I have been teaching algebraic fractions to my S4 IGCSE class. I am about halfway through the unit, and I wanted to take some time to reflect on how it was going so far.
I started the unit by getting students to answer a serious of questions on cancelling fractions and equivalent fractions. I generated these using my website as I wasn't concerned about the order of questions as it was all stuff they had done before. However, in reviewing this, I decided to talk explicitly about what we are doing when simplifying a fraction. I was very deliberate with my wording "We are looking for common factors in the numerator and denominator that we can cancel". In saying this I broke the numerator and denominator into a product of factors, as shown below.
We then extended this, with me saying "To simplify algebraic fractions we find common factors in the numerator and denominator which we can cancel."
Following this I ran through a series of Example Problem Pairs (shown below). I have started doing this using my visualiser so students can see the layout I use in the booklet they have to copy the examples into. I broke up the numerators and denominators into the factors, then cancelled them (I have always taught spotting these before). There are 9 example problem pairs, showing different types of simplifications they may come across. As this was quite a lot of examples, I didn't want to spend ages going through them all, so I decided to do it quickly.
For each example, I called students attention to the board ("Eyes on the board" - I have been working with them for 6 months on how to behave during examples, so this is all that is needed most times), and then I worked the example on my copy of the booklet. They are watching the screen, not talking and not writing. In this case I did not feel the need to narrate after doing the example. As soon as I was finished I said "Your Turn", and all students copied down the example and attempted the corresponding problem. I then Cold Called a student to give the answer to the Your Turn, making an effort to congratulate them for getting it right. In one instance the student wasn't sure, so I asked them to tell me the first step, which I wrote down in the Your Turn box (Your Turn 5), and they finished the explanation by themselves without further prompting.
I then sent students to my big whiteboards that I have around my room to work in pairs answering the following set of questions from the textbook. After a few minutes, they had all made really good progress (much better than when I have them work at their desks). During this time I had been able to quickly see the work of every student as they were all on display to the whole class.
This was followed by another quickfire round of example problem pairs, where there were factors in brackets. I made a big deal to refer to them as factors throughout.
Then back on the whiteboards for some more practice.
I followed this same pattern through simplifying algebraic fractions that needed to be factorised first, and here is where my over emphasis on the need for factors paid off. Another set of example problem pairs on linear expressions, with time on the boards, and then quadratic expressions. Students spent 25 minutes working individually in their books on the questions for these, as I explained to them that they were more difficult and they would need to give it their full attention. For a class that are generally quite chatty, they worked pretty much in silence for 20 straight minutes on difficult algebra problems. I was in shock (in a good way!)
I did make one error here when addressing prior knowledge. Although I got them to factorise a few quadratics so I could check if they were OK with this, I obviously didn't give them enough time on it, especially those with a coefficient of x2 which was not 1. This ended up distracting from the simplifying fractions, and I decided to ask them to skip a few questions to focus on the skill we were looking at today. I need to come back to reviewing more complicated quadratic expressions, and build in some algebraic fractions too.
The following lesson we moved on to multiplying and dividing algebraic fractions, and followed a similar process, though with only 2 example problem pairs (they didn’t need any more by this point). I also started this lesson with a set of questions on multiplying and dividing numeric fractions, and showed the class how to cross cancel before performing the operations (which some had never seen).
I then had them work on a challenging exercise from one of the old textbooks recommeded by Jo Morgan at Resourceaholic.
I particularly liked question 10, and once we had found the answer, I was even more convinced of the beauty of this expression. I explained to the class why it was beautiful, and challenged them in the last 5 minutes to create their own beautiful expression that simplified down to an integer.
I have now got to teach adding and subtracting algebraic fractions, and then dealing with complicated equations with algebraic fractions too. I am going to try to follow a similar process throughout the rest of the unit.
I have found the quickfire example problem pairs to be a great way to keep my lesson moving at a good pace, as I have found in the past that these can really slow the lesson down. Although sometimes that extra time is really needed to dig deep into the example and details, in this case, I think seeing lots of different examples close to each other allowed my students to make connections for themselves.
All students were able to perform in the lesson, fully able to deal with all the stuff we have covered so far. I am happy that I have laid a groundwork of success for the students before we now move into the slightly more tricky bits of the unit. Hopefully that feeling of success will run through, and students will continue to believe they can do it.
I know that them performing in the lesson is not a good indicator of their long term learning (these will be cropping up in starters and quizzes over the next few weeks and months to guarantee that), but I have never taught algebraic fractions before when all students left the room able to do it (except when I have taught the top set Additional Maths class). And this is definitely a good step in the right direction. After each of the lessons in this sequence I left the class feeling like ALL my students knew more than when they entered my classroom. On top of this, ALL the students were working for the full 40 minute period, fully engaged by both the quick pace of the example problem pairs and the use of the whiteboards for answering questions.
