Engineering a 21st Century Education

As a game developer turned teacher, the one most difficult part of the transition was paper. I'm not even talking about the thousands of copies of handouts and worksheets for students. I'm talking about the paper that compromises the layers and layers of bureaucratic processes that any sufficiently large organization develops over time. The forms. The reports. The mundane paperwork that must be done to uphold the laws that govern the operation of a school.

I get it.

It's stuff that must be done.

The problem isn't that it exists, it's how schools deal with it. The school has limited resources available so it needs to get the most out of the resources it has. That makes sense, right? The school administration has ready access to a large number of highly trained, adaptable, resourceful, and intelligent individuals on hand with a wide range of skills covering every discipline imaginable. It has teachers.

I'm always more than happy to help when needed! I just get frustrated when I'm asked to perform work that could be reduced or eliminated by technology.

In my last post, I talked about school being a game and the need to meta-game it. One of the first issues that I think we need to talk about are "opportunity costs". Every hour that teachers spend on administrative tasks is an hour that is not being spent on teaching. Furthermore, these costs are recurring. If schools could automate 10 minutes worth of administrative tasks each day from a teacher's workload, they would save each teacher about 30 hours of work over the course of the year. That's a lot of time that teachers could reallocate towards improving instruction.

Teacher time is a valuable resource and finite one. Education needs to be engineered to get the most out of that time. Based on my short time as a teacher so far, here are some of the systems that I think could be optimized:

We need a complete "Electronic Individualized Educational Plan Record" system overhaul. The current generation of "Student Information Systems" is grossly insufficient to deal with the complexity of our educational legislation. Schools need to keep documented records of adhering to a student's legally entitled accommodations, and a significant amount this documentation is still being done on paper. We have the technology to design an educational record system that is secure, fault tolerant, and efficient. It would take an substantial initial effort, but imagine the time that it could save school staff in the long run.

We need a better "asset management system" for school property used by students. It's very frustrating to me as a teacher when I need to fill out carbon copied checkout lists for textbooks by hand in the year 2016. When a student doesn't return the textbook, I'm required to fill out another carbon copy form, manually address an envelop to the student's home, and put it in the mail bin for processing. Why isn't this process electronic yet? I should be able to snap a picture on my phone, press a button to assign it to a student or document its return, and everything else should be taken care of by a computer program. We clearly have the technology to do this.

We need a "behavioral intervention tracking and diagnostic system". The school keeps paper records of certain student behaviors such as tardy slips and misconduct reports -- which again are filled out by hand on carbon copy paper. There are also some cases where the teacher is expected to intervene in certain ways such as contacting the parent. The issue is that there are so many different rules that I need to keep track of and responses that I need to take to that data. We need a system that that can track behavior data from multiple sources and suggest interventions based on a statistical models of what has and has not worked for that student.

On top of moving from antiquated "pen and paper" systems, we also need to improve interoperability between the educational software we already use. There's some good ideas happening with the Tin Can API, but the support from technology providers just isn't there yet. I love to see new ideas in educational software! The problem is that some of these applications seem to neglect the teacher's experience with the product. We need to set higher standards for educational software.

Whenever my students complete a learning activity on the computer, it should automatically go into my grade-book. The grade-book should automatically flag any items that need to be manually graded, and the process of providing feedback to the student should be as stream-lined as possible. More detailed information about the student's performance should be stored into a database for later statistical analysis.

The other problem is the lack of standards regarding assessment items. For example, my students love Kahoot. I would totally use it way more if it were easier for me import multiple choice questions from an existing database. If I could program randomly generated questions in MyOpenMath, export them to a standardized format, and then import them into Kahoot, I would be one happy teacher.

I don't think any of these technologies are unrealistic. It's not like I'm asking for facial recognition software to replace hall passes or an artificially intelligent grader (although those would be kinda awesome too). If schools want to instill "21st Century Skills" in their students, they need to lead by example. In the "21st Century", knowing what processes can be automated by technology is a crucial skill to have. To do otherwise is a disservice to both teachers and students.

