This week I had a breakthrough on how I could teach transforming functions to my IB AA SL students, which as with many of the best ideas, happened almost completely by accident!
The lesson was on combining different transformations to draw complicated functions. The end point for the lesson was questions like this where you have a function f(x) and have to draw something like g(x) given below (f(x) was defined earlier in the example, and is shown in the graph).
But from the previous lesson I knew that several students were still having issues with vertical and horizontal stretches, especially when the factors are negative, so I wanted to practice these first. I decided to use my IB Key Skills question generator to create 3 questions of increasing difficulty. We started with the ones shown below.
This is nothing new. I often do this when I know there are problems for some students. It gives us a chance to discuss as a class the approach to different questions.
But what I realised whilst doing this was that I could ask students to annotate on the screen to show their answers. I can't believe it has taken me this long to think of this, but it was equivalent to getting students to the front to draw on the board! Anyway, after kicking myself for not thinking of this earlier, I realised I could do a lot more with it.
For starters, I had three students working at a time, and I chose students to work on the level of difficulty that I thought was appropriate for them based on the previous lesson. I asked the other students to work out what their answers would be.
When a student had finished their question I asked the rest of the class to use the stamps built in to the Annotate function of Zoom, to either tick, cross or ? each answer. This gave me a feeling of what the class thought (and because I could see the names as they annotated, who was right or wrong). The ? was good too, as it allowed students to show they still weren't sure. For those that disagreed with an answer I asked them to explain why.
After we had all three answers done, I pushed the answer button to show the answer. And what worked REALLY well, was when I then scrolled down, the annotations don't move. Usually this is a pain, but in this instance it was perfect, as the graphs they had drawn slid on top of the answers, clearly showing if they were right or not (all of them were by this point).
This short recording gives an idea of the whole process. I made it after the lesson, so it looks like I am annotating, but in class it was students names that appeared.
Of course, the benefit of randomly generated questions is that then I could create 3 more instantly and get 3 different students to have a go (this time choosing those who I knew had struggled last lesson, and had intentionally avoided in round 1). I only needed to do this twice, but I could keep going if I needed.
Then with a quick change of settings we got these questions.
After a few of those, we pushed into combined transformations with these questions, and I showed them the answer for the first one. I asked them to put their answers in the chat, and they all got it correct. We had to talk about the importance of the order of the transformations later, but that wasn't on my mind just yet.
Then we moved on to look at this example together as a class (available in the lesson sheet for this topic).
Which I colour coded as below
Then I sent them off to try two Your Turn questions in pairs, suggesting they use the annotate function to communicate with each other as they worked through the problem.
Finally, as the lesson came to a close, I wanted them to quickly check their answers, so I whipped up a desmos file to reveal the answers.
Put yourself in these situations:
A colleague is talking at lunchtime in the staffroom about having trouble with a particular student. They do the bare minimum, but won't do any extra even though they could do really well in the subject if they just applied themselves a little bit.
After a day visiting your parents, your partner is upset about something your father said. This is not the first time, they have had a tumultuous relationship at best. You just want them to get along.
You are a middle/senior leader, and an area of concern/improvement in your area of responsibility has been identified from some data.
What is your default reaction to situations like these?
Mine has always been to go into problem solving mode. Certainly as a Mathemacian, that is what I do: solve problems. But recently I have noticed that I do this in all areas of my life, both professional and personal. If I get involved in a problem, then I am aiming to solve that problem.
I took stock of this position a couple of times over the last few years. The first time I really thought about it was when I started training as an instructional coach following the Impact Cycle by Jim Knight. I was fortunate enough to go to the Instructional Coaching Institute in 2019 to be trained by Jim himself, and it was am incredible experience, not only to be surrounded by 99 other people who were coaches, but to listen to Jim himself.
There was so much to take away from that week. But one thing that really hit me was that a coach is not there to solve the teacher's problem, but rather to understand the problem and help the teacher solve their own problem (though not in a facilitative way assuming the teacher knows what to do, but in a constructive conversation as partners).
