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.

The concept of substituting a point into an equation to find missing constants is one I have been pushing as it has wide applications across different functions types. This is the method we use to find the equation of a straight line when we know points. So it seems natural that this is also the preferred method. I also like the fact that it helps students realise the connection between the coordinate and the equation. Whilst going through this example I picked up on the difference between -4² and (-4)² and why it is the former in this case. At the end I also said there was a third different equation we could have used, and asked what it was. A couple of students spotted that they could have used the vertex coordinate instead of the roots. |

Similar to above (in actuality it is just another version of a simultaneous equation), but making use of the formula for the equation of the axis of symmetry (they are given this in the formula booklet for IB Mathematical Studies), one student noted that as they know the value of a, this is fairly straightforward. But they got stuck at finding the value of c, which they had worked out by the time we went through method 1. |

This links to the work we did previously on Root Form, but many students could not recall the details of how to do this. I modeled the answer, firing a few questions on bits I knew they could do. I told them this is the method that I personally would choose to answer the question, but recognised that it relied on confidence in algebra (something they do not have). |

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.

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:

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:

- Stick a question from the IGCSE paper 4 (these are longish questions broken down into steps) on each of the six whiteboards I have around my room. Students work in pairs to answer the question. Then they rotate and the next pair checks the work of the first pair. Markscheme is given out to check the actual answers, and students discuss any they got wrong. Groups rotate again to a new question. I chose questions on topics that appear regularly in this exam (percentages, trigonometry, graphing with their calculator, cumulative frequency, etc).
- A relay on the IGCSE Paper 2 (short non calculator exam, with 10-12 questions in 40 minutes). Each pair of students has to answer the first question, bring it to me, and if correct, they move on to the next question. To promote accuracy over guessing and speed, I used a factor of 3 if they got it correct on the first go, a factor of 2 on the second go, and a factor of 1 on the third go. I applied this to working marks too, so if they got 1 working mark in the first try and then the second mark on the second go, they would get 3x1 + 2x1 = 5 out of a possible 6 marks.

Review Homework/Quizzes

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.

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 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 blogged about a new approach I took to teaching quadratic functions to my IB Standard Level class, which was in parts both a bit of a disaster, and then successful after the second lesson.

Eight Tips for Learning Maths

I posted a link to a document I made last year which incorporates some ideas from cognitive science in how best to study maths.

I posted a link to a document I made last year which incorporates some ideas from cognitive science in how best to study maths.

T&L Newsletter Issue 6

I published the sixth issue of our T&L Newsletter, which now has a new design thanks to our design team. It includes how I am planning to make use of my Question Reflection Sheets (inspired by Ollie Lovell on the Mr Barton Maths Podcast) after the mocks. I also put together a three page overview of how human memory works in terms of sensory, working and long-term memory. This was checked by our Head of Department for Psychology, so I am thankful to her.

I published the sixth issue of our T&L Newsletter, which now has a new design thanks to our design team. It includes how I am planning to make use of my Question Reflection Sheets (inspired by Ollie Lovell on the Mr Barton Maths Podcast) after the mocks. I also put together a three page overview of how human memory works in terms of sensory, working and long-term memory. This was checked by our Head of Department for Psychology, so I am thankful to her.

Desirable Difficulties

I read this very short paper by Robert A Bjork on Desirable Difficulties, and summarised it in the following image. In our Collaborative Project group we have decided to shift focus this term from Cognitive Load Theory to looking at desirable difficulties, in particular interleaving.

I read this very short paper by Robert A Bjork on Desirable Difficulties, and summarised it in the following image. In our Collaborative Project group we have decided to shift focus this term from Cognitive Load Theory to looking at desirable difficulties, in particular interleaving.

Tree Diagrams

For the second year I taught tree diagrams in a more atomised way. I started by just focusing on drawing tree diagrams without calculating probabilities of events. We started with just simple 2 by 2 diagrams where the probabilities are the same, built up to 2 by 2 conditional trees, where the probabilities change on each branch, and then moved on to more complex diagrams with 3 outcomes and branches that end. After students were confident in drawing the tree diagrams themselves, then we looked at how we could combine the probabilities to work out probabilities for certain events.

This approach worked really well, and students seem much happier with being able to draw tree diagrams. Now I need to interleave this with some other probability questions which require other diagrams, such as Venn Diagrams or Sample Space diagrams.

