has a related measure of success. For example, “reduced need for developmental education” can be measured by placement test scores and “reduced time to complete developmental education” can be measured by student success in developmental classes. One interesting statistic that Dr. Jovanovich mentioned was the following: only 1/3 of developmental math students that don't pass reenroll in the course the following semester, of those, only 1/3 pass the second time, but those that do pass through the developmental program successfully have a 80% of graduating or transferring. So while success rates for the courses are grim, there are long term payoffs for the students who do succeed.

Dr. Wood returned at the end of the session for some closing remarks. The steps for the DMRT program are to have the curriculum approved by the Dean's course committee and to find out how the modularization of developmental math will affect enrollment services and financial aid.

For more information, see the VCCS Developmental Education home page.

# VCCS Reengineering Initiative

The second event I attended was a presentation from VCCS Chancellor, Dr. Glenn DuBois. The Chancellor began with an overview of the goals for the Reengineering Initiative, many of which are spelled out in the Achieve 2015 publication. The goals are to improve access, affordability, student success, workforce and resources. He noted that the VCCS is experiencing an increased number of students that register for classes, and increased number of these students are unprepared, a decrease amount of public funding, along with a call for more public accountability and more college graduates. Currently, about 50% of high school graduates require developmental education and only 25% of them go on to graduate in four years. He made the case that there is bipartisan support for improving the quality of education, using President Obama and Virgina Governor McDonnell as two examples. President Obama has stated that he wants to see 5 million more graduates in the US, while Governor McDonnell has stated that he wants to see 100,000 more graduates in the state of Virginia. This is the heart of the Reengineering Initiative: improving student success with sustainable and scalable solutions. Some of the funding for the Reengineering Initiative has been made possible by Federal funding, as well as the Lumina & Gates foundations.

In order to improve the 25% success rate of developmental education, the Reengineering Initiative is implementing major changes to the developmental math program. First is the opening of different paths for different students. Second is a revised business model which replaces a “test in/test out” philosophy with a diagnostics and short modules intended to improve mastery. To accomplish these goals, the Virginia Community Colleges are moving in a direction of more shared services, in areas such as Financial Aid and distance learning. The VCCS is also looking for ways to help local high schools better prepare students for college, such as making the placement test available to high school students and developing transition courses.

# Best Practices in a Changing Developmental Education Classroom

The last event of the first day was a keynote presentation from textbook author Elayn Martin-Gay. Elayn's first major point was about the importance of “ownership” for both teachers and students, and how language can affect the feeling of “ownership”. For example, instead students' grades being “given”, they should be “earned”. She seemed very positive about the Reengineering Initiative, saying that it was “good to be doing something, even if it's wrong, [so that] you can tweak it and continue”.

She then proceeded into more classroom oriented practices, saying that it was important to monitor student performance and catch students “at the dip”. If a drop in performance can be corrected early, this can prevent the student from getting too far behind. She also talked about the importance of students keeping notes in a “journal”. This encourages good study skills, giving students a source to go to when it comes time for the exam. She suggested that teachers should “learn the beauty of a little bit of silence”. Teachers should not always jump right into a solution to a problem, but that waiting a extra three seconds longer will dramatically increase the number of student responses. She also said that teachers should “raise the bar and expect more from students”, and that “they will rise to meet it”. She recommended that disciplinary problems occurring in the classroom should be taken care of immediately, to maximize time for learning later.

After these classroom practices, she moved into some of the larger social issues affecting developmental education. She noted that the supply of college degrees has gone down, while the demand for experts has gone up. She jokingly called the first year of college “grade 13”, noting that many college freshmen have yet to decide on a long term plan. She cited seven current issues affected new college students: lack of organization, confidence, study skills, attendance, motivation, work ethic, and reading skills. She argued that reading is often the biggest barrier to earning a college degree.

As some ways of addressing these issues, she presented a number of graphs relating college experience with employment and income. She said that she often presents these graphs at the start of the semester as a means of encouragement. She has students covert the statistics from annual income to an hourly wage so that they can more closely relate with the figures. She also included some ideas for asking “deeper” questions in math classes. One of the examples was “Write a linear equation that has 4 as the solution”. The trivial solution to this is “x=4”, then we can build off this to find others “2x=8” and “2x-3 = 5”. She says that students will typically solve these equations step by step each time, by the time she asks students to solve something like “-2(2x-3)/1000 = -10/1000” they start to look for an easier method – realizing deeper properties about equality in the process.

