How to be a great mathematics teacher - what the research says

Being a teacher is hard work. Being an effective teacher is even harder. It is surprisingly difficult to find clear advice on how to improve classroom performance - or rather, it is surprisingly difficult to find advice that is pedagogically sound or not advocating some form of educational bandwagon. To the rescue comes a series of pamphlets produced by the International Academy of Education - an organisation with the aim of producing “a syntheses of research on educational topics of international importance”.  Despite the somewhat weighty title of “Effective pedagogy in mathematics” they have produced a highly readable, highly relevant booklet containing some principles of effective mathematics instruction.

According to the authors of the booklet,  Glenda Anthony and Margaret Walshaw, both associate professors at Massey University and also directors of the Centre of Excellence for Research in Mathematics Education, the traits of effective mathematics pedagogy can distilled to;

1. An ethic of care
Caring classroom communities that are focused on mathematical goals help develop students’ mathematical identities and proficiencies. “Teachers who truly care about their students work hard at developing trusting classroom communities.”
2. Arranging for learning
Effective teachers provide students with opportunities to work both independently and collaboratively to make sense of ideas.
3. Building on students' thinking
Effective teachers plan mathematics learning experiences that enable students to build on their existing proficiencies, interests and experiences.
4. Worthwhile mathematical tasks
Effective teachers understand that the tasks and examples they select influence how students come to view, develop, use and make sense of mathematics.
5. Making connections
Effective teachers support students in creating connections between different ways of solving problems, between mathematical representations and topics, and between mathematics and everyday experiences.
6. Assessment for learning
Effective teachers use a range of assessment practices to make students’ thinking visible and to support students’ learning.
7. Mathematical communication
Effective teachers are able to facilitate classroom dialogue that is focused on mathematical argumentation.
8. Mathematical language
Effective teachers shape mathematical language by modelling appropriate terms and communicating their meaning in ways that students understand.
9. Tools and representations
Effective teachers carefully select tools and representations to provide support for students’ thinking.
10. Teacher knowledge
Effective teachers develop and use sound knowledge as a basis for initiating learning and responding to the mathematical needs of all their students.

The booklet is well worth reading and expands upon the extracts presented above.

There is little contained in the publication that will shock educators with an interest in mathematics teaching who have ventured beyond the use of standardised worksheets or textbooks.  However, there are some really reassuring aspects to this booklet. What pleases me most is that an ethic of care is mentioned first - caring for both the student as a learner of mathematics but also as a person.  This reflects the adage I first heard decades ago when I was training; “Students  don’t care how much you know until they know how much you care.”  Mathematics tends to have a dry and dusty “skills based” reputation so it is reassuring to see such a significant body placing emphasis on the teacher-student relationship as being of fundamental importance to effective teaching.

When we care about our students as much as the subject good things tend to result.

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Those with an interest in improving mathematical pedagogy might like to read a related post dealing with the work of Alistair McIntosh - Improving numeracy with the 7Cs.

Those with a general interest in mathematics might enjoy the maths page on this site which collects a range of posts dealing with mathematics.

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What you see is what you get - literally

This is a very well known optical illusion - it seems that everyone has seen it.  Everyone knows that it is either two faces or a vase.  Wrong.  It is both. It is two faces and it is a vase - but you can’t see them at the same time.  It is a classic demonstration of attention equals perception - we see what we are looking for.

Over time, we tend to wear lens’ that frame our vision, that shape our perceptions. We  learn from experience and form our world view according to those experiences.  However, once our world view is established the reverse seems to happen - our expectations and beliefs  actually shape our perceptions;  in other words, we see what we expect to see. Literally.  No less a figure then Einstein wrote about this - and he believed that we not only tend to see what we expect to see but that we ignore what doesn’t fit our expectations.   This holds true, not only for psychological perceptions - but also for our physical bodies. What we think determines what we feel.

For educators this is significant. We’ve all heard studies of self-fulfilling prophecies where teacher expectations predict student achievements.  (Strangely enough we tend to recognise this as a theoretical consideration but rarely seem to acknowledge it in our own practice.)  In short - we tend to see a child as a slow-learner, smart, a behaviour problem...and ‘lo and behold the child performs to our expectations.  The child who is perceived to be a behaviour problem tends to become a behaviour problem,  or at least is perceived to be one.  Perception does indeed become reality.

This means that children may become locked into our version of  reality … which in turn becomes their own. Our view of students may become their version of themselves.

Perhaps an answer is in training ourselves to look for the things that surprise us, for the things that don’t follow a pattern or meet our expectations.  We need to train ourselves to see what is really happening rather than think in mental cliches.  In practice this is not as easy as it seems.  The notion of  observing students, really observing, is important.  Perhaps the increasingly popular notion of the “focus child” offers some help.  During this time, as well as learning the strengths and areas for further development, perhaps teachers should try to discover something that surprises them about the student, to find out something that they did not know about the child, to take the chance to remind themselves that this student is also a person.

When we approach our students with a deficit model we limit our perceptions to what they can’t do.  Shifting our focus to what they can do, perhaps adopting a strength based approach, and helping them to build on that might just provide  the shift in emphasis that is needed to re-engage those students who can’t see any relevance or purpose in schooling.

It’s an idea worth looking at.

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Related posts dealing with “The more I practice the luckier I get - Mindset and Carol Dweck”  and another dealing with the importance of attitude - “The second most important word in education” may also be of interest.

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The second most important word in education?

It has been said that the most important word in the language is a person’s name.  And it is probably true - we all respond well to the use of our name, we feel that the speaker knows us, which makes us feel valued.

But, at  professional development session recently with Dylan Wiliam, author of “Embedded formative assessment”,  he mentioned a word he ranks as the most important word in education.  “Yet.”
It is a simple word that carries a powerful message.
Consider the child who says “I can’t do this.”  The educator’s response is “Yet."


The message is clear - this simple word sends powerful messages;

• You may not be able to do this task now, but with effort and practice you will be able to.  

• You have the capacity to do this.
• You can improve.
• You can get better.
• You can and will learn.
• Making an excuse is not an escape -you can and will learn this thing.

What a simple way to deliver a powerful message - possibly the most effective way of sending this message I’ve come across - yet.

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Readers might like to read a longer and more detailed post on a similar theme - “The more I practice the luckier I get - mindset”  here.

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If schools are gardens of children - I want to plant seeds

Schools have been likened to gardens. It’s well known that the German educator Friedrich Frobel coined the term “kindergarten” - which means “children’s garden”.  If that’s the case, then teachers can be likened to gardeners.  The metaphor has much to recommend it - but recently I’ve been asking myself do I do anything but weed the garden? Do I actually plant any seeds?

For classroom teachers the answer to this is easy - of course teachers plant seeds. They do it every day, in every lesson.  But the answer is not so obvious for educational administrators. Sure, we work hard to ensure that the garden has everything it needs, and we support our “gardeners” to the best of our ability.  But it is the “gardeners” who actually plant the seeds.

So today I left the paperwork for 15 minutes  and sat on the floor while a grade one child read me a story.   I did a lot of other work today - including some significant weeding and supporting of the “gardeners”. I dealt with a lot of students - but mostly “weeding” or perhaps “pruning” might be more apt. I did a lot of good things.  But as I walk towards the car park thinking of the highlight of my day it was the 15 minutes I spent on the floor hearing a child read.

I had planted a seed.