Since then I have been fortunate enough to have many coaching conversations and have worked for an extended period with several teachers as their instructional coach. I feel that in these conversations I am actually quite good at not trying to solve the problem. But it seems to be mostly limited to the structure of a coaching conversation.
In reading Kicking the Solution Habit recently, I was suddenly confronted with a behaviour I exhibit most of the time. I try to find solutions. In the post, Matthew Evans basically has one big message: before trying to find a solution to a problem, make sure you understand what the problem is.
And this is the sticking point for many people, in my experience. It is quicker and easier to make your own interpretations of the problem and solve those, than to actually spend the time investigating the true causes of the problem and addressing those. The quick fix is easy in the moment (even if it doesn't last), whereas actually solving the problem takes a lot of time and energy to explore what the true problem is.
So, whilst I find myself able to do this in the confines of a coaching conversation, it is the structure of that conversation that acts as my cue to behave that way.
When in another situation, be it an impromptu chat with a colleague or a conversation with my wife about our children, I fall back on my problem solving ways, trying to fix the problem instead of understanding it first.
One area where this has been very evident is in my role as lead for teaching and learning. Early on in my time in this role, I wanted to enact quick solutions: an inset on this topic, a collaborative professional development day. But as I have gained experience, reflected and gotten better at the job, I have realised that if you want to implement anything, you have to take it slow, not just to get buy in (though that is important), but to make sure you are actually addressing the real problem, and not some surface detail that is really just a symptom.
I want to improve at this. I want to be better at uncovering the real problem, and listening intently to people before trying to solve the problem. But I have a ways to go. I need to change a lifetime habit, and that is difficult. I need to work out some cues for myself to put me in the right frame of mind. I know I can do it, I just have to transfer what I do in a coaching conversation to other situations.
But that is difficult. It isn't a quick fix.
I am making headway. I have spent time identifying what the real problem is (I like to problem solve) where in the past I would have put the blame for failed fixes on the other person (they clearly didn't do it right). I am making progress. But I need to keep analysing the problem.
Are you a problem solver? Are you always looking for a solution, rather than trying yo understand the problem?
2020 will go down in history books as the year that COVID-19 swept around the world, disrupting every aspect of life as we knew it. The impact was different in different countries, as governments decided upon how extreme the measures they needed to take. For many this year was the worst ever. But for me, there were many positives to be taken from 2020.
I have previously blogged about some of the activities I use to help students to understand what differentiation tells us (that is what the derivative is), but today I had a great lesson on introducing the actual process of differentiation.
After exploring the idea of the derivative, I explained that differentiation is an algebraic way to find the function, rather than a graphical way.
I started by using a set of examples and asked students to use the Reflect-Expect-Check idea from Craig Barton. I showed them the first couple, then I asked them to reflect on what had changed in the question, expect what the answer would be and then check when I wrote the correct answer. I also made the different parts a lot more explicit than I normally do, as you can see below. The full set of questions is on pages 2 and 3 of this document.
As we went through we stopped at various points, talked about expectations, talked about the "obviousness" of the answer to y=3x (the derivative is the gradient, which is 3) and y=7 (the derivative is the gradient, which is 0) and that these fitted in with the patterns they had already spotted.
One thing I did differently in writing this sequence compared to normal is starting with the general case and showing y=x^2 as a specific case within this.
After this we then did loads of practice, but where I would normally do this via mini-whiteboards in class, since we are remote teaching, I had to find a technological solution. For most things this year, typing in the chat in Zoom has been enough, but I wanted to see the full written derivative from students.
Desmos comes to the rescue. I set up this assignment called My Whiteboards (copied one of the Desmos templates and added a few extra of my own). Then I paced them to the second slide so they were typing Maths. I decided to do this as they need to practice writing in Maths Type before they do their coursework next year. I then projected questions through sharing screen, and students wrote the answer in the desmos, deleting each time to write the next one.
This way I could see their answers as they wrote them, give immediate feedback and see who was participating and who was clearly unsure at any stage.
I used my website to generate the questions, starting at
And ramping up to
Slowly increasing the complexity by adding fractional coefficients, then multiple terms, then negative powers. We ran out of time to get to fractional powers, but will bring those in next lesson.