One of the things Elayn said that resonated strongly with me was that “students would rather be in charge of their own failure than take a chance on [asking the teacher]”. As math teachers, the general feeling of the audience was that study skills are not our focus, but as Elayn pointed out, those study skills can have a powerful influence on student success. By providing students with the skills necessary to “learn math”, those students can in turn take charge of their learning experience.

# Next time: VMATYC Day 2

Stay tuned as I collect my notes from Day 2. Day 2 events include: “Online Developmental Math on the Brink: Discussion Panel”, “Developmental Mathematics SIG Roundtable”, and “The Mathematical Mysteries of Music”.

## 2011/03/07

### #MathChat Recommended Reading

Filed under: Education,Math — ryan @ 12:54

This week's topic on #mathchat was "What books would you recommend for mathematics and/or teachers, why?". I offered several suggestions in Thursday's chat, but wanted to go back and explain in more detail "why". I've also added a few to help round out the selection. These books are listed in approximate order of "increasing density", with the more casual titles at the top and the more math intensive titles near the bottom. Of course, this ordering is my own subjective opinion and should be taken with a grain of salt.

Disclaimer: The author bears no connection to any of the publications listed here, nor was the author compensated for these reviews in any way.

### Lewis Carroll - "Alice's Adventures in Wonderland", "Through the Looking-Glass"

Recommended for: all ages, casual readers

Charles Dodgson, perhaps better known by his pen-name "Lewis Carroll", authored a number of children's books including the aforementioned titles. What makes these books so special, is that Dodgson was also a mathematician and embedded numerous mathematical references in these works. Most people might be familiar with the many film adaptations of these works, but I'd highly recommend reading the originals with an eye towards the logical riddles and mathematical puzzles hidden in these classics. You can find these books online at Project Gutenburg. For little a taste of the mathematics involved, you might start here.

### Charles Seife - "Proofiness: The Dark Arts of Mathematical Deception"

Recommend for: teens and older, casual readers, who don't think math is relevant to daily life

This book focuses on what I consider to be a important topic in the current socio-political climate. Ordinary people are repeated bombarded by "deceptive mathematics". Whether the source is trying to sell a product or push a political agenda, the inclusion of numeric figures or fancy graphs can go a long way to make a claim look more legitimate than it really is. Proofiness spells out some of the common warning signs of deceitful mathematics, so that the reader can be more aware of these practices. While somewhat lighter on the mathematical content that more advanced readers might expect, I think this book sheds some much needed light on an important social issue and was an enjoyable read. If you like this, you may also like How to Lie with Statistics by Darrell Huff

### Apostolos Doxiadis, Christos H. Papadimitriou, Alecos Papadatos and Annie Di Donna - Logicomix

Recommended for: casual readers, comic book fans

Technically this is a graphic novel instead of your typical book, but that doesn't mean it doesn't cover some important mathematics! Logicomix presents Betrand Russell as the antagonist in a series of historical events that took place in the early 20th century, culminating with Kurt Gödel's Incompleteness Theorem. which shook the very foundation of mathematics. Logicomix makes superheroes out of mathematicians in an epic story, while exposing the reader to some amazing mathematics. Ties in nicely with Gödel, Escher, Bach below.

### David Richardson - "Euler's Gem"

Recommend for: casual readers curious about topology

I was looking for a casual introduction to topology and found this little "gem". This is the book that I wish I read while studying topology in college! It covers everything from the basic principles of topology to the recently solved Poincaré Conjecture. Don't let all this mathematics scare you away from this title! The book is still written in a very approachable manner. It chronicles the life history of Leonard Euler, presenting the development of the field of topology in context that even the casual reader can enjoy.

### Douglass Hofstadter - "Gödel, Escher, Bach: An Eternal Golden Braid"

Recommended for: semi-casual readers with diverse interests

When someone asks me for "a good math book", this is my go-to recommendation. This book has a little of something for everyone. Math, music, art, language, computers, biology, and psychology are woven seamlessly into a humorous and playful narrative, reminiscent of Lewis Carroll. It goes deep into mathematical concepts where appropriate, and uses visual material and metaphor to bring complex concepts down to Earth. I listed it as "semi-casual" due to the depth of mathematics involved, but a casual reader can skip some of the more math intensive parts and still get a nice overview of the general principles.

