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).
Regions on Graphs
I recently taught sketching regions on graphs from inequalities, and decided to provide a scaffold to support the initial acquisition of the skill. I describe the lesson in more depth here.
The Final Nail for Inquiry Learning?
In this post I reflect on a short clip of John Hattie talking about the low effect size of inquiry learning, and give an example of how this might be interpreted in the context of me learning photography at the moment. Over the last couple of years I have really swung from pushing discovery based methods to a more explicit approach, but I am aware that I have probably swung a little too far in the other direction. There is definitely value in the ideas of inquiry, but it is where it comes in the sequence of learning that has changed for me.
What does great teaching look like?
We are in the process of creating a document defining what great teaching looks like at our school. I go into a bit more depth, and give the first draft of 12 principles here.
Seven Myths About Education
I recently reread this classic by Daisy Christodoulou, and took notes on my main takeaways. You can find a blog post linking to my notes here.
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?
I recently saw this clip of John Hattie speaking about Inquiry Learning, and why it has such a low effect-size in comparison to some of the other aspects he has looked at in Visible Learning. This is something that has intrigued me since first reading his work, as we clearly want our students to develop into effective inquirers, so why is it that using this approach is not more effective?
I think he perfectly sums up one of the main problems with teaching solely through inquiry, which is that in order to inquire about something, you have to know something to start with. This links to some of the other reading I have done, particularly on the ideas of Critical Thinking and Creativity being domain specific skills, and that in order to develop these skills you need to know a lot about the domain in which you want to apply them.
For example, I am a good critical thinker in mathematics, and am a pretty creative mathematician. I am able to use methods to solve problems that those with less knowledge of maths would not be able to do, even if they knew the methods. But I am a novice photographer (something I am learning at the moment). I am unable (at this stage) to think critically about the lighting of my photos, and be creative with my compositions, when taking photos, even though I know this is what I want to do. It is not through lack of trying, but rather that my knowledge is still relatively low in the domain of photography, and I am having to think about the technicalities of the photography, which would be automatic to an expert photographer.
The same is true of inquiry. I am very capable of inquiring and discovering new mathematical ideas, and I am quite quick at being able to apply these ideas to solving other problems. But in photography, my attempts at new styles are often disappointing until I have some instruction in how to approach them (usually from a Youtube video, or blog post). Even though I have a macro lens, I have never been able to take a macro shot that is a good photo, because I have not invested the time in being properly instructed in how to use it, nor have I then practiced enough at this skill to become better at it, and I will improve very slowly if left to my own devices to play around with the lens.
This chimes with the recent findings from the PISA 2015 data that show that the "sweet spot" for teaching is using teacher directed instruction in most to all lessons, and inquiry in some lessons (https://www.mckinsey.com/industries/social-sector/our-insights/how-to-improve-student-educational-outcomes-new-insights-from-data-analytics).
And this brings us back to what Hattie was saying in the clip. No, the low effect size for inquiry learning is not telling us to never do inquiry based learning. It is saying that we should save inquiry for the right time in the learning journey. And this is not at the start, but rather after we have developed a strong foundational base of knowledge and skills. At this stage, inquiry can help us extend and consolidate our learning in an area, but relying too heavily on inquiry in the initial stages of instruction can lead to more problems later on.
Perhaps counter-intuitively, the best way to develop students as enquirers is not to give them lots of practice at inquiry, but rather develop a strong foundational knowledge base, from which they can then base further inquiry.
Perhaps in a few years I will be able to develop new photography skills "on the fly", by trying things out. But for now, I will continue to rely on some instruction from my internet sources!
Suggested Further Reading
Putting Students on the Path to Learning by Richard Clark, Paul Kirschner and John Sweller.
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.
Quadratic Functions 4 Methods
We explored 4 different ways to solve a problem on quadratic functions which I detail in this blog post.
Chi Squared Break It Down
I really enjoy teaching Chi Square tests to my IB Mathematical Studies students as it is something they have never encountered before, allowing them a fresh start with an area of Maths. For many of these students, they have a difficult past relationship with Maths, and this is a nice topic they can all access. This year I decided to really atomise the process, and describe how it went here.
Newsletter Issue 8
The eighth issue of our T&L Newsletter is going out to all staff next week. You can find a copy of it here.
