reading/learning – In Pursuit of Great Mathematics Teaching

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?

Maths Jokes

Here are my current three favourite maths cartoons. (Click for bigger images.) What are your favourite funnies?

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?

Links Picnic #2

Links Picnic is an opportunity for us to share things from around the web that help and inspire maths teachers. Here are my picks. Please add yours in the comments section. You can leave a link to your own Links Picnic!

–A nice world clock that can be used between lessons or during registration. It includes world statistics that are continuously updating.

—WiseStamp for an interactive email signature.

–The $2 Interactive Whiteboard. I’m calling this the medi-whiteboard, since I already use mini-whiteboards.

–An article I’m reading by Alfie Kohn about homework, entitled Do Students Really Need Practice Homework?

–Some ideas for displaying student testing data in the classroom. How motivating is this, I wonder?

–I am looking for an app or program that helps me organise my to-do lists over my mac at home, PC at work, and phone (Android). Pocket Informant does look useful, but is just for my phone. Do you have a suggestion?

What links have impacted you recently?

Links Picnic #1

Are you making any of these four mistakes with your differentiation? Two of these items make me think setting students by ability might be wrong. (And those thoughts are making me nervous.)

Using Evernote as a filing system for public speaking. I wonder if this would work with teaching.

A new blog I am following is Under Ten Minutes. The videos posted there show you how to use an item of learning technology in less than 10 minutes. Today I learned how to use pivot tables for student data analysis.

The Centre for Innovative Mathematics Teaching (CIMT) pages contain a complete scheme of work, texts, and activities for 5-18 year old learners! What a treasure trove.

What links have impacted you this month?

Ways Students Learn

Do you remember how you learned maths in school? And do you remember if it was effective? As I teach I test out many different strategies, activities, and ideas. I am looking for methods that help everyone learn maths. Are some methods better than others for everyone?

There are certainly some students who learn well by listening to explanations and copying examples. I know that students can be trained to work this way and I have worked in schools where this got very good results. But is everyone able to learn this way? Can all students be trained to learn this way? Ollerton and Watson think not, in their book Inclusive Mathematics 11-18, which I am reading at the moment. (Yesterday I had read to page 7 and was already learning!)

“It is commonly believed now,” they write, “that all learning involves the learner interacting with the environment through experience and making sense of those experiences personally and through communicating with others.” So my job evolves from example-giver to activity-designer. I need to provide sense-making tasks for students to experience and communicate their ideas.

I struggle sometimes with knowing how to make these activities. I enjoy making different types of activities, but often I just pick an activity I like, somewhat randomly. But the key is to look at the underlying mathematical structure. Maths topics are often about classifying–and so sorting and classifying tasks will expose those ideas. Maths is frequently about justifying, so I need to ask questions that ask students to back up their ideas. I hope that Ollerton and Watson discuss task design in more detail. (I am still only on page 10.)

What thinking drives your task design? How do you decide what activities to give to students?

Maths as an Art

I am reading Inclusive Mathematics 11-18 by Mike Ollerton* and Anne Watson (my two biggest maths education idols?). I’m only on page 7 and my ideas are being challenged and shaped. There are a lot of fthings they say that I believe in principle but find hard to apply in the classroom.

Maths (as a school subject) should be something to do, they argue. I agree completely. Often maths is viewed as a body of knowledge that others have created that we need to learn (and memorise). In this way maths skills can be seen like a checklist to master. Some of my lessons have this theme: the aim of the lesson is to master the skill of finding one amount as a percentage of another. But Ollerton and Watson contend that thinking mathematically and doing mathematics are what maths is all about. Memorising (“learning”) techniques is somewhat useful–because of future maths study or work-related skills or everyday numeracy. But the goal of thinking and doing mathematics, I believe, is to have a more mathematical mind, to be more logical, and to be a beautiful thinker.

In this way studying maths at school could be similar to studying art or music. Students take music, art, or drama in order to appreciate these realms of life. They broaden their minds through artisitic expression. All students are painters or actors who have something to learn in school performing arts classes because they are developing their creative talents. Only a small proportion of the students go further with art or music–and the same is true of maths (though perhaps the percentage is higher?). But the arts and maths are still valuable to those students since their minds and creativity are improved.

So how can I teach maths to be more like an arts subject? I can do this by focusing on maths as a process to be explored rather than memorised. I can also give problems which allow students to investigate and apply their mathematical ideas. Ollerton and Watson say that students “sometimes learn more when plunged into a complex mathematical situation”–so I can avoid giving five simple examples then asking for skill practice. I can develop activities that make students think about why things work.

Would you say you teach maths like a list of skills to be mastered? Or do you focus on “doing” mathematics?

*Ollerton seems to have no website. Take me on as your apprentice, Mike, and I will create your website in return!

“Expose the Scaffolding”

Students aren’t mind readers. They can’t hope to understand what I’m looking for in homework, classwork, or group work unless I help them. I’ve received some funny homeworks when I haven’t made my expectations clean; mostly tiny amounts of work that a student thinks is the minimum they can get away with. For example, a sentence when I was expecting a paragraph, or one calculation when I was expecting a whole page. And who can blame them, if they don’t know what to do or why they should do it.

A colleague lent me a booklet of photocopied articles he’s reading about student motivation. “Teachers should spend more time explaining,” one of the articles expounds, “explaining why we teach what we do, and why the topic or approach or activity is important and interesting and worthwhile.” This reminds me of the advice of Paul Muir, one of my first teaching mentors: always “expose the scaffolding”. Make clear to your students why you are doing what you do. For example, he advised me to explain my marking strategies. Or to say why you are giving homework and what purpose it will serve. Explain where the course is going or how today’s work will fit in. Paul said this over and over again–it has been engraved into my memory now.

If we let students in on our planning and organising secrets, they are more likely to complete tasks well. They will know what is expected of them and more often hit the target. And they will catch some of our enthusiasm when we talk about maths.