Education ∪ Math ∪ Technology

Month: January 2013 (page 1 of 2)

Two views of mathematics

Beautiful fractal image
(Image credit: DanCentury)

As usual, there is an argument going on Reddit on mathematics education. There is a statement from that argument that I would like to highlight here, and a related discussion on Reddit with a related comment.

“I solemnly declare that no kid ever learned math by watching a video OR by reading a paragraph, since math is an action [emphasis mine], not an exposure.” deadletter

In a related discussion on resources for a 4th grade student wishing to explore mathematics further, a commenter made this bold claim, in response to someone suggesting that the Khan Academy would not be a good resource:

“Why not? If it’s to pass state Core Standards, it’s more than enough. If it’s to give his daughter who loves math more to learn, it’s more than enough. If it’s to showcase the power and beauty of math, it does that, too.” misplaced_my_pants 

These are very different views about what it means to learn mathematics.  One person holds that mathematics is an activity that people undertake, the other believes that mathematics is a specific set of knowledge that ones gains through exposure. These are very different definitions of mathematics, and have very different consequences of what would be required to learn mathematics.

I tend to lean toward the mathematics as activity definition, but understand that my responsibility as a teacher is to ensure that my students also know some specific set of mathematics. It’s not a line I’m particularly comfortable straddling and I feel a lot of tension as a teacher as a result. Whenever I have some freedom from the curriculum, I lean heavily toward explorations of math, either as independent activities, or as a group activity.

At the very least, I want my students to know that there are two views of mathematics (which some may consider to be opposing views), and that they should have the ability to make an informed choice between them (or to choose both, if that is even possible).

Alternate definitions of technology

Some alternate definitions of common technology:

  1. Email:

    A technology through which anyone in the world can add to your to-do list.

  2. Smartphone:

    A device designed to interrupt your thinking on a semi-periodic but slightly random basis with a buzzing or ringing sound, making completing lengthy tasks requiring significant concentration nearly impossible. Users can even customize the sound with which they will be interrupted! Added bonus: discourages face-to-face conversations when in use.

  3. E-book:

    A lot like a regular book except it requires an expensive electronic device to be viewed, can’t be shared, and can be potentially removed from your ownership by the book seller without warning. Oh, and if your electronic device for reading the e-book has a dead battery, you can’t read the e-book.

  4. Television:

    An electronic baby sitter for your children that can potentially introduce your children to all sorts of unacceptable behaviours.

  5. Computer:

    The most powerful computational device ever constructed by humans. Mostly used for finding and sharing pictures of LOL cats and music videos.

  6. Air conditioner:

    This machine cools down people’s houses, which results in people spending less time on their front porch getting to know their neighbours.

  7. Digital camera:

    Now everyone can take bad photos of places they have been, which they then share once on Facebook and then never look at again.

Build a better website for learning math

I’ve been thinking about what I think a truly great mathematics education website would look like. Dan and David have produced some awesome mock-ups of the future of mathematics textbooks, and I love their work, but I can think of more features I would add.

  1. There should be space available for students to ask and answer questions, just like MathOverflow. It would probably need to be moderated, and perhaps seeded with people with some knowledge of mathematics willing to attempt to anwer questions, but I suspect many of the interactions would be peer to peer. The level of mathematics discussed on MathOverflow and the sometimes snarky responses to people who ask lower-level questions lead me to believe that this type of discussion space should be have a different community standard – one that expects children to be participating in it.
  2. There should be a mixture of styles of problems from the directed-style problems with embedded, mediated peer to peer interactions that Dan and David are dreaming up  to open-ended problems and/or puzzles wherein students choose what tools they want to use to try and solve the problems.
  3. There should be a library of exploration spaces available for students. I’m thinking Logo, origami, and other types of similar resources would be available here, along with discussion space to share any discoveries and/or projects students develop.
  4. At least part of the site should be dedicated to sharing some of the wonder of mathematics. Perhaps this looks like a blog where some of the most fascinating and elegant mathematical ideas through-out history could be shared. The primary purpose of this section of the site would be to inspire, not to teach.
  5. The site could include a toolkit of different mathematical techniques students could use to solve mathematics problems. Alternatively, solving specific problems on the site opens up new tools in the toolkit. Students could also have a toolkit of skills they have developed themselves, and bookmarked to remember for later, much like programmers do as they build their own code libraries.

I can imagine that trying to introduce this all at once might be overwhelming or too challenging, so these features would have to be introduced over-time, but the ability for students to connect with each other to have discussions about mathematics would have to be front and centre from the very beginning.

What other features would you include?

Internet as cMOOC

How I learn about mathematics education

I’ve noticed that my experiences in #etmooc very much parallel my learning experiences on a regular basis, except that they are now branded and slightly more focused on a different topic – connected learning.

  • I participate in weekly chats on #mathchat whenever I have the time.
  • I follow hundreds of blogs on mathematics education.
  • I watch videos on mathematics education and infrequently participate in webinars related to the same.
  • I read books on mathematics education.
  • I work with my colleagues to improve our mathematics instruction on a daily basis.

None of the learning experiences I’ve had through #etmooc have been different (except for the focus). So it leads me to wonder, why don’t more people have these types of learning experiences on a regular basis given that they are freely available?



PS. I know that people complain about not having the time to participate, which I get. I’m busy too. I have to balance my desire to learn more against my family life as well, and lately, family life has been winning, but I always carve out some time each to learn a little bit more about my field each day. I couldn’t give up learning more about education any more than I could give up breathing.

