The Reflective Educator

Education ∪ Math ∪ Technology

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Month: December 2010 (page 1 of 2)

Learning through Guided Inquiry

My son has started to learn how to ski. He tried last year, and failed miserably, in fact he gave up in the first five minutes of the lesson, which ended up being a pretty expensive day for a 5 minute skiing lesson. It wasn’t his fault the lesson failed, he wasn’t ready for it. He was probably too young, and had strong expectations about what he should be able to do when we started skiing.

We had been to Science World in Vancouver, where they have a simulator that lets you pretend to be a professional skiier. You can race down alpine slopes at frightening speeds, but whenever you crash, the simulation resets and pushes you into the right direction. The problem with the simulation is that my son tried it, and at his age, that’s what he thought his first experience of skiing would be, and when he wasn’t immediately racing down the mountain, he got upset, and his lesson ended.

Now he’s a year older and we’ve tried a different tact. We bought skis and ski boots for him and put them in his play area. Periodically he’s put them both on and walked around our little livingroom. He hasn’t played the simulation at Science World in a long time, and so his expectations are different. We’ve talked about the need to practice to get good at something, and he’s learned a lot of patience. We set up a private lesson for him, instead of the group lesson which failed so badly last year.

He’s learning through what I would call guided inquiry. If we had just put the skiis on him and set him loose, he wouldn’t learn very much about skiing because there are some subtle things which are not obvious, like how to stop or turn. On the other hand, we can’t tell him everything about how to ski, he has to learn through practice and trying it out for himself. His private instructor, a friend of ours, guides him instead of instructing him. He doesn’t go through an experience with the other ski instructors in the big group which I liken to the factory model of education, instead his instructor spends most of the time skiing backward and asking Thanasis to "come here" without telling him too much about how to do it. Periodically she would give him pieces of advice and feedback, but by and large he figured it out himself.

More of our education system should be like this. Guided inquiry as opposed to factory instruction.

Is it possible?

A couple of nights ago @PeterVogel told me (and the rest of his followers) about the transit of the ISS or International Space Station. This amazing contraption floats above us in space and is manned all the time by astronauts from eight different countries. It is has been in operation continuously for the past 10 years, and has had 196 different individuals float around our world in it. It is a marvel.

My son and I went outside at 5:33pm as per Mr. Vogel’s instructions, and looked toward the South. Up in the sky we saw what looked like a star shooting zipping across the sky. It was moving fast, faster than any airplane could possibly move. We hollered and everyone else in the house came out and took a brief look. Honestly, it looks like a light moving across the sky, it’s not that thrilling, except when you remember what it actually is.

The ISS is a marvel of our world. It is a reminder of what is possible when you put the resources into a project and let lose the engine of human ingenuity. It is amazing. It allows humans to survive in an environment so hostile, nothing quite like it exists on Earth itself.

It was also a good reminder to me that our world has problems, but equally that we are able as a species to solve these problems. The problem of the educational transformation we require to continue to make our education systems relevant is a complex and difficult problem. It will require the cooperation of many different countries and possibly millions of individuals to make it happen, but it will happen, we will resolve this educational crisis.

I know we will because we built the International Space Station. If we can do that, we can solve any problem.

Are computers dangerous?

Socrates would definitely think that computers are dangerous. He would see them as weakening men’s minds so that the learner needs to know nothing and can rely on the box in front of them to provide all answers. After all, he thought the same thing about writing.

"…this discovery of yours will create forgetfulness in the learners’ souls, because they will not use their memories; they will trust to the external written characters and not remember of themselves."

Socrates as quoted by Plato in Phaedrus, 500 BC.

Obviously our minds have not become weak. Instead what has happened is that our society has transformed and writing has become an invaluable tool for remembering what past discoveries have been made. It allows us to build upon previous generations’ knowledge and experience and to move into uncharted territories. Our minds have become strong as we have to now digest centuries worth of written knowledge in order to be able to add to the new pool of knowledge. In fact the specialization required to add to the knowledge pool in some disciplines is so great that most people will never achieve it.

As Kurzweil points out, our total knowledge is increasing at an enormous rate, an exponential rate, and that we are nearing a singularity after which we have no idea what our species will even look like past this point. We have no idea what lies beyond the singularity. 

We do have a choice. We could decide that this unknown future is too frightening and uncertain to entertain, that we should abandon our current rate of progress and put aside our tools. If we all agreed that we know enough, that our world has moved far enough, then we might slow down this race toward a possible oblivion as a species.

