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.
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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.
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 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 oneoff 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/justmedoingwhatido/) 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/mybestplanningpart1/) by Kris Boulton. Below is an example for our first unit of Year 10 on Percentages. 
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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January 2019