### Jean-Pierre Changeux and Alain Connes - "Conversations on Mind, Matter, and Mathematics"

Recommended for: semi-casual readers, with interest in philosophy

This book spans several conversations between a Mathematician and a Neurologist on the Nature of Mathematics. One of the central questions is if mathematical ideas have an existence of their own, or if they exist only within the neurology of the human brain. Both sides present some fascinating support for their side of the argument. The material can be a little dense at times, making reference to advanced research as if it were common knowledge, and might not be appropriate for more casual readers. However, a reader willing to dig in to these arguments will reveal two very fascinating perspectives on the philosophy of mathematics.

### James Gleick - "Chaos: Making a New Science"

Recommended for: semi-casual readers, preferably with some Calculus experience

Chaos takes the reader on a historical journey through the emergence of Chaos Theory as a mathematical field. An amazing journey through the work of numerous mathematicians in different fields, who came upon systems exhibiting "sensitive dependence on initial conditions". This book serves as an introduction to both Chaos Theory and non-linear dynamics, while shedding light on the process behind the development of this field. Some experience with differential equations would be beneficial to the reader, but more casual readers can get by with assistance of wonderful visual aids. A nice complement to A New Kind of Science below.

### Roger Penrose - "The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics"

Recommended for: more advanced readers with interests in physics and artificial intelligence

Roger Penrose is a well established mathematical physicist, and The Emperor's New Mind offers an accurate and well written overview of quantum physics. However, what makes this book interesting is that Penrose takes this physics and mathematics to mount an attack on what Artificial Intelligence researchers describe as "strong AI". Penrose makes the case that Gödel's Incompleteness Theorem implies that cognitive psychology's information processing model is inherently flawed -- that the human mind can not be realistically modeled by a computer. Whether you agree with Penrose's conclusions or not, his argument is insightful and is something that needs to be addressed as the field of cognitive psychology moves forward.

### Stephen Wolfram - "A New Kind of Science"

Recommended for: more advanced readers with interest in computer science

Don't let its size intimidate you. If you made it through the titles above, than you should be ready to make headway into this giant tome. The central theme of A New Kind of Science is that complex phenomena can emerge from simple systems of rules. This is different from Chaos Theory (described above), in that this complexity can emerge regardless of initial conditions. A New Kind of Science takes the stance that we can learn a great deal about mathematics through experimentation, and makes the case that perhaps the vast complexity of the universe around us can be explained by a few simple rules.

## 2010/10/16

### A Rebel Math Curriculum

Filed under: Education,Math — ryan @ 22:07

One of many insightful educators I follow on Twitter, Tom Whitby, wrote A Modest Blog Proposal asking for bloggers to post educational suggestions on October 17th, 2010. He proposed the acronym REBELS for “Reforms from Educational Bloggers Links of Educational Suggestions”. I found the idea of a "rebel" education very intriguing and if there's one place where educators need to resist authority, I think it's the mathematics curriculum.

Before we get into my proposed curriculum, it's important to have an idea of where we're starting. Paul Lockhart describes the existing system quite concisely in A Mathematician's Lament:

The Standard School Mathematics Curriculum

LOWER SCHOOL MATH. The indoctrination begins. Students learn that mathematics is not something you do, but something that is done to you. Emphasis is placed on sitting still, filling out worksheets, and following directions. Children are expected to master a complex set of algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part, and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables are stressed, as are parents, teachers, and the kids themselves.

MIDDLE SCHOOL MATH. Students are taught to view mathematics as a set of procedures, akin to religious rites, which are eternal and set in stone. The holy tablets, or “Math Books,” are handed out, and the students learn to address the church elders as “they” (as in “What do they want here? Do they want me to divide?”) Contrived and artificial “word problems” will be introduced in order to make the mindless drudgery of arithmetic seem enjoyable by comparison. Students will be tested on a wide array of unnecessary technical terms, such as ‘whole number’ and ‘proper fraction,’ without the slightest rationale for making such distinctions. Excellent preparation for Algebra I.

ALGEBRA I. So as not to waste valuable time thinking about numbers and their patterns, this course instead focuses on symbols and rules for their manipulation. The smooth narrative thread that leads from ancient Mesopotamian tablet problems to the high art of the Renaissance algebraists is discarded in favor of a disturbingly fractured, post-modern retelling with no characters, plot, or theme. The insistence that all numbers and expressions be put into various standard forms will provide additional confusion as to the meaning of identity and equality. Students must also memorize the quadratic formula for some reason.

GEOMETRY. Isolated from the rest of the curriculum, this course will raise the hopes of students who wish to engage in meaningful mathematical activity, and then dash them. Clumsy and distracting notation will be introduced, and no pains will be spared to make the simple seem complicated. This goal of this course is to eradicate any last remaining vestiges of natural mathematical intuition, in preparation for Algebra II.