450,728 reasons to blog


I used a tool recently which counts the number of words I’ve posted on my blog. I’ve written a total of 339,254 words on my blog (as posts and comments), and other people have written a total of 111,474 words as comments on my posts.

This level of engagement and thinking has got to make a difference on my understanding of education, and at least indirectly on my skill as a teacher. When I say that blogging has made me a better teacher, I mean it.

The future of learning

A few years ago, as an assignment for my Master’s degree, I created a video on the future of learning. I was never very happy with the quality of the final product, nor do I completely agree now with the vision I presented then. That being said, here are some of the slides I used. It looks like some of the text in the comics is uncited, but I have citations near the end of the video I created.

Some of the slides below offer some ideas as to how technology may change how we learn in the future. Some offer quotes from other thinkers in the field, in case you want to explore more.



Education reform ineptness

"If you are absolutely no good at something at all, then you lack exactly the skills that you need to know that you are absolutely no good at it."

John Cleese

Perhaps this is part of the reason that so many education reforms that are attempted fail so badly? Could it be that at least some of the people involved in education reform are just so completely inept that they are not even able to judge their own performance at all?

The good news is, if you know you are bad at something, then at least you are not completely inept, because if you were, you might think you were good at it. The bad news is, people who are completely inept are unable to judge or recognize their ineptness, which may make them push ahead in ways which are harmful to the rest of us.

Learning outside of school

I realized that I have done a lot of learning outside of school as a kid. Here are some examples.

  • I took apart almost every piece of electronics we owned (except the television as I was told this was dangerous) and then put it back together, most of the time.
  • I explored every inch of the island I grew up on within 5 kilometres of our house, usually on bike, often with my dog.
  • We had a library of many hundreds of books in our house, and I read most of them.
  • I taught myself how to program, and I created a computer game in grade 10.
  • I (re)discovered the sum of triangular numbers formula, and the quadratic formula on my own.
  • I planned a trip to Mars in as much detail as I could, right down to where the bathrooms would be located on my rotating space-ship.
  • I collected a sample of every different kind of grass I could find on my island. I ended up with over 50 different types.
  • I played a lot of games, including Dungeons and Dragons and many exciting computer games.
  • I watched way too much television (with a strong preference for science fiction). I don’t know how valuable a learning experience this was, but in the interest of being honest, I thought I should include it.
  • I went for regular jogs around town. One time I finished my run, wasn’t tired enough, and so I went for another loop.
  • I wrote poetry, sometimes in the middle of the night.

I’m not offering this list to brag about my childhood, but simply to suggest that there are plenty of ways kids can learn outside of school, but they must have the time, space, and support to do so.

A factoring success story

I covered a couple of my colleague’s classes yesterday so he could attend a math conference. The afternoon class was a somewhat boisterous grade 10 group. I was asked to teach students how to find the greatest common factor, and if I had time, introduce them to more general factoring techniques.

I decided that the greatest common factor is a topic students find relatively easy, and so I just showed some examples of how to do it (actually, I drew the "how to" out of the class by asking them questions, but this is my standard technique) after verifying that they understood the distributive principle. I then assigned some practice problems, which then each student wrote their solution up on the board, and we discussed. I then showed students a couple of different techniques for multiplying binomials (like (x+2)(x+3) for example).

Next, I put up the following 4 questions.

1. x2 + 7x + 12

2. 2x2 + 7x + 3

3. x2 – 25

4. x3 + 8

I asked students to try and figure out how to write these expressions as one set of brackets times another, just like with the example from before, but I suggested to them that what we are trying to do is undo the distributive rule.

I went around the room and encouraged students, gave them hints when they needed them, asked them questions to prod their thinking, and observed their problem solving strategies. Students were engaged in the problem solving activity for a good 30 minutes. Once some of the students’ attentions started to wane a bit, I gave them a sheet with a description of how to do factoring by grouping and some problems to work on the back.

A group of students though really dove into question 4, which, as you may notice is actually quite a bit more difficult than the other three problems. I ended up having to give students two hints: I told them that the expression broke into two factors, one of which was (x+2) and the other of which was three terms long. The group of students worked feverishly on solving the 4th problem for a good twenty minutes, and then all of a sudden, one of the girls in the group leapt out of her seat and screamed, "I GOT IT!! YES!!" I circled around to see if she had the right answer, asked her how she was so sure it was right (she had multiplied everything back through using the distributive rule), and then gave her group x3+27 to solve (which she did quickly) and then x3 + a3 to solve.

At 5:30pm that night, I received an email from the girl, excitedly telling me how she had an inspiration while she was on the bus home on how to solve the general question, and had then figured out the general formula for how to factor a sum of cubes.

I emailed her back and congratulated her on becoming a mathematician.

Investigation into scoring systems

I played ultimate tonight, and we usually keep score with shoes. Our normal scoring system is to count in base 5. Tonight, I tried to use binary, but at half-time I switched back to base 5 when most of our team struggled to read our score quickly.

I took some pictures of the arrangement of shoes during the game (when I wasn’t playing).








I can imagine some investigations could be made out of these photos.

  • Given the numbers associated with each photo, try and determine how to count in this number system,
  • More challenging: From these photos, try and determine the missing numbers.

If you want a project that might take a while:

  • Design a scoring system using shoes. It should be easy to maintain and not require too many shoes.