Category Archives: pedagogy

Just follow these steps and you’ll be OK at Enlargements

Here’s a statement which I don’t think will be too controversial – I would have thought maths teachers the world over would agree with this:

I want my students to gain a deep understanding of the mathematics, not just follow a procedure to get the right answer.

This is our aim. We don’t always get there. We have different ways of getting there. I have recently re-read this seminal paper by Skemp from 1976.

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In it he talks about Relational Understanding, which I have generally thought of as understanding of concepts, and Instrumental Understanding, which I think of as understanding of procedures.  In my teaching I have been inclined to build conceptual understanding first, and then see what methods make sense from there.  However, I’m starting to think that it’s not that simple.  There are some situations where some instrumental understanding might come first and act as a foundation on which to build relational understanding.  Ultimately we want both, but the order in which we achieve this is not always the same.  We should not dismiss a didactic approach that provides a clear sequence of steps and worked examples as a part of the journey to a deeper mathematical understanding.

I recently observed a colleague teaching the topic of Enlargements to a Year 11 revision class, who are entered for the Foundation paper.  He used a method which I don’t think I had thought of before.  It was heavy on the Instrumental Understanding, but it worked and the students were doing some more tricky fractional and negative enlargements with centres of enlargement not at the origin.  About as hard as it gets for these types of exam questions (thanks for Maths Genie for these examples)

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So, what was this wonderful method?  Well, it might be nothing new to you dear reader, but it used vectors.  It relied on students being secure with describing the translation between two points using vector notation.  Given that this is how Translations are described and that we typically teach Translations (along with Reflections and Rotations) before Enlargements in a topic called Transformations, this should be a build on / consolidating of what was learned a few lessons ago.

The steps go something like this.

  1. Label the vertices of the shape you’ve been given (say A,B,C,D, something like that)
  2. Circle the centre of enlargement, CoL (helpful to distinguish it from a vertex later)

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3. Find the vector that moves you from the CoL to each vertex.
4. Multiply each vector by the scale factor

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5. Now use those vectors to plot the new points, starting from the CoL again. Connect the vertices to form the shape.

6. Finally draw in some ray lines to convince yourself that you have not made any mistakes.

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I think that’s quite a neat method that enables us to go straight to what might be seen as the most difficult example. The only real difficulty here, however is multiplying a (pretty simple) negative fraction by an integer.  Something which should be secure by the time this topic is being taught. And if it’s not secure, well here is an opportunity to practise it without detracting too much from the main objective.

Another benefit, is that it might be easier for students to plot points by counting squares rather than draw accurate, extended ray lines, as pointed out by Mr Blachford on Twitter.

Once students have done a few examples, we can draw attention to some things, for example:

  • If the scale factor is >1, it gets bigger, if it’s <1 it gets smaller
  • The scale factor applies to each side length of the shape (but not the area…)
  • Negative scale factors always place the image the “other side of the CoL”
  • All ray lines must go through the CoL. This is how it is constructed.

I would use Geogebra (as in fact I did to create these images) to examine what happens when we move the CoL. (Note: Geogebra uses the US terminology Dilation rather than Enlargement – which actually is more descriptive of what we are doing, isn’t it?!)

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And from there we can go onto questions that show the enlargement and ask for a description.  I would use Geogebra for that bit as well as it is very easy to create images that can then be used to ask learners to describe what they see.

Just for fun, I had a go at a 3D version in Geogebra.  I’l leave you to decide whether or not it adds anything. You should be able to access it here. This is what it looks like:

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This is an example.  It’s one way of doing it. For other topics, for example fractions, I would prefer to spend a decent chunk of time on building conceptual understanding before focussing on algorithms to get right answers. But that’s for another post…

 

 

 

 

“I noticed” vs. “I liked”

I am hugely privileged in my role this year as a Teaching for Mastery Lead having had the opportunity to join various Teacher Research Groups (TRGs). This, combined with the work we are doing in Year 7 (see this article), and mentoring trainees has meant I have spent nearly as much time joining other teachers’ classes as I have taught my own classes. Whilst that makes me feel slightly guilty, I hope that this will pay off next year and in years to come –  I am learning so much in the process. It would be great if all teachers had a sabbatical year, 4 or 5 years into their teaching career where they teach a 50% or less timetable and spend time observing others in their own school and other schools nearby including cross-phase (i.e. primary-secondary). A pipe-dream maybe, but it could go some way to alleviating the retention problem at that crucial stage in a teacher’s career when it should be getting more manageable but often doesn’t.  There were some encouraging signs of this in the recent Education Select Committee report on Recruitment and Retention of Teachers.