I used to think schools needed more games

I love games! I love playing them. I love making them. I love theorizing about them. They're an essential part of who I am as a person.

I used to think schools needed more games.

I was working as a video game developer and was fascinated by "tutorial levels". You know, that part of the game that is designed to help you learn how to play the game. Some games neglect their tutorial level and it comes off feeling like a dry lecture. Go here. Push button. Repeat. However, I've also been completely awed by some games that take their tutorial levels to a completely different level. Games like The Elder Scrolls and Guild Wars for example. The experience is so seamlessly integrated with the "game" that you don't even realize you're playing a tutorial. You just play. By the time you've completed the tutorial, you were totally immersed in the game and knew exactly what you needed to.

I used to think schools needed more games.

There's an certain authenticity to this learning that I never really experienced as a student. I thought if I could design the perfect "tutorial level" for math, then everything else would just fall into place. The students would have fun. They'd learn real mathematical concepts in a natural environment. They'd grow and develop as individuals and as a group. I'd be like a "math teacher" and "guild leader" all rolled into one (although I probably wouldn't run IWAY).

I used to think schools needed more games...

...and then I started teaching.

The problem is not that school doesn't have enough games, it's that school has too many games. Now, I'm not talking about the latest web app: Kahoot, Quizizz, Manga High, etc.. Those are certainly a type of game that has a place in school, although perhaps the number of apps is getting overabundant as well, but I'm talking about the games that are school. School itself is like a "Live Action Role Playing Game". Everyone invents their character, acts out their role, cooperates with some players, competes with others, and are rewarded or punished in accordance with the game master's rules.

Now school being RPG isn't a problem on its own. The problem is that there are a whole bunch of mini-RPGs being played simulatenously, and all of them have conflicting rules. Here is a short list of a few games that might be going on at a given time:

  • The students are playing a game with each other. They compete with each other for social status while cooperating against outside threats to their system.
  • The teachers are playing a game with their students. The teachers are trying to maximize student learning while the students are trying to minimize the work they have to do.
  • The administrators are playing a game with their teachers. The administrators are trying to maximize test scores while minimizing teacher burn-out.
  • The school board is playing a game with their administrators. The school board is trying to maximize community approval while minimizing school funding.
  • The parents are playing a game with their school board. The parents are trying to maximize the quality of education while minimizing the amount of attention paid to local elections.

Within these games, temporary alliances are made to accomplish mutual goals. Teachers and parents might cooperate to get students in for extra tutoring. Administrators and school boards might cooperate for better community awareness. Sometimes these alliances help the system as whole and sometimes they detract from it. It's one of the most complex network systems I've ever seen.

I used to think schools needed more games...

...and now I think schools need to have a closer look at the games that are already being played there.

In most of these games, competition is the dominant strategy. Students that are competing for limited scholarship funds have little incentive to help one another. Schools that receive funding based on standardized test scores have a very strong incentive to focus on instructional strategies that produce short-term results over long-term retention. School boards are underappreciated as a position of political power and tend to just "fly under the radar". Until we fix the reward systems so that they encourage cooperation, the games will continue to be frustratingly difficult for everyone involved.

We need to start meta-gaming school. We need to look at how the rules affect the relationships between players. We need to look how those rules can be changed to encourage more co-operation and less competition between the parties involved. Until we have these conversations, we're never going to win.

What I've discovered, learned or shared by using #mathchat

This was a #mathchat topic in July of 2012 that I really wanted to write about but didn't quite get around to at the time.  This happened partly because I was busy juggling work and graduate school, but also because I felt a bit overwhelmed by the topic.   I've learned so many things through my involvement in #mathchat that the idea of collecting them all was daunting.   It also kind of bothered me that my first attempt at a response to this prompt turned into a lengthy list of tips, books, and links.  This type of content makes sense on Twitter.  It's actually the perfect medium for it.  However, to turn this into a blog post I needed some coherency.  I felt like there was a pattern to all of these things that #mathchat has taught me but I just couldn't quite put my finger on it.