The resources I used for this unit can be found here.

For the second year I taught tree diagrams in a more atomised way. I started by just focusing on drawing tree diagrams without calculating probabilities of events. We started with just simple 2 by 2 diagrams where the probabilities are the same, built up to 2 by 2 conditional trees, where the probabilities change on each branch, and then moved on to more complex diagrams with 3 outcomes and branches that end. After students were confident in drawing the tree diagrams themselves, then we looked at how we could combine the probabilities to work out probabilities for certain events.

This approach worked really well, and students seem much happier with being able to draw tree diagrams. Now I need to interleave this with some other probability questions which require other diagrams, such as Venn Diagrams or Sample Space diagrams.

The resources I used for this unit can be found here.

Hummingbird

As spring seems to be starting in Lima, the tree behind my classroom has started to bloom, and a hummingbird has decided to visit the tree three or four times each day to feed. It really is beautiful, and I managed to capture this video.

As spring seems to be starting in Lima, the tree behind my classroom has started to bloom, and a hummingbird has decided to visit the tree three or four times each day to feed. It really is beautiful, and I managed to capture this video.

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:

- y = ax² - what happens as a changes
- y = x² + c - what happens as c changes
- y = (x + p)² - what happens as p changes
- y = (x + p)² + q - what can we say about this function
- y = (x + m)(x + n) - what can we say about this function

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/).

Making use of the new whiteboards I have around my room, I had pairs of students make a prediction of what each graph would look like, before revealing the correct answer. Prior to this lesson we had covered sketching parabolas from the graphical calculator, labelling the important points, so I emphasized that students must label these is possible for each graph they sketched.

After each "batch" of types of graphs, students returned to their seats, and I projected some example graphs for them to sketch in their notes, and some questions for them to answer. One such of these slides is shown below.

Those students in the group with a stronger prior knowledge of quadratics definitely took more from this lesson, however, there was still lots of confusion amongst the class. Many struggled to predict the y-intercept. Others got hung up on the idea of translations, which whilst being correct was in this instance getting in the way of them identifying the important points.

My guess is this was creating a large cognitive load as they were having to turn an equation into a translation (and sometimes stretch) and then draw this sketch.

At the end of the lesson I felt frustrated. Although to an observer this may have seemed like a good lesson (students were engaged and working for most of the lesson, they were being relatively successful, and it was clearly challenging them), I knew that they had not learned much from the experience. We did not get to look at graphs in Root Form as there was so much confusion around the prior examples, and some in the class could not even perform at the end of the lesson by stating an equation of a given graph.

I decided that I needed to reteach this in a different way.

I decided to go along another path set out by Craig Barton, the Pattern Spot (https://variationtheory.com/pattern/).

In the following lesson I just showed a blank set of equations in Vertex Form, with three columns next to them, as shown below.

Instructing the students to just watch in silence, I proceeded to fill in each row, pausing slightly between each entry. You can see the final filled in slide below.

I then revealed the following set of equations, which I showed side by side with the examples, making use of the dual page display option in SMART Notebook.

For those that finished, I challenged them to try to sketch the graphs of the quadratic functions, making use of the information they had just found.

During the process, one student asked what the axis of symmetry was. I decided not to tell them at that stage (not wanting to overload their working memory), and told her that I would explain that once they had completed the questions as I wanted them to focus on how to find it first. She was happy with this answer, and was then the first to complete the questions. The same student then said to me that this was much easier to understand than the graphs we had done previously. I asked if she was now able to make connections between this and the sketching we had previously done, to which she responded yes. I know student self-report can be misleading, but I thought it was an interesting comment in the moment.

After all students had completed the questions, I took the book of one student and projected their answers using my visualiser, and asked if anybody disagreed with any of the answers shown. There was one disagreement, which I clarified on the board.

I then used the whiteboard beside my SMARTboard to sketch one of the graphs, and show students what the axis of symmetry was, and straight away one student noted that it went through the vertex and that was why the value matched the x coordinate of the vertex. How before why definitely seemed to help (at least that) student develop a better understanding.

Then we formalised what we had found in written notes, as shown below.

We then repeated the whole process with the Root Form of a quadratic, with the examples I did and the questions they then did shown below.

