Year: 2012 (page 5 of 14)

August 31, 2012 / 0 Comments

Math in the real world: Train tracks

This is another in a series of posts about how one could find mathematics in the world around us.

Train tracks

My son loves to play with train tracks. A few days ago, while playing with his train tracks, he observed, "Daddy, I can’t turn a train around." I asked him what he meant. "No matter which way I go on this track, I can’t get my train to start facing in the other direction. I’d have to pick it up, but that’s cheating." (Note: I’m paraphrasing here)

Observations like this are mathematical observations about the world. He has abstracted from his train tracks to a property of his train tracks, specifically the direction his train is able to travel. He has then attempted, and I watched him do this, to verify this statement is true by running his trains around the track in every possible comination.

My wife and I spoke about this later, and she came to the observation that in order to be able to turn around his train on the track (without "cheating" by lifting it up), he needs a closed loop with a single entrance and exit point included in his track somewhere, and this entrance and exit point has to connect to the rest of the track in a certain way. So I asked the question, does he have the right track to be able to create a closed loop? If you look at the picture above, you may be able to answer this question yourself.

The area of mathematics that deals with these kinds of issues is called graph theory, and it was invented by Euler for a very different purpose many years ago. It is unfortunately not in most school curriculums, but it is certainly an interesting area of exploration, and one which is accessible to students.

August 17, 2012 / 1 Comment

They have call centres now

While I was at my mother’s house the phone rang so I picked it up.

"Hello?"

"Hello Sir," said a female voice on the other end of the line, with a slight accent, "We are calling you from an independent computer security company. We want to let you know that we have received numerous reports that your computer has downloaded viruses and malware, and we would like to help you fix your computer." In the background, I could hear the unmistakable background noise of a busy call centre.

"You know that’s impossible, right?" I responded.

"What’s impossible?" she responded.

"You can’t possibly, especialy as an independent computer company, know the phone number associated with a specific computer, even if you were somehow able to scan my mother’s computer remotely without her permission. You are trying to scam her. It won’t work this time. I teach people how to use their computers. I’ve taught my mother about you. You cannot scam my mother. I will record the phone call the next time you call, and forward it to Interpol. Leave my mother alone!" I said firmly. (I doubt Interpol would be able to do much about this scam, but hey, empty threats sometimes work.)

Click.

Warn your parents, your relatives, and anyone you care about who may be taken in by this scam. My mother got caught the first time, but with some help from me, we recovered her money, and I have hopefully helped immunize her from the scammers.

August 7, 2012 / 2 Comments

iPad apps for elementary school

We are introducing 30 iPads to our elementary school next year, and we are currently exploring what apps to put onto them. We have some suggestions as a place to get started, but I’ve been tasked with coming up with a list of useful apps for the iPads. I’m currently looking through my list of resources I’ve bookmarked for the iPad and deciding which of these apps I’ve seen will be most useful in our school’s context. (The screen-shots below are taken from the linked iTunes page).

Move the Turtle

This app both teaches kids how to use it (through a series of interactive puzzles) and allows students to create their own projects. It is based on the Logo programming language, but written using a sequence of commands chosen from the menu, rather than by typing. It works on the iPhone as well, but unfortunately (due to an issue with the Apple TOC) programs created with it cannot be shared.
Tinkerbox

This app allows students to build equipment to try and solve puzzles (which are based on physics concepts). I’ve not tried it out myself, but it is free, and so I’m going to at least try it out with the students.
Show Me

This app allows students to draw and record their voice while drawing, letting them create a voice-over narration. It could be used for student created tutorials, stories, and animations. According to the description of this app, the videos created can be uploaded and shared via the ShowMe.com website.
Motion Math

The series of apps Motion Math makes for the iPhone and iPad are excellent, because they are more than just the typical flash card apps that are all too common in the app store for math. I’ve played with the fraction one myself (so has my son) and enjoyed using it, and seeing how it creates an alternate representation of fractions. This representation is hardly sufficient for students to completely understand fractions, but I’m sure it helps.
Sketchpad

I’ve not used this app myself, but it comes recommended from Trever Reeh. From conversations with the mathematics teachers who work with Sketchpad, and from my time spent using Geometer’s Sketchpad a few years ago, I’m pretty sure this app will be useful. On the app description page, they note that they have activities built into it, which is encouraging.
DragonBox

DragonBox is a puzzle-game which tries to teach algebraic reasoning. It replaces algebraic symbols with visual representations (which are still themselves an abstraction of some arithmetic concepts) and then allows students to manipulate the symbols to try and solve the puzzles, which are all equivalent to standard algebra problems. It has a PC version which runs in the browswer and I have tried out as well. This is not a flash app so students can practice algebra – this game will try and teach students.
Shuttle Mission Math

I’ve not tried this puzzle-game out myself, but I have used the paper and pencil version of the types of puzzles presented in this game with my students, and I found them very useful. Through solving the puzzles, students will have to employ (and learn) algebraic reasoning skills, which are explicitly described on the support page for this game.
Scribblenauts

This game allows students to type in words, get presented with images that represent those words, and use the images to solve a puzzle presented. The students have an enormous amount of freedom in what words they choose, and what images result. My son has loved playing this game, and spends his time playing it constantly asking us how to spell words, which he seems to be able to (mostly) remember for the next time he wants that particular item.

