I have just finished reading A Compendium of Mathematical Methods by Jo Morgan. It is a book directed at Maths teachers and has a simple purpose: sharing different methods that are used to perform some common processes that we teach. For each of 19 topics, spanning the whole of secondary maths, Jo goes into depth on a variety of methods, always using 2 well chosen examples to show some of the subtleties you might otherwise miss. Accompanying this are some of her own notes, and excerpts from historical textbooks to show how these were approached in the past. Jo stays neutral throughout the book, never saying one method is the best, but rather presenting them as they are. A few concerns about some methods which rely on following a procedure rather than developing understanding are raised, but not in a judgemental way. The tone throughout is one of trying to start a conversation about mathematical methods. When we come out of lockdown, I am going to take some of the chapters to my department to discuss. I think it is a great idea to talk about the merits of different methods, and looking at ones we don't use will help teachers develop their own subject knowledge too. I am also a fan of being consistent across the department in the main method we teach. I think this has benefits when students change teachers, and allows for more continuity. As we are a 318 all through school, we could even extend this to the primary school to discuss how we teach the foundation skills. In terms of sharing methods with students, it is also nice to have a few other methods "up your sleeve" for those situations when they do not understand the primary one you use. Or with those who need an extra push, asking them to see if they can understand why different methods are actually the same can push their understanding. Perhaps using a method comparison example like Emma McCrea discusses in Making Every Maths Lesson Count could be used. One thing that the book has made very clear to me is that we need to move to an area model of multiplication. It is a versatile and easy to understand method for multiplication, that can easily be extended to more complex topics such as algebraic expansion and factorising. I will be taking this to our department soon as I think this is something we should be consistent about. It is a great read for any Maths teacher. It is not something that you need to read in one go, and perhaps is better read by chapter when you want to look at a particular topic. And I am with Jo. Let's talk about methods.
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In Part 1 I looked at why I have started to use Booklets in my teaching. In Part 2 I looked at how I go about making a booklet for a unit. In this part I want to have a look at how I use the booklet, once it is made. There are two parts to this: how I use the booklet to plan the actual lessons; and how I use the booklet in class. In planning With a completed booklet (either from a previous year, or just done for an upcoming unit), I will set aside two copies for myself. The first is used in planning the lessons. The first thing I do is complete the booklet how I would expect a student to have theirs completed at the end of the unit. This includes doing each of the examples and your turns, writing out the notes, completing some of the exercises (depending on the skill I will do more or less of these based on how difficult they are), and the practice activities. There are three benefits to going through this process:
Alongside completing the booklet, I also add notes to myself, usually in green pen. These might be things like:
Finally, in a blue pen, I add extra questions that I want to ask along the way. This could be as part of a worked example to draw attention to a particular part, or they could just be oral questions that I don't want to include in the booklet itself. This is a process I have worked up to over the last couple of years, and I find that all three stages of the planning process with the booklet help dramatically reduce my cognitive overload during class. This allows me to give my attention to other things, such as focusing on checking for understanding and dealing with misconceptions and being more aware of behaviour issues and acting promptly. The next stage for me is to start thinking about who I am going to ask the questions to. This post from Doug Lemov explores a lesson preparation process he observed that follows a similar outline. In lesson When the lesson comes around, I am well prepared to get going. I now teach using a visualiser/document camera, and make use of the second booklet that I put aside. For the notes section of the booklet, I started by writing notes on my blank copy and getting students to copy this down, but more recently I have moved away from this. Now I go through my explanation (penned in green on my copy), and will then direct students to fill in the notes. For example, when looking at the magnitude of a vector (one of the notes examples in Part 2), I would explain with a semiscript: "A vector is a quantity with both direction and magnitude. The direction is given by where the vector points. The magnitude is the size of the vector. It is the length of the line segment that represents the vector. If the vector has a direction parallel to the x or y axis, then the magnitude is easy to count. For other vectors, we can draw a right angled triangle where the vector is the hypotenuse and so we can use Pythagoras' Theorem to calculate the magnitude of the vector. If the vector is given in component form, then we can either draw a diagram to help us, or use the formula a = sqrt(x^2 + y^2) for a = (x,y)." Then I would direct students to the page in the booklet, and ask them to fill in the notes based on what was just said. This acts as an immediate form of retrieval practice for students, and gives them a second exposure to the idea. Whilst students write it down, I will circulate and check what they are writing over their shoulder. After time to do this, I will cold call on a student to read out their notes, and ask all students to check theirs against this. This gives students a third exposure to the notes in a short period of time. I also use the visualiser for doing the examples. I will ask all students to put their pens down, and look at the board. I will then go through the example completely, using silent teacher as described by Craig Barton. I will then ask students some elaboration questions to get them to explain what I did at each stage, and why. If I think there is a step that the class will not know why it is there, I will narrate it instead. Then I will give students time to copy the example down (exactly as I wrote it), and they can move on to the Your Turn. As they do this, again I circulate the classroom, with my complete booklet, looking at their work and checking it against mine. If it is a very short question that requires little work, I would probably not do this. Then I will cold call a student to give their solution, talking through the whole solution, not just giving the answer. At the end I will ask the rest of the class if they agree. The student I cold call may have given an excellent answer or made a misconception (that I picked up on when circulating). The other way I use the booklet is when students are doing practice on the big whiteboards. I have found that having students hold their booklet as they do this just gets in the way, so I will project the activity on the board for them to refer to. As I go through the lesson I will also make further notes in my completed copy (with black pen) as to any issues that arise with the booklet, or anything I want to change for the next year. Perhaps a note to add more examples, or remove some, or how to make the explanation clearer, etc One thing I have not been doing, but am planning to start doing, is to also take notes of students who struggle with a particular part (I guess a purple pen for that!). This would give me something to refer back to when assessments and quizzes come up. After the unit After the unit is complete, I scan the 'live' copy of the booklet (the one I worked on under the visualiser) and upload it to the class website. This acts as an aid to those who missed any lessons as they can see some bits of notes and also all the examples. This is the reason I don't write the your turns in this booklet, so that students can download a new copy to do the your turns again if they wish, whilst having the examples. Do you use booklets in your teaching? How do you use them effectively to plan for lessons? How do you use them in class?