Originally I had included the vertex in these slides, but decided to remove it to lower the intrinsic cognitive load of the task. As students completed the set of questions, this time I challenged them to find the vertex, using what we had discussed about it a few minutes before.

When they had finished I sketched two parabolas and showed the axis of symmetry. With some encouragement, a student pointed out that it is half way between the roots, and then I linked this to the midpoint formula which they are given in their formulae booklets. We also discussed how we could find the vertex once we knew the axis of symmetry, as the x coordinate is the same (as we had previously discussed), and then use the equation to find the y coordinate (though we will be returning this in more depth soon).

Again we formalised the discussion in their notes with the following slide.

As the bell drew closer, I sketched a few graphs on the board and Cold Called students to say what the equation would be. In future we will be looking at extending this to include finding the value of the coefficient of x² for a given graph, and then making use of this Geogebra applet (http://www.interactive-maths.com/quadratic-graphs-activity-ggb.html) I made a couple of years ago for this very reason.

Overall, I felt the second approach worked really well. All students, even those who are currently failing the course, were able to be successful in this topic, which is one of the most difficult on the syllabus. In future, I think I would still do both of these activities, but in the opposite order, only getting students to sketch the graphs after being successful at finding the points. I think this will lower the intrinsic load of the activity, as they can master finding the points first, and then put this into practice when sketching graphs.

]]> I have produced a poster/stick in with 8 tips for students on how to learn Maths. My plan is to refer back to these regularly. They are Maths specific ideas from some of the research in cognitive science. Feel free to use, and feedback. I originally got this idea when observing a colleague in the Music Department, where they have a similar document on how to practice in Music successfully. |

Over our winter break I had most of my display boards replaced with extra whiteboards. I now have 6 big whiteboards around the room. This week I have made the most of them on several occasions.

My IB Maths Studies class did the Buildings Around the World task (https://www.tes.com/teaching-resource/surface-area-and-volume-of-buildings-6428047) where I printed two building per page and stuck them to the middle of each board. In pairs students had to find the volume and surface area of the two buildings, and check with me. Once they got the correct answers, I swapped the sheet for the next one.

Following this, we looked at compound volumes, and I projected some shapes on the projector, and again students worked in pairs to find the volume and surface area on the whiteboards. This time I controlled the pace a bit more as all pairs worked on the same one. When one pair was struggling, they could easily get a "hint" from another pair by looking around the room.

With my S3 class I projected this fantastic Show That activity (from Catriona Shearer available in the last two slides here), and got them to again work in pairs to do each one.

In both classes I found that the students maintained focus on the task for a significantly longer period of time than if they did the same thing in their books, even if they were working in pairs.

My favourite thing about the whole process, though, was that I could easily see there work (as could their classmates). This made it really easy for me to pick up on mistakes and misconceptions, but also to help students improve their layout, something I have long wanted to improve.

I am definitely going to be making use of this new workspace as much as I can, both for student work and also my own instruction (see below).

Quadratic Equations with S3

We have just finished our 3 week winter break, and upon coming back I needed to spend a little more time looking at Quadratic Equations with my S3 (year 10) class. We had already looked at solving them, but still had to look at solving problems involving quadratic equations.

We have just finished our 3 week winter break, and upon coming back I needed to spend a little more time looking at Quadratic Equations with my S3 (year 10) class. We had already looked at solving them, but still had to look at solving problems involving quadratic equations.

In the retrieval starter, I asked students to name the three methods for solving a quadratic equation (for the IGCSE this is factorising, formula and graphing using the GDC), and then to solve an equation using them.

We then recapped the need to rearrange equations into the form ax²+bx+c=0 (unless using the GDC in which case they can just graph the left hand side and the right hand side and find the intersection).

Finally we moved on to looking at solving problems.

I made full use of my new whiteboards, as I left each stage on the boards, as shown in the images above. This meant that students could refer back to them, but also made it very clear that they were only adding one extra stage each time. I could constantly refer back to the previous work as it was still visible. I think this is what people mean when they talk about the Japanese method of boardwork.

This worked really well as a way to recap what they had already done before the holiday, and build it up into the problem solving that was the aim of the lesson. I have been trying to focus more attention on the incremental build up to complex processes, and felt this approach worked well in the moment. We will see how well they remember it next lesson!

]]>I have been reading a lot about the science of learning lately, in particular around cognitive science. This explains empirically what works in terms of memory, which obviously impacts learning. I have boiled it down to 6 key points that I think all teachers should be aware of.