For the T&L Newsletter I wrote a summary blog post for the article Test-Enhanced Learning which can be found here.
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.
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.
I have previously shared this padlet of links to blog posts on a host of different areas of Teaching and Learning. Hopefully it will prove useful to some people when trying to find posts from the edu-blogosphere. If you have any suggestions of headings, or blogs for me to include, then please do let me know (might be easiest to do this through twitter).
But I have now created a second padlet which focuses on research articles and books. For each article or book that I read I am challenging myself to write a one page summary sheet (one page per chapter for books). This is to act as a quick reminder to me of the key points, but also as a way to help my staff find the time to interact with research a little more (a one page summary is quicker to digest than a 10 page article). For some articles I have also written summary blog posts on our T&L blog, and these are also linked to, along with the original article. I will be adding to this as I read more articles (and updating the ones I read before I started the summaries at some point too).
Quadratics from Graphs
I have been trying to extend the use of example problem pairs to more of my classes, and have started to use them with my first year IB students. We were looking at finding the equation of a quadratic from a graph, having already covered sketching graphs given in root (factorised) and vertex (completed square) form.
We started by recapping the two forms and some of the things they tell us.
After this I jumped into a series of example problem pairs of the different type of questions that can be presented. As you can see from the screenshots below, I have been working on my use of colour coding the examples that contain multiple steps. I find this helps me think about the different steps, and also helps the students identify the steps.
The examples are taken from our textbook, and the your turns were created using Autograph. I then made use of my Quadratic Graphs Activity that I created a couple of years ago (using Geogebra), to test them on some more examples. I will be making use of this activity again during the coming lessons to induce retrieval and spacing of this complex skill.
Revision with S4
I am a firm believer that exam groups need to get lots of practice of past exam questions in the final run up to the exams. Often this will involve doing lots of past papers in class, but to keep it a little bit varied I have done these two activities this week:
With my first year IB students I have been having a few issues with some of them doing the homework to an acceptable standard to help them recall. Too many were copying from friends/notes, rather than retrieving. So now I have started to do a quiz based on the homework. They hand in the homework (which is a double sided sheet of approximately 5 exam style questions from topics they have seen previously) at the start of the lesson, and, as before, they do the starter (an past exam question). I will then teach some new content. In the middle of the lesson I give them the quiz (it is a 80 minute double period). This quiz is the same as the homework, but with different numbers. I collect the quiz version, which I mark after class, and they self mark the homework.
We are only a couple of weeks into this, but it seems to already have had the desire effect of getting students to pay more attention to their homework, and many students have commented on how they like this to "help them remember" (which I interpret as them liking retrieval).
P6 Cover Lesson
I was put on cover for a P6 (11-12 year olds) Maths lesson. The lesson was on the unitary method of proportion, and I was taking the second half of a double lesson, which had already been covered by another Maths teacher in the first period. I walked in to a class struggling to apply the unitary method to an indirect proportion problem (12 workers take 20 hours, how many workers needed for it to take 15 hours). Although they had solved the problem, they were unable to explain why it worked, and my colleague was struggling to see how to apply the unitary method to this particular problem (being a cover lesson, he was caught a little off guard!). This naturally led to a bit of team teaching with my colleague (who has a very different teacher personality to me), with him working on a direct problem on one board and then passing to me to show the indirect problem on another board. The light bulb moment came for most of them when my colleague suddenly shouted "I got it! If it takes 12 workers 20 hours, then there is 240 hours of work! If my 11 friends didn't turn up, then I would have to do 240 hours of work all by myself".
The rest of the lesson I just chose questions from the set worksheet, put them on the board and challenged students to work them out in groups. But what I kept doing was linking it back to the words UNITARY, DIRECT and INDIRECT, and making comparisons between the problems as they went up on the board.
What's the first thing we have to do?
Find the unit!
How do we know it is direct?
As one goes up, so does the other!
How do we know it is indirect?
As one goes up the other goes down!
What was particularly fun about this lesson was that I "adopted" the personality of my colleague. This took to a way of teaching I haven't done before, and I can certainly see the benefits. Making a big deal out of things really helped it stick in their heads. It comes so naturally to my colleague, but this is certainly something I am going to try to pinpoint what he does, and incorporate it into my teaching a little.
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.