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Math of the game Portal

A few weeks ago, I spotted one of my students playing an interesting looking game on his computer, so I asked him about it. Turns out the game he was playing is called Portal (created by a company called Valve), and it’s still a fairly popular game today.

The basic premise of the game is that you have a special kind of gun which can create two portals, and your character can use these portals to travel instantly between two locations in the level. Each level of the game is a 3D puzzle that you have to solve. I decided tonight to look to see if I could find this game online, and I found this interesting 2D version of Portal, built in Flash. To really understand how the game works, I recommend playing it.

Step 1

 

What I noticed, as I played the game, is that the puzzles are very much logic puzzles. You need to both strategize what a good move will be in the level, and also experiment a fair bit to figure things out. Every time your character dies in the game, you get to try over again to solve the puzzle, and so you get as many chances as you need to try to figure the puzzles out. Some of the puzzles involve a bit of reflexes, and some of them just involve some reasoning.

Step 2

 

I also noticed that the portals themselves introduce a little bit of topological reasoning to the game. Once you start playing, you quickly realize that two positions in the game are equivalent, if there is an easy way to generate a portal between them. You also learn some tricks like the "infinite loop", where you create two vertical portals which you can fall through endlessly. Sometimes, I felt a little like I was playing Towers of Hanoi (itself a fairly mathematical game) because I would have to plan my moves ahead and choose the order of my portals carefully.

Step 3

 

The perspective of the 2D version of the game is quite distorted (to allow for more surface area upon which to place one’s portals) and this got me thinking about perspective. "What’s wrong with the perspective in this game?" I thought. "Oh right, that wall and that floor are at the wrong angle with each other." The line of site of the portal gun also reminded me of intersecting lines, and I found myself visualizing the intersection of a wall and a line from my current position, and wondering if I was "going to hit that wall or not."

Step 4

 

Timing is fairly critical on some of the levels. I found myself occasionally timing how long it took me to do an action, or a series of actions, so that I could time myself to be "in good position" to avoid a deadly (to my character) obstacle in the game. This is a little bit like algebraic reasoning, wherein I work backwards from one time and attempt to calculate (at least roughly) a good time to begin the sequence of actions.

Step 5

 

I experimented a fair bit, and would systematically move one of my portals a little bit each time, so that I could see how this changed the outcome of my movement through the portals. This is similar to a strategy used to solve some problems (and is a little bit like the scientific method) in mathematics. Sometimes to find a pattern, you have to build up representations in carefully thought out sequences, and the same is true in this game.

Step 6

 

Portal is challenging like interesting problems in mathematics are. In the game, you can keep trying to work on the puzzle for as long as it takes. The only feedback you get from the game is the progress you have toward completing your goal, or (as often happened to me) your character’s death.

Level completed

 

You could formalize some of the mathematical ideas that are part of this game, much in the same way that formalization of the use of Angry Birds to teach physics has been done.

I’d be interested to hear of anyone has some others on how this game could be useful in a math classroom, so please share any ideas you have.

Math in the real world: Balloons

This is part of a series of posts I’m doing on math in the real world.

Balloons in an office

 

The first question I thought of when I saw these balloons in my colleagues office was, how many of those would I need to be able to float? Clearly, this is a math problem, and one students can actually test themselves (I would recommend using inert ballast to test student guesses, rather than actual students). Students would first have find out the amount of weight one balloon can lift, and then use division to determine how ballons would be required to lift their weight.

If you want to make this problem much more complicated (and more of a calculus problem), you would point ouf that the density of air decreases as the balloon lifts, lowering its buoyancy, and putting a limit on how far the balloons will actually lift the student.

The shape of the balloons in this picture is also mathematically interesting, as is the shape of other balloons. Why do balloons form the shape that they do? How do the manufacturers of balloons know in advance what shape the balloons will have before they fill them up with helium?

Paypal and password security

This afternoon, I had to change a Paypal password. I went to Paypal, got to the screen to change my password, and after an attempt to choose a new password, I was confronted with this screen.

