Knowing and Understanding in Mathematics

In order to investigate the current debate about knowing and understanding sparked by David Didau's post, I want to examine one small part of mathematics, which I happen to be teaching to a Higher maths class at the moment: finding the point which divides a line segment in a given ratio.

One way to approach this is to teach a formula:

The position vector of P, where P divides AB in the ratio m:n, is given by p=(na+mb)/(m+n)

If you know
how to convert a position vector to a coordinatethe convention that capital letters represents points and bold lower case letters represent corresponding position vectorshow to multiply or divide a vector by a scalarhow to add vectors together then can probably now solve a problem such as:

Given that the point P divides S(3,4,-1) and T(5,8,11) in the ratio 3:1, find P.

At this point, a student knows how to find a point which divides a line segment in a given ratio. They may have no idea why this rule works. They may have no idea what a position vector …

My Leadership Story

I have volunteered to "share my leadership journey" for ten minutes before leading a discussion with other middle leaders at a SCEL event in Edinburgh.  This blog post is a rehearsal of those ten minutes, and I would gratefully appreciate any constructive feedback.
I have a leadership story rather than a leadership journey to share. This is the story I tell myself about how I got to where I am now as a leader, and about where I might go next. It is very subjective and selective. Nonetheless I think it is worth sharing, because this is the truth I inhabit. You also have stories you tell yourself, and you inhabit your stories every day of your professional life. It is sometimes easy to recognise these stories in others - the colleague who sees themselves as the victim of unreasonable burdens regardless of changing circumstances or another who sees themselves as blessed and lucky no matter what misfortunes befall them. It is much harder to identify the stories we tell ourselves,…

Planning for Learning or Planning for Behaviour?

Several months ago I found myself sitting beside a quite senior staff member from one of our colleges of initial teacher education, who complained to me that her student teachers all seemed to be obsessed with behaviour management.

I was flabbergasted, but failed miserably to put together a reasonable argument for why the students were quite right.

That conversation came back to me today as I taught a maths cover class, who were learning about the graphs of linear equations.  The class had some lively characters in it, and they were clearly struggling with the central concept: that a line on a coordinate diagram represents all the points where a particular linear equation involving x and y is true.

As they explored this concept through a series of activities, many of them experienced confusion and frustration. These are normal, healthy emotions for learners. We were able to stick with these challenging experiences partly because the pupils were operating in an environment in which mis…

And what if we don't achieve our dreams?

"If A then B" is logically equivalent to "If (Not B) then (Not A)". For example "if a shape has three sides, then it is a triangle" is logically equivalent to "if a shape is not a triangle, then it does not have three sides". The second statement in quotes is called the contrapositive of the first statement in quotes. Logically speaking, they are identical statements. If one is true, the other must be, and vice versa. In this case, they are both true statements.
A commonly touted inspirational message we deliver to young people can be distilled down to "if you do all the things right that are within your power, then your dreams will come true". Consider its contrapositive.

NQT Professional Enquiries

Our school's NQTs presented their professional enquiries to staff at an in-service session this afternoon, with each one at a different table in the canteen. I was humbled and thrilled by the depth of their thinking, and by the tone and level of their discussions with staff.

I picked up the following key insights from the three enquiries I heard about:

female pupils still often appear to defer to male pupils in group discussions. Randomised groups aren't good enough.pupils prefer to work with friends, but the quality of their discussions might actually better when they are with peers they know less well. With the right support they can realise this themselves.we hold enormous power as teachers, and the language we use with pupils can make a huge difference to how they feel about learning and about themselves as learners. About the ensuing discussions at the canteen tables, I was particularly delighted that: many teachers hung around in conversation with the NQTs long after the s…

Classroom Rules: Imagine it's the last lesson of the year.

This tweet

Drawing up class rules collaboratively should be compulsory in every class! #edchatie — Bríd (@BridCarlow) June 13, 2016 led me to this video

in which Tom Bennett says that teachers should not negotiate classroom rules with students, because they may produce rules which are not sensible.  I'm not sure if I agree, but it struck me that I should share what I do.

I start new classes by asking them to imagine it is the last lesson of the year. They are walking out of the door, and I have a wee tear in my eye, because they have been the best class I have ever had, and am really going to miss them. I ask them to tell me what they think they will have to do during the year in order to make that happen.

I am not pretending that they are in control, but having generated the list of behaviours (which has always been pretty good, even with some very challenging classes) I ask them if they think they can live up to this list. They have always said yes to me. I must say I only began…

An exercise to explore school improvement

1. Consider this list of "changes" that one might make as a teacher in order to improve one's effectiveness: "write a new course", " read about and implement a different approach to marking homework ", " change one's beliefs about the potential of one's students", "reflect on the language one uses with students, consider the extent to which it empowers or belittles them, and make changes", " go to a sharing-good-practice session and implement something from the session in your classroom", "invite a colleague to observe a lesson and give you feedback about your 'way of being' with the students", " flip your classroom", "reflect on your body language in the classroom and improve it"...

2. Add some more to the list yourself.

3. Write each of the "changes" on post-it notes, and give each a mark out of ten for the potential they have to improve outcomes for learners in yo…