ALGEBRA II. The subject of this course is the unmotivated and inappropriate use of coordinate geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety of standard formats for no reason whatsoever. Exponential and logarithmic functions are also introduced in Algebra II, despite not being algebraic objects, simply because they have to be stuck in somewhere, apparently. The name of the course is chosen to reinforce the ladder mythology. Why Geometry occurs in between Algebra I and its sequel remains a mystery.

TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and obsolete notational conventions, in order to prevent students from forming any clear idea as to what the subject is about. Students will learn such mnemonic devices as “SohCahToa” and “All Students Take Calculus” in lieu of developing a natural intuitive feeling for orientation and symmetry. The measurement of triangles will be discussed without mention of the transcendental nature of the trigonometric functions, or the consequent linguistic and philosophical problems inherent in making such measurements. Calculator required, so as to further blur these issues.

PRE-CALCULUS. A senseless bouillabaisse of disconnected topics. Mostly a half-baked attempt to introduce late nineteenth-century analytic methods into settings where they are neither necessary nor helpful. Technical definitions of ‘limits’ and ‘continuity’ are presented in order to obscure the intuitively clear notion of smooth change. As the name suggests, this course prepares the student for Calculus, where the final phase in the systematic obfuscation of any natural ideas related to shape and motion will be completed.

CALCULUS. This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned. To be taken again in college, verbatim.

So if Educational Rebels could have their way with the math curriculum, what would it look like instead? Certainly it would be different from what goes on in the Educational Empire.

Rebel vs. Empire terminology geekily borrowed from Star Wars. Image obtained from Wookieepedia.

Within the Educational Empire, there are Official Imperial Standards which teachers must adhere to or they will be fired and annual multiple-choice tests that students must take as if their lives depended on them. As Imperial teachers routinely state, students who do poorly on these tests will die poor and lonely, and students who do well on these tests will go on to an Empire approved Private Academy where they will accumulate massive debts which must be repaid to the Empire through decades of hard labor.

The Rebels would do away with all of this. Instead of multiple-choice choice tests, students would have authentic assessments where they create products they can be proud of. Teachers would have educational goals in mind for instruction, but the nature of the material covered is directed by the students. Students are free to learn about the things they are genuinely interested in, and often go home telling their parents how they “can't wait to go to school tomorrow”. Upon graduating from public schools, students are well prepared to start pursuing the career of their choice. Those who want to further the field of knowledge in their respective area of interest can go on to to a Rebel college for free, where they are guided in conducting effective scientific research.

The Rebels would need to rewrite the curriculum from scratch to accomplish all of this, starting with the math curriculum. The new curriculum would be nonlinear and individualized for each student. The students would be the ones to direct learning, and the teacher would be there as an experienced learning guide. The following tale describes the math curriculum of some hypothetical Rebel students:

# Rebel Mathematics Curriculum

## Elementary School

Student exposure to mathematics begins with games. Students start with simple games in the beginning, and the games get more complicated as the students progress. These particular students start with games like Rock-Paper-Scissors, Go Fish, Tic-Tac-Toe, Hide-and-Seek and Dodgeball. In the context of these games, students learn how to play within a set of rules and learn the basic language of logic and sets. Students learn what it means when the rules say “Do this and this”, “Don't do this or that”, “Move this from this group to that group”, “Combine this group and that group”, “Separate this group into multiple groups based on some quality”. These concepts form a solid foundation for mathematical thinking. As students progress, they get into games where counting becomes more important. Students play games like Sorry!, Chutes and Ladders, Hi Ho Cherry-O, and even sports like Tennis. As students become more familiar with counting, they get into games with more complex scoring methods like Yahtzee, Blackjack, Monopoly, and Risk where they further develop their arithmetic skills.

Dice games like Yahtzee can be used to introduce basic arithmetic skills. Image obtained from Wikipedia.

## Middle School

In Middle School, students continue to play increasingly complex games. Students play board games like Chess and Go, card games like Poker, and video games like Sim City and World of Warcraft. Students are encourage to engage in meta-cognitive processes as they play, by talking about different strategies for optimizing how they play. The basic concepts behind Algebra and Probability emerge naturally from these discussions. Students start developing games of their own, beginning with board and card games and moving into programming simple video games. That's right, all students are encouraged to start programming in middle school. After the students develop a prototype game, they play-test the game, collect data from the play-test, analyze that data, and use the information they discover to revise the rules of their game. The middle school hosts a Game Convention at the end of the school year where each class puts on a demonstration of their game and the process they used to come up with it. The parents are invited to come play their kids' games and see how the students' critical thinking skills have developed.

