In Pursuit of Great Mathematics Teaching

Teaching to the Test

While reading today, I discovered an unexpected side effect of the practise of teaching to the test. (At least it was unexpected to me; perhaps you will not find it so surprising?)

First, the “usual” problems with extended coaching[1] for exams:

the exam is no longer a good indicator of what students understand since they have only been narrowly trained to do specific question types
the exam is no longer a useful way of selecting students for further courses since we don’t know if they really understood the mathematics they have “learned”
the possible ranking of schools (for example, league tables in the UK) by student grades is not accurate since some schools have used extensive coaching while others have not
employers can not be sure that students’ ability matches their test score

All these arguments are ones I have heard before about coaching students to pass an exam. But one undesirable side effect of exam coaching stood out to me:

younger children see older ones being coached to success and they learn that at just they right time, they, too, will be spoon-fed what they need to know

As a result students learn that they do not need to take responsibility for their own learning, they do not need to study seriously or make an effort to understand and connect what they are learning.

Wow! I was shocked by this. One of my main goals is to make sure students understand what they are learning and can connect their mathematical ideas. I don’t want younger ones to learn that they don’t really need to work hard until the exam coaching begins.

My school only partly subscribes to the idea of coaching for exams. But I fully subscribe to the idea that students need to take responsibility for their own learning.

What exam preparation strategies do you use?

[1] I am talking about the repetitious, algorithmic coaching that happens in the lead up to standardised tests. Students, especially those who are near a grade level borderline, are taken (forcibly, at times) though many, many revision resources.

Ken Robinson: Escaping Education’s Death Valley

I was watching this TED talk today in which Ken Robinson talks about the characteristics of successful education systems. He mentions three main things that are needed:

1. An understanding that education needs to be made up of individualised teaching and learning experiences, since all children are different

2. An investment in the best people as teachers and in their support and professional development

3. Authority for making educational decisions is devolved to the school level

Sir Robinson puts a lot of stress on the human aspect of education and rejects the idea that schools are an industrial machine for transferring knowledge. Instead, children’s curiosity needs to be nurtured. Teaching is an inherently creative job.

Leadership in education, therefore, at regional and school levels, needs to be focused on creating a climate of possibility.

In my current role (coordinator of IB DP maths), this means that I need to foster teachers’ creativity (and my own). As part of the maths leadership team I can encourage us all to teach more inquiry-based lessons, which is one of our goals this year. As I try new ideas myself I can share my experiences. And I have lots to learn from what the others around me are doing.

A Peek Inside My Resource Cupboard: How I Organise My Card Activities

My students use a lot of manipulatives, sorting cards, and activities. I store them all in the tall cupboard at the back of my classroom.

Each of the boxes contains a class set of cards. I like to have seventeen sets so that there are enough for the students to work in pairs plus a few extras (because I always lose a few random cards). Each set of cards is packed into a little snack-sized zipper bag. (I import these from Canada in bulk when I visit.)

The seventeen sets of cards are then packed into a plastic box. They are just Chinese food takeaway boxes. (My husband and I have eaten a lot of takeaway food over the years!)

I’ve been using sticky notes on the shelves to divide the boxes into number, algebra, calculus, shape/measurement, and data/probability activities. This helps my colleagues, because they sometimes pop their heads into the cupboard to borrow something.

I’m always interested in organising strategies. How are your hands-on activities organised?

Standard Form (Scientific Notation) Sorting Cards

Each card in this set of sorting cards shows a conversion between a very big (or very small) number and its equivalent in standard form. However, quite a few of the conversions are incorrect. The students need to sort the cards into which are correct and incorrect. Then they need to correct the standard form conversions.

They could also order the numbers from smallest to biggest or find further uses for standard form to write very big or very small numbers. Students could also make up several questions and make deliberate errors for their classmates to find.

Download the cards or the instructions and answers slides.

Procrastination

Late in the afternoons when the maths office is quiet, I like to set aside the emails, form-filling-in, reports, lesson planning, and admin jobs, and just do some reading for the pleasure of learning something. Sometimes I read about maths teaching theories. I like to read scholarly papers. Or I read an article from Mathematics Teaching (the ATM journal) or Mathematics in School (the MA journal). I also usually find lots of ideas for things to read from my Twitter feed.

