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).
Over the last few weeks I have been leading a small group of teachers as we look to define what Great Teaching looks like at our school.
The document will form the basis of our future INSET, as well as part of the instructional coaching programme I am looking at implementing over the next couple of years. My hope is that we can agree on a set of "safe bets" (as Tom Sherrington likes to call them), that all teachers can make use of to improve their teaching, and ultimately, the learning of our students. This would not be a prescribed checklist of things we expect to see in every lesson, but rather things we expect teachers to think about, and that probably would be seen over a period of time.
Another aim of this process is to have a set of words and phrases that we all recognise what they mean, giving the school a unified language when it comes to talking about education and learning.
The process started in August when I sent a survey to all teachers, students and parents asking them this exact question. I wanted to get a range of responses and see what different groups within our community had to say.
Before the first meeting I put together a small booklet with the following resources in it for each member of the groups:
After a brief discussion, I asked the team to read through the booklet before the next meeting, where we brainstormed all the words and ideas that came to mind from what we had read. In this meeting we reiterated the point that we were looking for things that describe great teaching not great teachers. Although subtle, I believe there is a difference between these two. I also made it clear that we want a list of practices that can be seen.
The end goal is to have a one page document that summarises the points, but also an accompanying document that explains what is meant by each in more detail, and provides examples of what this might look like in the classroom. If we can't explain what it would look like to an observer, then I was reluctant to include it in this document. I would also like to include references to places for further reading on each aspect as appropriate within the document.
We are nearing the end of the process now, and it will need to go to SMT to be approved. At this stage we have 12 points, given below.
I would be really interested to hear what others think of this list. Are there any glaring absences?
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.
As a starter for my first year IB students I use a random question based on past exam style questions (my generator of these is here http://classes.interactive-maths.com/mssl-exam-qs.html). In this lesson I was using question 25, a topic that students find really difficult. Below is the actual question that appeared.
After some fumbling around, a few students were able to answer parts a and b, but they were all pretty stuck on part c. A couple of students had some initial ideas, so I asked them to share what they thought. I then informed them all that I could see at least 4 different methods for solving the problem using methods we had seen, two of which were put forward by students.
I asked the students to come up to the boards and start their solutions, even if they could not finish them. They both started correctly, and I continued the method, with questioning to the final answer.
Method 1 - Simultaneous Equations
Method 2 - Axis of Symmetry
Method 3 - Root Form
Method 4 - Vertex Form
Although this starter ended up taking 30 minutes(!) it was a useful discussion to have. For the students, they got to see four different methods that would get them to the answer, so helped them realise there is not just one correct way to do things. And for me I got to see which methods they preferred, and which they found difficult.
I will be revisiting this question in a few lessons time to review it and see how well students can recall how to do it, and I will be challenging them to remember all four methods. This is one of the reasons I love randomly generated questions, as I can do the same skill but with different numbers really easily, and it also gives me a quick starter activity (with answers) for each lesson.
This is the process of how I have approached developing attacking exam style questions with this course, and it seems to have been working pretty well. When I teach the content, all the practice is on the basic skills, focusing on developing fluency and some understanding. Then after a couple of weeks I will use a starter exam question on that topic. This introduces some spacing and retrieval, and has given students some time to consolidate the new learning. By then returning to exam questions periodically for that topic, students get regular retrieval opportunities, and each time they are able to further develop their conceptual understanding, having had time to process the information.
In IB Mathematical Studies students have to recognise the Vertex (Completed Square) Form and Root (Factorised) Form of a quadratic function. Although they have a graphical calculator to help them sketch the graphs, they need to know the links in order to find the equation of a given graph. This is utilised in analysing data and creating models that follow a parabola.
In previous years I have taught this through a guided investigation which has students use technology to discover what happens in each of these situations:
This year I decided to try something a little different, following one of the ideas from the amazing variationtheory.com, the activity type that Craig Barton calls Demonstration (https://variationtheory.com/demonstration/).
I started logarithms with my (second to bottom) S4 class this week, and I think I managed to introduce it in a way that really helped the students to understand what a logarithm is. First I started the lesson with this recap set of questions on indices.
As the students were completing it I realised my error in including 4^(1/2) as this can lead to misconceptions that an index of 1/2 is the same as halving. I probed this after the class completed the questions by asking what 9^(1/2) is, and most of them correctly recalled that it was 3.
In an attempt to make use of retrieval practice (also known as the testing effect), I have tried several strategies in my classes. In this post I am going to talk about why I have started using weekly review quizzes, how I run them, and reflect on the successes and failures.
The evidence that retrieval practice is an excellent way to learn is vast (http://markhamtl.wixsite.com/teaching-learning/single-post/2017/11/07/Retrieval-Practice). But retrieval is simply the act of brining something to mind, so do we need to use quizzes? Well, the answer is no, we do not need to, but I think, for Maths at least, they are the most effective form of retrieval.
Things like brain dumps are an excellent informal way to make use of retrieval, but in making a quiz I can tailor it to the topics that I know students need to review, and also, the individual aspects of the topic. They also allow me to comment on things such as mathematical layout, the importance of working and other key skills which cross all topics.
Another aspect of quizzes that I like is that I am hoping they will help students disassociate tests with grades a little. If they are doing low (or even no) stakes quizzes regularly, then it just becomes part of life, rather than something high pressure to worry about. I am not sure this has taken effect yet, but I am still hopeful!