Game record of a Go match between Honinbo Shuusaku and Gennen Inseki in 1846. Image obtained from Wikipedia. The game of Go is immensely complex, and one of few games where Artificial Intelligence research has yet to reach the level of professional human players.

## High School

In High School, the class of students that worked as a group in middle school starts to diverge into different groups based on individual interests. Students with an interest in sports might have a math course that is focused on Geometry and Spacial-Reasoning, with a little bit of Game Theory on the side. Students with an interest in Music might combine Trigonometry and a little computer programming to produce new and interesting sound effects. A group of photography and art students start programming new filters in GIMP to create original effects for their images. A group of students interested in journalism learns about web programming as they put together a professional looking web site. A group of students that developed an interest in Racing games, is introduced to physics and some real-world automotive engineering. A group of students with an interest in programming starts learning about Calculus as they write their own First-Person Shooter. Another group of students uses a Lego Mindstorms kit to build a robot that sorts a line of objects by size, learning a variety of math and engineering skills in the process. Students graduate from high school with more than just a diploma, but a portfolio of work demonstrating their mastery in their subject area of interest. Students show off these portfolios at a convention where the local employers are known to stop by to identify potential job candidates. Most of these students will move straight into a job in their field of interest, but some will go on to pursue further research opportunities in college.

Products like Lego Mindstorms can be used in high school to develop practical engineering skills. Image obtained from Official Lego site.

## College

All Rebel students have the option of furthering their education in a publicly owned and operated Rebel University. This education is provided at no cost to students. Exceptional students conducting research at the Rebel University may even be paid for their contributions. Rebel society recognizes the value of academic research, and considers the value of the knowledge resulting from student research to more than compensate for the costs associated with running the Rebel University. Students don't just go to college to learn, they go to further the existing knowledge in their respective fields.

## Conclusion

In this Rebel Education, gone are the days of Algebra, Geometry, More Algebra, Trigonometry, and Calculus. Gone are the days of lengthy multiple choice tests. Teachers assess students by analyzing the products they create and encourage the students themselves to critically reflect on their own creations. Students are not pressured to meet Imperial standards, but instead are responsible for setting their own goals for improvement each semester. The students don't feel like they are competing to score higher than their classmates, but instead learn to recognize that each of their classmates has a different set of skills and that by cooperating they can achieve things that they could not do alone. While the Empire is pumping out clone after clone, the Rebels are producing a diverse array of students with varying sets of knowledge and skills.

Which students do you think would be happier and more successful in life? Those with their Empire prescribed cookie-cutter education? Or those from the Rebel academies?

I must confess that there has not been enough research to predict what the long-term effects of such a Rebel education would be. However, I do think there is a substantial amount of evidence indicating that the traditional Imperial curriculum is failing. Educational research provides incremental improvements to the existing curriculum, but perhaps the system's assumption that everyone should have the same education is fundamentally flawed. At some point in the future, it may become necessary for "Educational Rebels" to overthrow the "Educational Empire" and challenge this assumption. The mathematics curriculum proposed here may not be perfect, but it might provide a starting point that educators can revise and improve over time.

This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License. You are free to share and remix this work, provided that the source is properly attributed and derivations are released under a similar license.

## 2010/10/13

### (d/dt) Nature Of Math = 0?

Filed under: Education,Math — ryan @ 14:20

Over at the Republic of Math, Gary Davis launched a preemptive strike on Thursday's upcoming #mathchat topic: "Does the nature of mathematics change as students get older or is it only the teaching methods that change?". His conclusion? Neither. I would tend to agree with this, but I'm going to play devil's advocate here and argue a different perspective.

Looking at this from the overall historical perspective, the "nature of math" has not changed very much over the years. The concepts at the core of mathematics, like quantities and patterns, are the same as they were thousands of years ago. The teaching methods for encouraging development also remain fundamentally the same as students age. Teachers seek to identify a student's present level of understanding and design learning activities that will bring that understanding to the "next level". From this perspective, neither the nature of math nor methods of teaching change over time.

However, there's something interesting in how this #mathchat topic is phrased. The question is not "does the nature of math change over time?" but rather "does the nature of math change as students get older?". If we concede the former point about the nature of math remaining the same over time from a historical perspective, the phrasing of the #mathchat topic hints at an alternative interpretation. Redirecting this question to the reader:

Did the "nature of math" change as you got older?