- Learning is not the same as performance (Soderstrom and Bjork, 2015)
- Learning is a change in long-term memory (Kirschner et al, 2006)
- Learning is built upon prior knowledge (Willingham, 2006)
- Learning is effortful and requires spaced retrieval (Bjork, 2018)
- Memory is the residue of thought (Willingham, 2010)
- Working memory is limited (Sweller et al, 2011)

I will be writing a more in depth blog post about this in the upcoming weeks.

My plan is to incorporate these into the staff training programme for the next year, but I have not decided how best to go about this just yet.

My plan is to incorporate these into the staff training programme for the next year, but I have not decided how best to go about this just yet.

References

Bjork Learning and Forgetting Lab (2018) Research [ONLINE] Available at: https://bjorklab.psych.ucla.edu/research/. [Accessed 4 July 2018].

Kirschner, P. A., Sweller, J., and Clark, R. E. (2006) Why minimal guidance during instruction does not work: an analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching, Educationsal Psychologist, 41(2), 75-86.

Soderstrom, N. C., and Bjork, R. A. (2015) Learning Versus Performance: An Integrative Review, Perspectives on Psychological Science, 10(2), 176-199.

Sweller, J., Ayres, P., and Kalyuga, S. (2011) Cognitive Load Theory, Springer.

Willingham, D. T. (2006) How Knowledge Helps, American Educator, Spring.

Willingham, D. T. (2010) Why Don't Students Like School? A Cognitive Scientist Answers Questions About How the Mind Works and What it Means for Schools, Jossey Bass.

Bjork Learning and Forgetting Lab (2018) Research [ONLINE] Available at: https://bjorklab.psych.ucla.edu/research/. [Accessed 4 July 2018].

Kirschner, P. A., Sweller, J., and Clark, R. E. (2006) Why minimal guidance during instruction does not work: an analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching, Educationsal Psychologist, 41(2), 75-86.

Soderstrom, N. C., and Bjork, R. A. (2015) Learning Versus Performance: An Integrative Review, Perspectives on Psychological Science, 10(2), 176-199.

Sweller, J., Ayres, P., and Kalyuga, S. (2011) Cognitive Load Theory, Springer.

Willingham, D. T. (2006) How Knowledge Helps, American Educator, Spring.

Willingham, D. T. (2010) Why Don't Students Like School? A Cognitive Scientist Answers Questions About How the Mind Works and What it Means for Schools, Jossey Bass.

T&L Newsletter 5

I published the latest issue of our T&L Newsletter last week, which can be found here (https://drive.google.com/file/d/1RGSA7beQAJ0zDGl_78h2x2Akm3y1liwZ/view?usp=sharing). I have enjoyed putting this together, and staff seem to be looking at it. I print copies to leave in the staffroom for people to have a look at, as well as emailing it round to all staff.

I published the latest issue of our T&L Newsletter last week, which can be found here (https://drive.google.com/file/d/1RGSA7beQAJ0zDGl_78h2x2Akm3y1liwZ/view?usp=sharing). I have enjoyed putting this together, and staff seem to be looking at it. I print copies to leave in the staffroom for people to have a look at, as well as emailing it round to all staff.

Introducing Logarithms with S4

I blogged about a series of lessons I did on Logarithms with my S4 class here.

I blogged about a series of lessons I did on Logarithms with my S4 class here.

Class Website

I finally got round to updating my class website to include all the resources for the IGCSE course (http://classes.interactive-maths.com/igcse.html), so that students can use it in preparation for their mocks. For each unit I copy the objectives from our scheme of work, and link to a video for as many as I can. Then I include the work booklet that I print and give to all students, and the PowerPoint lessons, which include solutions to exercises, and the notes that students have to fill in on the work booklet.

I finally got round to updating my class website to include all the resources for the IGCSE course (http://classes.interactive-maths.com/igcse.html), so that students can use it in preparation for their mocks. For each unit I copy the objectives from our scheme of work, and link to a video for as many as I can. Then I include the work booklet that I print and give to all students, and the PowerPoint lessons, which include solutions to exercises, and the notes that students have to fill in on the work booklet.

