Author: skm.aldous@gmail.com

The Plenary Pyramid

While teaching my year 8 class yesterday I used the last ten minutes on this plenary activity. Our lesson starter asked them to draw lines from their equations (for example, y = 2x + 3), and I managed to draw out some ideas about perpendicular lines and reciprocals. Then in the main part of our lesson we were working on a Maths Trail concerning systematic working.  Students had to use a logical system to draw as many reflection patterns as they could, following a set of restrictions.

Here are some responses from the Plenary Pyramid.

I was surprised that:

  • There were so many possible reflection patterns.
  • Working systematically makes my head hurt.
  • Multiplying two reciprocals always gives the same answer.
  • Reciprocals are cool!
  • There were less than 100 but more than 50 [reflection patterns].

Questions:

  • What makes reciprocals so important?
  • How many reflection patterns are there?
  • How do we know for certain?
  • How much time would it take?
  • When are reciprocals used other than straight lines?
  • How to know the equation of a line.
  • Who made up the idea of perpendicular lines?

I learned:

  • What perpendicular means.
  • What systematically means.
  • That a rotated triangle does not count as a separate shape.
  • Shade triangles in a smart way.

This little activity for the end of a lesson helps me discover what went well and the students get to reflect on how successful they were. It lets me praise students for all they have learned. And I love it when students come up with interesting questions; it shows they are thinking mathematically.

What methods do you use to get feedback from students?

Giving Options to Students

I have been reading More Good Questions by Marian Small. One differentiation strategy she recommends is to give students 2 options. The questions are related and the follow-up discussion engages students who did either option. For the question above, we talked together after a few minutes.

  • What is the area of each of the full circles in your picture?
  • How can you test that the answer is reasonable?

I set these cheese questions for some year 11 students who are revising for their IGCSE Foundation exam. After a few minutes we talked about the two answers using these questions.

  • Which measurements did you use?
  • Did you look up any formula?
  • What units did you use for your final answer?

These two questions are available here.

Teaching Activity for Double and Half Angle Formulae

Here is a classroom activity I used with my IB Maths HL students to practice the double and half angle formulae. It is a set of dominoes that the student has to connect from the Start card to the Finish card by solving problems with the double and half angle formulae. The question on the right (above) needs to be connected to its answer on a another card. The answer on the left fits with a different question. I know that the students have completed the task correctly if all the cards are used in a chain from Start to Finish.

I made this activity with Formulator Tarsia, brilliant free software for teachers. I have used it to make matching cards, dominoes, and hexagon puzzles. It’s designed for maths and it can handle maths notation including fraction, exponents, roots, integration, matrices, and so on. I converted the Tarsia file to a pdf using Cutepdf.

Wisdom of Crowds

At the 21C Learning Conference this past weekend, I heard James Surowiecki speak. He wrote a book called The Wisdom of Crowds. He shared many examples of circumstances in which a diverse crowd can give more accurate answers than any individual in the group. For example, a large crowd’s mean guess for the number of jelly beans in a large jar is often within 5% of the correct answer. Frequently the mean guess is better than any individual guess.

image source: The Great Decide

Surowiecki said that the key to a wise crowd is that it is diverse. When there are lots of differing guesses, the mean is more effective. But Surowiecki took this idea further and said that when a group needs to make decisions, a diverse group does better at this, too. On the other hand, a group of experts often all think the same as each other, and so do not always come up with creative solutions.

Surowiecki tried to make connections from his ideas and to issues facing the teachers at the conference. One useful idea is that a personal learning network (PLN) is more useful if it contains many different types of people. For me, this means I need to follow (on Twitter, for example) and talk to teachers from many different backgrounds. People with diverse skills will teach in different ways – and I should tap into this. Beginning teachers, for example, may not be experts but they have lots of new ideas to share.

But I was more struck by an idea that concerns teaching diverse groups of students. Surowiecki said that non-correlated mistakes don’t affect the wisdom of a group. In other words, as long as everyone is not making the same (type of) mistake, their work together will be greater than any individual contribution. I draw from this that I need to encourage a classroom atmosphere where everyone feels free to share ideas with none censored and all taken as valid. If students can share vastly differing ideas, then they are valuable even if they are wrong.

This made me think further about teaching groups of homogeneous students. A popular strategy in maths is to stream or set children by their ability. I can see that Suroweicki’s opinions can be used to argue against the wisdom of this. A more diverse group of students should be able to help each other learn better. Suroweicki didn’t go as far as saying this, but I believe this is a reason to teach in mixed ability groups.

Mixed ability teaching has been much on my mind lately – since arriving at West Island School I have had to teach more mixed ability classes than previously. Our year 7s are in mixed ability classes and our year 10s and 11s are in broadly mixed ability groups. I’ve (usually) enjoyed coming up with ways to differentiate work for these groups. But now I see that not only can students work together effectively in mixed groups, but perhaps they can even spur each other on to greater accomplishments because of their diversity. I can help this along by allowing (and requiring) discussion of ideas and methods among students of differing abilities.

Where do you stand on mixed ability teaching?

Square and Cube Numbers

After learning about indices recently, I asked my year 10s to respond to this question for homework: Among the natural numbers, are there more square numbers or cube numbers?

I want to teach them over the course of two years to express their mathematical thinking in words. I asked for a paragraph for homework. Some students wrote me a single sentence, others a few sentences. Some included examples.

The next lesson this led to a really interesting discussion about the size of an infinite set. How many items in the set of square numbers? Could you count all the cube numbers? If there is a cap on the size of the number, then there are certainly more squares than cubes. But when you let the numbers go to infinity, it’s a different story altogether.