 

Paypal and password security screenshot

 

I definitely had at least eight characters in my password. I didn’t use my name or my email address. I used a mixture of upper and lowercase letters and numbers and symbols. Paypal just refused to change my password. I decided to test a longer password, specifically, InfinityIsCool4321! (I’m not actually using this password, so it’s safe to share it here) which according to this script would take 12.13 trillion, trillion centuries to break. Paypal still refused to accept my password, presumably because it contained some common words.

I’ve written about passwords before. It’s annoying that Paypal would rather that people created passwords they will forget (unless they write them down, kind of negating some of the security of a password) than to use some simple tips to create a secure password.

This is part of the reason people get frustrated with technology. When developers build forms which are broken like this, it makes the casual user feel like technology is something magical and incomprehensible.

New Math equals trouble, education expert says

The CBC just ran an article on the problems in our current math system which was terribly one-sided and an example of the worst kind of fear-mongering journalism. They are quoting an article by Michael Zwaagstra, an "educational expert" writing on behalf of the Frontier Centre for Public Policy.

First, let’s examine the article written by Zwaagstra.

A solid understanding of mathematics, also known as numeracy, is an important component of a well-rounded education. The ability to perform basic mathematical computations is a requirement of many entry-level jobs. In addition, careers in fields such as engineering, medicine, finance and all of the sciences require a solid background in higher-level university mathematics, including calculus, statistics and linear algebra.

The first thing to point out here is that the basic mathematical computations … for entry level jobs are much different than the higher-level university level mathematics needed for engineering, medicine, finance, and the sciences.

I have to agree with Zwaagstra that a solid understanding of mathematics is an important component of a well-rounded education, but his assertion that mathematics equals numeracy is definitely false, as I have had pointed out to me on a regular basis. There are many mathematicians, engineers, doctors, economics, and scientistis who struggle with basic computational math, but are fully capable of doing higher level mathematics, and this has been true for a long time; far longer than the new math has been used in schools.

Because math is such an important skill, schools have an obligation to ensure that students learn key math concepts. Unfortunately, schools are largely failing in this regard. First-year post-secondary students are increasingly unprepared for university-level mathematics, and this has led to a proliferation of remedial math courses at universities across Canada. Many parents choose to enroll their children in special tutoring sessions with organizations such as Kumon and the Sylvan Learning Centre to fill in the gaps left by the public school system. Unfortunately, many cannot afford extra tutoring, and this creates a two-tiered system that unfairly penalizes children whose parents cannot pay for extra math lessons.

Now Zwaagstra points out that remedial math courses are on the rise in universities, but he doesn’t mention a couple of key facts. First, under the old system of mathematics instruction, around 50% of students failed first year math courses, which were often included in programs as a tool with which to weed people out of university. Could it be that this issue has always been around, and universities are simply now doing something about the problem? What about the increase in students seeking a university education? Could these two issues be connected? Zwaagstra has assumed a correlation between the number of remedial math courses, and the effectiveness of k-12 math education, without actually finding research which supports his conclusion.

Further, he talks about parents enrolling their kids in after school tutoring programs without discussing the reasons why parents are doing this? Are parents increasingly enrolling their kids for extra tutoring because they are dissatisfied with their kids current educational attainment? Or do they have other reasons for paying for these tutoring services? We don’t know, and Zwaagstra doesn’t provide us with any evidence for the reasons for parents to choose tutoring, he just cherry-picks this fact because it seems to support his argument.

Although there is solid evidence supporting the traditional approaches to teaching math that involve mastering standard algorithms, practising skills to mastery and introducing concepts in incremental steps, most provincial math curricula and textbooks employ a different approach. Constructivism, which encourages students to come up with their own understanding of the subject at hand, is the basis for this new approach to teaching math. As a result, there is very little direct instruction of important mathematics algorithms or rigorous practising and memorization of basic math facts.