Oblivion is a possible future, as the alarmist theories present in such modern day nightmares as the Terminator series suggest. If you look at many of the science fiction movies that exist today, you see examples of dystopian futures which surely emanate from legitimate concerns about the future of our race and planet. However, fear sells. Maybe these nightmare futures are just created to because they scare us into watching them…

Maybe our species has a chance. Perhaps the progress we have made, and yet to make is worth the risk? We have not yet solved all the world’s problems. We are too reliant on fossil fuels for energy, and our social justice, although rapidly improving, still lingers in many parts of the world. We need to solve the environmental, economic, and political problems previous generations have foisted upon us, and find a better way for the future.

It is not hard to argue that our use of computers is enabling the current acceleration in change. They allow us to compute faster, more collaboratively, and analyze information in ways never before possible. In effect a single computer, connected to the Internet, has access to vast quantities of the world’s information. In effect, you can know anything you want within seconds at the least, and possibly minutes depending on how open the data structure is. Access to a computer allows us to calculate quantities millions of orders faster than the human mind can do alone.

A computer should be the ultimate disaster for thinking in the framework of Socrates. After all, one does not even need to remember which book a piece of knowledge lies. Google and other search engines can organize the world’s information for us, allowing even the task of cognitive categorization to disappear. However, it is through the computing devices we have that amazing discoveries are made. People can collaborate from around the planet, and the ability to communicate with each other about any issue is unprecedented. Perhaps it is this communication that turns the crutch into a tool? 

Writing is after all a form of communication. It is through the writings of his student Plato that we know anything of Socrates at all. Writing has allowed us to communicate across generations. Computers and the world wide web allow us to remove other barriers to communication. It is the ability to communicate what we know that makes these devices transformative for our culture.

Where Socrates was right is that writing does weaken our memories as we rely on the crutch and so too will computers. What he failed to recognize is that it greatly strengthens our collective mind, which is far more valuable than any individual could hope to be. Computers aren’t going to destroy our society, we will, through our indifference, our inaction, and our inattention to the ample warning signs of a planet in distress.

Finding examples of reasons to learn math is easy

Here’s an example from the maintenance committee for our housing coop. Basically, our budget was off by thousands of dollars, and we were trying to figure out why. The explanation is below in the screen-shot I took of our email exchange.

Emails about budget discrepancy

"What a difference a dot makes."

Finding examples for using mathematics in the real world is this easy. We should find more examples like this one. Imagine you gave the spreadsheet to students (in paper form or otherwise) and said, "Okay, you find the error," and see what the kids come up with.


Essential tools I use for teaching mathematics

I’ve moved country a fair number of times and had to bring my teaching supplies with me. I started collecting my own supplies when I taught in NYC, and my collection has grown over the years. What follows are some of the essential tools I use when teaching middle school and high school mathematics.


A graphing calculator

These devices let students quickly generate graphs, do calculations on lists of numbers, and a huge amount of other mathematical operations. I tend to focus on the things we can do with the mathematics we know, and treat mathematical calculations as tools for solving problems. As far as I know, so far no single computer program is as easy to use and does as much as the modern graphing calculator.

Graphing calculator


Decks of cards

You can use these in probability simulations, learning mathematical card games and tricks, or just choosing which group is going to present next.

Decks of cards



We use dice for more probability simulations, choosing partners for groups (occasionally, I usually let students choose their own group members), and playing board games. Yes, we play games in math class.

Graphing calculator



Whether you are using the string as a tool for more interesting mathematical phenomena (like pendulum motion), using it to tie parts of a project together, or studying the pattern of knots tied in the string, this is an invaluable tool in mathematics class.

Graphing calculator


Scissors and rulers

I have a class set of these very useful devices. Being able to construct models from shapes cut out of paper helps bring mathematics alive for students.

Graphing calculator


Compass and protractor

I also have a class set of a compass and protractor. I ask students to buy one of each of these things as part of their supplies for class, but provide them for students as well. I don’t want my lesson floundering because some 13 year old forgot his supplies for class…

Graphing calculator


Golf (and other similarly bouncy) balls

You can bounce them, roll them down inclines, and throw them through the air. All of these result in interesting mathematical models for students to inspect and analyze.

Graphing calculator



These are great for counting, using as markers, keeping track of positions of frogs jumping past each other on lily pads, and loads of uses I haven’t even thought of yet.