Overall, all students were successfully differentiating functions like the one shown by the end of the lesson, so I am happy with the progress they made. Unfortunately we only have one more lesson before the end of our school year, so will probably have to review a lot when we go back in March.
In a recent lesson with my IB Maths AA SL class, I set them this indices question to simplify as part of the starter activity.
They struggled. A lot.
Since teaching via Zoom, I don't usually go through these retrieval based activities in class straight away, rather opting to take in their work to check through all their work, which gives me a better idea of what they can and cannot do (in a classroom it is different as I can see their work live).
Very few of them got anywhere near close to solving the problem. In the following lesson, they asked me to go through it, so I did.
After showing them how to do it, I made a passing comment about enjoying doing this kind of problem, even finding it relaxing.
Even via Zoom, I could tell the reaction of my students. Some of them even turned their cameras on to show their disbelief. How could I enjoy solving this kind of problem? How was it even remotely relaxing?
This provoked a rather interesting discussion where we talked about the things we find enjoyable. My point was that when you can do something, but it requires a bit of work, that is normally what we find fun. That is, learning is fun if you know enough to be able to learn.
We discussed how some people enjoy music or sports, and the reason why is normally they are relatively good at it. And then they enjoy getting better and doing more difficult aspects of that course.
It is the same with Maths (and anything else really). If you are constantly failing at it, you will not enjoy it. But if you can do it, with a bit of effort, normally you will enjoy it.
They could understand this point. I don't think they had ever really thought about why they enjoy some things and not others, and it helped them see how I could enjoy solving a Maths problem. I told them that my job was to help them know enough that they could enjoy solving Maths problems.
Now I just need to live up to that!
Due to COVID-19, we are unable to run our IB and IGCSE exams this November. Local restrictions make it impossible. It has been tough for students and teachers to have 2 years worth of work count for nothing. Our students will not get an IGCSE set this year (as an international school, no systems in place like there were for the UK). The IB students will have grades awarded purely on coursework.
The announcement that there would be no IB exams this year was made the day before I was due to finish teaching the syllabus, and my IB HL class were keen to finish that last little bit of vectors. But we still had two weeks left of term left, and with no exams, the usual rush of exam papers was pointless.
So between me and the other HL teacher we decided to offer two separate options: she taught the calculus option (we did a different option, but many students were interested in this) and I did a series of lessons on random mathematical diversions.
Here I will share those diversions, along with the resources, in case anybody ever feels like using them.
We started by looking at this problem.
I blogged about this problem in the past here when a student brought it to me. I ran the session basically as an open problem, with students in breakout rooms in Zoom, and me popping between them.
Next we took a look at Benford's Law.
I asked students to think about a set of data. They could choose anything, but I gave some ideas like
Once they all had an idea of their data set, I asked them to think about the first digits of each data point in the set, and to decide what the probability distribution would be for them. That is, what is the probability if you choose a random point in your data set, the first digit is a 1 (or 2,3,…)
We had a brief discussion about this, with the first answer being the expected 11% each as they are all equally likely. One student suggested that they would be clumped around a number (probably the mean) value.
After a brief discussion, I told them to go away and find the data set they had thought about in the first place.
We entered them all into this Google Sheet.
And then I added the data sets one by one to Autograph.
Obviously, with any activity like this you are open to it failing dramatically, but below you will see all the data sets plotted together, and Benford's Law falls out beautifully. There is even an excellent non-example.
The non-example was of weights of NBA players, and we discussed why this did not fit the pattern (weights will all fall within the range 70 - 120 kg approximately, so the first digits will be 7, 8, 9, 1, and the mean is in the 80s, so more 8s as this is a normal distribution.
But what about the other data sets? Why do they follow this same pattern?
I finished the session by explaining Benford's Law, the percentages it predicts and the formula, and how it can be used to spot people who have made up a data set.
Exploding Dots was a part of the Global Maths Project a couple of years back, set up by James Tanton. It is definitely worth checking out the website.
However, I prefer to teach it a bit more actively, and so created a version that I can present to students, with questions for them to do along the way. You can find a blank PDF of this here.