There is also solid evidence showing that the longer that people are out of school, the less likely they are to use the algorithms they use in school, but the more successful they are at solving mathematical problems they encounter, as Keith Devlin points out in his book, The Math Instinct. In other words, traditional school math seems to be a hindrance to people being able to actually solve real world mathematical problems. It’s worth pointing out that Devlin’s research is reasonably old, and most of the participants in the research learned mathematics in the traditional method. Is it even worth pointing out that Zwaagstra doesn’t actually include any of the solid evidence in his paper, and the footnote here (see the original article) leads to a definition of the word algorithm?

Our students deserve better. Pupils who are not taught math properly are being unfairly denied the opportunity to enter careers in many desirable fields. The public school system has an obligation to ensure that every child has the opportunity to learn the mathematics required for university-level mathematics courses.

It’s pretty important to note that the new math is not being taught evenly, and that when teachers are given proper training in how to use the new math materials, their students’ understanding improves. To say that the problems in our math education system are entirely due to the introduction of the new math curriculum, is pretty irresponsible, given that any number of other factors could be contributing to the problem. Further, many schools use the International Baccalaureate program, which itself relies on the "new math" with a focus on students understanding mathematics and being able to communicate their understanding and these students are highly sought after by universities. If the new math was so destructive, wouldn’t we see these students being turned away by universities in the sciences?

Zwaagstra then goes on to bash the results of the PISA examinations, citing an article (claiming it is research) written that suggests that Finnish students are not as good at math as the PISA results would claim, and that by extension, neither are Canadian students.

There is a strong consensus [emphasis mine] among math professors that the math skills of these students are much weaker than they were two or three decades ago.

Zwaagstra links to two articles (neither of which is a research study) that state that some professors have found a drop in numeracy skills (again, these are associated with mathematical ability, but are not equivalent), and the other of which makes no mention of math skills at all. In this case, Zwaagstra is completely misrepresenting the articles themselves. He then points to two professors who have done research on the computational abilities of graduates and noticed a decline, but he does not clarify whether or not this is correlated with a decline in their ability to do university level mathematics.

Zwaagstra continues by bemoaning the lack of standards and emphasis on accurate calculations by the National Council of Mathematics Teachers (NCTM) and the Western and Northern Canadian Protocol (WNCP). Clearly the research these two organizations have done for decades is not sufficient for Zwaagstra, especially considering Zwaagstra’s credentials (Hint: He’s never been a math teacher, nor has he any credentialed expertise in mathematics education Update: Apparently, Zwaagstra spent 7 years as a middle school math teacher, so I’m retracting at least this part of my response).

However, there is a big difference between demonstrating a conceptual understanding of mathematics and actually being able to solve equations accurately and efficiently. Just as most people would be very uncomfortable giving a driver’s licence to someone who merely demonstrates a conceptual understanding of how to drive a car, we should be concerned about a math curriculum that fails to emphasize the importance of mastering basic math skills.

To extend Zwaagstra’s analogy, we should similarly be afraid of giving the keys to someone who has no real world experience driving. If someone has spent all of their time in a flight simulator, but never actually driven a car, should they be allowed to do so? Does an emphasis on the mechanics of driving a car (or the mechanics of mathematics) turn someone into who is capable of driving a car (or able to use mathematics)?

Zwaagstra’s solution to improving math education is to move "back to basics" which is as unoriginal an idea as I’ve heard, and it is arrogant of Zwaagstra to assume that this approach hasn’t been tried before. Perhaps Zwaagstra could instead address the issue of elementary school teachers often lacking support and training in how to teach math? Zwaagstra points out (correctly) that having mastered one computation, students are then better able to learn another computation, but this leaves students learning a series of computations, and not spending any time actually using them.

JUMP math is mentioned in Zwaagstra’s article as an antidote to the problem, but he doesn’t talk about the issue of the associated training, or the lack of diverse assessment used in the JUMP math system. I think that the training manuals which go along with the JUMP math curriculum, for example, actually address the misconceptions of the people teaching the math (mostly elementary school teachers) rather than itself being a significantly better system. As one educator has told me, JUMP math is pretty useless without the training materials for teachers.

Just as someone who does not practise the piano will never learn to play well, someone who does not practise basic math skills will never become fluent in math.