Graphing calculator


Long measuring tape

I have 8 of these. At one point I had 10, but they are such an incredibly useful device, they sometimes go missing. I use these to test the Pythagorean theorem out on the soccer field, or apply trigonometry to finding heights of things, to testing that our measures of distances using parallax are accurate. I can’t imagine not having these measuring tapes, they’ve been to 4 different countries with me. When my wife (while we were packing for Thailand) asked me if I really needed these, I just gave her the most incredulous look. Of course!

Graphing calculator


Paper clips 

It’s amazing what you can do with paperclips if you see them as something other than a paper clip. Sure, I use them to hold pieces of paper together, but I have also used them to pick locks on filing cabinets when the key went missing with the previous teacher, construct 3d models, and twist into interesting mathematical shapes.

Graphing calculator


iPhone (video camera)

I love the fact I always have a regular camera and a video camera on me at all times. It also doubles as a way to retrieve information from the Internet which is handy when having a discussion where you really want to be right. Being able to capture moments from my classroom as they happen is awesome.

Graphing calculator


My laptop

I’ve had access to a computer in my teaching since the very beginning, and I wouldn’t have it any other way. I used it at first for research for lesson ideas, and for collecting resources for class, but am now using it to connect to other mathematics teachers from our global community. The fact it also includes Google Apps, Geogebra, video and image editing, and a host of other applications is just bonus.

Graphing calculator


What essential tools are in your teaching kit?



Technology Scavenger Hunt

I’ve created a scavenger hunt. The idea is that each clue is embedded in some technology which you will have to solve. Some of the clues are very hard, some of them are easy. Start at and watch the first clue.

Merry Christmas to all of you technology geeks out there that will enjoy doing this. Feel free to share this scavenger hunt with others or to reuse the idea at your own school. Comment below if you think there are any errors or problems with the scavenger hunt, or if you have ideas for how to improve it. One immediate thought is that we could create a bigger scavenger hunt and host the clues all over the web…

Note: I’m unable to make all of the clues accessible in this scavenger hunt. Please work with a partner if you are unable to access a particular clue with your screen-reader.

How is Stratford Hall using technology?

Reposted with permission from my school’s monthly magazine, the Imprint.

How is Stratford Hall Using Technology?

We are in the middle of an exciting transformation this year. During the past month alone, I noticed teachers and students using video cameras, podcasting radio plays, working collaboratively via Google Docs, blogging, tweeting, and using some of the most useful educational technology out there. The use of technology in the classroom has exploded in the past few months.

Podcasting, video creation, Google Docs, blogging, and Twitter change the writing process. They are becoming literature in their own right, but fundamentally they are about communication. Of all of the things our students will learn at school, communication is among the most important.

When students create an audio podcast, or edit a video in post-production, they are learning the most crucial skill of writing, which, of course, is editing. As they learn how to shuffle and extend their digital work, they are also learning the basic framework of text editing. Students who would otherwise struggle to put 100 words together on a piece of paper can create a three-minute radio play and learn many of the same skills.

The process of planning the podcast or video is important too. In writing, we call this the outline; in the podcast it is called the script. Learning how to look ahead, organize your ideas, and search for supporting resources are all vital to the writing process. In podcasting students have to both choose the script they follow, and the other digital resources they use to complement their writing.

Google Docs is an online word processor that lets students collaborate real-time in the process of writing. Instead of passing paper back and forth and taking turns editing their work, they can work together simultaneously on the same document. One of the most important parts of learning anything is the feedback you receive. Google Docs lets students receive nearly instantaneous feedback on their work.

Through blogging, our students learn that writing has an audience, that it is for someone else. We never write for only ourselves; even a diary is for a future, transformed you. Our students are learning how to make their writing interesting and engaging, and how to deal with the criticism of a public audience. Do you rewrite your thoughts according to the thoughts of others, or act independently and stand up for yourself?

Tweeting is making the writing process mobile. Our students — those who have cellphones — have some of the most flexible and powerful mobile computing devices ever built. More than any other technology our school uses, the mobile device has the power to be a game changer for learning. Now, 24/7, use-it-whenever-you-need-it, access-it-anywhere learning is ubiquitous.

We still have a ways to go in order to learn more productive use of educational technology as a school. During the next months, teachers will be experimenting with technology and learning how to use it. The journey has its challenges but our world has changed, and so must we.

