Alice Keeler

I have been fortunate enough to visit the BETT show on many occasions it was here in 2018 that I met Alice Keeler and although I'm not afraid to say that I hadn't heard of her before, I am ashamed that I hadn’t! I think most maths teachers should know about her brilliant work. Although based in America the core principle of what she is trying to achieve is relevant even over here in the UK. How often when teaching in maths are you so busy helping and solving problems and assisting students that you don't have time to spend the quality time that is needed to deepen the student understanding? Alice's approach completely transformed my teaching and took me out of my comfort zone. What I'd like to do is tell you about an example of how I embraced Alice's principles in my teaching. Not all my lessons follow this model but, for those that do, the power it has greatly improved my student feedback and their experiences.

Alice talks about getting students to discover their own learning, "why pretend that Google doesn't exist" is one of her famous quotes. As an adult if you don't know the answer to something you would just Google it. It's the same as when students say, I don't need to know how to work it out, I have a calculator on my phone for that in the real world. We can debate that argument another day! Alice is all about changing our teaching to embrace the technology that is available. Alice asks how can we teach differently in this blog here: https://alicekeeler.com/2014/04/10/we-have-to-do-things-differently/

Reflecting on the inspiration Alice had given me I began planning for doing this differently. I looked at the units coming up and I bravely thought it's now or never! The next day I taught transformations in a completely new way. In my planning I reflected on how I usually teach transformations and asked myself what value was I adding? I demonstrated transformations then got students to do some of their own and make some notes. I asked myself, why was I there? What was I adding to the lesson? Honestly not a lot! With the help of Google they could probably have achieved the same outcome outside of my classroom. I think it’s important to ask now... how often do we teach in this way? A Director I work with is always asking for the return on investment, if you invest time into something make sure you are getting value back. I'm not sure I can honestly say that I have added value to every maths lesson I have ever taught and in these lessons I couldn’t see my return on investment. So I dared myself to do something differently inspired by Alice Keeler.

I must explain I work in a GSuite for education department and am a big fan of the product, as is Alice Keeler. I am a Google Certified Trainer. However, whatever technology your education establishment has could apply in this example too. To teach transformations I created a Google slides presentation with tasks to complete throughout the lesson. A basic example is here:

It asks students to research translations then to curate their list of helpful sites. Curation is becoming an ever important skill and I really encourage it at every opportunity. Students then answer a GCSE question. The presentation continues asking students to evidence revision techniques for revising translations as well. I am too controlling to let them leave without evidence in their learning by attempting an exam question. This may be wrong on my part but I am who I am!  Interestingly across my learner's there were some very different approaches to the task. My dance students would insert video clips, songs, sounds and raps that they'd found on YouTube of people talking about translations and transformations. My sports students would cut and paste images from the internet into their work. My art students would write notes and make interesting infographics. I wrestled with the amount of cut and paste, is it ok that they were just cutting and pasting things from the internet into their work? Does this evidence that they have understood what they have researched? I didn't have the answer to this but I would discuss it with colleagues and it was a brilliant student teacher that was working with me who said well what do they do normally? Well normally, I answered, they copy down my notes from the board. It was then it struck me... I don't normally have any evidence that they have understood what they have copied from the board so what difference does it make that they have copied something from the internet and inserted it into their work? At least this way it was their choice of what worked best for them in terms of making notes. Feel free to disagree, I'm still undecided as definitively where I stand on this, but I accepted the cutting and pasting was ok in this lesson. 

I then would ask them a GCSE question, typically a describe a transformation question and I would give them feedback on Google slides. If you are unfamiliar with using Google slides and Google classroom and the tools that it has for giving feedback and I will discuss these in a later blog post. What happened in every lesson that I did this activity in, all the learners were sat quietly working happily on a computer. Yes computers are not always available, but they could access this on their phones. Students would sit quietly working...brilliant! This was just what we wanted! I was then able to, as Alice does, spend quality time with students in a one to one situation. I could go over previous work, the topic of transformations, general revision, whatever the student needed. There is a risk in this approach that some teachers may choose to sit at the desk twiddling their thumbs but there is a risk that some teachers would take the easy option in every lesson. In teaching transformations this was able to spend 10 15 minutes, a really lengthy period of time, going over things with them. We did the full four transformations in this way so we had a series of lessons where the rest of the class were learning while one was out at the front with me. 