Similarly, someone who has not had time to play with a piano, to improvise, and to perform music for others will never develop an appreciation for the instrument. Zwaagstra is suggesting that we should discard the extra parts of math education, like problem solving, and focus on computations, which is the musical equivalent of only learning scales, and never getting to perform music.

No one would stand for that in music education, so why should we accept it in math education?

 

Update: Here’s another good rebuttal to Zwaagstra’s article.

Moebius Noodles

A couple of weeks after I posted some resources for parents looking to teach their young kids about math, Maria Droujkova has introduced the Moebius Noodles project which is intended to build a book and a support site for parents who would like some support teaching math to their children.

In her own words, the reason she started this project is:

  1. There are very few materials and no community support for smart math for babies and toddlers. Just try to find anything that is not about counting or simple shapes! Mathy parents create opportunities for their own kids, of course. But without support and resources, it’s very hard even for the rocket scientist mothers and fathers. We want to change that!
  2. Peer-to-peer learning, research and development groups in mathematics education need a process for crowd-funding their projects. We are the trailblazers for other fabulous communities that want to make open and free math materials with the support of their members, such as the group developing materials for learning mathematics through music, the play math network, and the math circle problem-solving depository project.
  3. We are creating OERs – Open Educational Materials. It means people can access, use, modify and share the materials for free [emphasis mine]. Imagine the project you support translated into any language in the world, and used freely to support young kids everywhere!
  4. The activities are sustainable in many senses. You can use everyday household items and recycle materials for Moebius Noodles games.
  5. If you are a parent or teacher who loves arts and crafts but is afraid of math, the book will help you teach your kids mathematics through your talents. If you are a math or science geek who envies other families always doing neat art projects, the arts-math bridge in the book goes both ways!

You can donate to her cause by clicking on the image below. At the time I posted this entry, Maria is about $4000 away from her goal.

Moebius Noodles Fundraiser Badge

Disguising flash cards as a game is deceptive

I’m reading The Connected Family by Seymour Papert, and ran into a quote which I found appropriate.

"…learning multiplication facts by putting flash cards on the screen is not a new way of learning math. It is a polished-up version of the old ways and promotes to greater heights their worst and most mechanical features. Moreover it is often done in a spirit which I see as dangerously dishonest: Disguising [emphasis mine] flash cards as a game introduces an element of deception that undermines two fundamental educational principles.

First, learning works best when the learner is a willing and conscious participant. Second, deception and dishonesty in the teaching process make a mockery of the idea that schools should develop moral values as well as knowledge of math or history." ~ Seymour Papert, p19, The Connected Family, 1996

It was timely, because just this morning, I saw this tweet from Jason Klein:

Most Education category apps (iOS, ChromeOS, Android) are low-level, fact-based. Just give me Internet & Creation apps.

Just searching now on my iPhone for math in the App Store, these apps showed up.

Top five math apps on the iPhone

All five of these applications are based on learning math facts and arithmetic, and two of them even have the word "flash" in their name. Of the top twenty five math applications that I saw, 23 of them are essentially flash cards disguised as games. Two of them aren’t, one is called "Equation Genius" (it solves algebra equations), the other is called "Motion Math" (which lets students learn the relationship between fraction as symbols and visual representations of those fractions).

Could we please get more educators programming these apps? (If someone would donate me a Mac to work on, I’ll happily do it myself.) 

Heading to the Computer Based Math conference in London, England

So I just got confirmation (and have paid for registration and my airfare AND found a place to stay – mostly) that I get to attend the Computer Based Math conference happening London, England on November 10th and 11th. I’m very excited about it!

I’m flying out of Vancouver on Tuesday, November 8th (after being in workshops all day with my colleagues), and arriving in London midday on the 9th. I’m at the conference on the 10th and 11th, and flying to Toronto on Sunday, November 13th, where I’ll be attending the Mindshare Learning Canadian Edtech conference on Monday, November 14th.

I have a place to stay arranged for my time in London (actually many offers of places to stay) but I could use a place to stay for the Sunday night I’ll be in Toronto. I’m trying to make this trip more economical for my school (since they are footing the bill) by staying with friends.