My answer to this is "Yes". The "nature of math" became more and more refined as I was exposed to new mathematical ideas. In particular, my preconceived notions about "the nature of math" were shattered into pieces when I learned about this guy:

Around the turn of the 20th century, Russell, Whitehead and Hilbert were attempting to build a solid foundation for mathematics using logic and set theory. This is more or less how I thought of the "nature of math" prior to college. I thought that if I only understood the basic rules of the "math game", I could figure out anything I needed to from applying those rules in a logical manner. It turns out I was wrong.

In 1931, Gödel published his famous Incompleteness Theorem. This result proved that the kind of system that Russell, Whitehead and Hilbert were attempting to create would either be incomplete or inconsistent. Likewise, this theorem single-handedly destroyed the concept of "the nature of math" that I had built up over the years. In its place I started to form a new concept about the "nature of math". Gödel had taught me to start looking at mathematics from outside the box.

Some other works that altered my notions about the nature of math were Turing's proof that the Halting Problem is undecidable and Cantor's Diagonal Argument. There are many parallels between all three of these results. What really resonated with me was the metacognitive component to these proofs. The nature of math had shifted from thinking about how I could play within the rules of the game, to thinking about how those rules could be bent or broken. Instead of "thinking about math", I started "thinking about how I thought about math".

This change in my perception of the nature of math not only influenced my future learning, but also changed how I thought about previous topics. The Republic of Math article mentions trigonometry, which is one of those areas that I was forced to revisit with this new perspective. As Gary notes in this and previous posts (here and here), there are lots of problems with trigonometry's reliance on triangles. Under my new "nature of math", the assumption that triangles needed to lie in a Euclidean space was no longer a safe assumption to make.

My background as computer programmer also altered my conception of trigonometry. Because trig functions and square roots can be computationally expensive, I developed a habit of avoiding them whenever possible. Most of the time I can get the data I need from a dot product instead of working with angles. Instead of using the distance between points, I'll often use the square of the distance as my metric. I'd learned to not just solve the problem, but to reflect on how I was solving the problem and try to optimize that process.

This brings me to the second part of the #mathchat prompt, which is "do teaching methods change as students get older?". Earlier I discussed the notion that the teacher should identify a students level of comprehension and guide them towards the "next level". With this trigonometry example, we can see that this model is overly simplistic. Math is not linear. Instead, the teacher must not only identify the student's current conception, but also which path that student is following so they can encourage them in that direction. If students are starting to think about trigonometry in terms of vector dot products, guide them towards linear algebra. If students are starting to think about triangles on spherical surfaces, guide them towards non-Euclidian geometry. Experienced educators like Gary are probably very adept at this. However, I think the reality of the situation is that most math teachers are not.

From the perspective of this student, the "teaching methods" did not change as I got older. With one exception, my math teachers adhered rigidly to the following procedure:

1. lecture to blackboard for entire class period
2. assign dozens of homework problems
3. test on material
4. rinse and repeat

How many math classes have you taken that followed this pattern? How many did not? There is an urgent need for the kind of teacher training Gary describes, where the focus is on personalized student development. Too many teachers are caught up in teaching the content, when they should be facilitating student learning instead.

In conclusion:

Does the nature of mathematics change as students get older or is it only the teaching methods that change?

Yes, a student's model of the "nature of math" can change as that student grows older and discovers new results from the field of mathematics. I also think it's arguable that the "nature of math" is not necessarily static, as presumed above, but that Gödel's Incompleteness Theorem fundamentally changed the "nature of math" by using mathematics as a tool for analyzing itself -- giving birth to metamathematics.

No, teaching methods do not change as the student grows older, but they do vary from teacher to teacher. In general, I think the "typical math teachers" need to take a cue from Gödel and start thinking more outside the box. The mathematics classroom needs to shift from its lecture/homework/test/repeat cycle, where math is essentially taught using an "argument from authority", to an experimental environment where students are encouraged to question the information they receive.

## 2010/10/11

### Reflections on #mathchat: Mathphobia

Filed under: Education,Math — ryan @ 18:06

Today's #mathchat was a repeat of last Thursday's discussion on “Mathphobia”. One of my questions in Thursday's chat prompted a very insightful commentary from Gary Davis, a.k.a. @RepublicOfMath. With this new evidence in mind, I tried to observe today's #mathchat with a fresh perspective. I couldn't quite condense my thoughts into 140 characters, so I'm taking this opportunity to summarize what I learned from the experience.

