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.

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)

So, what was this wonderful method? Well, it might be nothing new, 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.

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

3. Find the vector that moves you from the CoL to each vertex.

4. Multiply each vector by the scale factor

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.

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: If you are the in the default US language setting, Geogebra uses the terminology *Dilation* rather than *Enlargement* – which actually is more descriptive of what we are doing, isn’t it?!)

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:

This is a specific pedagogy for this topic. 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…

First thing to say is that that method definitely isn’t a “procedural” method. Or rather, it needn’t be. One of the problems in teaching mathematics is that every method can be either procedural or conceptual depending both on how it is taught and how it is received.

Let me start by explaining why I say that this is conceptual. It is because it links to a topic that we only just touch on in ordinary A’level maths (there’s a bit more in FM) but which is crucial in degree level maths: vectors. Having taught vectors for many years in universities, I would have been overjoyed had any of my students met this kind of exercise when studying transformations. To see the circle of ideas between points, vectors, translations, and transformations makes so much more of mathematics connect together.

The second thing to say is that I don’t see the dichotomy that you allude to between procedural and conceptual approaches to mathematics. Understanding and capability go hand-in-hand and it varies from person to person and from topic to topic as to which is in the lead. There are some topics where it is possible to have a sufficient understanding and overview before one starts learning procedures, but there are others where actually one should become au fait with the methodology in order to gain a conceptual understanding. See that the method *works* often leads to insight as to what’s going on underneath.

It continually frustrates me as an educator to see this attitude of “I won’t try until I fully understand every step”. This leads to the “Where do I begin?” problem when attempting more intricate problems: the student hasn’t developed the habit of just starting. They understand that to solve a complicated problem they need to break it down, but don’t realise that one often doesn’t break it down at the outset but rather chips off piece after piece until it all falls apart.

A related frustration is when I’m explaining a method to a student then they don’t *write things down* as we’re going along. They expect to be able to hold the entire thing in their head and then reproduce it when I’m gone. That’s not how mathematicians work! Part of the point of developing mathematics was to avoid having to think too hard at any one stage!

Anyway, there’s my initial thoughts. To sum up, I think this is a very good method for teaching scaling (the term “enlargement” annoys me!) to any ability of student.

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