It starts from the very beginning of school level Maths, with counting (in different bases), followed by the four operations. It introduces the idea of zero pairs to perform subtractions, and then builds up to unknown bases: that is polynomials. Within an hour you can take a group from counting to performing polynomial divisions.
I have done this with other classes before, and it went down well with this class too. By the end of the first double period we were answering questions like the one below (admittedly, they do this in the course, but we did the more traditional long division).
We took this further to polynomial division that creates infinite polynomials, and the students wondered what would happen in other situations.
It was so popular, that we did a second double period on Exploding Dots, and this was all new stuff that I hadn't done with classes before.
We looked at decimals and fractions in the exploding dots model, which allowed us to look at fractions in different bases.
Then we looked at fractional bases, and explored what they might look like.
I love the exploding dots model. The students all commented on how visual it was, an how easy it was to understand what is going on. I really need to make more of this in 'normal' teaching, and not just as an enrichment activity.
We study this in the course, but a couple of students asked to look at it in more detail. I was fairly lazy with this one, relying on some excellent resources available online.
First I sent students to this page to read the examples. We discussed the importance of the size of the population, and then did a few of the questions at the bottom of the page.
Then we watched this excellent video from 3Blue1Brown which visualises the whole thing beautifully.
I taught D1 once before leaving the UK, but it has been a while since I have done any decision maths. I thought this was a wonderful opportunity to take a look at the Chinese postman problem. I based the lesson on the plan from the Standards Unit, and turned it into a Desmos Activity. It was a very discursive session, so I paced them through the activity to start with, and also talked about the ideas whilst demonstrating and collating their ideas on a whiteboard.
Based on a couple of articles from nrich, I put together an activity on continued fractions. We started with evaluating them (like the one below).
We quickly moved on to look at infinite continued fractions
And generalised this
We took a quick look at how we can write any rational number as a continued fraction by using reciprocals.
And then how we can use continued fractions of surds to get pretty good rational approximations for them
Finally we took a look at some of the continued fraction representations of other irrational numbers such as pi and e.
It was fun to get to explore some different aspects of mathematics with the class. I really must try to build it into my teaching more often, and not as enrichment, but as an integrated part of teaching Mathematics. If you have any go to activities like this, I would love to hear about them.
Yesterday was the birthday of my Grandad. When I was young this was always a day that the whole family got together, usually to have a BBQ which he would do in his garden. Me and my cousins would play around outside, maybe in the paddling pool. The adults would play some cards as the day wore on (probably start with Running Out of 7s, followed by pairs Cribbage). I am 7 years older than my next cousin, so I also remember playing along at cards. The memories of the bank holiday weekend are always good ones for me, and since his passing in 2015, it is the weekend when I most remember him.
This weekend was no different, and this year it really got me thinking about the people we lose in our lives, and the impact they have on us. In this post I want to share the impact some of the important people in my life, who I have now lost, have had on me as a person.
Ray Tointon was my Grandad, and he will always be in my heart.
Pat Tointon (nee Isaacs) was my Nanny Pat, and she will always be in my heart.
Thankfully, both Grandad and Nanny Pat were able to make it to my wedding, and were both healthy for it. When my wife and I married in August 2013, they were both there in St Andrews with us, and the wedding photos are one of my final memories of them both.
One of the saddest parts of their passing when they did though is that they didn't get to meet their great-grandchild who was born in 2017.
I have written before about The Hardest Time of My Life. Briefly, before the birth of our son, my wife and I went through 4 miscarriages. It was not a good time. Nanny Pat was one of the few family members who knew. We were not at a point to be able to share that with others at the time (something that, looking back, was a mistake).
But those four babies were losses for me as much as any other family member who has passed. They taught me to be more human, and treasure the human connections we make. They taught me to be more resilient, to keep trying, to push through it. They taught me that life is not always roses, that we all have to go through difficult times. They taught me that family is more important than work.
And they taught me to be more open about how I feel. I am not a particularly open person when it comes to my feelings and what I am thinking, but going through that time made me realise that I have to be open with the people I love, and particularly my wife. I am not perfect at it, and I still find it hard to share my feelings. But I try. I want to be more comfortable doing it.
