I have done a few sessions with students recently on the importance of sleep, as lack of sleep has become a chronic problem for our sixth form students. They are staying awake into the small hours of the morning to complete work that they have left to the last minute, thinking that sleep is a luxury that they can do without. 

But that is simply not true.

Based on my reading of Why We Sleep by Matthew Walker, I have been being more explicit about the dangers of a lack of sleep, but also the benefits of getting enough sleep. I have been particularly focusing on the way sleep relates to learning, but have also been talking briefly about the other health concerns.

I have started with a brief introduction to the importance of sleep for the learning process, and I have cobbled together an analogy that I think sums it up quite well.

Our brains act like a sponge, absorbing lots information throughout the day. As they day progresses the sponge gets more and more full. When we sleep, it is like squeezing the sponge into a bucket: the new learning is safely stored into our long term memory (the bucket). Day after day we keep adding more stuff to our bucket. Of course, when we squeeze the sponge some of the water splashes out and misses the bucket, and we lose that water, but most of the water is makes it into the bucket. When we wake up, the sponge has been completely emptied, and is ready to absorb a new set of learning the following day. 

I have been using this analogy to highlight the two benefits of sleep towards learning: firstly it helps consolidate our learning from the previous day by transferring ideas from our pre-frontal cortex into our long term memories; and secondly, it leaves our brains in a more receptive state to learn the next day.

Of course, when we do not get enough sleep, we do not fully squeeze the sponge, so not everything makes it to the bucket, and we do not have as much ability to soak up new knowledge the following day as the sponge has not been fully emptied.

Or if our sleep is of low quality (such as when we drink alcohol or take sleeping pills), when we squeeze our sponge it is like having a shaky hand, and much less of the water makes it into the bucket.

This analogy works for the deep sleep cycles, and seems to get the point across. 

But it does not work as well for the importance of the REM cycles. This phase of sleeping is also vitally important to learning, as this is the time when our brain starts to make connections. As many mathematicians know (and I am sure many from other walks of life) when we are stuck solving a problem, one of the best things to do is to go away and come back the next day. Part of this is due to the power of the REM cycles of sleep, where the brain continues to work on the problem, accessing that deep bucket of knowledge you have stored away in the daily sponge cleansing.

Having talked about the importance of sleep for learning, I then share some of the startling facts and figures that Walker shares:

• A 24 hour period of no sleep leaves you in the same cognitive state as being legally drunk. So pulling an all nighter before an exam would be equivalent to walking into the exam drunk. I think all students would agree the latter is not sensible, so clearly the former is not either;
• Going 10 days on only 6 hours sleep a night also leaves you in the same brain state. Many of our students fit in this category, and I make the point to them that they are walking around drunk at school if this is the case. Of course they can't learn or work as effectively and efficiently if this is the state their brain is in;
• When Daylight Savings Time deprives us of one hour of sleep each year, the global incidents of heart attacks increase by 24% on the following day. When we gain an hour of sleep, the incidents of heart attacks reduce by 21% the following day;
• A lack of sleep reduces the effectiveness of the blood cells that fight and kill infections, and as such the World Health Organisation has listed night-time shift work as a "probable carcinogenic" based on the fact that night-time shift work disturbs the natural sleep cycles.

Many students bemoan that they just don't have time to sleep as they have too much work to do. I point out to them that if they were getting enough sleep then the work would take far less time as they would not be trying to do whilst cognitively drunk, and that, more importantly, their health is far more important than any piece of work.

I finish with some recommendations for sleeping better:

1. Be consistent - have a sleep routine and stick to it every day. Go to bed at the same time, and get up at the same time, every day (weekdays and weekends);
2. Screens away before bed - the blue light that screens emit trick our brains into thinking it is daytime, and so stop them from naturally shutting down to sleep;
3. No caffeine/alcohol before bed - both drugs have a negative impact on sleep (caffeine lowers quantity, alcohol lowers quality), and caffeine in particular stays in the system for over 6 hours;
4. Keep it cool - our natural sleep cycle is kicked off by the cooler night time air, so keeping the bedroom cool (around 18 celsius) will help the body drift into a natural sleep, and maintain a good nights sleep.