Sitting and watching (actually I rarely manage to sit still for long, the urge to get up and engage with students is too strong!) someone else’s lesson only gives half the picture, however. Going hand-in-hand with the lesson is the shared reflection on that lesson afterwards between teachers. And this is the point.  It is not a lesson “observation” in the traditional, pre-2014 Ofsted sense.  I am not there to evaluate the teacher in any way. I am a fellow professional who has another perspective on the learning happening in that room.  Because I am not leading the lesson, I should be able to notice things, and I may notice different things than the teacher who is leading the lesson.

A lesson observation is traditionally is followed by “feedback” which is more often than not a one-way conversation between the observer and the observed.  Usually it is a very polite affair which starts with a lot of “I liked…”, “I thought … was lovely” – the WWW.  All nice to hear, but do you ever get that feeling that these are platitudes and really you are waiting for the EBI? The “I thought maybe you could…”, or “In the past, I’ve tried…”  I’m not saying that this style of feedback is not useful, especially when the observer has many more years experience that the observed.  But I would say that anyone with more than a few months’ experience in the classroom has something to offer and that the conversation should start off very differently.

Earlier this year I was invited by Danny Brown to join a lesson of his. It was last thing on a Friday and after the lesson I also joined his staff meeting.  The lesson was fascinating, but it was the staff meeting that has really stuck in my mind since.  It wasn’t a department meeting as such, but a voluntary gathering to reflect on a lesson that had been given by one of the department and observed by others.  The focus was on something that was “noticed”.  It was not an attempt to analyse everything that happened in the lesson, but a focussed discussion on something that was interesting for some reason and that we can all learn from.  There was a high degree of respect and trust between these teachers and the discussion became deep, insightful and .

I have been practising this ever since in discussions following lessons, be they informal “feedback” with colleagues (I dislike this term because it implies a one-way flow of traffic) or more formalised TRGs as part of my Teaching for Mastery work.  It takes some practise.  Commenting on something without evaluating it can be tricky.  You sometimes feel like you aren’t really making a point.  But actually just clarifying what happened at a particular point can then open into useful conjectures as to why that happened.  This is where different insights from different people in the room can become really powerful and is the essence of a fruitful TRG discussion.

We need to see a major culture shift in our schools. For too many years, lesson observations have been about scrutiny and accountability and not about close collaboration of a team of professionals seeking to improve their practice.  This has led to a culture of fear in schools where many teachers still would rather not have someone “observe” them because it causes anxiety as they feel they are being judged.  I would warmly welcome anyone into my classroom at any time and would always want to know what they noticed, but I recognise that is not a common attitude amongst teachers.  We need to practice how we share these noticings with each other so that they are truly supportive, non-judgemental and lead to fruitful discussions.  And we need to be open and receptive to these discussions and realise that they are about mathematics and learning.

The approach can be very time effective.  We don’t need to sit through an entire lesson to notice something interesting, 10 minutes might be enough.  One noticing might spark a couple of useful insights on a short post-lesson conversation.  I might call this “noticings-lite”.  It’s not a huge investment of time, the bigger challenge is the shift in culture.

If we are serious, though, we do need to organise this and having more than one adult seeing the same lesson can generate the range of perspectives. This is what the Teaching for Mastery programme is achieving this year and for once it is coming with funding to enable teachers to be out of class.  In my experience watching each other and carefully analysing lessons is simply the most powerful form of CPD there is, far more beneficial than most whole-school INSET.  I hope it continues to grow in a funded sustainable manner to increase the skill levels of all teachers.

Fraction images

Imagery is so important to help with conceptual understanding of fractions and I have seen some really powerful uses of imagery in lessons recently.  So I thought I would create some kind of repository of fractions images that can easily be used when designing lessons involving fractions. The Windows snipping tool, Smart Notebook Screen Capture toolbar or Shift-command-4 on a Mac will all come in handy to quickly pull these images into your lesson.

A really quick way to create fraction images like these is on Excel (or Google Sheets).  It’s much easier and more accurate than trying to create boxes in Powerpoint or on Smart Notebook. There are a random collection of these in this spreadsheet, all very easy to adjust by changing shading and/or borders of the cells as required.

You might be looking for something a bit more pictorial:

There is a large and also rather random collection of these in this PowerPoint. Many thanks to Declan Byrne from the London SE Maths Hub for agreeing to share these which have been compiled from his lessons.

Finally, there are some handy websites that enable you to create images which again can be quickly cropped into your lesson.