I don't think many families can boast having 5 generations alive at the same time.
Nanny Pop died this year after a couple of years battling old age. She made it to 98.
She was always the person in the background, listening and learning about people. She didn't do anything with the information, she just knew that being there for people was important. She was the embodiment of trust. Anything you said to her was safe. I do my best to be somebody that people can trust. I am trying harder to not engage in gossip. I want people to know me as somebody they can talk to, without fear of that information being repeated.
Dora Farman (nee Neil) was my great-grandmother, and she will always be in my heart.
I have just finished reading The Teaching Delusion by Bruce Robertson, and it hit all the right notes for me. I found myself nodding along, lapping up what Robertson says, constantly thinking "This is exactly what I think, but said so much more eloquently." In fact, I am thinking of copying a few extracts to give to people when I can't put into words my own thoughts!
I jest, of course. There were plenty of insights in the book that I had not thought about before, and a couple of things I disagreed with.
The main premise is that no matter how good teaching is, it can always be better. This has been a point I have made at the start of each new school year since getting the job of T&L Coordinator, and my most recent phrasing has been "It is both our right and our duty to continue to improve our teaching". I use this wording carefully, to instil the idea that it is our right to want to continue to improve ourselves, get better at our jobs, and become better teachers. This aligns with Robertson's idea of a Professional Learning Culture. On the other hand, we serve a community of children and their parents (who, in my case, pay a fair amount of money for our services), and it is also our duty to them to do the best job we can, which includes continually improving our teaching. Our duty to the parents who pay, yes, but mainly our duty to the young people we have the pleasure of working with, whose future depends so much on what we say and do, how we make them feel, and what they learn from us.
Robertson asserts that The Teaching Delusion is made up of three factors:
End of a Semester
We are approaching the end of our first semester, and so have a two week holiday coming up. After teaching online since March, with only a week break, I have to say that this is very much welcome. We (teachers and kids) are all tired and having two weeks away from Zoom will be a beautiful thing. We are not allowed to do any travelling as kids under 14 are still quarantined in Peru, so it will be spent at home and going for walks, but I am glad to have some time away from work for a little bit. I have been getting to spend more time with my son as I work from home, which is amazing, but I am looking forward to doing this without nagging feelings in my head about work that needs finishing.
IB Key Skills
I have been using my new IB AA SL Key Skills Generator to create retrieval starters for my lessons. I am using this in conjunction with the Spacing Spreadsheet which tells me which skills to do each week.
With the lockdown still continuing here in Peru, we switched all classes to double periods, and so I only see the class twice a week, so I do a key skills like the one above on a Monday, and then a shorter definitions/facts recall on Thursday where I simply ask them to define the important words and concepts, and state some important facts.
I have also just finished working on the Binomial Expansion questions, which I am quite proud of.
In particular, I went for two different ways for presenting the solutions, based on a conversation on twitter.
I listened to the recent Mr Barton Podcast with Daisy Christodoulou last week, and one thing that intrigued me was Anki. I had previously downloaded it to my phone as I had heard Ollie Lovell talk about it, and thought it was an easier way to go than the old fashioned flash cards I was using to learn some Spanish, but I never actually got round to setting it up. I will get on this in our two week break that is coming up.
But what really caught my attention was using this with kids. I am thinking about setting up an Anki deck of the key terms and skills that I was recording in the spreadsheet, and then sharing this with kids. They can then do this as the starter, giving them recall practice that is a bit more individualised to what they need. Hopefully this will also get them using Anki to study other things. And then each lesson I can get them to add new stuff to their deck too.
As Craig said at the end in his reflection, it would be great to be able to combine this with randomly generated questions. This is definitely not something I am up to coding myself (mine is largely just a hobby), but it would be interesting. I am thinking a workaround could be to use my IB Key Skills Generator and get students to put a reference to a question in their decks. Then, when it comes up, they go to the site, do that question, and then mark it as right or wrong on Anki as they would a normal flashcard. I think you can even insert a link directly into the card, so that could take them straight to the page.