My hope is to find out more about how different people are using technology in math education specifically at the Computer Based Math conference, and to be part of the team trying to build a curriculum for math based on the assumption that students have computation devices with them whenever they need them. The big questions I have are, what does that kind of curriculum look like, and would it be effective for teaching math?

It will be strange to be at these conferences as it will be the first time in 2 years that I’ve attended a conference, and not presented. Maybe I’ll find a way to get to talk about some of what I do at one of these conferences anyway… even if I’m not officially on the schedule to present.

I’m posting this to let the people in my PLN know, and I’d love to connect with anyone else heading to these conferences that I’ve met via Twitter.

Questions about the flipped model of instruction

I’ve been reading a lot about the flipped model of classroom instruction, where students watch instructional videos for homework, and then do the practice and problem solving during class time. Here’s a video of the process being explained by Aaron Sams.

 

Some of the questions I have are pretty much the same as the ones posted as responses to the YouTube video so I’ll just quote them:

I’m curious as to what you do with kids who don’t have the internet or a computer at home? I see someone else asked this question below, but I don’t see where that was answered. This seems to be just another way to divide classroom success socioeconomically. Rubyfreckles78 

What if you don’t believe in homework? What if you believe a child’s time outside of school should be their own, to explore the other adventures life has to offer outside the formal academic arena?katiramom 

"What to learn, how to learn it, when to learn it and how to prove to me that they learned it". I can see that the times of learning has changed but is it not still teacher-cetred in this respect? Yes, a different modality – online and video (great!) but what underlying structural changes in terms of power and student-centredness? "We’ve changed the place in which content is delivered". In what ways are the pupils negotiating content?audhilly

This is great, Aaron. Unfortunately, some school districts–like that one I work in–do not allow their teachers to access Youtube.l2spanishteacher

It seems to me that there is no good answer to the first question. Students without parents at home, who are homeless, or who do not have access to technology at home to view these videos are out of luck. They’ll have to stay at school to watch the videos in the library.

As for the question about homework, this to me is the biggest question I have about the flipped model. It assumes that the time kids spend outside of the classroom should be taken up watching videos. In essence, the flipped classroom model assumes that the instructional time schools are given is insufficient for kids to learn the material. Perhaps we there is simply too much content for kids to learn effectively?

The third question is mostly about pedagogy. Should kids learn in a teacher-centred way, or a student-centred way? While our curriculum is bloated and filled with content, it seems impossible to switch to a more constructivist model, particularly in the upper grades. Constructivist teaching methods take more time than more traditional methods of teaching (but hopefully lead to deeper understanding). 

The fourth question is similar to the first question as that both of them are about access. Clearly the solution here is for the school to self-host the videos, but if this becomes a common instructional strategy at your school, the costs incurred to host what could be thousands of videos is enormous. Now we have an issue that the schools with the money to afford the hosting (or at least the policies in place to allow YouTube and other video hosting sites) are a further advantage to the poorer schools.
 

Some more questions I have are:

  • What does this approach look like for someone who is a novice to teaching?

    One of the valuable pieces of feedback a novice teacher gets about their instruction is the questions students ask during class. Students will often share misconceptions they have about whatever is being taught, which helps improve the teacher’s delivery for the next time. While I think an emphasis on lecture based instruction is not the best possible pedagogy, it certainly is an easy place for novice teachers to start during their career. Flipping the classroom could reduce the feedback the teachers get on their instruction, but see my next question.
     

  • How do students ask questions?

    Students need feedback during learning as well. One of the points of practice problems, and of problem based instruction, is to maximize the number of opportunities for feedback during learning for students. Lecture based instruction typically fails in this regard, and so many instructors have switched over to discussion based instruction. The flipped classroom model, without a way for students to actively ask questions, moves instruction back to a purely lecture based format. One way to counteract this a bit would be to provide space for students to ask (& answer) questions underneath the video lecture as comments, but then the job of the teacher will be to moderate and join into these discussions. While students can obviously record the questions they have (which is a useful learning strategy), this requires organizational skills and self-management skills not every student possesses.
     