I hope to do more activities like this with them this year. They should be able to express and justify their opinions.

Reading and Writing in Maths Class

In light of the new report-style IAs in the IB classes, we are thinking of introducing more mathematical reading and writing in our courses. I am even thinking of having younger students do some reading and writing mathematically. For example, our IGCSE students regularly do investigations in class (from a book called New York Cop) and so I am thinking about developing some writing up lessons for them. Also I would like to help students read some mathematical material. For the younger (KS3, 11-14 year olds), I have been thinking about the Murderous Maths books (by Kjartan Poskitt). They are highly entertaining and very readable, yet also quite mathematical.

I have been reading a book called About the Size of It: The Common Sense Approach to Measuring Things (by Warwick Cairns) that I think I could use with 14 or 15 year olds. And a more advanced book is Math Through the Ages (by Berlinghoff and Gouvea), that older students could read, with short “sketches” of about 5 pages from various eras and areas. Now I have still to find some student-accessible mathematical articles of the kind that report on an investigation or problem. I suppose what I want to achieve is similar to the goals of teachers who show their students sample IAs.

Do you do mathematical reading or writing exercises with your students?

Investigation: Things that Worked

I left my year 1o class feeling really positive about how it went and so I want to record a few ideas to remember later. It’s a very small group (eight) of lower ability students and at times I find them hard to motivate. I’m not sure if it was easier today because I was away on training during their last lesson. They looked genuinely pleased to see me–perhaps their cover lesson didn’t go very well? It turns out they had a non-specialist supply teacher. They worked through the sheet I left them–an investigation called Crossed Lines from New York Cop. But they found it hard as expected, and so I wanted to review it with them.

The first good part of the lesson was asking them to use the mini whiteboards (MWBs) to draw some of the patterns. They seem less afraid to make mistakes when using the MWBs and it also lets me see more quickly what they are thinking.

Another good thing was the way I felt quite upbeat (from coffee?!) and smiley. I kept encouraging them throughout the lesson to try new ideas. We recorded our ideas as a group on the board. One student seemed to me, in my caffeine-assisted state, to be more communicative than usual. And I was glad I mentioned to her (after class) how pleased I was with her contributions.

When I asked them to talk to their partner I heard more ideas being shared than usual. I tried hard to stay out of the conversations. One boy turned to me to share his idea with me, and I replied, “I am not your partner… Talk to your partner, please.”

I managed to explain the transition to algebra better than usual. We do these little investigations every fortnight; it’s the same sheet for all nine classes of year 10 students. (This means all the other students, in the higher ability classes, are also doing the same work. We try to stay realistic about doing as much as we can–which is usually about half the sheet.) Each time the investigation leads the students to find a rule for the patterns they see, whether number patterns or shape patterns. My students know this question is coming, so we are starting to anticipate it: “Write your rule in algebra.” Now that we are on the fourth one, they are more willing to look for and express the rule. They are not as scared by the n appearing! What went well today was they agreed that the algebra was easier and more elegant to write than the word formula. Success for Mrs A!

I have realised that I am the one responsible for making sure the correction notation is used. I am happy if they can describe the algebra they want to write and then I make sure it is correct. In time I think they will be responsible for the algebraic notation on their own, but while their algebra skills are still a bit weak I want them to see correct algebra from me.

What went well for you today?

Should we give homework?

A chapter from Alfie Kohn’s book The Homework Myth is available online and I devoured it yesterday and today. In it, he asserts that practice homework (especially in maths) is, at best, useless, and at worst, damaging to students. For students who understood how to do a procedure they were taught in class, he argues, practicing the same thing at home twenty or 40 times is boring and useless. For students who didn’t understand in class, Kohn lists at least four reasons why practicing at home is damaging: it reinforces their mistakes (40 times), reminds them they don’t get it, makes them want to cheat or pretend they understand when they don’t, and makes them think they are not meant to understand maths, just do it mindlessly.

I was reading some of the article today while sitting at a table of year 11s during their lesson. The three of them were doing matrix multiplication and I was sitting on the fourth chair and piping up with hints and tips whenever they looked my way. As other students around the room were asking for help, I was in and out of the seat, visiting other tables. Near the end of the lesson, the boy I was sitting next to asked, impishly, “So, Mrs A, I see you are reading about homework…. Are we getting any today?”

I paused. Great question. I find myself won over by many of Kohn’s arguments. I agree that rote practice at home doesn’t create understanding for students, and may, in fact, discourage them. Kohn opines that rote practice is damaging in the classroom, too. Take, for example, this statement, “Terrific teachers generally refrain from showing their classes how to solve problems.” I agree with this, to a certain extent. Students need to explore, discuss, and discover for themselves. But I often provide technical explanations when students are ready to formalise their mathematical ideas. And I explicitly teach things like matrix multiplication.

I asked the other year 11s sitting with us what they thought. “Do you think practising at home is worthwhile, or worthless?”

“When I have an idea how to start, then the practise is useful,” a girl offered.

“I can teach myself how to do the harder questions,” added another student.

“But it can get a bit boring if the questions are all the same.”

I think these students do gain from the homework I set them. I allow them to skip repetitious questions (both in class and at home). Furthermore, I want them to be exposed to as many types of matrix questions as possible, so I assigned work for them at the end of the lesson. But if they left the room with a faulty understanding, then the homework won’t likely do much good.

What do you think? Do you assign practice homework?