The Reflective Educator

Education ∪ Math ∪ Technology

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May 13, 2012 / 6 Comments

Teaching probability

My colleague found an activity to do with his 5th grade class, similar to this one. Basically, he gave the students 10 coins each, and asked them to put the 10 coins on a number line (with numbers from 1 to 12) with a partner. Each round they roll 2 six-sided dice, find the total, and remove a coin from their number line if it matches the roll. They keep going until one of the two students has no coins on the number line.

 

At first, most student’s starting positions looked like this:

Student 1 - flat distribution

or this:

Student 2 - another flat distribution

 

At the end of the lesson, we played a 5 coin version of the game, and one student’s paper (after 1 round) looked like this:

Student 3 - All 4 coins on number 7

 

Unfortunately, although most of the students did notice that some numbers came up more frequently than others, as evident by their distributions looking a bit less flat, and bit more centred on 7 on the number line, many of the students still had obvious misconceptions of probabilty. Students made comments like:

"If I spin around twice before rolling, I get a more lucky roll."

"I got a few 11s last game, so I’m going to put a few more coins on 11."

"8 is my lucky number! I’m going to put 3 coins on 8."

"I need to spread out my numbers so I have more chance of getting a coin taken on each roll."

Our plan for next class is to have students switch up groups, discuss insights they’ve had on the game, play a couple of rounds, switch them up again, while we walk around and see what strategies they use. Hopefully we’ll see less students spinning around in order to improve their dice luck…

I also asked for resources on Twitter for using a probability game as part of a lesson on probability, and had the following 5 games recommended (all of these look good because they are relatively easily produced and used in an elementary school classroom).

 

Update:

I wrote a simulation to test to see what distribution of coins is the best. I am somewhat surprised by the results. Check it out here.

Why am I surprised by the results? My intuition about how this game works failed me. I thought that the likelihood of each number coming up was the most important factor in deciding a good strategy for the game. It turns out that one has to balance out the knowledge of the likelihood of each number being rolled with having a selection of numbers available. It may be that the probability of a 7 being rolled is 6/36 but the probability of a 6 or a 7 being rolled is 11/36.

In fact, it turns out that having a selection of different rolls available is fairly important. I updated the simulation so I could choose the number of coins to place, and with 3 coins placed, choosing 4, 5, 6 is better than having all three coins on the number 7.  With 2 coins, it is better to choose 6 and 8 than to put both of the coins on 7 (although 6, 7 is better than 6, 8 – but only slightly).

There are three messages I get from running this simulation.

  1. One should, in general, not make too many intuitive assumptions about how probability works, particularly with somewhat complicated examples.
  2. One should be careful how one uses even simple games to teach probability. All of the students saw that 7 was the most common number, but their intuition of "choosing a wide spread" is a valuable one for this game, but it doesn’t help us get at the idea of some numbers being more likely than others. I think we’d need a different game for that.
  3. It is probably a good idea to build the simulation before you play the game with students, if at all possible.
May 13, 2012 / 0 Comments

PISA results from 2000 to 2009 for Canada

I noticed through this blog, that the CBC had published the PISA results for Manitoba (released as charts) for 2000, 2003, 2006, and 2009. I wanted to verify the results they had posted, especially the mathematics data, so I went and looked up the data for myself on the Stats Canada website (which you can access yourself here, here, here, and here). Using this data, I created th graph below, which shows the scores in math for Canada and for each province (get the raw data here).

I’m not sure what this data shows, although I can see some trends. Of course, if I change the scale, the overall trend seems more clear.

It looks to me like overall the results have been somewhat stable, at least at this scale. While the trend in Manitoba definitely looks like a downward turn for the last few years, and this trend is probably statistically significant, overall for Canada, it looks like the results have moved somewhat randomly, as one would expect from year to year.

May 12, 2012 / 0 Comments

Culture and counting

Not convinced that there are cultural nuances in how we understand and define math? Watch the following short video (see http://www.culturecognition.com/ for the source) in which a child explains the number system his culture uses to another child.