The learner feedback afterwards was amazing they all said they thoroughly enjoyed the activity, it was a nice change from what we normally did and they were really appreciative of spending quality time with me. It's like your kids at home you know yourself if you really devote the time and do a quality activity with your children at home that carries a lot more weight than a quick McDonald's on the way back from the shops, well it does in my house anyway! 

I was feeling very pleased with myself, I had done something different, the learners have learnt and had quality time from me.  The verbal feedback was positive but then I asked in a typical four in a bed channel 4 programme style would you do this activity again? Would you prefer more lessons to be done in this way? And overwhelmingly the response was no! I was heartbroken! I had spent time on it, what was my return on investment? Being really brave I asked some of the more chatty students why they didn’t want any more lessons like this. They said they missed me, my input and my explanations. 

Reflecting on the student feedback I kept asking myself, how could they be missing me when we were spending quality time together? Was it me they were missing or was it each other and the general classroom atmosphere they were missing? Alice promotes collaboration. Had I missed a trick by not getting them to collaborate? Would they have been able to do the work collaboratively on this topic? Did they need teaching how to work collaboratively first? I could go on but I still have so many questions that as yet I don’t have the answers to. Next time I plan to add in a more collaborative element to see if it’s the class rapport that was missing. I enjoyed the experience and so did my learners. I think the return on investment was fair and enjoyed spending quality time with my students. Could I teach a whole GCSE programme in this style? Probably not. Could I teach more lessons in this way? Definitely yes!

Ratio

One of my most favourite teaching positions has been as a one-to-one tutor in a multi academy trust. This role was exactly as it says on the label. Working one-to-one with learners getting ready to sit their GCSEs. What I loved about this was that I naturally inherited every teachers style and approach as the students were all taught by different teachers across the trust. I could often be presented with two students studying the same topic but tackling it in completely different ways. This kept me on my toes. One day we were doing ratio and I had a group of six in an after-school club and three of the six wanted to use the bucket method. The bucket method was new to me and it as I've only ever seen it in this multi-academy trust. It looks a little bit like this. 

It's particularly fun when you were dividing large quantities a £10,000 into the ratio of shares...I was concerned at the size of the buckets we would need, but students were quite adept at using crosses to represent 100 circles to represent 1000 triangles to represent 50 and so on! It worked and it was brilliant! The learners in the multi academy trust could tackle any ratio sharing question using this method. When I moved into further education and resitters I was in the same situation in that I had a group of learner's and they've all been taught differently. Ratios one of those topics where once you get it, you get it! It is quite a powerful one to unlock as well. With a bit of humour and a tongue-in-cheek I often ask students do they use a bucket or do they use Adam to solve ratio problems cue puzzled looks! I say to all my learner's I don't care how you answer it as long as you answer it correctly can explain why you do things that way. As I have mentioned before, I say to all my learner's that if you have a way that works for you then that's the way you use! What I can offer you is an alternative way if you don't have a way to work things out that is solidly your own. Adam came about similarly to the Pythagoras SASSYLASS when I was looking for a quick win to help learners easily remember the steps of a ratio problem. I googled, as I always do and checked great maths teaching ideas and resourceaholic and tes and I can't remember where I saw ADAM but he has been with me ever since. Adam is nice because it includes a check as well to check the answers are correct! 

It’s a nice way to remember the steps on how to solve. Generally students do these steps independently but most in my class enjoy having ADAM as a check that they have done it correctly. First step is to add the ratio parts together (doesn’t matter if it’s 2 or 3 part ratio!) then you divide the amount by your total. Third you multiply the parts back in and finally you add to check take a look at this example. 

In my experience I have seen higher ability learners take to ADAM easier than lower ability. The opposite being true for bucket method. However, I don't try to categorise learner's into high and low ability, I just find a way that works. Adam is an ever-present force in my classroom and he’s not only displayed on my wall, I like to insert famous Adams into my lesson to remind them that they need Adam to solve this problem so for example we will be reviewing a complex problem solving paint and area problem it will require ratio at the first stage (you know one of those six mark questions) and as we're going through I will randomly pop up Adam Sandler or Adam Woodyatt or a famous Adam just to really stick it in their minds that they need to use Adam to solve this problem.

