# What were they thinking??

I do an increasing amount of work in primary schools and with primary school trainees developing maths subject knowledge and pedagogy. Over the last few years I’ve managed to blow my own mind many times just imagining the beginnings of concepts being developed in children. At secondary level (11+), most concepts at their basic root are pretty much embedded and understood or at least taken as ‘the way it is’. For example, a square is called a square, and the number after ten is called eleven. When I wrote the book ‘Yes, But Why?’, I really enjoyed putting myself in the position of a student being told all these strange and mystical maths facts and I tried my best to explain them to make them make sense. At a younger age, concepts must be even more mind boggling at first. An example I’ve used before is counting.

We count in base-10 (ie we use 10 symbols (0-9) and then start combining them. Imagine learning this for the first time, and getting to the number 9, then being told we start to use them all again. It’s weird right? And why do we use ten symbols? Why not five, or sixty? Does everyone count using the same system? (spoiler: no). These are thoughts we as adults probably don’t have, but they fascinate me. Who decided to call a square a square and not a quadrangle or a a tetragon? What were they thinking? Why are we using 360 degrees, and why are they called degrees? Where do these weird mathematical words come from? We’re used to using terms like polygon and acute angle, but these words are weird! “I’m going to find the mean of this data set”. The what?? Who decided to call it that? I “can’t subtract 3 from 1 so I need to borrow…” – you’re doing what now?? Can I borrow from elsewhere? Can I just move everything around? Also who decided that we should stack numbers like that to add or subtract them? What were they thinking? How did that idea become so popular?

Trying to think ‘like they do’ is a good first step into empathising with how abstract even the little details can be in mathematics, and how easy it is to misunderstand or misinterpret things. Furthermore, it’s a great exercise in testing your own subject knowledge, and figuring out what might be beneficial to incorporate into your own teaching and explanations to prioritise what must be prioritised – making maths make sense.

# Scaffolding

I saw an interesting lesson about ratios with a low achieving group last week which got me thinking about scaffold. The students had successfully been able to answer a number of questions in the form “A and B share £ in a ratio of x:y, how much do they each have?” and the teacher wanted to move on to a slightly more difficult variation in the form “A and B share £ in a ratio of x:y, if A has £, how much does B have?”.

For clarity, we’ll look at one specific example:

“Ben and Andrew share in a ratio of 4:3. If Ben has £420 how much does Andrew have?”

I’ve carefully chosen £420 here as it’s a nice round number and it fits into a number of potential misconceptions nicely too (hint, it’s a multiple of 4, 3, and 7). But this is less about that and more about ways to scaffold… so…moving on…

The students initially struggled with adapting to the new type of question, and i scribbled some notes about how I should write about how you could scaffold it. So here we are.

This is an exhaustive list, and I wouldn’t recommend doing *all* of these ideas in a single lesson necessarily. This is intended more as an insight into ways that may be necessary to help scaffold tasks to make them accessible.

The aim here is two-fold: make the maths understandable/accessible (usually by stripping it down), but also get to a point where the question as written is approachable – ie don’t strip it down without building it back up otherwise you’re missing the point.

Finally, there’s no specific ordering of ideas here necessarily.

Back to the question:

Ben and Andrew share in a ratio of 4:3. If Ben has £420 how much does Andrew have?

The first thing to note is that there’s a lot of text there, and two separate pieces of information that are particularly useful. A simple tactic here might be to simply break up the text:

• Ben and Andrew share in a ratio of 4:3.
• If Ben has £420 how much does Andrew have?

It may seem trivial, but you’re highlighting the two components to the question, and reducing/eliminating the need for the student to be able to discern the two things on their own (remember that your eventual aim is to get them to do this themselves).

You could further the support by adding a suitable diagram for them to complete:

• Ben and Andrew share in a ratio of 4:3.
• Draw a suitable diagram to represent this ratio
• If Ben has £420 how much does Andrew have?

Note the subtle difference below, the diagram has even more support in that the names are filled in on the diagram:

And if you’re concerned that they might not be able to draw the diagram correctly independently, then you could potentially ask a question like this instead:

Note that the scenario I gave was that the students could already confidently answer some ratio questions, so it’s likely that they can already draw suitable diagrams based on the previous types of question they were answering.

Below is a slightly different approach where the assumption is that they can draw the diagram, but that they might confuse the two names / ratios

Below is an example of less scaffold, in that the question has not been split up, but there is still guidance which could be considered more general to this type of problem:

• Ben and Andrew share in a ratio of 4:3. If Ben has £420 how much does Andrew have?
• Task 1: Underline the ratio
• Task 2: Draw a diagram to represent the ratio

Note that the two steps don’t solve the problem, they’re just a starting point. By underlining the ratio you’re getting students to draw attention to the important information instead of getting lost in all the words or rejecting the question because it’s “wordy” (recall we’re talking about low achieving students, not top set!).