 

 

There are other areas in which we understand mathematical concepts differently depending on our culture. For example, this recent study suggests that something like ‘numbers come in a certain order’ may be a cultural representation, and not one of which most of us are aware.

One wonders, if we can see such dramatic differences between different cultures in terms of understanding something fundamental like number, how likely is it that there are other differences within our own culture?

My wife, for example, tends to rely on landmarks for navigation, but I tend to rely on an internal map based on the names and numbers of the roads. She and I therefore have a different understanding of how one should navigate. I can remember meeting people who could not read a map (but who were otherwise able to navigate with ease) suggesting that our representations of geographical information may differ greatly between different people.

How does this influence how we should teach?

May 9, 2012 / 3 Comments

We didn’t do any math yesterday

Practice makes perfect comic

 

Yesterday, I was covering a colleague’s math class at the last minute, and he had made photocopies of a chapter 1 to 7 review. I looked at the review sheets, and the grade 10 students in front of me, and decided that it was unlikely that the review sheets were going to be useful. I handed them out, and then started putting puzzles up on the board.

 

Seven Bridges problem

The first puzzle I put up was the Seven Bridges of Königsberg problem. Within  a couple of minutes, every student was trying to figure out the path across the 7 bridges that doesn’t cross any of the bridges more than once. Before the students got completely frustrated with this problem (since it is deceptively simple to state, but "difficult" to solve), I put up a couple more problems, including a gem from Dr. Gordon Hamilton. I added the frog hopping problem to the board, and taught two students the game of Nim.

Each problem had some students who were working on it intensely. Every student found some problem which was interesting to them, and almost all students were working in small groups on the problems and puzzles. Eventually, a small group of students gave up on all of the puzzles and worked on the review sheets while the rest of the students continued to work on the puzzles until the end of class.

Some students asked for a hint on the bridge problem, and I led them (through questioning) to Euler’s formulation of graph theory. From this, we discussed that there could be at most one starting spot, and one ending spot, and that only a starting and ending spot could have an odd number of paths leading in and out of it. I then put up the 5 rooms puzzle, which one of the girls said within seconds was unsolveable by applying Euler’s analysis to the graph.

A group of boys worked on the frog problem, and went from struggling to even find a single solution to the 3 frog problem to being able to generalize a solution for n-frogs on either side (and a formula for determining the number of moves for each frog puzzle).

The next day, I spoke to my colleague, and asked him if he was okay that I had not done the worksheet with the students. As expected, he was fine with it. I asked him what the students said. He said that students said that they enjoyed the day before, but one student had said, "We didn’t even do any math yesterday."

I’m not sure I agree with that student, and I’m slightly distressed that he didn’t see the problem solving activities we did as being part of math. What do you think? Are problems like these important in mathematics? If so, why aren’t more of them in our curriculum?

May 8, 2012 / 5 Comments

Interesting ways to use Google Apps in the math classroom

I just found this presentation from more than a year ago on some interesting ways to use Google Apps in a mathematics classroom. I noticed that it had been edited slightly, so I did some more edits and thought I would share it here.

You can help edit and curate it here. I could imagine that Google+ would be useful, and that some of the file sharing options through Google Drive have improved, neither of which has made it into this presentation yet.

 

May 7, 2012 / 9 Comments

On motivating mathematics education

Here is a funny comic from the Fake Science blog.

Fake science - Use a ruler to find the third side of a triangle

 

The problem is, there is a kernel of truth in this satirical comic. Given most problems we will encounter in life, we would use a ruler to find the third side of a triangle. Obviously I think that there are good reasons to learn the Pythagorean theorem, but for most real life applications, one could draw a careful scale diagram (an incredibly useful skill in itself) and apply ratios to your measurements of your diagrams to find the missing length.

So why do we teach the Pythagorean theorem? Is it because of the power this abstract idea has? Are there other abstract ideas which have equal value? Could you imagine a mathematics curriculum which includes lots of rich abstract ideas, but happens to not include this theorem? How important is this theorem anyway?

