Author: David Wees (page 28 of 97)

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?

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.

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.

I’m assisting our 4th grade teachers in finding resources for their upcoming unit on division, and I’m hoping to avoid either of these situations from arising (see the videos below). Actually, these videos could be great hooks to see if students can figure out what mistakes the people in them are making…

A key problem in both of these videos is that these (fictional) people are making key mistakes in their use of a standard algorithm which they clearly do not understand, nor do they seem able judge how accurate their solutions are. Now, my suspicion is that students will not make such egregious mistakes, but they may make more subtle mistakes based on the same principles.

The key components of any division instruction should help students by:

• emphasizing understanding what they are doing when they use a division algorithm,
• providing tools for students to estimate their answer,
• encouraging them to use their estimation skills to check the reasonableness of their answer,
• suggesting alternate techniques to double check the accuracy of their answer,
• giving time to see the connection between different division algorithms,
• embedding the use of division in a problem solving context.

If I were going to teach division, (after confirming that students understand multiplication very well) I might start with a problem about equal sharing for students to do that every student can access. Next I might gradually increase the size of the numbers in the problem until students start to see that their "naive" strategies may work, but are very inefficient. One obvious problem I can foresee happening here is that some of the students in many grade four classrooms already know a division algorithm, and may be reluctant to come up with their own strategies as a result.

Next, I would compare different strategies that students use in a whole group setting. We would spend some class time discussing the strategies and talk about the benefits and drawbacks of each. We might also explore some other alternate strategies not yet shared, perhaps using a gigsaw technique where each group explores a specific strategy, perhaps suggested by a video.

One interesting activity for students might be to watch the videos above and try and figure what mistake the people in the video are making when they calculate.

Here are some division strategies that your students may not have seen before:

• Double division
• Galley division
• Chunking
• Division "by parts" (see the third method for division listed on the page)

I’m also looking for projects and/or puzzles which require the use of division as a strategy for solving the problem/puzzle.

An aside: I am not convinced that these algorithms have much value for our students. More time spent understanding what exactly what division is would be more useful, I think, for our students in the long run, and then just give them a calculator. However, that is not my decision to make, it is up to the BC Ministry of Education to decide.