Two more variations are below:

Again, the difference is subtle, and you know your students. Sometimes a slight change can be all that’s needed.

Now consider the question below:

• Ben and Andrew share £420 in a ratio of 4:3. How much does Andrew have?
• Ben and Andrew share in a ratio of 4:3. If Ben has £420 how much does Andrew have?
• Task 1: How are these questions different?
• Task 2: How does it change your approach?

Here the shift is more to help students recognise how the new style of question is different. In Task 1 there isn’t really any maths to do, it’s more of a literature exercise – but it’s important. Without these discussions it’s likely the students will jump into question 2 using a question 1 approach.

Below we have assumed that the class are competent with determining the right diagram and drawing it accurately, so we move onto the second (new bit) part of the question:

I like this multiple choice element, which helps identify misconceptions about what the diagram is telling us, and where it’s telling it. Note that I’ve highlighted the part of the question we’re looking at with a different colour, to avoid anyone thinking we’re still talking about the first bit of the question. Note also that we’re giving away a bit here, in that we’ve inferred that one of A,B or C represents £420.

You can take that bit of help out quite easily:

Another approach to help students identify “what bit represents what” is to correctly or incorrectly complete it like this:

Again, note that we’ve given even more support in that we’ve specifically stated it’s wrong. Hence below is slightly less scaffolded:

Indeed, you could go the whole 9 yards and do something like this:

Note again that the explicit mention of how those numbers are derived is support you can add or take away.

Moving onto the final step in the question, students are likely to assume they have finished simply because they have reached a kind of end to part of it:

More structure:

Now remember that the ultimate aim is to take all the scaffold away eventually. When that’s done is of course at the discretion of the teacher. It may be that scaffold like the examples above only occurs in the teaching part, or it may be that independent questions have some of these kind of structures in place, but eventually you’ll want to move back to how the question was originally presented, something like this:

And then, remove the guidance altogether.

So there you have it. A brief guide to scaffolding a task in various ways. You can be as explicit / implicit as you like, but remember it’s about what makes it accessible to them, not how easy you can make it – and always aim to get to the point where the question as presented originally is approachable.

# A Maths Event in London #LateMaths

I’m very pleased and excited to announce that on Saturday October 27th I will be taking part in a maths inspired evening open to all in London. I’ll be launching my latest book, ‘More Geometry Snacks’ published by Tarquin Publications alongside my amazingly talented co-author Vincent Pantaloni. It’ll be the first time I’ve actually met Vincent after working on two books together from afar! If Geometry Snacks missed your radar, you can read about it in the Guardian here or Aperiodical here, or most recently, this lovely article posted a few days ago by Sunil Singh.

Anyway, the followup is coming out in time for Christmas, and we’ll be including a free signed copy to people attending the event.

The Venue:

The event will take place at a book-themed bar called The Fable in Farringdon, London. Easily accessibly by public transport:

52 Holborn Viaduct, London, EC1A 2FD

The evening will start at 6pm and include a variety of talks and activities, as well as food and drink! Both Vincent and I will be talking about some of our favourite things about mathematics, and we’ll be joined by the wonderful Ben Sparks of Numberphile and Maths Inspiration fame!

Activities will also be provided by the fabulous Think Maths team, Matt Parker, Katie Steckles and Zoe Griffiths

Excited? You should be! It’s an amazing line up and I can’t wait!!

To book your places, click here. Tickets are £29.50 + booking fee and include a free signed book and food as well as all the provided entertainment.

The event will be hosted by Jo Morgan (@mathsjem) who has an excellent dedicated webpage for the event here. I hope to see you there for an amazing night of maths-based entertainment and fun!

#LateMaths

# Year 7 Maths – A Missed Opportunity?

One thing I’ve noticed more and more recently is how out of sync the transition is for mathematics students leaving primary (grade?) school and starting secondary (high?) school. I’ve worked in, and visited, a large number of schools at this point, and although certainly not always the case, I’d comfortably say the majority of schools I visit follow very similar, and in my opinion, flawed, year seven programmes for their eleven year olds.

Typically (I should be cautious using that word), students arrive on day 1 and are placed in mixed groups for the first few weeks. This period can last anywhere between two and six (!) weeks. This period is effectively a holding cell until departments determine the ability of each child according to their own internal tests and assessments, despite the fact that students are coming in with a lot of maths data attached to them from primary. The reasoning behind this is often cited as standardising across groups who have come from different schools, or protecting against students performing well or poorly in their final assessments in primary school, but not performing consistently at that level. Cynically one might say that this is all really about not trusting the data they come in on…

Once the students are finally re-assessed in Year 7 – which could be anything from a single test to three or four, students are then placed into sets – which could be anything as strict as a class-by-class hierarchy, or (my personal preference for what it’s worth) loose setting with three ‘bands’ that are essentially mixed within a couple of grade boundaries.  Now I should reiterate that this is just what I’ve experienced, and certainly does not speak for the entire English school system, but as I said, I’ve seen a lot, and they usually fall into some kind of version of this structure. I’m really interested to hear about alternative approaches in the comments.