Make It Stick

I recently finished the book Make It Stick, and wrote a summary of the main ideas for our school T&L Blog (http://markhamtl.wixsite.com/teaching-learning/single-post/2018/07/11/Make-It-Stick).

]]>I recently finished the book Make It Stick, and wrote a summary of the main ideas for our school T&L Blog (http://markhamtl.wixsite.com/teaching-learning/single-post/2018/07/11/Make-It-Stick).

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.

Next I used the idea from James Tanton's Take on Logs where I wrote some on the board like this

power2(8) = 3

power5(25) = 2

Your turn!

power3(27) = ___

power10(100) = ___

power2(8) = 3

power5(25) = 2

Your turn!

power3(27) = ___

power10(100) = ___

I didn't use the examples from his essay, but rather used ones that linked to the questions from the starter (the answers were still projected), and wrote them from scratch on the whiteboard. Students could easily see the link between the two. As they grew confident I started to use some that were not from the starter. Towards the end I threw in a couple of impossible situations such as power1(73) = ? and power5(-1) = ?. As James suggests, I then made a big deal out of changing power to log, and explaining that we just use a different name for the function. As we went through these examples I stuck to a particular routine. So, for example, for the question power3(27) = the back and forth went like this:

Me: "Maria, what is the actual question being asked?"

Maria: "3 to the power of what is 27"

Me: "And the answer is?"

Maria: "3"

Me: "Maria, what is the actual question being asked?"

Maria: "3 to the power of what is 27"

Me: "And the answer is?"

Maria: "3"

I used cold call from Teach Like a Champion 2.0 throughout this process, writing a question up and asking a student. The class came to the phrase "3 to the power of what is 27" as a group rather than me telling them.

At this point a student asked "Why do we need logs". Fortunately I had this slide from Dr Frost Maths ready, and launched in to talking about how logarithms are the inverse of exponentials and that we need them to solve equations where the unknown is in an index.

We then did an example problem pair, just to reiterate the process, and again I asked what the question was saying.

Then I set them this exercise that I put together in the style of the ones from the amazing variationtheory.com by Craig Barton. I designed it to try to get students thinking about the connections between each question, though I admit that I still need to work on this aspect of running an exercise like this. However, the questions did bring out some interesting ideas, and many of them were able to spot the impossible ones.

In going through the answers we once again used Cold Call, and rattled through them pretty quickly, again following the same dialogue as above. That was the end of the first lesson.

In the second lesson I started with a few simple questions as retrieval from the previous day, and then students filled in this stickable from Sarah Hagan, with the slide with information given below.

We then did these excellent place the log on the number line activities from the Mathematics Vision Project.

And finished with students writing their own set of Two Truths and a Lie cards (again from Sarah Hagan) based on logarithms. I collected these in, and in the next lesson we started using these, which I projected through the visualiser. Students then used this ordering activity from Susan Wall.

After this I will be going on to teach the laws of logarithms and solving simple exponential equations using logs.

You can find my folder of resources on this topic here. It includes a work booklet for students, the powerpoints I used, along with some other activities.

Continuing to share the ideas of Cognitive Load Theory has been an important part of our collaborative project. We have now started to share it more widely, doing 10 minute sessions in tutor time with our S3 students (14-15 year olds). This is a very brief introduction to the idea of our working memory being limited, and the need to prevent overload if we want to learn. I talk a little about being able to complete a task but not learn anything from it, the fact that practice helps to reduce the intrinsic load, and also the different extraneous loads they might experience. At the end of the session we decided to set them the challenge of reducing the extraneous load of talking with friends in class. The tutor is doing a follow up session a few days after our session, where they look at some common examples of myths around cognitive load, with a focus on "distractions".

We are now half way through the tutor groups, and although students are engaging in the session, and seem to understand what it is we are saying, the only way it will become habit is with continued reminders. This is the next challenge for us as a group, to come up with how we could extend this past our own classes.

Boosting Achievement with Messages that Motivate

I read this article and wrote a blog post on it for my school's T&L blog.

I read this article and wrote a blog post on it for my school's T&L blog.

Science of Learning Week 5

I completed the Science of Learning course last week, and my reflections for the final week can be found here. In a couple of weeks I plan to reflect on the whole course in a little more depth and what implications it has for my future teaching.

I completed the Science of Learning course last week, and my reflections for the final week can be found here. In a couple of weeks I plan to reflect on the whole course in a little more depth and what implications it has for my future teaching.

