Consulting mathematician
Here is a large square of Lego bricks.
What do you notice?
What do you wonder?
Can you see any smaller squares in the large square?
Can you see any half-squares?
If this was the fourth pattern in a sequence, what might the fifth one look like?
Each day I will update this post with the next challenge. To find out what GeoGebrAdvent is all about, check out this post. To see people’s creations, look for #GeoGebrAdvent on Twitter.
December 1st: Create a star
December 2nd: Create a snowman
December 3rd: Create a snowflake with sixfold symmetry
December 4th: Create a paper chain
December 5th: Create some holly
December 6th: Make fairy lights/Christmas tree lights/holiday lights
December 7th: Create a tessellation
December 8th: Design wrapping paper
December 9th: Make a bell or some bells
December 10th: Design a ribbon or bow
December 11th: Create a wreath
December 12th: Create a Christmas tree
December 13th: Make an angel
December 14th: Make a Christmas scene using only triangles
December 15th: Christmas dinner
December 16th: Gingerbread
December 17th: Trees – not Christmas tree but wintery tree scene
December 18th: Make a Christmas card
December 19th: Create a winter scene using only circles and arcs
December 20th: Make a bauble
December 21st: Father Christmas!
December 22nd: Create a (Christmas themed) optical illusion
December 23rd: Make a shooting star
December 24th: Make a present or gift
It’s Advent! As well as opening the first door of my chocolate calendar, I wanted to do an advent challenge this year, and as you can never have too much GeoGebra in your life, I thought a GeoGebra challenge a day from now until Christmas would be a nice fun thing.
For each day of advent, I will post a challenge to complete in GeoGebra. You can share pictures or animations on Twitter (or Instagram if you prefer) using the hashtag #GeoGebrAdvent. I’m hoping people will join in to create beautiful geometrical art, improve their GeoGebra skills and share any cool and ingenious methods they come up with! The challenges are designed to get harder during the course of advent, but you are of course encouraged to attempt them at whatever level of difficulty you like, and to adapt, mix up, mash up, and let your imagination run wild – these are prompts rather than commandments.
Big thank you to Sam Hartburn and Becky Warren who gave me ideas for some of the challenges, and who are both queens of GeoGebra creativity. If you’re not already following them on Twitter, they are @SamHartburn and @becky_k_warren.
So, onto today’s challenge: Create a star. It can be as simple or as complicated as you like. It can be two dimensional or three dimensional, static or animated, coloured or monochrome. Don’t forget to share your star on social media with the hashtag #GeoGebrAdvent.
Suppose you want to find the area of a dodecagon inscribed in a circle of radius 1. One way to do this is to divide the dodecagon into 12 isosceles triangles with two sides of length 1 and an angle between them of 30°. Then we can use the formula for the area of a triangle $$A = \frac12ab \sin C$$ to give us an area of $$ \frac12 \times 1^2 \times \sin 30 = \frac14$$ Multiplying by 12, the area of our dodecagon is therefore 3 square units.
There’s another way to get this result which doesn’t rely on knowing that $\sin 30 = \frac12$. I first came across this dissection of a dodecagon when working at NRICH and looking for ideas for the Wild Maths site, which was about exploring mathematical creativity. Take a look at the GeoGebra construction below.
Move the slider, or click the play button to start the animation. There’s a little bit of thinking to be done to convince yourself that the pieces fit nicely together with no gaps, but I hope you agree it’s a very pleasing way to show that the area of the dodecagon is 3 square units!
I’m finding it strange to believe that it’s been just over two months since my last day at NRICH, and the start of my fully freelance maths communication and education career! Two months in seems like a good time to do an update, so here goes…
Late August and early September was taken up with some question vetting work. I hadn’t done this sort of work before, and I have to say, I really enjoyed it! It gave me a lot more insight into the process of writing exam questions, I spend a big chunk of time just working through questions looking for snags, and the best bit was thinking about how students would approach the question and trying to be as imaginative as possible about how the question might break, before coming up with possible fixes. I am looking forward to finding opportunities to do this sort of work again in the future.
In September and early October I worked on a teacher guide and some lesson plans for the brilliant Mathigon website. If you haven’t checked out Mathigon you are missing a real treat – I keep discovering more and more “wow” moments as I click around the site. The little bit that I worked on can be found in the Polypad section – check out the teacher guide and lesson plans with suggestions on how to use the Polypad digital manipulatives. I’m looking into other areas of the Mathigon site that I can contribute towards – it’s so great to be able to play even a small part in creating “The Textbook of the Future”.