One thing I am a big fan of is an open door observation policy. We build observations up to being something serious and formal when actually it is a snapshot of what goes on in our classroom every day. Ideally before I observe someone I invite them into my classroom but all too often this happens afterwards instead. I have been fortunate enough to observe many maths teachers over the years, coincidentally many have been on ratio. Recently, in an observation of a newly qualified teacher, not all learners were making the required progress within the ratio lesson (this GCSE resit class hadn’t been streamed). I invited the NQT to observe me teaching ratio the following week. They were unaware of the bucket method and saw learners using 2 different methods in my lesson. Their main question to me was how do I manage 2 different methods. I explained that experience plays a part in that and I asked the NQT, what methods did they give their learners? The NQT took away ADAM and bucket and taught both methods to their mixed ability ratio class the following week and the learner feedback was amazing. So much so that one student passed me in the corridor and said “you know you saw us in maths and I couldn’t get it? Well we did it again and I ADAM it now, it’s well good!”

Pythagoras

“Today we are going to look at Pythagoras”

Cue whoops and cheers...no? Deathly silence instead?!

It's a tough one, “my mum could never to Pythagoras so I can't” is my all time favourite line from a post 16 student.

Having been based with dance students re sitting GCSE maths, often not for the second time, I had to become more creative in my approach. The benefit of teaching post 16 resit is that the misconceptions are there, and students are not shy about explaining and sometimes oversharing why they can't do things or what has gone wrong in the past. From this I learnt that when my students looked at Pythagoras questions they could label the hypotenuse easily but not the opposite and adjacent. Then they could recall beautifully the Pythagoras proof of cutting squares out and making right angles triangle activity they did in year 10 but couldn't answer a missing side question. This is where we were and it’s my job to piece together a way forward.

Let's start with labelling, from experience making things easy to remember and build on the knowledge that's already there is a good plan. So we label the hypotenuse first. Always opposite the right angle. Learners in my experience are often OK with this skill. I discuss labelling in the blog post about trigonometry if you want to know more about this. What we were struggling with when looking at Pythagoras was that we didn't know when we were adding and when we were subtracting. Over the years I tried many many many approaches and I began to think about how we often think of mnemonics and acronyms to help learner's get along and that's when I came up with SASSY LASS. 

As you can see all it is is the steps that you need to do in the right order:

Square, add or subtract, square root, for the longest you add and the shortest you subtract. As I teach resit students only now, I know students have previously been taught a2+b2=c2 . I must explain that the starting point activity of this lesson aims to establish if students can recall a2+b2=c2 and apply it correctly then that is what we use. I only use SASSYLASS as an option for those who cannot use a2+b2=c2 correctly. 

Once we can apply SASSYLASS correctly I like to then look at rearranging a2+b2=c2 and extending learners to using a2+b2=c2. I am not limiting them to only using SASSYLASS. SASSYLASS is merely a starting point for them to get into Pythagoras and in my experience may give learners the confidence to tackle harder questions once they have practised the skills involved. Aiming for conceptual understanding is always key but as educators we often have to find a path through the fog and SASSYLASS affords me that.

Tremaine is a talented dancer and studies level 2 dance. However, he hasn’t got a grade 4 in maths and has to resit whilst studying dance. As much as we have a positive relationship Tremaine really doesn’t want to be in maths class. He would rather be dancing, the canteen, the library, anywhere but maths! We began to look at Pythagoras and our starting point starter cue Tremaine “Eh no, I ain’t doing to Greek squares and cutting and that!” 

Tremaine clearly remembers the proof lesson he must have had in his school career! After some calming down I asked Tremaine what this was: a2+b2=c2

“I aint squaring nothing. I hate this triangle (swear)”

This is our starting point. Tremaine isn’t alone, many resitters have the fear of Pythagoras. 

I explained that to sort out these triangles we needed to have a little bit of sass. I suddenly had Tremaine and everyone’s attention.

We start by squaring the numbers in the question. We then eliminate in the LASS, are we looking for a long or a short? In this example we are looking for a long so we add the squared numbers. Then we square root.

In this example we square the numbers. We are looking for a short so we subtract (circle the final S in LASS) and then we square root.

Working through examples of SASSY LASS the whole class was ready to start work. Some were using a2+b2=c2 and some were using SASSY LASS. All thankfully ended up at the same answer and within our 90 minute session we even managed to reach 3D problems. Tremaine ended the lessons by saying “You know that triangle (swear)? Bossing it now aren’t I?!”