How I run the quizzes
I am currently doing this with my S3 and S4 classes every week. They know that they will be having a quiz each week, and at the start of the year I explained my reasons for doing this. Firstly, there is writing the quiz. It is split into three sections: This Unit; Last Unit; Further Back.
In this post I am going to share some of the ideas I use in my day to day lessons. These aren't lessons that I have prepared specifically for an observation, or those one-off lessons designed to engage/challenge students beyond the curriculum. These are the bread and butter of my teaching. The things I do every day and every unit. Some of these are things I have been doing for a while, some are relatively new, and a few are actually brand new (I have started them in the last few weeks). This post was inspired by this post (https://teachinnovatereflectblog.wordpress.com/2017/12/29/just-me-doing-what-i-do/) by Ben Gordon.
Planning the unit
First off, I think it important to acknowledge the bigger picture for any lesson, and begin by planning the unit as a whole. Before anything else, I identify the different objectives I need to explicitly teach, as well as the prior knowledge students should have to be successful in this topic. Identifying all the individual items that will need to be covered in a given unit helps me clarify exactly what it is I need to do, and what I want the students to be able to do by the end of the unit. This is something I have always done, but the process has become much more rigorous since reading this excellent blog post (https://tothereal.wordpress.com/2017/08/12/my-best-planning-part-1/) by Kris Boulton. Below is an example for our first unit of Year 10 on Percentages.
I started to flip some of my classes two years ago. I started with my class just starting on the IGCSE in the first year, and after some positive feedback from students, as well as evidence of good progress (both academically, but possibly more importantly, in students independence), I decided to expand slightly last year. I continued it with my IGCSE class, did it with my Year 10 class sitting the IGCSE and Additional Maths simultaneously, and also with my IB Standard Level class.
So what have I discovered about this method of teaching? Some practicalities on the homeworks first...
Firstly, it is a lot of work. I have been using videos for the homeworks, and for each objective, I need to find or make a video. I have been trying to produce my own videos for the IGCSE content, as I have found that the students react better to material produced by me than by other teachers, but that is very time consuming. If I do not make my own video, I need to find one that I like and that teaches the material in a way consistent with my own teaching. That means watching it first, which is also time consuming.
Secondly, early on I realised I needed some kind of accountability for the students. There needed to be a way for me to check they had watched the video. I started with questions in class, but this proved difficult to guarantee they had watched it. I then started to use Google Forms to create 2 or 3 questions for students to complete on the video before class, so I could check their basic understanding, and have something concrete to know they had done something. This worked well, but was also time consuming to set up, and still didn't guarantee they watched the video (copying homework in our school can be a big problem). Eventually I found EDpuzzle, which is an excellent tool, I talk about below.
Thirdly, find a way to set the videos. Most schools have an online system now where you can set a link direct to the video. This works well. But EDpuzzle also takes care of this. You can import a video directly or from YouTube (or any number of other video sites), and cut it to only be the bit that you want (cut out the long introduction or the finale). You can then add annotations to the video which pop up as the student watches the video. These could be voice notes or typed notes. Best of all, you can add questions (open or multiple choice) throughout the video. Best of all, students sign up to a class, and then you can see exactly how much each student has done (how much of the video they watched, did they skip bits, their answers to the questions). All this is visible in real time, so you can check before the lesson if they have done the work, and identify any misconceptions. It also tells you when they did it (I have had to talk to a couple of students about sensible working times when it was registered at 3am).
But the homeworks are only half the story. You also have classtime, so how does this work in the flipped model?
Other than developing independence in students, the main benefit of this method of teaching for me is the time it opens up in class for students to do maths. I can give students more challenging problems as I and their peers are there to discuss the problems with. For those who need more practise of the basic skills, they have the safety net of being able to ask whilst they are doing the question. For those more confident, they can move to more challenging questions more quickly.
I always start the lesson with a starter based on the video. Sometimes this is one of the questions I attached to the video, if several students struggled with it. I then get one of those who got it right to explain how they did it, or get students to discuss their methods in their pairs. If there were a lot of problems arising from the video, I will get students to discuss these, and I will always review the key points, usually going through a final example based on the video, asking the students how to do it. Also, at the end of each video I include a question asking if the students have any questions on the content. I use this time to talk to individuals about these, or sometimes discuss them with the class if they point to a key misconception.
This is followed by jumping straight into questions. For exam classes, this has proven a great way to get them practicing more questions, especiaclly moving on to exam questions more quickly.
And what about for the students' learning?
Well this very much depends on the student. As I have mentioned, this method is really good at developing student independence. We have moved through the course significantly quicker than before, which allowed more time to do revision at the end, but in future I would make sure to do more practice in class at the time, with a larger variety of tasks. For my additional maths class, this has given me a lot of scope to get through the material for both courses in the time allocated, something we have struggled with in previous years.
I would not say that I have evidence that the results are better, but they are certainly no worse than those classes I have taught using the traditional method. With the thrown in benefit that students are visibly more independent, and have a better work ethic, I think this method has its advantages. It also provides the students with a good set of revision resources.
Some key points that I have learnt:
This year I am going to continue to use the flipped classroom with my Additional Maths class in Year 11 and my IB Standard Level class in Year 13. I am not going to use it with my IB Higher Level class in Year 12 since it is the first time I will be teaching this course, and want to teach it through once first, but next time I teach the course, I would definitely strongly consider it. Similarly, in my Year 7 class, I want to use a more traditional approach, though I will probably use elements of the flipped classroom through the year (such as the odd homework).
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.