Is my afternoon reading habit a form of procrastination? I choose to believe that it is not. It is more of a break for my brain and a chance to recharge. I always feel really refreshed by learning something new. I often pick up a new idea about how to teach a tricky topic. Quite often I read about productivity and gain a bit of motivation to work more efficiently and effectively. Recently I have been reading about leadership in connection with my Links course.

If I’m honest, sometimes I do procrastinate by reading. It’s a chance to avoid things I don’t like doing for a while. At least I get something good out of it at the same time.

Today I stumbled upon this article about 15 things that good leaders do automatically every day. One of them was that they don’t procrastinate! I was really struck by the idea that good leaders are proactive and keep their progress moving by not avoiding jobs, even unpleasant ones. Good leaders know that “getting ahead in life is about doing the things that most people don’t like doing.” I feel a renewed sense of purpose after reading that. I want to be the kind of person who approaches life with determination, from the most important jobs (planning for great lessons) to the most mundane (administrivia). I’m glad I took my late afternoon reading break to gain that sense of momentum again.

Do you procrastinate? Do you have any procrastination tips?

Fractions, Decimals, and Percentages Number Line

Sometimes my students struggle with all the interconnected ideas about fractions, decimals, and percentages. I really want them to understand that these interchangeable representations of the same number. And to know that fractions can be placed on a number line, just as they are used to doing with other numbers.

2013-03-15 10.21.17

I asked my students to cut out this set of cards. Then they had to add to each card so that it had three representations of the same value. Finally, they had to draw a number line and place the cards on the line.

Both my students and I learned a lot from this activity. For my part, I learned that a few of these these particular students found drawing the number line troublesome; their first attempts weren’t evenly spaced or long enough. I learned that my class are able to translate among fractions, decimals, and percentages, though for some this is still a stuttering process. My students learned that they are able to move between the three representations for any number, even unfamiliar fractions or decimals.

The cards for this activity and a slide of instructions
are available here.

Laws of Logarithms

My year 11 students are nearing exam time and the last item on our course is an introduction to logarithms. They had just learned the laws of logs and so we finished the lesson with this activity. I put some pink papers around the classroom, each with a requirement.

Each group of students had to use their sticky notes to add one or more expressions to each pink poster. I was quite impressed by their responses and it was clear they had understood our lesson objectives.

Classroom Decoration and Design

From this NY Times article, I learned that humans are more productive when they work in well designed, well decorated spaces. A study on people in hospitals, offices, and schools has shown that a window view with a landscape outside helps them be more productive and healthier. Also, humans like to look at fractals (and fractals of a certain density), and this helps them perform better, too.

The implications for teachers are clear.

1. Try to get a classroom with a view!

2. Decorate your classroom with shades of green and landscapes.

3. Put fractals up – not too sparse and not too dense.

Intersections of Curves Treasure Hunt

My year 13 class were ready to practice finding points of intersections of two curves using their calculators. Instead of a boring textbook exercise, I made this treasure hunt for them.

There are ten orange cards stuck to the walls, windows, chairs, and surfaces of my classroom. Each one has a question. Students travel around the room in pairs, solving the questions. Then they look for the answer on another card.

The students know they have finished when they have visited all ten cards and solved ten problems. They need to tell me the card numbers of the loop they followed, which allows me to check their work.

Treasure hunts are easy to make and are a clever way of turning a worksheet or textbook exercise into a more social exercise that students enjoy. The cards are available here as a document and as a pdf.

Coaching and Leadership

My head of department was on a three day performance coaching course. He came back really pleased with what he had learned. (I think he had worried it might be “life coaching”.) He said most people go to their managers with problems or ideas and just want a sounding board, not advice. There were three of us in the office and the other two role played how the coaching could work.

Teacher: “I’m having a problem. So-and-so is second guessing my decisions. What should I do?”

Head of department: “Aha, that sounds frustrating. What is your current thinking?”

T: “Well, what I want to do is… [removed to protect the innocent]… but I know that’s inappropriate.”

HoD: “Yes, so do you have other ideas?”

T: “I was thinking that I should go talk to him and discuss…. [etc]”

HoD: “And what do you think would happen next?”

And then the follow-up questions:

“What are the advantages of that?” “What are some other possibilities?” And so on.

I know that I frequently go to my head of department to get his opinion on my ideas and all I want is for him to agree that I am making a good decision. (Sometimes I really do want his advice, though! So I hope he doesn’t coach me too much.)

Do you like to be coached (or to coach)?