These are some of the apps I’m looking into. I’d also like further recommendations. I’m looking specifically for iPad apps which:

Are not just skill practicing / flash card apps. There are thousands of these, so finding them is easy, should our teachers want to use them.
I’m hoping to find apps which will actually help teach concepts, rather than just review existing concepts. I’d like this teaching to be of the non-explaining-type teaching style, and more of the discovery-it-yourself-inside-a-guided-framework style of teaching. I can find things like the Khan Academy for myself fairly easily, but finding apps which support our inquiry-based teaching program in our elementary school is a bit more of a challenge.

July 31, 2012 / 3 Comments

Images to make one think about technology

Here are some images I’m collecting for future presentations on technology. Each of the images is intended to ask a question, and to have people reflect on their own use of technology, and our society’s use of technology.

Technology transforms us
(Image credit: Andy Hooper)

(Image credit: Wikipedia)

Laptops in schools
(Image credit: T Coffey)

TV Sunset
(Image credit: Michael Leung)

People texting instead of talking
(Image credit: Susan Sermoneta)

Cars on a highway
(Image credit: Dom Nozzi)

Kids playing games outside
(Image credit: Geek of the day)

Volvo and IKEA
(Image credit: Erik Johansson)

MRI machine
(Image credit: Daniel Martin Reina)

If you have any other images of various technologies that you think require their viewer to ask questions (or which easily lead to questions), please share them with me.

July 25, 2012 / 0 Comments

Education is not the Olympics

Students First (I won’t link to their website, you can find it yourself) has published a new video comparing the ranking of the US education system to a potential performance in the Olympics.

There are a huge number of issues with this analogy, but I’ll just bring up two.

Countries which perform well at the Olympics outspend the other countries, have a larger population base, and more free political environments. So from this analogy, we can predict that the US will perform better in education if they spend more money, increase their population, and work on implementing a more democratic decision making system within their education system. Not what I think Students First is hoping for with their privatization agenda…
The Olympics accept only the most elite athletes in their competitions. The US education system is intended to educate everyone.

July 13, 2012 / 4 Comments

What do I know?

What do I know about the most of the people with whom I connect online?

Nothing.

I know very little about their families, their history, their relationships, their beliefs about God, their dislikes, their intolerances, their emotions, and who they are. What I do know, I know only through their statements about themselves, so I know really nothing about them except what they have confessed themselves. What I know is their manicured self, their selves with the make-up on, and I do not really know them at all.

And of course I know that people tend not to share the uglier parts of themselves, and this is especially true in print mediums. I know that people tend to find it very difficult to view themselves objectively, and they (myself included, of course) have a tendency to exaggerate some parts of their personality, and ignore other parts. So what do I really know about the people with whom I connect online? Nothing.

And who knows about me? Who knows that I hated school until it became an escape from the bullies? Who knows that I once believed myself to be bisexual (and no longer do), mistaking a desire to connect and form relationships with the people around me with sexual desire? Who knows that I love to play games, often because the game is enjoyable, but mostly because I like to win? And who knows that right now, my favourite thing to do (and something for which I do not have sufficient time) is to wrestle with my oldest son, and gaze into the curious eyes of my youngest son as he observes the world? Who knows that I love my wife fiercely, and would do anything for her?

No one.

If you are reading this, even you cannot be sure these things are true, since you have not observed these in me yourself, you have only my (written) word for it. We cannot really know things about other people, except what we can predict is true about them from our observations of them. I know from my observations, for example, that John has a deep relationship with his God and his family, and that Mary Beth is passionate about coops, and that Shelley thinks her dog is the cutest dog in the whole world, but I know these things are true because I have talked with each of these people in person and I can see these things are true from the actions and words I’ve experienced while with them.

These are still shallow characteristics of who these people really are. John is not defined completely by his love of his family and God, Mary Beth is more than her passion for coops, and Shelley is deeper than just being a dog-lover. If I want to really know who these people are, I need to spend time with them, experience the highs and lows of their life, and this kind of relationship takes years to build.

We must be careful not to mistake interactions with the words of people online as friendships, and we must further be careful not to take away from the already precious time we have to build real friendships with the people around us. We must balance our desire to know more about the world that is away from ourselves, with building deep connections with the people surrounding us.

July 10, 2012 / 2 Comments

A thought experiment

Here’s a thought experiment for you (h/t to Dan Meyer for the sports analogy).

Imagine you start learning the game of basketball by learning how to shoot free throws. At no point are you told what the point of shooting free throws is, or how being a good free throw shooter will help you play the game of basketball, or even that there is a game of basketball. Worse, you are occasionally asked to shoot as many free throws you can in a minute, and then judged against your classmates based on your performance.

You are at some point asked to start practicing all of your free throws blindfolded, possibly after unsuccessfully learning how to shoot free throws earlier. If you are lucky, your coach tells you how many free throws you sank, and how many you missed. You finally have some understanding that free throws are important in basketball after years of not having a clue why you were practicing them but no matter how many times you practice, you never seem to get better at free throws.