In part 1 of this series I discussed why I have been won over by the humble booklet. In this post I am going to expand on how I design my booklets and what I include in them. I will include images from some of the booklets in the post, but I am not able to share whole booklets as I use some material that I do not have permission to share. I will reference to the main sources I use for each section of the booklet, and give some images of the types of resources used. You can find one full example on Coordinate Geometry here.

The front page is fairly simple, with the unit number and title, a space for students to write their name, and the video numbers linked to the topic on www.corbettmaths.com. I also have a back page to all the booklets which has references to websites I use to put them together.

Within the unit I start by breaking down the objectives into individual skills that students will need to master. So, for example, in the advanced trigonometry unit there is a skill for sine rule, one for cosine rule and one for identifying which one to use. Within each skill I will break them down into smaller sub skills if necessary. So sine rule is broken down into finding missing lengths and finding missing angles.

So the final break down of skills and sub skills for the advanced trigonometry unit is:

• Right Angled Trigonometry (recap of prior knowledge)
• Angles bigger than 90 (includes labelling triangles and calculator skills)
• Sine Rule (split into lengths then angles)
• Cosine Rule (split into lengths then angles)
• Area of a Triangle (finding area then finding other missing information)
• Choosing the Rule
• Bearings
• Trigonometric Graphs
• Exact Values (the standard results)
• Other Exact Values (given value of sin x find cos x)

With each skill identified, I go about planning them following the same format.

Required Prior Knowledge
The skill starts with a short item on the required prior knowledge for that particular skill. Sometimes this is a recap of a prior skill from the current unit (eg factorising quadratics before solving them). Sometimes it is something from an earlier unit (eg solving equations before sine rule).

As can be seen in the examples below, these take a variety of forms. Some are simply questions on processes that need to be secure. Some are ideas that will lead into the current skill. One of the things I need to work on is developing these to cover ALL the prerequisite knowledge and skills that have not been covered already in the current unit. 

The point of this section is to help me and students identify if they can do the necessary skills required to do the new skill. If they can't do them, then the lesson will adapt to address those issues first, before moving on with the new skill. There is little point in teaching students to solve the Sine Rule, if they cannot solve equations with the unknown as part of a fraction.

It is worth noting here that these are not meant to be lesson starters. I use a retrieval starter of Last Lesson, Last Unit, Further Back as the 'Do Now'. In single periods I have started doing a single retrieval question rather than four, usually from last lesson. I do sometimes plan the Further Back question to address any required prior knowledge too, but this might be a couple of lessons ahead of teaching what requires it. 

Notes
Next there is a section for notes. In terms of teaching, this is when I will explain the new skill, and give any definitions, etc. 

The notes section is structured as a fill in the gaps exercise, usually with a sentence starter given, and then some space. I also have prepared powerpoint files to go alongside the booklets (though I have stopped using them as much) which line up with the notes section. I now prefer to say and explain the idea and give students time to fill in the gaps themselves. 

I have been thinking a lot about the use of non-examples at this point of the booklet, and although they are not embedded in them at the moment, I will be adding space for these in the next iteration of them. I am thinking of adding Frayer Diagram templates as well as a way to structure students notes on the definition, characteristics and examples and non-examples of concepts. I have used these a little at the end of units as a reflection activity, but I think they have potential to form part of the actual notes students produce as well, and would push me to think more deeply about non-examples.

Example Problem Pairs
The notes section is followed by sets of Example Problem Pairs. These largely follow the idea set out by Craig Barton here, though I have not been so careful with making them minimally different yet. Perhaps I will adjust these moving forward.

Printed are both the example and the your turn problem, as you can see below. This allows me to give students the questions (no time spent copying out questions), and include any images so they don't need to draw them. It also allows me to include graph paper when necessary, along with any other diagrams (for example they can write straight on transformation examples).