The start of October was Maths Week in Scotland, and I was delighted to have the opportunity to contribute to the programme together with my good friend and fellow maths-freelancer Fran Watson. Fran and I worked together when we were both at NRICH, and being able to continue that working relationship has been hugely supportive to me as I make the transition to freelance. We were delighted that several hundred people tuned in to our Maths Origami live streams, where we shared some folds that explored 3D shape and symmetry. Following on from this success, Fran has persuaded me to team up again for Maths Week England next week – this time, instead of doing a stream targeted at younger students we are putting on an after school event that will be ideal for teachers but also older students or any interested adult who wants to explore how maths and origami are related! We are running it using a “pay-what-you-can” model so that we can cover our costs but keep it open to as many people as we can – check out the Eventbrite page for more details.
I was invited to lead a session for the London ATM/MA branch on October 10th, so I put together a set of problems and musings on the topic “Using Patterns and Structure to develop Algebraic Thinking”. I really enjoyed thinking about what I wanted to say on this topic – it merits at least one blog post in its own right – but highlights were probably using it as an excuse to get the lego out and make some lego patterns for delegates to look at under the visualiser and speculate about how the sequences might continue! I do think that teaching Algebra in a way that links with patterns and structure should be part of every secondary maths teacher’s toolkit, and it’s something I’m very happy to offer teacher professional development on – if you’re reading this and you want to book me to work with you on this, get in touch!
I have also been very privileged this term to be able to work behind the scenes for the excellent Maths Inspiration online programme. If you’ve watched any of their shows this term, you’ll know they now have an excited interactive voting and commenting system – I have been helping out to make sure that all the votes open and close on time correctly, as well as reading all the comments to pass them on to the presenters. It’s a nice change of pace for me to be working on a live online event in a support capacity rather than presenting, and of course it means I get to watch some of my favourite maths presenters and pick up lots of tips and tricks! There are more shows coming up in November and early December so if you are a secondary teacher looking for enrichment opportunities for your GCSE and A Level classes, check out the Maths Inspiration website.
The rest of my time has been spent lining up projects for the next couple of months (as well as the usual admin and finance chores) – keep an eye on my Twitter feed if you want to know what I’m up to and can’t wait for the next update post. And of course I am still devoting some of my time to teaching – this term I have GCSE and A level students who I am tutoring online, though I do still have a bit of availability, so if you know anyone who is looking for a maths tutor, drop me a line!
As part of Maths Week Scotland, Fran Watson and Alison Kiddle are delighted to offer two free online interactive origami webinars aimed at 8-13 year olds. 
The first event, 3D Shapes and their Properties, will take place on Thursday 1st October during the school day and is designed for whole classes to join in with their teachers.
The second event, Understanding Symmetry, will take place on Saturday 3rd October and is designed for families to participate together.
These free events are open to children and young people anywhere in Scotland, thanks to support from the Maths Week Scotland small grants fund. Both webinars will be hosted on YouTube, and there will be opportunities to submit comments and questions to the presenters via an online form, as well as sharing photos of your origami creations!
For more details about each webinar, and to sign up, please visit the booking pages.
3D Shapes and their Properties
Thursday 1st October, 9:30-10:30
Understanding Symmetry
Saturday 3rd October, 11:00-12:00
If you have any questions about the events, get in touch!
If you’re not in Scotland, and sad to be missing out, don’t worry! We are hoping to be able to run the same event again for other audiences very soon!
Last weekend, it was Big MathsJam, and among the mathematical delights I sampled, there was a talk by Christian Lawson-Perfect about stacking cups. You can see a really quick version of his talk on the Aperiodical website. Before reading on, you might like to think about the problem a bit…
At the weekend, I went to a lovely party, and on finding out I was a mathematician, one of the other attendees shared a weight puzzle that I think I have come across before but never tried to solve. I have been thinking about it on and off since then, and I promised to the other party-goers that I’d share my solution once I got there, so here we go!
First of all then, the problem. You have 12 objects which look identical. 11 are the same weight; one is different, but we don’t know if it’s heavier or lighter. You have a set of balancing scales, and you can do up to three weighings. How do you determine the odd weight out and determine whether it’s heavier or lighter?
You might want to have a think about the problem for yourself before reading on.
Continue reading “A weights puzzle, and reflections on how I solve problems”
There are various words I would like to ban, or rather, as I am generally quite liberal in my views and don’t tend to go in for banning, words I would caution against using carelessly. I may blog more about such words in the future (Ha! Who am I kidding? When did I last blog?) but for today let’s talk about the word… easy.