  1. National Strategies Virtual Manipulatives – there are a bunch of these that were created as part of the National Strategies and now hosted on eMaths.  You can quickly and easily create bar fractions like these and add or remove the fraction, decimal, percentage and ratio alongside.screen-shot-2017-02-12-at-23-11-02
  2. NRICH Cuisenaire Environment – A simple to use tool for displaying Cuisenaire rods on screen with a grid.

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3. Clip Art Kid – some images on this site which might be useful

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4. fractiontalks.com

A range of more complex fraction shapes including Tangrams:

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If you have any more suggestions for places to find or ways to create fraction images, please let me know by leaving a comment below and I will update this blog post.

Angle as a measure of turn

Angles in parallel lines is a topic that doesn’t usually feel too tricky to teach, but I often feel that I am just telling them “this is how it is” without giving a good explanation.

Inspired by a Twitter discussion on this Brilliant task, I’ve had a rethink:

I use Geogebra a lot. It’s a powerful tool which somehow seems more powerful when you start with a blank page rather than something that has already been created. And angles in parallel lines is quick and easy enough to create on Geogebra:

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Whilst doing this, I would want to move things around a bit to show that the two lines can move but stay parallel whereas the third line can move by moving the points.

This is the point where I previously might have just started measuring angles and showing what stays the same and changes as I moved the lines.

But it has struck me that this is a good opportunity to reinforce the idea of degrees as a measure of turn.  I often use a simple “Guess the angle” game like this.

I ask students to estimate the angle and show me on mini-whiteboards. Something as simple as this can cause great excitement when someone gets it exactly correct!  But it also reinforces the idea of degrees as a measure of turn from one line to another.

So, back to parallel lines, I am trying to show that the reason that alternate angles and corresponding angles are equal is because after turning one way an then back the other I end up pointing in the same direction.  And the reason co-interior angles sum to 180° is because I end up pointing in the opposite direction.

So, here is one that I did make earlier.  I have set it up so you can see the actual lines turning with the degrees increasing as they do so. I need to convince my learners that you can move from the first arrow to the second arrow by moving down the transverse line without changing direction.  Because of that, I will need to turn through the same angle to land back onto the parallel line:

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And here is the same idea for Corresponding Angles:

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I’d be really interested to hear any thoughts on this as a pedagogy – please leave comments below.

The Power of Boxes and Circles

Here is a presentation that I gave at today’s #teachmeet held in Oakwood School, Surrey organised by Paul Collins, @mrprcollins.

I have been using this technique a lot recently so I collated some example questions from Year 1 to Year 12 (!) really just to show how versatile this technique can be. Here is a sample:

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The full set are here as a pdf . For any Lyx.org fans out there, the original Lyx file.

 

 

 

 

 

Formal methods – the devil’s work!

I’ve been thinking a lot recently about “formal methods”, e.g. Column Addition or Long Multiplication.

As secondary maths teachers we don’t pay much attention to these.  I assess their ability to carry out the procedure and have sometimes attempted to teach long multiplication explicitly. But more often than not it is assumed that these methods have been learnt at primary school have been pretty well-practised and are therefore secure.

We have been doing a lot of number work with Year 7 so far this term including exploring in some detail the Laws of Arithmetic.

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Here is an example of the type of question we have been looking at:

127 + 54 + 73

Most of my students’ first instinct with a question like this is to draw up a nice column addition and solve it.  Would you agree that most students do this?

I want them to look at the structure of the numbers first.  To realise that addition is commutative and that it’s much easier if we add the 127 to the 73 first because that’s a number bond to 200.

I am having 2 issues with this approach:

  1. My students are not convinced that this is a valid exercise.  Basically they think it is contrived (which, of course all exercises are). They think I have chosen numbers to make this work, it will only work for those certain numbers and basically they feel like I have tricked them.
  2. Some students perceive the message that the formal methods they spent hours practising at primary school are now not the way we do things at secondary school. I have said that this is not the case, they are not the devil’s work, and that those methods still have a place. But we need to have smarter ways to work. The fact that we have spent many lessons not solving calculations using formal methods leads me to believe that they are confused about what I want them to do.

Fundamentally I want my students to rely less on the authority of the teacher and rely more on their own understanding.  I want them to see the structure in everything they do, to be certain that what they have done is valid because the mathematics tells them it is so, not because I tell them that they are right.

How should we bridge the gap between informal methods which require (and develop) depth of understanding and formal methods which are efficient and accurate?

I have an idea using manipulatives.  I am not sure if it will work, but if I get chance to try it with the class, I will write about it again later.

In the meantime, does this resonate with anyone else? How do you bridge the gap? I’d love to read your comments.