Of course this does introduce the potential issue of students rushing ahead and not paying attention to the example (I will discuss how I deal with this in Part 3).

The biggest issue I have found with this layout is that I need to make sure I include enough space for students to write their answers! They tend to require a lot more space than I do to answer a question (bigger handwriting is one problem, but also the fact they are novices so can't "see" the way forward as easily and so jot things around a bit more). This is one of the things I take note of when annotating my copy of the booklet for adapting the following year.

I have also just finished reading the excellent Making Every Maths Lesson Count by Emma McCrea, and recently listened to the Mr Barton Maths Podcast with Michael Pershan, both of which mention Algebra by Example. In particular they mention the use of incorrect examples, and this is something I want to explore further within the booklets. Getting students to review an incorrect example, or compare an incorrect with a correct example, sounds like a great way to get them thinking about the details a little more.

There is also incomplete examples, where students have to fill in the missing bits, as you gradually reduce the amount that is given to them. Booklets would be great for this as you can have them all printed a ready for students to write on.

The number of example problem pairs will vary depending on the skill. For example, in using the sine rule to find lengths, there is a single example problem pair. But in graphing regions using inequalities there are a total of 8 example problem pairs. These are included to go through the different variations of the types of questions that can occur. If a class is moving on fine though, I might push them to do the remaining examples themselves, rather than working through them.

Exercise
After the example problem pairs, there will be an exercise. This will probably be a fairly classic set of questions to practice the new skill they have just learned. I do sometimes make use of the sets of questions from Variation Theory, but also use CorbettMaths, Dr Frost Maths, 10 Ticks, exercises from our ebooks, Pixi Maths and my own site Interactive Maths. These are not the only ones I use, as I also get stuff from TES and Resourceaholic, and have found the old textbooks great for some of these too.

I like to include more than enough practice in here for students to do, so will generally have much more than I need. This is also the bit of the booklet where students write in their own exercise books to save space, so I can bunch questions up as much as possible. This allows me to choose what I want them to do based on how they are understanding the material.

For each skill there is also an accompanying powerpoint that has the answers to most of the exercises.

Further Practice
Sometimes I will also include links to even further practice for students. Usually this is to a page in their textbook or CorbettMaths, but also CIMT. We very rarely use these in class, and they are provided as extra for students to do in their revision outside of class.

Test Your Understanding?
These are not always included in the skill, but when they are I use them as a quick way to check before moving on to another subskill, or before more independent practice. These will usually be answered on mini-whiteboards, and are there in case students struggled with the your turn and need a little more guided practice before moving on to the exercise. They normally include 4 questions similar in style to the example problem pair.

Sometimes instead of including these in the booklets, I have used diagnostic questions as part of the powerpoint, which I project and students answer by raising the number of fingers that correspond to the answer they think is correct.

Another sub-skill?
Sometimes a skill is broken into smaller sub-skills. Rather than creating booklets with 20 skills for a unit (which I feel can be a bit overwhelming) I will not relabel these sub-skills, but rather incorporate them into the bigger skill. For example, in the skill of Sine Rule, there is a section on finding lengths and then on finding angles.

I also make a lot of use of these in the first skill of a unit when that is largely prior knowledge. For example, in the unit on Quadratic Equations the first skill covers expanding, factorising and use of the graphical calculator. I do not want to dedicate a whole skill to each, but this does mean there is some material on these if I discover I need to reteach some bits of it.

Each new sub-skill will follow largely the same layout as above. I am more likely to use Test for Understanding instead of an exercise if there are lots of sub-skills that build up, and then include the exercise at the end of the whole skill.

Activities
I sometimes include activities like matching activities, or odd one outs. These will often cover the whole skill and so will be included at the end of the skill. Sometimes this is just a blank space with a title to stick in the cards. Other times there is more structure. It depends on the activity. This is where I still get to include some of the great activities you find on TES.