Trigonometry

Often I have worked in a maths and science department within schools and colleges. I have zero science knowledge and the thought of even being mistaken as a science teacher terrifies me. In the spirit of doing something different and stepping out of my comfort zone I observed quite a few science lessons when working in an academy. I saw speed, distance, time being taught. I felt comfortable with this as we also teach the topic, although I admit I teach it badly. The teacher gave each student a triangle of SDT, you know the ones. 

She said cover over the one you are working out, if it's on top it's a divide, if its side by side it's a multiply. The speed (no pun intended) the students were able to answer questions was astounding. Part of the ‘bad’ part of the lessons when I teach speed, distance and time is getting students to set up the calculation correctly. Forever encountering divides when we were after multiplication! Yes the students in this science lesson had calculators but they could set the calculation up correctly, my mind was blown! After some pupil feedback it became clear that this is the way speed, distance and time (SDT) was taught throughout their school lives. It made me wonder, why do we try to do things differently in maths then? How many times had I previously taught SDT with algebraic manipulation under the algebra topic rather than a stand alone SDT lesson? Why had I never used these triangles before? Was I comfortable with students being able to solve SDT in this way without knowing how to rearrange equations? I think I was willing to give it a go as an alternative for those who struggled with algebraic manipulation. I wasn’t planning on lowering my aspirations but sometimes when students just can’t access it, they need to be shown another way. Could I start with triangles and then explain the algebraic manipulation afterwards? I think I had found a way.

It got me thinking, not just about SDT but what else we could use these magic triangles for. I stumbled upon someone's kind free resource on TES that introduced trigonometry in the same way. I will forever be grateful for the contributor to TES for these magic trigonometry triangles. 

In the same way as I saw SDT taught. If you are calculating a missing hypotenuse you would divide the opposite by the Sine of the angle.

Bravely (the same day!) I set about teaching trigonometry to my bottom set year 10s (this was in the old spec days where pretty much everyone did higher tier in the schools I worked in. We can debate the morals of that widespread practice another time, I offer this information to give context to this example) my bottom set year 10s knew that I believed in them and they had developed a level of resilience where they would be willing to give it things a go. True to their reputation and our relationship they gave the triangles a go. The magic triangles were a hit! They could calculate with trigonometry! Bottom set year 10 could find missing sides of triangles!

Over the next few lessons we even got to inverse operations and calculating missing angles. Reflecting on the book work we still had a problem in that we couldn't label triangles correctly. Once labelled correctly we could answer missing angle, missing side almost all types of question, it was just how to label triangles that was letting us down now. I then began to think about how can I help learners accurately label triangles? I realised they could do the opposite with ease (the one opposite the angle with writing on!). Then they could label the hypotenuse (the one opposite the right angle!). Finally the missing side would be the adjacent. We had cracked it, they could label triangles accurately!

Reflecting on my success I happily sat down to mark and realised we were still making mistakes. Students were using the adjacent when it should be the opposite. They were using all 3 sides of the triangle regardless what the question was asking. I decided to have a chat with my students and see if we could work out why they were using all 3 sides. Ellie immediately became frustrated and said “just tell me what information do I actually need Miss?” this got me thinking, we don’t need  three sides to solve the problem. We only need two sides. Once Ellie and I worked through this she flew! Immediately Ellie began eliminating a side, always correctly. I asked her how she knew which one to cross out and she said “Miss, there’s always one without any writing on!” She was right! 

I had great success with trigonometry this year. Was I to ask a student a trigonometry question after the exam would they be able to answer it? Probably not, but if we have them a magic triangle could they? Probably yes. Reflecting on my practise though, did the students learn trigonometry or did they learn how to use a magic triangle? We all know what the answer to that is. This was quite a few years ago and keen to develop conceptual understanding I now I teach trigonometry quite early on. It wins students over in the early weeks. Trigonometry is often a struggle and with magic triangles students seem to be able to attempt it easier. I then move into algebraic manipulation from the magic triangles rather than the other way round as it was before I taught Ellie and bottom set year 10. Last year a resit student of mine in her late 50s said, “ I didn't know why things change from a multiply to a divide and that they undid each other, but it's just magic triangles isn't it?” I replied “yes it is!”

Students can often recall sohcahtoa with ease, with a slight change in the way you say it, making the o in soh and the a in cah with the o in toa slightly higher when you say them you can write them like this naturally making magic triangles. Meaning you can always grab a magic triangle to help you get started!