“Let’s start with something easy!” “I know it looks hard, but don’t worry, it’s easy!” “If you can do x you’ll be able to do y because it’s much easier!” Familiar? These sort of phrases trip off the tongue, particularly if you are an educator who wants to make your learners feel safe. They are all messages designed to make the listener less anxious, and more capable. They are intended to empower! Unfortunately, I know from personal experience that such messages can be the opposite of empowering.
You see, “easy” is not a property of a task or a concept. It is a relationship between the task or concept and a person. There is no such thing as an easy question, because it depends on whom you are asking. (Don’t even get me started on political interviews in which someone is badgered to answer a “very easy question, yes or no” where actually a more nuanced answer is necessary and neither “yes” nor “no” is a satisfactory answer).
In some cases, it is glaringly obvious that “easy” is not a straightforward absolute concept. For example, if I were to ask an A Level Further Maths student to find the values of x such that x2+5x+6=0, I would hope they would agree with me that it is an easy question. If I asked my 11 year old niece, she would find it very hard. If I asked my friend’s toddler, he would find it impossible to even understand the question.
I can see two problems that may arise when using the word “easy”. Firstly, using the word glibly without knowing your audience. This can happen when teaching or presenting to a group you do not know well, or a group where you make assumptions based on their prior knowledge, achievement and experience. You start off with an icebreaker, something everyone will be able to handle, and you introduce it as such. Then you find out that you’ve massively misjudged the situation, and people are stuck on your easy task! Or, perhaps worse, everyone does find it easy, except some poor soul who is then left behind (or hides the fact they don’t understand and just feels utterly rotten). This can be mitigated against by using “Low threshold, high ceiling” activities where literally everyone can get started and you can assess what “easy” means in the context of the group in front of you. And if you introduce the task with “here’s a thing” rather than “here’s a lovely easy thing”, so using neutral language, you’re not setting people up for failure if they don’t get it straight away. The flip side of this is that if you introduce something that many people might find difficult, but with neutral language, you’re not in danger of setting up a self-fulfilling prophesy. I remember teaching the technique of completing the square to a group who were not expected to tackle such questions because they were in one of the lower sets. I didn’t tell them it was a “hard” topic until we’d finished. Their response? “But that was EASY, miss!” It wasn’t often I heard that class say THAT!
The second problem is more subtle. This can happen when you know someone well, and make assumptions about what they will find easy from what you already know they can do. The problem with this is that there isn’t a nice linear spectrum from easy to hard with everything in the same order for everyone. This one has bitten me in both directions. It has taken me decades to understand that just because I find some things very easy that other people find hard, it doesn’t mean I won’t find hard the things they find easy! For example, I am pretty good at solving STEP maths questions, and I am terrible at recognising faces or noticing when people have changed their appearance. There have been times when people have made me feel awful by saying things like “but you can do x, of course you must be able to do y!” I am pretty sure I have also made other people feel rotten by assuming that they would find something trivial based on my knowledge about other things they could do. (Sorry! I really will try harder in future not to do this! If you catch me doing it, call me out please.)
In general, I think as educators we should use the word “easy” with caution. There are better and clearer ways to express the meanings we are trying to capture, and if we allow learners to make up their own mind whether something is easy or hard, and listen to what they have to say, perhaps they will become resilient and resourceful, rather than feeling rotten.
Why do we teach maths in schools?
a) To create the research mathematicians of the future
b) To empower ALL of our children to take their place as mathematically literate members of society
c) To instil in our citizens an appreciation of mathematics as a thing of beauty and truth
If we create a mathematics curriculum that allows everyone to reach a minimum standard of mathematical understanding (functional numeracy, perhaps) but also allows a generation to leave school without any appreciation for the wonder and pleasure of doing mathematics, then we have failed. But at the same time, if the system identifies and nurtures superbly talented mathematicians who go on to win Field’s Medals, while allowing some children to slip through the net and leave school innumerate, we have also failed.
If we work towards c) however, and see the job of school mathematics lessons as teaching all children to think mathematically (and to understand what we mean by thinking mathematically), I think we will go a long way to achieving the other two objectives – they needn’t be mutually exclusive. In classrooms where high-level mathematical reasoning is the norm, a good level of mathematical literacy becomes the currency for convincing others of your ideas, so pupils are given a motivation for wanting to become more skilled in mathematical procedures. If thinking mathematically is the expectation, those children who enjoy the pedantic* process of convincing themselves and others of the truth of a conjecture will discover themselves to be mini-mathematicians and will be more likely to embark on the process that could lead them to fame, fortune and Field’s Medals. As soon as our curriculum aims to do anything other than exposing young people to mathematical thinking, we risk doing at least some of the children in our care a great disservice.
*I do not use this term in a pejorative way. I took great delight in being exceedingly pedantic throughout my secondary school career.