 

May 6, 2012 / 1 Comment

Unidirectional instructional mediums

 

Derek Muller has done research on the effectiveness of science videos. To summarize his research in brief – when you present only the correct information in a science video without the possible misconceptions that students may have, students learn less (but feel better about the experience) than if you present information in a science video and include the misconceptions.

Of course, we should recognize that this effect probably does not depend on the medium of instruction, only on the nature of the medium. Videos are a unidirectional medium because they present information without the ability of the learner to ask questions. One might guess that any unidirectional medium may have the same effect. So textbooks, lectures, and other unidirectional mediums may suffer from this same deficit; without common misconceptions addressed in these mediums, the learners learn much less than if those misconceptions are addressed.

 

 

Dr. Eric Mazur shares essentially the same message – unidirectional instruction (in his case lecture) – has flaws. He relies on peer instruction and student response devices (clickers) to change the nature of the instruction so that it is more bidirectional (from each student’s perspective). The key here is that he has embedded more opportunities for feedback to reduce the chance that students incorporate the new information they are receiving into their existing misconceptions.

Textbooks (another unidirectional instructional tool) rarely present misconceptions and address them. Most students rarely use their textbooks as a learning resource (at least in k – 12), prefering to rely on the bidirectional instruction their teacher (or parent) provides. This means that the vast majority of information presented in a textbook goes unused. There are some changes to the textbook I’d like to see, which would allow for them to be a more bidirectional learning tool.

While it is clear that the medium of instruction influences the type of cognition that occurs, as Marshall McLuhan has pointed out, it should also be clear that different mediums have similarities in how they affect cognition or learning. If we find out that failing to address misconceptions in video instruction results in poor learning of the concepts, we may be able to transfer this finding to other modes of instruction. If that is the case, then we need to look at our instruction carefully, and ask ourselves, how much opportunity do we give students to address their existing models and resolve conflicts between their misconceptions, and the models we suggest?

May 2, 2012 / 2 Comments

Not a math person

Original blueprint

 

Someone I know produced the diagram above in her planning steps to produce the shelves seen below.

 

Finished shelves

 

This person describes herself as "not a math person." What do you think? Is she a math person or not? It worries me that we have all these people walking around thinking they aren’t "math people" when in fact, they quite obviously are. We need to do a better job of explaining the difference between every day mathematical reasoning, which quite a lot of people are good at, and the formal systems of mathematics that have taken generations to develop. 
 

May 1, 2012 / 0 Comments

I did professional development all wrong

Last year, I presented a lot on the need to improve mathematics instruction. I had pictures, I had questions, I had effective arguments, and my audience was engaged. I could present like the best of them on some of the ways that we can improve mathematics instruction. What I did not have was effective teaching.

The role of someone involved in professional development for teachers is to help the audience, teachers, improve their practice. It may be that they take part of what you do and use it, and it may be that they attempt to copy your method exactly. The problem is that the typical presentation does little to improve someone’s practice. It may inspire them, it may anger them (I’ve done both), and it may provide some helpful tips, but effective change in practice does not come from someone presenting on their practice. The best you can hope for from a presentation is small, temporary, surface level changes.

Improving one’s practice requires thinking. It requires time spent looking at the context of one’s school, on the way that one approaches one’s own teaching, and on what other practices one can incorporate into one’s own pedagogy. It requires discussion so that the learner can take the ideas they are assimilating and seek clarification and direction.

So instead of spending the entire time I present talking, I give participants much more opportunity to talk. Instead of participants sitting around listening, I give them opportunities to do. The last few workshops I’ve done have been more about conversations. They’ve involved rich, mathematical problem solving activities. They’ve involved teachers having insights, and sharing those insights, often things that never would have occurred to me. I’ve learned much more from my workshop participants than when I was a presenter.

I spent an afternoon talking with my colleagues about computational thinking, how computational thinking really is mathematical thinking, and how if our students get opportunities to program, then they are doing mathematics. My colleagues were working on a particularly challenging problem, and one of them stopped and said, "Okay, I get it. Solving problems is hard. I can see why the kids struggle with this stuff." This kind of insight, not directly related to my objectives, was probably the most valuable insight to come out of that workshop. It never would have happened had I not given participants a chance to think and to do.

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