Overall, this is definitely something I am going to build into my teaching on a more regular basis.
New School Year
We have just started our new school year in Peru, and the kids have been back for a week, after an INSET and Preparation week. It has been tough to get back into the routine, but it doesn't take long to get back in the habit.
In the INSET week I launched our new Principles of Great Teaching document with two sessions with our staff: introducing the document in general and then focusing on the first standard, which is having an activity for students to do as they enter the classroom (inspired by Doug Lemov's Do Now). It has been accepted well by colleagues, who have commented on it being useful to track their own development, and as a guide towards great teaching. Most people were even on board with the Do Now, though we will have to see how that goes as term progresses.
In general, people were very positive about the whole week, and I have had a couple of comments about it being more useful this year than usual, even though staff had less time to prepare for their classes.
My personal focuses for this year that are taken from the Principles are:
In terms of teaching, I have started my classes with more clarity on my expectations for lessons with me, by producing this document which I have on my walls, and have given copies to my students.
Introducing Direct Variation with S4
In the first lesson of the year I introduced my S4 class (taking IGCSE this year) to the concept of two variables being connected by a constant factor, and linked this to previous work on proportion. I ended by showing them the notation for a is directly proportional to b.
We continued this work in the double on Tuesday. I started with the traditional Last Lesson, Last Unit, Further Back, but I have added a new section this year: Drill. The idea is to drill some of the most basic skills so they do not occupy space in working memory when solving more complex problems.
After reviewing a couple of bits from this, I moved on to a silent example problem pair. I used Show Call on the Your Turn problem, choosing a couple of pieces of work I noticed while circulating, one using 0.75 and the other using 3/4.
Previously I have written the examples on the SMART board, but this year I have decided to do it using the visualiser live on a copy of the booklet I give to students. This worked well as it gave them more guidance on how to lay it out (although I needed more space to keep the vertical structure I would have preferred).
I then had students complete a short exercise in pairs on the big whiteboards I have around my room. I much prefer this way of working than using mini-whiteboards, as it is easy for me to see everybody's work, and to draw attention to important points. For example, one pair started with x=ky whereas everybody else started with y=kx. I used this to identify that their values of k were different, but the final answer would be the same. As it was on a big board, it was easy for everybody to see.
After a couple of groups had finished the 10 questions, we moved on, and I had the class generate the 5 steps for the process. I insisted on them using proper vocabulary (substitute not "put in"). I followed this with a quick Pepper round (also Lemov), asking them quickfire questions on their squares, cubes and roots. This energised the room nicely before seguing into some more examples-problem pairs, where we have proportional to squares and roots. I actually decided to have students attempt Example 4 by themselves before I showed them, and several got stuck by the fact that it was asking for the value of the subject of the formula. When I then showed them, they had had a chance to think about it, and this step made more sense.
Then I used a few Diagnostic Questions to check for understanding of notation and steps. We finished with some independent practice questions taken from Dr Frost.
Next lesson we move on to Inverse Proportion.
I downloaded the 5 Minute Journal App towards the end of the holidays to start developing my own reflective processes. Each day I have to come up with 3 things I am grateful for, 3 things I will do to make the day amazing, and then in the evening reflect on 3 things that were amazing, and 1 thing I could have done to make it even better.
I am finding it difficult to come up with new gratitudes each day, but it is slowly becoming easier. I have built it into my morning and evening routine now, so I am remembering to do it. I am starting to feel generally more positive about situations, and I am hoping that this will help me become more charitable in my interpretations of what people say, as well as making me more aware of my own good fortunes. It is very easy to focus on the negatives, and this will hopefully stop me from doing that so much.
My colleague and I who are both teaching IB Mathematical Studies, have decided to combine our classes and team teach this year. It is the second year of the course, and we always have a few students leave school at this point (it is only compulsory until 17 in Peru), so we both have small classes (less than 10). We figured this will allow us to provide more support, especially during independent practice. We are going to see how it goes, and I will reflect more on it as we progress, but I am excited to give this a go.
Sets Notation Matching
We started the new year with a unit on Sets and Probability with the final year IB Mathematical Studies classes. Sets is something they have encountered several times before, and it is on the IGCSE that we do. However, the notation is always something that causes confusion. To start the unit we decided to do a matching activity with the symbols, the names and a brief description.
The activity was done in pairs, and there was lots of good discussion and questions about some of the notation and words. They had not seen the words Order, Disjoint or Ellipsis before, so these threw a lot of them, even though they had an idea of what the symbols meant. This gave us a pretty good idea of which were the problem areas (subset vs proper subset).
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.
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.