Challenge
At the end of the skill there is usually a challenge section. What I mean by challenge is that it is not your "ordinary" style questions. This is where I include things like Maths Venns, stuff from Don Steward, Clumsy Clives, Arithmagons, stuff from nrich, UKMT questions. This is not something I included from the beginning so not all booklets have them yet, but I will be adding as I find new things and adapt them the next time I teach the unit.

More so than other sections, this is the one I find most useful to have available at any time, as I can push students who have demonstrated a basic understanding on to these tasks to develop their understanding further. 

Unit Review
At the end of the booklet I like to include a unit review section. This will always include a Unit Review Worksheet which basically has a two questions on each of the sub-skills from the unit. It is meant to be used by students to assess themselves on what they can and cannot do.

There are also sometimes activities that cover the whole unit, though these really do depend on the unit in questions.

Exam Questions
For some units I also included a section of exam questions on the topic at the end of the booklet. We already have a set of documents of exam questions by topic, so this is not something I have done religiously as they already existed. However, I am starting to think that including them at the end would remind me to make use of them more often, and would truly enable students to have everything in one place. 

Concluding Remarks
In part 1 I discussed 10 reasons why I have been won over by the use of booklets. Some of these are determined by the way I make the booklets (e.g. having everything in one place). Although there is definitely an initial time commitment to putting these booklets together in the first place, in following years you only have minor tinkering to do for a whole unit, which allows you to focus on how you can best use the booklet in your teaching, and how you can use that extra time made available to teach better. In part 3 of this series I will be exploring how I go about using the booklets I produce, both in planning and in class.

Learning Map
In High Impact Instruction, Jim Knight talks about what he calls Learning Maps. These are designed by the teacher to show the learning of a given unit, but also emphasise connections. The completed Learning Map is a revision resource for students, but also lends focus to lessons. He suggests starting lessons by referring to where you are on the map, and finishing by updating the map with what you have learned that lesson. 

As well as being a tool for in the classroom, I particularly like the way they can help me plan out a unit. So I decided to give them a go in the recent unit I taught on Differentiation to my IB HL class. ll as being a tool for in the classroom, I particularly like the way they c

Before planning the unit I started by drawing up a rough version of a map that showed all the things they needed to learn, grouped under some headings. With this, I started planning the Lesson Sheets for the unit, working my way through the objectives listed on the rough map. Once the sheets were finished I then drew a neat finished map, with more details rather than just headings. The point was that all things covered would be on the map.

To start the unit we drew the outline, with just the headings. I did this under my visualiser and students made their own version. As we went through this I summarised the key points verbally. Anecdotally, I felt that this gave them a big picture of the unit. It allowed them to see what we were going to be doing, and where we were going. Given that many in the class had done Additional Maths at IGCSE, they had already seen the basics of differentiation, so giving this big picture showed them there was a lot more to it than in Add Maths!

Then, every few lessons, we would take out the map and add to it the details of what they had learned. This included key points and definitions.  As the unit progressed, the map grew in detail, and became more like my version.

Of course, there were a few things that I missed off my version of the map (using induction with differentials, some common derivatives) that we actually did talk about in class, and we added these to the map too. If I missed something off that I had mentioned in class, students were quick to say it. Often these were things said in passing, that I hadn’t planned, but next time I will know to plan them more specifically into my lessons.

The end result is shown below (and available as PDF here).

The students said it was a useful exercise to see the topic grow organically, and also liked the fact that they had a one page summary revision sheet. One student commented on how she was a little scared at the start as the map was so big, but that filling it in helped her realise the connections between the different things we looked at.

Overall I was pleased with this first attempt, and I will continue to use them with this class. I am not sure if I will start referring to it every class as Jim Knight suggests, but we shall see.

Differentiation Increasing Activity
After listening to the Mr Barton podcast with Chris McGrane, I have been thinking more about task design. I really like OpenMiddle problems, and MathsVenns. But the More/Same/Less idea is one I have not used much. After trying a logs one from OpenMiddle, I decided to give making one a go for my IB HL class.