So what do students *do* in this interim period? Well, again, it differs, but often it’s looking at the basics of numeracy – number operations, proportion work, fractions etc. This in part is to help prepare them for the assessments, but also acts as a testing ground to see if students stand out as being higher or lower achieving than their data suggests. It can be completely detached from any kind of scheme of work or curriculum. I hear a lot about students ‘going backwards’ or ‘standing still’ in this time, simply because they (most of them) aren’t learning anything new, and much of what they’re being taught, or are revising, is stuff they did to death, successfully, at primary school.

At this point I should add a little balance. I’m not bashing teachers, and I understand that there is some truth when I’m told “they *should* know that stuff, but often they don’t, or they’ve forgotten it”, so perhaps this time is not wasted as such. In fact, in my own teaching experience I have frequently found that to be true with a lot of children, (but also false with others!). Certainly, if this entire process takes place over a week or two, then the implications of ‘not moving forwards’ are fairly light – the end justifies the means and all that. However I also feel that with all the massive overhauls of the primary curriculum, and the remarkable work primary teachers are doing (under impossible pressures I might add) with the new maths programs everywhere, that times have simply changed. Is a review needed? I think so.

What I’m seeing and hearing now is that Year 7 students coming into Secondary schools are, in general, much stronger at maths. They have a better conceptual understanding and ability with number. Of course not *all* of them, and of course there will be some areas where this difference is less obvious than others, but I do think the changes are having the desired effect. We could argue about the costs of those changes to everything else, but that’s not what I’m focusing on here.

With those changes in mind, the second thing I’ve noticed is that the entire Year 7 scheme of work is often blind to the new primary curriculum, which has been in place a number of years now. I don’t want to keep using anecdotal evidence here, so let’s take a look at an actual scheme side by side with the primary programme of study.

The picture above is taken from a scheme of work for Year 7 which is freely available on TES. A few disclaimers: anyone can put anything on TES. This was uploaded in 2010, but was updated in 2014. It has reviews from as recent as a month ago, so it is being used. It has been downloaded 20,000 times. There are more detailed breakdowns of the topics, differentiated etc but this is the broad overview.

From September to January there is literally nothing on there that isn’t taught at primary, several times over six years. Furthermore, almost all of it was taught at primary (several times across the six years) BEFORE the curriculum changes.

One could argue that we need to reteach things all the time (I agree), but typically we’d reteach it and add more content, go deeper, expand the concept. The above is taken from the Year 5 programme of study. It involves composite shapes, and is therefore arguably more advanced than what is listed in the Year 7 scheme of work (point 9).

Above is part of the Year 6 programme of study. It seems a lot more advanced than fundamental concepts of arithmetic using fractions (point 10), which is taught in January in Year 7 (for the scheme I’ve posted).

I won’t keep comparing, you get the point. The Key Stage 1 & 2 Maths Programme of Study is available here.

Many schools have fantastic schemes of work for Year 7, I have no doubt about that, and if you’re reading this thinking ‘yeah but we dont do that, this is rubbish’ then great, clearly you’re one of those schools.

All I’m trying to put across is that times have changed, and the Primary curriculum should, I think, feed directly into the Secondary one. The gap between Year 6 and Year 7 is no bigger than Year 5 and Year 6, so students aren’t forgetting anything more than they would normally. If you’re in secondary and aren’t familiar with the new (not that new now) programme of study, it’s well worth a quick read – particularly the Year 5 and 6 parts.

Any interesting contributions about how you structure your scheme of work (and your groups) in Year 7 are most welcome in the comments.

# Garfield’s Trapezium

James Abram Garfield was a member of the United States House of Representatives when he submitted a proof of the Pythagorean Theorem to the New England Journal of Education in 1876. He of course went on to become the President of the United States. His proof (shown below) was unusual in that it used a trapezium (trapezoid for US readers).

It’s a straightforward proof to follow, and can be used quite easily in the classroom with a little guidance. But a trapezium structured like this offers so much more than just (just!) a proof of the Pythagorean Theorem.

Take the following configuration, which also shows the enclosing rectangle:

This allows us to investigate angles and determine some properties of the inverse tangent function, arcTan:

In fact you can use Garfield’s Trapezium to derive a whole host of trigonometric functions. Take the example below, which enables you to fairly easily find the tricky trig functions of 15 and 75 degrees:

Fun! As a challenge to you, can you use Garfield’s Trapezium to derive the addition and subtraction formulas for Tan?