By way of analogy, this is almost exactly how addition and multiplication facts are taught to students. They spend the earliest years learning addition and multiplication facts with only a superficial explanation of how these facts might be useful later, and most do not learn how addition and multiplication fit into mathematics as a whole and they certainly never get to experience "the game."

In their later grades, their teacher (although lacking the time to give students feedback on their "basic skills") expects students to work on their higher level mathematics without a calculator or any aid of any kind for their foundational numeracy skills. The premise behind these calculator-less classrooms is that students will likely forget their addition and multiplication facts if they get to use a calculator, and so the use of a calculator is banned. Unfortunately, in most of these classes, very little time is spent reteaching addition and/or multiplication facts, and almost no feedback is given to students as to whether they have even done their addition and/or multiplication correctly, so if you are a student who never understood addition or multiplication in the first place, this further practice without support is unlikely to be useful.

If you are going to ban students from using calculators in your class for basic arithmetic operations, then you must at least take the blindfolds off of your students and help them improve their arithmetic skills. On the other hand, I prefer not to ban tools, but instead find ways that these tools are used productively (and unproductively) and change my teaching to compensate.

July 7, 2012 / 3 Comments

Things good schools do

This is a bit of an experiment in collaborative writing. How it works is that you copy this entire post verbatim, and add one thing to the list below. If you put this on a blog, please tag this post with "goodschoolproject" if possible to make these posts easier to find later.

Good schools focus on the learners, not the system.

You are free to share and modify this post, but whomever you share it with must enjoy the same freedom.

July 4, 2012 / 6 Comments

Connected educator

I see a lot of people discussing the need to be an online connected educator. The only problem I’ve noticed is the why is sometimes missing or weakly argued in these discussions. Why connect? Why bother? What can I learn from educators a thousand miles away in a completely different context which is useful for my teaching now?

"We are in great haste to construct a magnetic telegraph from Maine to Texas; but Maine and Texas, it may be, have nothing important to communicate." Henry Thoreau, Walden, chapter 1, p. 67 (1966)

We often forget how important context and local dialect is when communicating with each other. It is not obvious in our daily face to face communications because we have come to a common agreement on the meanings of words we use.

When I say the words "mathematics education", I have discovered that the words "mathematics" and "education" themselves not only have region-specific meanings, but meanings quite specific to the speaker. If I do not take the time to unpack the meanings of these words, I can end up having a terrific argument with someone because of the lack of understanding on what we have both assumed are common definitions of terms.

I support becoming a connected educator because I know how many opportunities to learn more about education, teaching, and learning I would have missed without the opportunities to learn from those people with whom I am connected. However, I strongly urge those encouraging others to join our ranks as connected educators to warn their colleagues of this pitfall, this potential trap of online communication.

June 26, 2012 / 1 Comment

Technology influences cognition

Technocriticism Wordle

That technology can influence cognition should be painfully obvious when you examine our primary technology of communication – language. Someone who knows a language cannot choose to ignore that language when confronted with it. If you are a literate person, letters arranged together do not appear randomly placed, they form words. When someone talks, you cannot hear it as babble, you are forced by how this technology has influenced your thinking to hear words.

When one looks at language, it is nonsensical to ask if one has a choice whether or not to use this technology. Once proficient in a language, barring a severe brain trauma, one remains proficient in that language and has it forever more alter their thinking. You can’t choose not to use a language once you have it and are exposed to it anymore than you can choose your parents.

Different languages and the accumulation of culture (another technology) that goes along with them result in different ways of thinking. One of the reasons why translation is so difficult between cultures is because quite often cultures have concepts which are unique only to their culture, which tells us that differences in the technology of culture results in different types of thinking. In the same way, people who are proficient in digital cultures have their own thinking altered by their participation in those cultures.

It is possible that there are technologies which do not influence our cognition. It may be, for example, that we are not influenced by our cell phones (which incidentally, whatever their value may be, a device which allows almost anyone in the world to interrupt you no matter what you are doing) to the degree that they influence our thinking.

I do not think this is true though. When you look at cell phone use in particular, you will no doubt recognize that possession of a cell phone and knowledge in its use means that if you are planning to meet someone else, who also posses a cell phone and knows how to use it, you are less likely to carefully plan exactly when and where you will meet them. So just possessing and knowing how to use a cell phone changes your behaviour, and changes how you plan your life. If one possesses a smart phone, and are at all proficient in its use, one generally stops planning exactly how one will travel somewhere in advance.

My strong suspicion, although I cannot yet prove this, is that all technologies include different types of thinking which are a necessary part of using the technology, and that while the influences of technology are not deterministic — we have some free will in our use of technology, I do not think that someone who is not cognicent of the limitations of their technology will see those limitations.

The benefits of technology use are generally easily apparent. What are usually less apparent are the drawbacks. So instead of blindly using technology without regard to the potential drawbacks, we need to be considerate of its use, and be critical of how it has changed us. We need to be technocritical as users, and those of us who are experts in technology use must especially be experts in critical reasoning around its use.