A year and a half has passed since this topic came up.  It's now been 6 months since the last official #mathchat.  Despite this, Tweeps from all over the world continue using the hashtag to share their lesson ideas and thoughts about math education.  It's inspiring.  The weekly chats might have stopped, but the community continues to flourish.  Looking back on how things have changed on #mathchat helped put perspective on how #mathchat changed me.  I think I'm finally ready to answer this prompt.

What I learned by using #mathchat was that learning requires taking risks.

On the surface, it seems like this assertion might be obvious.  Whenever we attempt something new, we run the risk of making a mistake.  By making mistakes we have an opportunity to learn from them.  The issue is that we go through this routine so many times that it becomes habitual.   When learning becomes automatic, it's easy to lose sight of the risks and how central they are to the learning process.

Consider the act of reading a book.  For many, like myself, this is the routine method of learning new information.   In fact, it's so routine that the risks aren't readily apparent.  That doesn't mean they aren't there.  Have you ever read a book and found yourself struggling to understand the vocabulary?  For me, Roger Penrose's Road to Reality is still sitting on my bookshelf, taunting me, because I can't go more than a couple pages without having to look things up elsewhere.   Attempting to read a book like this entails a risk of making myself feel inadequate.  It's much easier to read a book that's within one's existing realm of knowledge.  By taking the risk out of reading, it becomes a recreational activity.  This isn't necessarily a bad thing -- we could all use some relaxation time now and then -- but it's not until we step out of that comfort zone that the real learning begins.  Have you ever read a book that made you question your own assumptions about the world?  It's not often that this happens because we're naturally drawn to books that reaffirm our own beliefs.  When it does happen, the impact can be quite profound.  The further a book is from your existing world model the greater the risk of that model being challenged by reading it, but the potential for learning scales in proportion.

I was rather fortunate to have discovered #mathchat when I did.  I had signed up for Twitter at approximately the same time I started teaching math.  Anyone that's ever been a teacher knows that learning a subject and teaching that subject are two entirely different beasts.   I'd been doing math for so long that most of it was automatic.  It wasn't until I started teaching that I realized I had forgotten what it was like to learn math.   As a result, I was struggling to see things from the perspective of my students.  I needed to step out of my own comfort zone and remember what it was like to learn something new.  It's through complete coincidence that my wife stumbled upon Twitter at this time and said, "Hey, I found this new website that you might find interesting".

I didn't join Twitter looking for professional development.  In fact, for a while at the start I didn't even know what "PD" stood for.  I joined Twitter purely out curiosity.  I was never really comfortable interacting socially with new people, and it seemed that this was an opportunity for me to work on this skill.  I called it "my experiment".   I didn't even use my full name on Twitter for the longest time because I was afraid of "my experiment" going wrong.    I started simply by looking for topics I was interested in, following people that sounded interesting, and speaking up when I felt I had something to say.  One of my saved searches was "#math" and I started trying to answer questions that people were asking on Twitter.  This lead to making some of my first friends on Twitter.   I noticed that some of those people that regularly tweeted on #math also frequently tweeted with the hashtag #edchat.  I started to observe these people would often post multiple #edchat Tweets within a short period of time and had inadvertently stumbled upon my first real time Twitter chat.  Once  I started participating in #edchat my network grew rapidly.  From there, it was only a matter of time before I discovered #mathchat.

My social anxiety was still quite strong at this time.  With each Tweet, I was afraid that I would say something stupid and wake up the next day to find that all my followers had vanished.  However, #mathchat provided a welcoming atmosphere and discussion topics that were relevant to my work environment.  This provided me with an opportunity to engage in discussion while mitigating  some of the risks.  I knew that each topic would be close to my area of expertise and the community was composed of people who were also there to learn.  There was a certain comfort in seeing how people interacted on #mathchat.  People would respond critically to the content of Tweets, but always treated each participant with dignity and respect.   I was experiencing first hand what a real learning community could be like.