First, I think its important to clarify what is meant by “mathphobia”. For the sake of clarity, I'll use the term “mathphobia” in the same sense as @RepublicOfMath's article. Mathphobia is a condition where an individual is terrified of mathematics to the point of feeling physically sick at the thought of math. I'll use the term “math anxiety” to refer to a lesser version of this condition, where an individual experience a fear of math that interferes with mathematical performance but is not as completely disabling as mathphobia. In general, moderate symptoms of math anxiety are highly prevalent in society. As @ColinTGraham noted, research studies have shown that simply telling adults that they're going to take a math test will cause their blood pressure to rise! I realize that this distinction between “math anxiety” and “mathphobia” is somewhat fuzzy, but for the sake of argument these labels will suffice for now.

Reviewing Thursday's #mathchat archive, I think you can see two different conversation lines taking place. One conversation about mathphobia and another about math anxiety.

With regards to @RepublicOfMath's proposal that mathphobia is the result of abusive teachers, this makes a lot of sense from the standpoint of classical conditioning. If a student repeatedly has painful experiences with mathematics instruction, then the student will gradually learn to associate the two. As a consequence, experiencing any subsequent mathematical instruction will automatically trigger a painful response.

With math anxiety, there are similar mechanisms at work. For example, the rise in blood pressure in preparation for a math test can be interpreted as a conditioned response to the need for an increase in cognitive processing. The high prevalence of math anxiety symptoms suggests that math anxiety can develop with or without “abusive teachers”. I think that a variety of the “mathphobia causes” discussed in Thursday's #mathchat may contribute to math anxiety in some form or another, but may not be a cause of the more extreme mathphobia as described above.

With today's #mathchat, I saw something a little bit different happen. The conversation took a turn towards “math avoidance” – the lack of participation in mathematical activities. Here I think we see the crux of the problem. When a student develops math anxiety or mathphobia, that student begins a behavioral pattern of math avoidance. This behavior is self-reinforcing because it allows the student to avoid the painful stimuli associated with math. In order to undo the association that underlies the math anxiety or mathphobia the student needs to be presented with stimulus-response situations that are positive, but when the student avoids math altogether this becomes a difficult task.

The other complication that math avoidance presents is that it becomes difficult to distinguish between students who suffer from math anxiety or mathphobia, and those who are avoiding it for other reasons. Those reasons might be a lack of perceived relevance, a negative social image of math, or a lack of self-confidence. Many of these issues were identified in Thursday's #mathchat, but the focus of today's chat really tied them all together for me.

In conclusion, I think we need to address math anxiety and mathphobia from two directions. First, the classroom needs to be a safe environment where students are free to make mistakes and learn from them rather than being punished for them. Secondly, the behavior of math avoidance needs to be addressed. In order to facilitate the extinction of the conditioned stimulus-response to math, students need to be exposed to math in a positive environment. At first glance, it may seem like this is “treating the symptom rather than the cause”. However, if teachers do not provide temporary relief for the symptom of math avoidance, it won't be possible to “treat the cause” of math anxiety or mathphobia.

Some questions for further discussion:

• Where does one draw the line between "math anxiety" and "mathphobia"?
• How can educators address "math avoidance" behaviors?
• What are the best practices for creating and maintaining an empathetic and non-threatening mathematics classroom?

## 2010/09/04

### Rationalize This!

Filed under: Education,Math — ryan @ 00:45

It's been awhile since I blogged, so I thought I'd take a moment to talk about couple interesting math problems that came up in a conversation I, @SuburbanLion, had on Twitter with @MathGuide, @RepublicOfMath, and @GMichaelGuy about a month ago. The topic of discussion was the role of rationalization problems in Algebra II and whether or not the current curriculum addresses the “conceptual core” of these problems.

A common example that one would see in an Algebra II course is:

• Rationalize the denominator of $\frac{1}{\sqrt{2}}$

Or simply:

• Rationalize $\frac{1}{\sqrt{2}}$

The expected answer for this problem can be obtained by multiplying the numerator and denominator both by $\sqrt{2}$ to get: $\frac{\sqrt{2}}{2}$. Students might be assigned dozens of such problems in Algebra II. The question to be asking is “Why?”.

I suspect that the obvious answer is historical tradition. These math problems have been passed down from generation to generation as “standard Algebra II problems”, and even the new Core Standards includes “rewrite expressions using radicals and rational exponents” as an objective. Instead of treating these problems as a “means to an end”, these problems have become something of an end in themselves. Algebra II students learn to rationalize expressions because that's what they're going to be tested on. End of story.