  • How much time does it take for teachers to make these instructional resources?

    Preparing for classes and assessing students are the two tasks, other than administrative paperwork, that take the most time for teachers during the course of their day. Preparing high quality instructional videos has certainly become much easier for teachers to do, but it is also time-consuming. Sal Khan might be able to create 8 videos a day, but teachers do not have their entire day available to devote to making videos, and would like to produce videos which include images and animations to clarify some concepts. We could rely on the videos from sources like the Khan Academy rather than making our own videos, but we’d need to search for and preview all of the resources we use, which in itself is time-consuming. There is also the additional time spent during our evenings responding to questions students might have about the videos.
     

  • Will class time be used more productively?

    Aaron’s video above shows some great examples of what I think should be happening in more science classes. The students look like they are getting more chances to experiment, and more chances to interact with and actually do science. Is this what happens in every flipped classroom? If students really understand the concepts being taught by the end of a unit, how can we tell if it was the instructional video, or the time spent actively experimenting that made the biggest impact on their learning? One comment I had from a student was that although his teacher assigned videos for homework, he rarely watched them, but made sure to actively participate and learn during class time. He loved the flipped model because "it meant [he] had less homework."
     

Although I have these questions, there are some things which I really like about the flipped model of instruction.

  • It forces teachers to really think about their instructional strategies and the potential questions students might have.

    You can’t create these videos without putting some serious thought about what you will be teaching for that lesson. This particular type of teaching is much more difficult than turning to page 27 in the textbook and selecting some questions for students to do.
     

  • Students can potentially access a variety of different explanations for different concepts from teachers all over the world.

    Not every student has access to a specialist in their subject area. In British Columbia, for example, there are many teachers teaching math outside of their specialty. I can remember tutoring math when I was in grade 11 in the PE teacher’s classroom (who was not a math specialist, or trained to teach math) and frequently helping the teacher understand the math he was "teaching".
     

  • It provides more class time for more student centred instructional strategies.

    This is the best reason to implement the flipped classroom model since many teachers aren’t ready to give up on teacher led instruction. Students need more time processing the concepts to which they are being exposed. If they do this at home, as is unfortunately too typical in many classrooms, they struggle. In the flipped classroom model, that struggle can happen with their peers and an expert facilitator.
     
  • Students can now more easily opt out of rote memorization.

    Richard Feynmann, one of the best physics lecturers of all time, investigated Brazilian science education, which was heavily dominated by memorization, and discovered that almost no one from this system actually understood science. Since students do not learn well from memorizing information, one can conclude that lecturing is not sufficient to produce students who understand concepts at a deep level.

Does anyone have any answers to these questions?

 

Math in the real world: Sound

This is another post in my series on math in the real world.

Vi Hart explains much of the mathematics behind noise in great detail, so watch her awesome video below. Thanks to @delta_dc for sharing it with me.

 

Notice her use of Audacity? I think we could quite easily turn this into a lesson plan… perhaps related to fractions, or to sine and cosine waves.

Math in the real world: Roller coasters

This is another post in a series I’m doing on math in the real world.

 

When my son and I were on the roller coaster, I was again in awe about how quickly even a small roller coaster like this travels, and how it doesn’t drive right off the tracks.

Roller coasters have to be constructed fairly carefully, and follow some mathematical rules in their construction. They need to first be concerned about how to make the roller coaster safe. They need to calculate exactly how fast it will travel through the loops and turns, and how much of an angle they will need to prevent the roller coaster from taking a dive during those turns. They need to watch out that they don’t cause the participants of the roller coaster to pass out during a turn as they experience additional forces on their bodies!

The various costs associated with a roller coaster need to be calculated as well. There’s the cost to build, maintain, and operate the roller coaster. There’s an additional cost to pay for insurance for the roller coaster, which means an actuary needs to examine the probability of a problem occurring for any given roller coaster. The operator of the roller coaster needs to determine, given the cost to operate the roller coaster, etc… what they should charge to make a return on their investment, and attempt to maximize their profits.

While you could use a roller coaster simulator to explore some of this math, it’s a lot more fun to experience it in person…