A frequent motif in these #mathchat discussions was Lev Vygotski's model of learning.  With my background in psychology, I was already familiar with the concepts and vocabulary.  However, #mathchat helped me link this theory with practice.   I became more and more comfortable with a social perspective on learning because I was learning through my social interactions.  While I had known the definition of terms like "zone of proximal development", I wasn't quite to the point where I could see the line separating what I could learn on my own and what I could learn with assistance.  I had always been a self-driven learner, but in order to be successful in learning I needed to limit myself to areas that were close to my existing skills and knowledge.  I needed to minimize the risks when learning on my own.  Learning in a social environment was different.  I needed to become comfortable taking larger risks with the reassurance that the people I was learning with would help me pick myself up when I fell.

The #mathchat discussions themselves were not without risks of their own.  Colin took a risk himself by creating #mathchat.  It was entirely possible that he could have set this chat up only to have no one show up to participate.  Indeed, many a #mathchat started with an awkward period of silence where people seemed hesitant to make the first move.  There's much lower risk in joining a discussion in progress than starting one from scratch. The risk is lower still by simply "lurking" and only reading what others have said.  As time went on, there was a growing risk that #mathchat would run out of topics for discussion.  This risk has since manifested itself and #mathchat has entered a state of hiatus.

I'm aware of these risks only in hindsight.  At the time, I wasn't really conscious of the shift occurring in my own model of learning.  What started to make me realize this change was the adoption of my two cats.  This provided my another opportunity to put learning theory into practice by training them (although it's arguable that they're the ones training me instead).  The smaller one, an orange tabby named Edward, responded quickly to classical and operant conditioning with cat treats.  The larger one, a brown tabby named Alphonse, didn't seem to care about treats.  It quickly became obvious that I was using the wrong reinforcer for him.  With his larger body mass and regular feeding schedule, there was no motivation for him to consume any additional food.  It's easy to forget that in the experiments that these concepts developed from, the animals involved were bordering on starvation.  The risk of not eating is a powerful motivator for these animals to learn in the experimental setting.  My cat Alphonse was under no such risk.  He was going to be fed whether he played along with my games or not.  I've since learned that Alphonse responds much better to training when there's catnip involved.

The key to successful training is very much dependent on being able to  identify a suitable reinforcer.  What functions as a reinforcer varies widely from subject to subject.   With animal studies, survival makes for an universal reinforcer as the reward of living to procreate is (almost) always worth the risk.  However, humans follow a slightly different set of rules because our survival is seldom in question.  We're also unique in the animal kingdom because we can communicate and learn from others' experiences.   In a typical classroom situation, the ratio between the risk and reward takes on greater significance.  We're faced with such an overabundance of information about the world that we can't possibly learn it all.  Instead of maximizing performance on a test, the desired outcome, a common alternative is for students to minimize the risk of disappointment.   It's often much easier for a student to declare "I'm bad at math" than to go through the effort of actually trying to learn a new skill.  Rather than taking the high-risk choice of studying for the test with only a moderate payoff (a grade), these students opt for a low-risk low-payoff option by simply choosing not to care about the exam.  When looked at from a risk/reward perspective, maybe these students are better at math than they're willing to admit.

The solution, as I discovered through #mathchat, is to lower the risks and adjust the rewards.  I've started working on making my courses more forgiving to mistakes and acknowledging them as an integral part of the learning process.  I've started working on increasing the amount of social interaction I have with students and trying to be a better coach during the learning process.  There's no denying that I still have much to learn as a teacher, but thanks to #mathchat I have a clearer idea of how to move forward.  For me to progress as a teacher, I need to more comfortable taking risks.  It's far too easy to fall into habit teaching the same class the same way, over and over.  I need to do a better job of adapting to different audiences and trying new things in my classes.  Fortunately, there's a never ending stream of new ideas on Twitter that I'm exposed to on a regular basis thanks to my "Personal Learning Network".