The real reason for having these problems in Algebra II goes deeper than that. The Core Standards hits on this reason (at least partially) with one of the additional objectives: “Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, and division by a nonzero rational expression.” While this objective is not specifically talking about radical expressions, the core concept is the same. More concisely, we might say that the important point of rationalizing radical expressions is to come to the conclusion that $\mathbb{Q}+\mathbb{Q}\sqrt{p}$ is a field.

Ironically, the 93 page Core Standards does not even mention the word “field” even though this is essentially the core concept that students are learning about. Students shouldn't be rationalizing expressions just for the sake of rationalizing expressions, they should be exploring the intermediate steps between the field of rational numbers and the field of algebraic numbers. By the way, the phrase “algebraic number” doesn't appear in the Core Standards either!

To make things interesting, G. Michael Guy presented a couple rationalization problems that are typically not included in Algebra II problem sets. These are actually really great examples of the potential complexity involved in rationalization problems.

Let's warm up with the first one, which is significantly easier:

• Rationalize $\frac{1}{1+\sqrt[3]{2}}$

This one has an elegantly simple solution using the sum of cubes factorization:

• $a^{3}+b^{3} = (a + b) \cdot (a^{2}-ab + {b^{2}})$

Making the substitutions a = 1, b = $\sqrt[3]{2}$, we can take advantage of this product to rationalize the denominator:

• $\frac{1}{1+\sqrt[3]{2}}\cdot\frac{1-\sqrt[3]{2}+\sqrt[3]{2}^{2}}{1-\sqrt[3]{2}+\sqrt[3]{2}^{2}} = \frac{1-\sqrt[3]{2}+\sqrt[3]{2}^{2}}{1+\sqrt[3]{2}^{3}} = \frac{1-\sqrt[3]{2}+\sqrt[3]{2}^{2}}{3}$

This solution is simple enough that this could almost pass for an Algebra II problem. Rationalization problems in Algebra II are something of a gimmick: the problems chosen are special cases designed to have an easy answer. The methods taught in the textbook will solve the given problems, but they don't generalize well to the larger class of problems. The techniques used in Algebra II provide very little help in rationalizing an expression like $\frac{1}{1+\sqrt[3]{5}+\sqrt[3]{3}^{2}+\sqrt[3]{2}}$!

Before we attempt something like this, lets go back to $\frac{1}{1+\sqrt[3]{2}}$ and come up with a method that will generalize well. This is where easy problems come in handy as a test bed for discovering the broader patterns. The general case of rationalizing $\frac{1}{x_{0}+x_{1}\sqrt[3]{2}+x_{2}\sqrt[3]{2}^{2}}$ can be reasonably done by hand.

The method used here is not likely to be seen in a typical Algebra II classroom, as it relies on concepts from Linear Algebra which typically aren't addressed until later. Seems a little backwards if you ask me. The trick here is to think of the product $(x_{0}+x_{1}\sqrt[3]{2}+x_{2}\sqrt[3]{2}^{2})(y_{0}+y_{1}\sqrt[3]{2}+y_{2}\sqrt[3]{2}^{2})$ as the product of a matrix and a vector:

• $M_{x}\cdot\overrightarrow{y} = [1, 0, 0]$

Solving this equation is then simply a matter of multiplying both sides by $M_{x}^{-1}$. This method extends nicely to even harder problems, including $\frac{1}{1+\sqrt[3]{5}+\sqrt[3]{3}^{2}+\sqrt[3]{2}}$. Inverting a 27 by 27 matrix is not something I'd want to be doing by hand, so this is where computers come in handy. Here's an algorithm for rationalizing $\frac{1}{1+\sqrt[3]{5}+\sqrt[3]{3}^{2}+\sqrt[3]{2}}$ in Sage. Compare this with Wolfram|Alpha's result.

I'm a strong believer that computing should play a larger role in mathematics education than it is presently. Not only should the curriculum be addressing the fact that the algebraic numbers form a field, but also that all algebraic numbers are computable numbers. By shifting the focus of discussion from solving problems to finding an algorithm for solving those problems, we can reveal a better picture of the mathematics behind the problem. Simple problems worked out by hand play an important role in the process of designing an algorithm, but an ability to generalize the solution should be the larger goal. If the rationalization problems that students are completing by the dozens do not lead the student in the direction of a general solution, then those problems are not doing their job. Perhaps some “harder” problems are necessary to encourage that generalization.

For the record, “computable numbers” are not referenced once in the Core Standards. It's hard to give students a 21st century education when the math curriculum is trapped in the early 1900s.

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