The activity created some excellent discussion amongst students, with them arguing with each other about answers. One girl took the particularly interesting route of sketching what each must look like, before finding functions that would work. Most of the groups missed the fact that boxes beneath each other can't just have f'(1) less than 0, but that all three boxes needed to be equal, but perhaps that is a less important part of the process, as they all thought hard about how to input functions in the right places.

I am becoming a big fan of this type of activity, and will try to build them into my teaching more often.

IGCSE Booklets Part 1
I posted the first of a three part series on using booklets with my IGCSE classes. This first post looks at 10 reasons why I love booklets. 

Ingredients for Great Teaching
I wrote a summary blog post of this excellent book for my school T&L Blog.

Over the last couple of years I have moved to a booklet model of teaching my IGCSE classes. In other posts I will detail how I put together a booklet and how I plan lessons using a booklet. But in this post I want to start by exploring what I mean by a booklet and why I decided to move towards using them, and why I am now won over by their usefulness.

I design booklets for each unit. They cover the different skills within a unit, building up to exam style questions. A booklet is designed to contain all the resources I might need whilst teaching that particular topic. That does not mean I will use everything in the booklet, but that I do have a variety of things available to choose from. Depending on the class, I will adjust what I use. 

So why do I use booklets? Here are some of the reasons I have come to really appreciate them.

It forces me to think about the whole unit (or learning episode)
The first huge benefit is that it forces me to consider the whole unit when planning, not just focusing on lessons. There has been a lot of talk recently about the lesson being the wrong unit of time to plan for, but when our time is split in that way, I find it difficult to not plan in those chunks. Using booklets has helped me break through that barrier.

In creating the booklet I have to do it before I start teaching the unit so I can give the complete booklet to students when we start. This means I have to think about all the individual skills that form a part of the unit, and how they connect to each other and build up to the big picture. It means I have to consider not just the order in which I will teach these skills, but how I am going to link them together. Rather than teaching a series of 10 lessons, I now teach the unit. Of course I plan what will go in each lesson, but this is really flexible as we can just pick up from where we finished last lesson. So if we get through it quicker than expected, we can move on, and if it takes a bit longer, there is no need to rush at the end of the lesson.

Initial time input but saves time in the long run
Putting the booklets together in the first place takes a long time. But now I have a set of booklets on 21 units covering the IGCSE, and I can reuse them again and again. In reality, I make adjustments each year, but the bulk of the work is done. In future I can plan a whole unit in about an hour as I just need to review the notes I made the last time I taught it, and make the necessary changes.

I can plan for interleaving and Retrieval of linked prior knowledge more easily
When planning lesson to lesson I always found that my focus was on the current bit of new learning, and rarely did I think about interleaving other topics in. But with a bigger picture of planning, I can add more interleaved exercises within the booklet. 

I don't currently do this, but you could also pre-plan retrieval of prior topics within the booklets. You could design an optimal spacing schedule and plan in these retrieval opportunities within other units.

No running for last minute photocopies
As everything is in the booklet and the booklet is printed for the start of the unit, there is no need to be running trying to get the worksheet copied just before the lesson. It is also cheaper on photocopying as I am not copying things that I end up not using, and there is little wasted white space within the booklet. Three separate worksheets might fit on a single double sided page, instead of 3 single sided sheets.

Changed focus on lesson planning from finding activities to thinking about explanations, what I will use, how I can supplement
In the run up to a particular skill, I no longer have to spend time finding/putting together a lesson/activity to use. I can focus my attention on thinking about how I will explain difficult concepts clearly, what visualisations I could use to enhance my explanations, and any other materials that might enhance the teaching of that particular skill.

I don't forget any skills
Perhaps not groundbreaking, but I can't forget to teach something. It is all there and in my face. I can't get to the end of the booklet without teaching everything from it. Of course, I could forget to include something in the booklet, though that is less likely. What does happen is that I realise I need to break a skill into more smaller bits, but I can just take a note in my copy of the booklet to refer to later.