I feel it's a crucial time for me to be sharing this perspective on the role of risk in learning.  There seems to be a rapidly growing gap between teachers and politicians on the direction of educational policies.  There's a political culture in the US that is obsessed with assessment. Policies like Race-to-the-Top and No Child Left Behind emphasize standardized testing and value-added measures over the quality of interpersonal relations.  The problem with these assessment methods is that they don't take the inherent risks of learning into consideration.  Risk is notoriously difficult to measure and it doesn't fit nicely into the kinds of equations being used to distribute funding to schools.

There was recently a backlash of (Badass) teachers on Twitter using the #EvaluateThat to post stories of how our assessment methods fail to capture the impact teachers make in the lives of their students.   Teachers are the ones that witness the risks faced by students up close.   It's our job as teachers to identify those risks and take steps to manage them so that the student can learn in a safe environment.  As the stories on #EvaluateThat show, many teachers go above and beyond expectations to help at-risk students.

While teachers struggle to reduce risks, policy makers continue to increase them through more high-stakes exams.  At times it almost seems like politicians are deliberately trying to undermine teachers.  Maybe what we need in education policy is a shift in the vocabulary. Lets stop worrying so much about "increasing performance outcomes" and instead focus on "decreasing risk factors".  Doing so would encourage a more comprehensive approach to empowering students.  For example, there's strong statistical evidence that poverty severely hinders student success.  By addressing the risks outside of the classroom, we can enable students to take more risks inside the classroom.

In Memory of "Doc"

Recently I went to write an essay and found myself focusing on an influential professor that I met during my undergraduate studies.  While writing it, I discovered some very heart-wrenching news.   What follows is an excerpt from that essay, and I hope you'll spare a few minutes to hear my story of this inspiring teacher.

When I went through my undergraduate study in mathematics, I faced something of an identity crisis in my sophomore year.  I had just spent the summer participating in a research program where we were exploring the application of topology to macromolecular protein folding.  Chemistry was hardly my forte, but seeing how these abstract mathematical concepts manifested in a practical application was an incredibly eye-opening experience.  I came back to taking pure mathematics courses in fall and it felt like something was missing.  I could follow the mathematics, but it seemed like I was missing the “big picture”.   I wasn’t satisfied with simply learning the mathematics anymore and wanted to understand how that mathematics fit in the broader human experience.

Around the same, I was enrolled in an intro psychology course as part of my general education requirement.  The professor, who answered only to “Doc”, had a very eccentric teaching style.  Intro courses at this school typically ran large, around 300 students per course, and you could go through most lectures without being noticed.  Not in Doc’s class.  His classes were part lecture and part Comedy Central Roast.  Doc would observe some small unconscious behavior by a student and proceed to make fun of them for engaging in it -- almost like a stand-up comedian going off on someone who answers their cell phone.  Then, without skipping a beat, he’d turn the subject around and tie the whole incident back to the course material.  Something as subtle as a girl tossing a wink to the boy three seats over would turn into an hour long discussion about the role of classical conditioning in human courtship behavior.  At first some students seems slightly offended by this approach, but Doc insisted that he wasn’t singling anybody out and promised to offend everyone in the class equally at one point or another.  Every lecture was brilliantly connected and it was impossible to tell whether he was improvising or meticulously planned the whole thing out in advance.

In addition to his razor sharp wit, Doc also had a gift for storytelling.  These stories were easy to identify since they seemed to always begin with Doc’s favorite pastime: drinking Budweiser.   He’d start with “So I was drinking a Bud on the porch and...” then proceed to talk about some seemingly ordinary occurrence like a bird visiting his bird feeder or a naive remark made by his 4 year old daughter.  To an unaccustomed observer these stories might have the appearance of small talk, but every single one of them had a hidden purpose that related to the previously assigned reading.  Every story had a basis in some behavior and every behavior had an evolutionary basis for its development.  The bird might be use an example of territorial behavior.  His daughter’s comment might characterize a particular stage in childhood development.  As a student in the course, there was a chance of being called out to identify this link.  Those that didn’t read up were subject to further roasting, so there was significant negative reinforcement to keep up with the reading.  One thing remained consistent though, and that’s that every story was eventually revealed to be an application of natural selection.

