Planning time and class size

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This is going to be a general rant about statistics, research and the workplace conditions for teachers.

I hear a lot of people (mostly those controlling the money or not working as a teacher) referring to research that indicates that class size doesn’t affect results significantly. For me, this is a great example of empty statistics and research removed from reality. Here’s what I believe:

  • A good teacher with a big class is better than a bad teacher with a small class
  • Class size standards differ greatly based on local culture and this is not always taken into account. Yes, in Japan they have far larger classes than in Sweden, but the students have a completely different attitude to school and learning there so it is ridiculous to make the comparison on number of students alone. This sort of cultural difference in attitudes towards academics even varies within a country. Just look in Sweden at the difference in student attitudes in a country side school compared to a school in a serious university city (I would say this is difference is even bigger for math than other subjects).
  • If you are only looking at average results on standardized test scores then you may miss lot of real differences

And here is what I know: I am personally a better teacher with a small class compared to a bigger one. People who have not taught have trouble understanding the energy it takes to simple stand in front of a class of 30 junior high students. Even if you are doing nothing but talking, there is this invisible energy interaction between you as teacher and the students, and it is exhausting. The difference between 30 and 20 students just on the simple psychological and physical toll it takes on a teacher is hard to accurately put into words, but it is huge.

With fewer students I have more time, energy and ability to get to know every single student personally. I have a much greater chance of learning their strengths and weaknesses on a real and deeper level and am much better able to find the appropriate way to challenge and reach them personally. Not just help them raise the average standardized test score, but help them gain a deeper understanding and (more importantly) a greater appreciation for math.

Standardized scores do not measure student motivation, interest and happiness.

With fewer students I can have more frequent graded assignments without killing myself. This helps get a much more detailed pictures of where potential problems lie and allows a lot more opportunity to help correct misconceptions. Constant formative evaluation is really helpful for both teaching, but also for learning. Even the students need to “test themselves” constantly to see what they really can do and not just what they think they can do. It is very hard for students at that age to develop an accurate understanding of their command of mathematics without a controlled testing environment. Doing this frequently also makes it not so stressful for the students, since they are no longer “tests” but “diagnostic assessments”. This is not possible with too many students without killing yourself as a teacher (been there, done that).

With fewer students I can plan more fun group activities. It is much easier to have classroom discussion with more students participating.

With fewer students I have more energy in general. I am less exhausted and less burned out after smaller classes. I can do more. Standardized test scores do not show the toll taken on teachers physically or mentally when they burn themselves out with big classes. The extra energy  I have from smaller classes translates into more motivation on my part and more creativity and better ideas.

And back to where I started – I am just a better teacher with fewer students.

Now on to planning time. This is the invisible part of our job that many who don’t teach don’t see or understand. The myth that once we have taught a topic we never have to prepare again. For bad teachers maybe, but not for those I know.

Each group of students I teach is different, which different needs and interests. I have to constantly adapt my teaching to each individual group. But more than that, I always try and evaluate every lesson – not just every topic, but every lesson. How could I have explained that better? Can I find a way to make that more clear? What were the problem and misunderstandings this time? And fixing that and creating lessons takes a lot of planning, over and over again. Refining and improving. Trying to always grow and develop as a teacher.

But this time is not recognized or given. We teach too many students and have to waste time with things like being a rastvakt (walking around the halls telling students to behave), or at our school being forced to sub for other teachers when they are sick for free during the little time we have to try and plan.

Ideally, I should be able to teach each group of students for more hours per week in math, but have far fewer groups to teach. This would make a huge difference in what I could do, who I could help, and how I would feel as a teacher emotionally.

Okay, end of rant.



It seemed easy to me – 3D geometry problem solving

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I find it interesting (and a bit depressing) that I still sometimes am surprised but which problems my students find hard. I recently made up a 3D geometry problems solving exercise that I thought was kind of interesting and not too challenging. I mean, I knew some students would have trouble with it, but I fully expected the majority of the class to get it, especially since it was group work. Boy was I wrong.

To be fair, the actual math content of the problem was not difficult for them. It was being organized enough to find a way to see the pattern that they had trouble with. The ones who did solve it made a table starting with smaller numbers and quickly saw the pattern.

Here is the problem. I would be interested in any comments any readers might have about it:

You have a large number of small identical cubes in front of you. The surface area of all of these cubes together is 23328 cm2. You start playing with the small cubes and realize you can put them together to build one giant cube with no small cubes left over. The surface area of the big cube is 1944 cm2.

  1. How many small cubes do you have?

  2. What is the area of one face of one of the small cubes?

  3. How long are the edges of each small cube?

  4. What is the volume of the big cube?

  5. What is the volume of each small cube?

A nice geometry and algebra problem

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Dan Meyer recently posted a nice geometry and algebra problem on his blog for discussion. Here is the original text of the problem:

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.

A lot has already been written on that problem on his blog and others. I wanted to post solutions to two interesting extensions of the problem in case anyone was interested.

The first solution is for dividing the line into an n-sided regular polygon and a circle:

The other solution is for dividing the line into two different n-sided regular polygons:

It’s worth noting that most people (me included at first) seem to solve these problems using quadratic equations, but that is not necessary (which is convenient since my students are not yet able to solve quadratics). There is nice way to solve them by just setting up a simple fraction based on the perimeters and setting the areas equal.

It was fun to check the limit as n went to infinity to see that I did end up with the circle solutions as well.