Constantly improving
And on that note, whilst teaching I can easily annotate my copy of the booklet. This means I can note anything that doesn't work, or works particularly well, as a reminder for next year. As some of my colleagues are also using the booklets, my hope is that they will start making suggestions too and the booklets will continue to improve each time they are used. No need to reinvent the wheel each year.

And because I don't need to focus on creating the whole thing each year, I can give my attention to finding/creating more interesting problems. This year, for example, I have tried to put more Open Middle and Maths Venns problems into the booklets.

Everything in one place means it is more efficient to navigate to content in lessons - means I can be more responsive in my teaching
With everything in the booklet it is easy to navigate as I just say the page number they need to turn to. No getting out different books, or finding the ebook. For most things they don't even need their exercise book as they can write straight in the booklet. This saves maybe 3-5 minutes every lesson, which over a few weeks really adds up.

The other advantage to having everything in one place is that I can be more responsive in the way I teach. If students need more practice, there is loads in the booklet so we can just carry on with that. If some students need to be pushed a little harder, there is a challenge question (available for all students, not just the 'high achievers'). If the whole class is ready to move on within a lesson, that's fine, we can just move to the next skill. No filling time as I don't have resources prepared.

Standardise the format
In Teach Like a Champion 2.0, Doug Lemov discusses the strategy he calls Standardize the Format. The idea is that I can save time and effort checking student work if they all answer in the same format. Booklets are perfect for this as they guarantee that all students will write in the same space. Walking around the classroom you can quickly look to see every response, as they are all in the same space, so you don't need to hunt for them.

Students can use them for revision
My students have been particularly happy with the booklets in the run up to exams. The booklet gives them a structure to their notes, clearly shows examples, and has plenty of practice questions for them to do. I provide an electronic blank version of the booklet too, so some students use this in their revision, printing it off and filling in the examples and your turns again. What a great way for them to practise the skills they need.

So there are 10 reasons I have grown to love the booklet. Many of these relate to workload issues, and many more relate to better teaching. I feel that by using booklets I have been able to focus more on my teaching (explanations, examples, models) and less on the activities. Moving towards using booklets has happened alongside my general switch to a more explicit teaching methodology. I love them. And my students are also overwhelmingly in favour of them.

In the next post I will be looking at how I actually go about making a booklet and what I include in them.

Reviewing Mocks
As a southern hemisphere school we have just had our mock exams, so I thought about way to return them for the biggest impact. I put the question to twitter and got some great responses (given that it was the summer holiday for UK teachers).

What I ended up doing was giving students back their papers and getting them to go through them and fill in the following template that I printed off. I got this idea from Blake Harvard's post From Unknown to Known in the Classroom.

As they went through the paper I wanted them to write down what their known known's were and, more importantly, what their known unknowns were. I then wanted them to correct their errors, making use of each other first (for most questions at least one person in the class got it right), or referring to my worked solutions.

I found the template useful, but did need to push students to be more thorough and specific in using it. Just writing trigonometry is of no use, they needed to identify what exactly was the problem in trigonometry. Was it spotting the need to use trig? Or applying the rules? Etc. I think for this to be really useful students need to become practiced at using it, and I need to model how to use it well. It is probably too late for my students doing their exams in a month, but for my S5 (Year 12) class with whom I do weekly quizzes, I am going to start building this into the following lesson for them to reflect on the quiz

Trig Relations
At lunch today, a colleague and I were talking about Trig, and one thing that came up in conversation was that if a+b=90 then sin(a)=cos(b). Not that this was something that we did not know, but it certainly was a case of the curse of knowledge as neither of us could think of a time we actually taught this to our students. We both felt this was a useful thing to explicitly teach them.

Implicit Differentiation
I had a good lesson (as far as it can be judged at the time) with my IB Higher Level class this week where we started implicit differentiation.