I will be having a play around with this when we come back in August.
The second book by Craig Barton (well, ignoring the non Maths teaching ones) is everything a sequel should be: it builds upon the greatness of the first, but has its own tale to tell. It gets into the nitty gritty of the story, focusing on one of the smaller parts of the first. And yes, it is a bit controversial.
I am not going to write a summary here. I thought I was, but that would not be able to do any justice to the book. If you are a Maths teacher, you should read the book. Craig is open throughout about not trying to tell you what to do, but rather telling what he does, why he does it, and provoking you to think about how you could adapt those things to work for you (if indeed you find them valuable in the first place). But even if you disagree with everything Craig has to say, then you will still learn a lot from reading the book. If nothing else, if you do all the sequences of questions he provides, you will be giving your subject knowledge a good servicing!
Here I am going to share a few of my main takeaways, and what I want to incorporate into the book.
In my teaching
In the very last chapter of the book, Craig gives some advice on "Making it work", and the first thing is to choose one thing you want to try. Well, I am going to ignore him on that! Well, not completely. I am going to choose one thing that is new, and 3 things that I currently do but want to adapt after reading the book.
Reflect, Expect, Check, Explain - this is the new one. I have been using some sequences of questions from Craig's www.variationtheory.com website, as well as making some of my own, but, honestly, they have really just been a set of questions. They have allowed me to direct student attention to certain things, but I have not been systematic enough in my approach to develop their mathematical thinking in the way Craig describes.
Since starting the book I have been adding some elements, in particular the reflect stage, but I want to make more of this. So I am going to try the full structure, and use the Prompt Questions that Craig suggests (available on his website: http://mrbartonmaths.com/booklinks/). I think I will need to use a template to help them structure the process too. My plan is to try this with my first year IB class, though I need to think carefully about what topic to do this with. We have some recap of indices and logarithms coming up, so that seems like a good fit. I will blog again on this when I give it a go. I have been using some sets of questions with them (such as the one below on Binomial Expansion), and have made short references to the ideas of reflecting on what has changed, but will need to be more explicit about this.
Another aspect that I have not been building in that Is important is the idea of Fluency Practice before the intelligent practice. I used to do too much fluency practice, now I am not doing enough for students to get the most out of these sequences of questions. For students to develop the mathematical skills, they need to be more confident with the method they need to apply first,
Atomisation - this is something I have been exploring, in particular with putting together the IGCSE Booklets and IB Lesson Sheets, but the systematic way Craig approaches it grabbed my attention. Going through the small atoms that make up a new idea and ensuring they are all secure first is something I want to look into further, but think that will definitely need to be a collaborative project. I am also thinking about how I could do some of that in the "flipped" model with IB classes.
Example problem pairs - just a minor adaption to my current process, but I need to find a consistent way to get all their attention on the example. I print examples and your turns in the booklet/lesson sheet, so the easiest way seems to be to get them to shut their booklet when I go through the example. Even go as far as put it on the floor if necessary. Then they can open their booklet once I am done to try the Your Turn. I also make scans of the sheets available to students after they are complete, so I might get students to NOT copy the example in class, and then get them to do the example as a homework, where they can check against my version. This would give them another exposure pretty soon after the first, and give them instant feedback on it.
Rule - I have been playing around with Frayer Diagrams for a year or so too, and the Rule sequence is a nice structure to lead into these. So far I have been using them to introduce the definitions, but this has not really been successful. Flipping this and getting students to fill them in themselves after seeing a sequence of examples, non-examples and boundary examples is a much better approach.
In out department
I also want to get my department thinking more deeply about the questions we offer students and the experiences they get of thinking mathematically. I am hoping to get some time with the whole department to get them to do some sets of questions over the coming months, and then start building some of our own sets. I think I will start with the Reflect, Expect, Check, Explain cycle.
It is difficult as we are still in lockdown from Covid, but I think we can make it work via breakout rooms in Zoom. Thoughts and plans are coming together…
I am a maths teacher looking to share good ideas for use in the classroom, with a current interest in integrating educational research into my practice.