Slow Education

Carl Honoré came to speak to us this week about the SLOW movement, and, in particular, slow education. The idea of the slow movement is one that has interested me for a while, and much like Carl himself, the birth of my son last year has been a catalyst in changing my priorities (more time at home, less screen time, etc). To find out a little more about this side of things, I strongly recommend watching Carl's TED Talk. I want to reflect on a few of the points he made specifically about education here.

Carl Honoré came to speak to us this week about the SLOW movement, and, in particular, slow education. The idea of the slow movement is one that has interested me for a while, and much like Carl himself, the birth of my son last year has been a catalyst in changing my priorities (more time at home, less screen time, etc). To find out a little more about this side of things, I strongly recommend watching Carl's TED Talk. I want to reflect on a few of the points he made specifically about education here.

Firstly, he talked about the pressure on students to be involved in a myriad of activities, and the "need" for these to improve university applications. This is something I have felt for a while, and that we sometimes push students to do too much. It is better to focus on one thing and do it well, and improve in that. No top athlete/musician/actor/businessman/anything else you like to think off got to where they are by spreading themselves thinly. They focused on improving in one thing, and becoming excellent at that. As Carl said, "We've forgotten how to do one thing at a time" and we should be aiming for "…not as fast as possible, but as well as possible". Should we be limiting students to one activity?

Carl also talked about the impact of technology on our lives, and specifically on education and learning. He cited an OECD study which found that there was a negative correlation between student computer use and learning outcomes. From a quick glance over the report it looks like a small amount of computer access is beneficial, but lots has a negative impact on outcomes. This is something I have been thinking about a lot recently, and finding this study is certainly of interest. I am looking forward to diving into it in a bit more depth.

Carl also talked about the digital native myth and the fact that we have not evolved in 20 years to learn differently than we did before. Though he did not labour this point as much as I might have liked, it was certainly refreshing to hear this from him. He also discussed briefly the myth of multitasking, and that this is actually just switching between things quickly.

Whilst Carl was talking there were lots of times when my mind drifted to the ideas of Cognitive Load Theory (and interestingly one of my colleagues made the same links). For example, the extraneous load of technology in the classroom, or the load caused by having too many external pressures from activities. He also talked about other ideas that are staples of good teaching, such as wait time (he called it the 5-minute warning).

Coming out of the session, it was clear that all teachers were thinking about the slow movement, and accepting that it is important to slow down at times. Although he did not quote Willingham, his idea of time to think links to "Memory is the residue of thought" very well. And if we are getting teachers to think about the thinking students are doing then that is great.

But, what about the slow movement for teachers? If we know it is more productive for our students to slow down, then surely the same is true for teachers? As a school we need to be careful that we don't take this to the extreme and push teachers so hard to incorporate it that they are always in a rush!

Logic in 6B

I am teaching Logic for the first time to my IB Mathematical Studies class at the moment (and I never studied it at University either). This is the second time in this course that I have had to teach myself some Maths that I have never studied myself (the first was Chi Squared tests for independence), and before that I had never been in that situation. There have been times when I have had to relearn something (when teaching Higher Level IB, Mechanics and Statistics at A-Level for example), but nothing that I have never seen before! It has been an interesting experience, and once I have finished teaching it, I will have to reflect on how to improve for next year.

I am teaching Logic for the first time to my IB Mathematical Studies class at the moment (and I never studied it at University either). This is the second time in this course that I have had to teach myself some Maths that I have never studied myself (the first was Chi Squared tests for independence), and before that I had never been in that situation. There have been times when I have had to relearn something (when teaching Higher Level IB, Mechanics and Statistics at A-Level for example), but nothing that I have never seen before! It has been an interesting experience, and once I have finished teaching it, I will have to reflect on how to improve for next year.

Graphs with S4

After the disaster I described last week, I managed to pull it back when looking at finding the constants of different types of graphs. I have been thinking a lot about try to use consistent methods for solving problems lately, to help students create links between related ideas. So I decided to teach finding the equations of lines, quadratics, reciprocals and cubics separately (in light of cognitive load theory), but using the same method (as shown in the summary image below). After doing a little practice on each, I presented the mixed examples, and then an interleaved exercise. I have found that using the colour coded way of linking the same steps is really helpful for students to see the connections (and I got them to do the examples in their books in a similar way).