# Geometry Snacks

I’m very pleased to announce that my second book, Geometry Snacks is now shipping from Tarquin Publications. The book is coauthored by Vincent Pantaloni, a fantastic French mathematician who helped design over 50 geometric puzzles as well as some of the creative solution approaches to each one.

Geometry Snacks is a mathematical puzzle book filled with geometrical figures and questions designed to challenge, confuse and ultimately enlighten enthusiasts of all ages.

Each puzzle is carefully designed to draw out interesting phenomena and relationships between the areas and dimensions of various shapes. Furthermore, unlike most puzzle books, we offer multiple approaches to solutions  so that once a puzzle is solved, there are further surprises, insights and challenges to be had.

As a teaching tool, Geometry Snacks enables teachers to promote deep thinking and debate over how to solve geometry puzzles. Each figure is simple, but often deceptively tricky to solve – allowing for great classroom discussions about ways in which to approach them. By offering numerous solution approaches, the book also acts as a tool to help encourage creativity and develop a variety of strategies to chip away at problems that often seem to have no obvious way in.

The inspiration for the book came from the responses to puzzles I have created here, and the different ways in which people solved them. Take the figure below for example:

(not included in the book!)

This puzzle requires you to find what fraction of the whole regular shape the shaded section represents (constructed using midpoints).

Here is one possible solution:

You can see that the pink area covers half of each vertical pair of rectangles, and that is equal to 4 of the congruent triangles. So the answer must be 4/12 = 1/3

But there’s more life in this puzzle yet! Can you see an alternative approach? Can your students?

Shearing the parallelogram into a rectangle, then reconstructing the entire shape into a quadrilateral! It’s a lovely approach, and one I wouldn’t necessarily think of myself.

By sharing multiple approaches both to the reader and potentially with a class, everyone learns from new insights and styles of problem solving in geometry – and each puzzle teaches us something new even if we can solve it!

As such each puzzle in the book includes at least 2 solution approaches.

If you have enjoyed my puzzles over the last five years, consider buying this book. It ships internationally, and I think you’ll like it a lot.

# Unit Conversion

I have often found that students really struggle to understand what is going on when we convert units of area and volume. It’s pretty straight forward to demonstrate conversion of units of distance: 1 metre = 100 cm etc just by using simple instruments like a ruler or metre stick, or measuring tape. Whilst some may initially struggle with this, really it’s just a question of scale / ratio etc.

When dealing with area however, it gets a lot more confusing. 1 m2 is frequently misunderstood as being equivalent to 100cm2. It’s understandable why: it feels intuitive, and maps nicely against existing knowledge that 1m = 100cm. Unpicking this misconception can be tricky. In what I’d consider to be the least appealing scenario, students are left just memorising that they multiply/divide by the conversion ratio ‘squared’ so 1m2 would be equivalent to (1 x (100)2) cm2 and so forth. This, as stand-alone information, feels a little empty – and of course I’m keen to expand on why this works. The most common explanation I’ve seen is using a diagram of, say, a square (below is taken from BBC Bitesize):

I have no issue with this explanation. It makes sense and it helps students overcome the somewhat counter-intuitive result upon first inspection, that 50000cm2 = 5m2

However, what I’ve often found is that even with this explanation, students resort back to making the same misconception/error regardless of demonstrations to try and make it feel more intuitive. Perhaps then, the issue needs to be tackled further by adjusting the way in which we approach these problems.

Take the following example:

“Convert 250cm2 to mm2

Students who get it right will (again, in my experience) typically mull it over, recall something about areas being a bit different to distances, and then decide upon a multiplier. They would literally write down 250 x 102, or 250 x 100, then arrive at the answer and move on.

I prefer a pictorial approach, which I’ve found tends to bring out more right answers from those other students, and maintains the intuitiveness of the answer being right.

Forget the conversion element of the question for a moment. We are dealing not just with a number (250), but an area of 250cm2. We can represent that as follows:

This gives us our required area.

Now convert the units for each side into our new units, millimetres:

Our rectangle is the same size, it has the same area, and the lengths are unchanged. We’re just using equivalent amounts in different units of measure.

So now the area, which is unchanged, is represented as 250 x 100 = 25000mm2

Students would literally draw both rectangles, one below the other like so:

Now, all is well and good so far, however, what if we get a question like this:

“Convert 17m2 to mm2

17 is prime, and less easy to put into a nice rectangle. Or is it? In fact, this gives us the perfect adaptation to the rectangle representations. Rather than continuously adapting the heights depending on the numbers we’re given in the question, just make them all of height ‘1’, forever, for every question:

“Convert 25mm2 to cm2

The idea can be further adapted for volume conversions:

“Convert 3m3 to cm3