Mathematical problem solving

Today I decided to record the process of solving a mathematical puzzle I found at the Project Euler website, in an effort to try and begin to analyze the problem solving techniques I use. My interest here is mostly in how the process unfolds, and the skills I use to solve these problems, rather than the actual problems themselves, although those are interesting. Below is the video I recorded when solving this problem.

Find the problem at

I think it would be interesting if we could create more examples of these videos in action, where we solve mathematical problems. I’m kind of inspired by Vi Hart, whose creative genius in mathematical doodling I cannot match.

Check out my discussion of a solution to the problem I solved below. Could you do this for another problem?

Playing to learn?

First watch this fantastic presentation by @busyness girl.

Okay, nuff said. Play is so important in learning and so little of it happens in schools. As a math teacher I do try and incorporate games into the lessons as much as possible, but recognize that the curriculum I’m supposed to follow leaves little room for play.

As I watched this, I thought about myself learning through games and how that my ability to type and to organize groups of people is related to the massive multiplayer text dungeon games I used to play. I also thought of how much my son learns through play. Actually he learns practically everything through play.

What can we do to actually in our schools to provide more opportunities to learn through play?

A Vision for 21st Century Education

The BC government just released a report from the Premier’s Technology Council (PTC) on the future of education in British Columbia which is a fantastic read. It’s like someone took the conversations we have on #edchat on Twitter and bottled it up into an official report.

The report starts with discussing the needs of a knowledge-based society1, which it describes as

  • Functional Numeracy and Literacy
  • Critical Thinking and Problem Solving
  • Creativity and Innovation
  • Technological Literacy
  • Communication and Media Literacy
  • Collaboration and Teamwork
  • Personal Organization
  • Motivation, Self-regulation and Adaptability
  • Ethics, Civic Responsibility, Cross-Cultural Awareness

They move onto some over-arching principles about what they feel a 21st century education should look like. Specifically, the document indicates "[t]he system must place greater emphasis on the learning of skills over the learning of content…[which] will have to evolve constantly, not only to remain relevant but so students are ready to deal with how rapidly information changes in a knowledge-based society."2 Of course information in our society is changing very rapidly. Kurzweil, in his article entitled "The Law of Accelerating Returns" said "[t]he Singularity is technological change so rapid and so profound that it represents a rupture in the fabric of human history"3 referring to his prediction that the rapidly approaching time when our ability to digest and create knowledge as a species will exceed our current ability to comprehend and predict the changes.

How would such a system operate? According to the document4 "the transformed system would have a flexible curriculum that allows for more in depth study…The system would have a mixture of face-to-face classroom and online learning." According to the Premier’s Technology council, the system would require:

  • A Flexible Educational Path
  • A Blended System
  • Access to Learning Objects and Teaching Tools
  • Open Access to Information Systems
  • Constant Feedback and Assessment

Many of these requirements will need additional infrastructure supports, and certainly the opponents of such a radical transformation will point to the lack of access to the Internet in rural communities, or the disparities and inequities5 in the necessary technology between communities as reasons not to move toward this educational system. The Premier’s Technology report does say that "a critical component of adding connectivity to the system is equity of access"6 so obviously they recognize this as an issue but the report does not make any recommendations on how to address this issue.

These are issues which we address daily in our discussions on #edchat. If you look through our conversation topics from the past year our wiki, you’ll find that they address the current problems in our system, and acknowledge a desire to move toward this system by many educators. The transformation of our system, as the PTC acknowledges, is not going to happen from a single report, but from the combined effort of educators, parents, students and other stake-holders in our educational community.

What is refreshing is to see that the very same ideas are being discussed at the highest levels in our (BC’s anyway, sorry to you US folks) educational system. Unfortunately there will be much opposition to these proposed changes. Many teachers in BC will naturally resist change, since it is easier to keep doing what one is doing than to transform one’s practice. The BC teacher’s union will resist these changes since many of them will require changes to how teacher contracts in BC are structured. I have hope however, that a majority of BC educators see that our system is not working and that it is not currently meeting of the learners it is intended to support. We must change, our children depend on our ability to be flexible and adapt to a rapidly changing world.


1. Premier’s Technology Council, (2010). A Vision for 21st Century Education, retrieved from on December 14th, 2010

2. Kurzweil, R. (2001). The Law of Accelerating Returns, retrieved from on December 14th, 2010

3. BBC News, (1999). Online Education Increases Inequality, retrieved from on December 14th, 2010