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 …

The Scottish Curriculum: A Beginner's Guide

There is no universally agreed meaning for the term "curriculum" as applied to the education of children and young people. In academic circles it is a highly contested concept, and covers everything experienced by children and young people, within or without formal education. To an average parent, I guess it means "the stuff schools are going to teach children and young people". From this perspective, Scotland has no national curriculum. Scotland has no national framework which spells out explicitly what knowledge schools should teach children and young people. We used to have one, called 5-14, but it was decided that this was too prescriptive.

Instead, the Scottish Curriculum for Excellence has a series of statements called the Experiences and Outcomes, which make vague, implicit references to the knowledge we expect young people to acquire, wrapped up in highly general statements about what we expect young people to be able to do, and what we expect them to exper…

Assessment Through The Looking Glass

The old 5-14 assessments used to give robust, consistent answers to specific questions. The questions were of the form "what percentage of P7 students have managed to score at least 15 marks in one of a specific set of nationally generated assessment instruments". We thought that the questions were of the form "what percentage of P7 students have passed level D mathematics", but of course that isn't what we were measuring. We were measuring performance in tests. The mistaking of test performance for evidence of "passing a level" was bad, but we were right in thinking that we compared like with like as we looked at performance data from across Scotland.

Now we have abandoned national tests, and have moved towards measuring "passing a level" based on a combination of a rich bundle of assessments and teacher professional judgement. But what does it actually mean to have passed level 3 in numeracy? CfE documentation talks about assessing breadth…

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…