One of the most influential moment in my college experience was visiting his office hour.  Doc’s previous lecture had started getting into game theory, but his charismatic style had allowed him to make his point without getting into the mathematical details. This gave me the exactly motivation I needed to visit Doc during his office hour -- a visit I’ll never forget.  We started talking about mathematics, psychology and the interrelation between the two subjects.  I could tell that I had stumbled upon an area that Doc as he smiled with youthful enthusiasm. We discussed everything from the role of mathematics in human development, to artificial intelligence and the abstract notion of consciousness.  Not only did Doc have a deep knowledge in a broad array of subjects, but he had an incredible passion for learning that was quite contagious.  He had this sense of awe and wonder about the world combined with an insatiable drive to make sense of it all.  It was impossible to leave his office hour without feeling inspired.

Before parting ways, Doc recommend that I read a book by Douglas Hofstadter called Gödel, Escher, Bach: An Eternal Golden Braid.  Never before had someone made a book recommendation that was so appropriate for that exact moment in my personal development.  It was as if Doc knew me better than I knew myself after only a brief meeting.  GEB managed to connect a diverse range of my interests through a common mathematical thread, and gave me the perspective I needed to make sense of the mathematics that I was previously struggling with.  Not only did Doc give me the motivation I needed to continue my mathematical studies, but he also shed light on how that knowledge could be applied in other subjects.  My interest in psychology continued to grow, and I would eventually graduate with dual degrees in mathematics and psychology.

One of the central themes to GEB is the concept of self-reference and tangled hierarchies of abstraction.  In the namesake example of Gödel’s Incompleteness Theorem, Gödel managed to use mathematics as a tool for analyzing mathematics itself.  The resulting “meta-mathematics” revealed a deep insight into formal logic and its limitations.  Doc’s unconventional lectures shared many of the same qualities.  He wasn’t just teaching psychology, he was actively using psychology to teach psychology more effectively.  Once I understood this about Doc, his lectures took on a whole new level of depth.  Class was an enlightening display of “meta-psychology” with new twists at every turn.  Doc had this zen-like understanding of himself, and set a model for students to embark on journey of self-discovery of their own.

During the last week of my senior year, I made sure to visit Doc’s office hour before graduating.  This time I made sure to bring a 6-pack of Budweiser along with me to properly say thank you.   He laughed and said “well I usually only drink Bud Lite, but I think I can make an exception”.  We talked about my plan to apply what I had learned mathematics and psychology to making video games.  He gave me some final words of encouragement then I went off to enter the working world. A few years later, I ended up changing careers and began teaching mathematics courses at a community college.  I tried to look Doc up, but the college directory showed him as having retired and didn’t list any contact information.

Recently I was saddened to learn that Doctor Dennis Mitchell passed away in August 2012 after a 5 year battle with brain cancer.  He was an exemplary teacher and I feel honored having known him.  Doc helped me along my way to becoming a lifelong learner.

Actually, it’s more than that.

Doc helped me become a lifelong meta-learner.

 

What can we learn from the Sci-Fi classroom?

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.startshiptroopers3

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.

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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.

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

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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.

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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".

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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.

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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.

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

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

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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.

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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.

voyager3

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.

voyager2

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.

accelworld1

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

accelworld2

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.

accelworld4

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!

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

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.

Bridges of Königsberg

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.

Pre-Calc Post-Calc

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?

Mathematics as a Foreign Language: a Tale of Two Classrooms

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...

VMATYC 25th Annual Conference: Day 1

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”.

#MathChat Recommended Reading

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.