]]>After the disaster I described last week, I managed to pull it back when looking at finding the constants of different types of graphs. I have been thinking a lot about try to use consistent methods for solving problems lately, to help students create links between related ideas. So I decided to teach finding the equations of lines, quadratics, reciprocals and cubics separately (in light of cognitive load theory), but using the same method (as shown in the summary image below). After doing a little practice on each, I presented the mixed examples, and then an interleaved exercise. I have found that using the colour coded way of linking the same steps is really helpful for students to see the connections (and I got them to do the examples in their books in a similar way).

We started the week looking at the plasticity of the brain, meaning that there is no set limit to our intelligence. Clearly there are links to the work of Dweck on growth mindset. In terms of the neurosciences, the size of the hippocampus can be increased through learning experiences. This leads to the idea that "learning begets learning".

This led on to a discussion about resilience, and in particular, that students' beliefs about learning can have an impact on how they respond to challenges. Believing that ability is plastic and can be improved through effort (as opposed to it being fixed) increases resilience, which helps students learn more. It is a virtuous cycle. By believing in a growth mindset, it is believed that students are able to engage their reward system through challenges.

For teachers, we need to understand the plasticity of the brain so that we do not place "restrictions" on what certain students are learning. But we also need to talk about it with our students, to help them understand the importance of hard work in developing ability.

Next we reviewed the EBC model, and were once again reminded that it is not a model for a three stage lesson plan, but rather all should be happening at various stages. We are encouraged to think about how engagement, building of knowledge and consolidation of knowledge fit in our teaching, and to think actively about these when planning lessons.

We were then given a very handy summary of all the key points, and asked to reflect on our own teaching by using the concepts to justify our teaching approach, and justify new things we will try in the classroom.

The last part of the course was aimed at further engaging in research. We were given links to Best Evidence In Brief (__https://the-iee.org.uk/what-we-do/best-evidence-in-brief/__), the Learning Scientists (__http://www.learningscientists.org/__), Tom Sherrington's blog (__https://teacherhead.com/2018/03/19/evidence-informed-ideas-every-teacher-should-know-about/__), ResearchED (__https://researched.org.uk/du__), Impact (__https://impact.chartered.college/__) and the EEF (__https://educationendowmentfoundation.org.uk/school-themes/__).

We were asked to think about the last time we changed our practice in light of evidence/research, and why we did it. We were then guided towards the EEF projects (__https://educationendowmentfoundation.org.uk/projects-and-evaluation/__) to think about which ones we might like to get involved in, either individually, as a school, or actually through the EEF.

Then we were given some guidance on doing an action research project, with the following steps:

- Choose a question or focus
- What research is already out there?
- Plan how you will carry it out
- Implement change
- Collect the data
- Analyse the data
- Reflect on your findings
- Decide who else will benefit from knowing about your results and share

Finally we reflected on our learning, thinking to how our definition of "What is learning" had changed over the course, and taking the post course audit to compare with the pre course audit.

The idea of plasticity of the brain is clearly important, and helping students develop a growth mindset is something we should be doing in the classroom, and the wider school community. However this does not mean just put up posters about Growth Mindset, but rather work with students to see that if they put in the effort then they will improve, and also to be very careful about the language we use when talking about ability/achievement/attainment.

Reflecting on the EBC model is a useful strategy, and I have identified the need to engage my classes a little more. By this I do not mean make the activities fun, but make sure their brains are engaged for learning. Yes, this includes creating some interest from them, but also teaching them about how they learn, and creating the right environment for that to happen. I am going to try to reflect on the EBC model as I plan lessons over the next few weeks, and we shall see how it goes.

In terms of action research, I am not sure now is the right time to jump into this right now, both for personal reasons (young baby at home doesn’t leave much time) and professional (there are changes happening at school that I need to work on). But the idea of measuring the impact is a part of our collaborative project system at school, and I am pushing staff to think about how they can do this. As I am in a project looking at CLT, we are also discussing ways we can measure how much impact the changes we are making are having. Not a full action research, but a step in the right direction.

I have really enjoyed this course, and feel that there was a healthy balance of ideas I knew about and those I didn't to make it of interest to me. I will be looking back over these notes over the next year or so to refresh my memory of what we looked at, and see how I can incorporate some of the bits into my teaching.

]]>