Next week we have our taster lessons for the new IB cohort. The IB syllabus has changed this year, and there are now four options that students can choose from.
I will be teaching the AA SL course next year, and for the taster classes we wanted to choose a topic that would show the difference between the courses. We (well I wasn't actually in the meeting, but I agree) chose to use proof. This will make good use of algebra skills, ensuring students know this is a prerequisite for this course, but also gets to the heart of Mathematics. In preparing for this I have put together a couple of documents. First is this learning map which shows an overview of proof as a topic. They will only need to do Direct Proof and Disproof by Counter Example in the AA SL course (though Induction and Contradiction are in AA HL), but I think giving them a broader picture is important. I will reveal this a step at a time, as the powerpoint builds up slide by slide (something like in the gif below). I also put together a lesson sheet for the topic. I have talked before about how I have started using lesson sheets in IB and booklets in IGCSE. This lesson sheet includes examples and plenty of your turns for the students to have a go at, to test their skills in proof. We will see how the lessons go next week, and I will post an update.
I recently read the excellent Making Every Maths Lesson Count by Emma McCrea. Like the original, it is an engaging read, but I really liked the subject specific ideas that this version had to offer over the original. In this post I am going to identify 1 idea from each chapter that I am going to work on over the next year or so. 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:
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 nonexamples 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 nonexamples 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 nonexamples. 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 miniwhiteboards, 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 subskill? Sometimes a skill is broken into smaller subskills. Rather than creating booklets with 20 skills for a unit (which I feel can be a bit overwhelming) I will not relabel these subskills, 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 subskill 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 subskills 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 subskills 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. Do you use booklets in your teaching? What do you include in them? Do you do things differently to me? If you don't currently use booklets, could you see any benefits in having a resource like this?
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 preplan 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 35 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. Do you teach using booklets? If so what are your reasons for using them? If not, have you ever tried it? Is it something you would be willing to try?
I have previously blogged about How I Teach and in more depth about my Weekly Quizzes. In this post I am going to go into a little more depth about the way I start my lessons, using what I call Last Lesson, Last Unit, Further Back. This strategy is based on the idea of spaced retrieval practice, which incorporates both the Testing Effect and Spacing Effect, two of the most well documented ideas in the science of learning. The testing effect says that we learn better by forcing ourselves to retrieve knowledge from our long term memory, as opposed to restudying it. The spacing effect tells us that we remember material better if we space out studying out over time, rather than cramming. Both of these ideas are also considered to be desirable difficulties by Bjork in that they make initial performance lower, but long term learning better. One of the important things with spaced retrieval is that it is most effective if done on the verge of forgetting. This is when it has the biggest impact on learning. However, the time taken to get to this point increases with each subsequent retrieval.
Each year, our students go away for a trip that incorporates some activities, service projects, and outdoor education. But these are done in half year groups, so half the year is away Monday, Tuesday, Wednesday and the other half are away Wednesday, Thursday, Friday. When S3 were away in Tambopata, I had two doubles with them, but in each double I only had half the class (and a few from other collapsed classes). In order to make the most of this time, I wanted to do some activities that would get them ready for the next unit we are starting after they got back, which was functions. Students have previously met the idea of functions, function notation and domain and range, and this unit will focus on composite functions and inverse functions. However, it has been a couple of years since they saw them, so I wanted to review the basics before moving on. I started with this activity asking students to write functions given in words as algebraic statements (taken from here  thanks to Jo Morgan for pointing me in the direction of this reference).
Our latest unit with S3 has been on teaching straight line graphs and inequalities. This covers the basics of finding equations of lines from graphs, drawing lines from the equation, finding equations from descriptions (eg gradient and a point), parallel and perpendicular lines, inequalities on the number line, solving linear inequalities and drawing and describing regions on the coordinate plane using inequalities. In this post I am going to talk a little about how I approached this last objective this year. First I checked that all students were able to draw lines from equations, and were relatively confident with this. This had been something we had focused on over the previous couple of weeks, with it popping up in the retrieval starters on a regular basis, so I was not expecting any problems at this point. All students were able to complete this task confidently. This year I have been focusing on giving appropriate examples (followed by a your turn question) and in trying to break processes down in to the constituent parts. To teach sketching regions given by inequalities, I took some inspiration from the excellent Math = Love blog, and created this template for students to use. For each example and your turn, I gave students a copy of this template within the work booklet that I print for them. The broken down structure helped the students to scaffold their thinking in the early acquisition of this skill, by prompting them in to each step. As students gained experience with answering the questions, the template was removed and they had to answer the questions from this Corbett Maths worksheet.
In IB Mathematical Studies, students have to be able to complete a Chi Squared Test to determine independence (or not) of two variables. The full process of carrying out a Chi Squared test is quite long, and so I decided to break it down into small steps, and get students to master each step before adding on the next one. The steps are listed in the image below, which I showed to students at the start of the unit.

Dan RodriguezClark
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. Categories
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