Maths Lesson Principles and Structures


This blog is going to attempt to pull together all the different ideas in my own head and through my summer reading to produce a structure/set of principles for lesson creation. These are thoughts and is an attempt to tidy my head up after the summer!


Having worked for a number of years with the 4 sections of a lesson Connect, Activate, Demonstrate and then Consolidate which is connected to the accelerated learning cycle and I still see that as the base on the general start of my lessons, its the how those parts are delivered that I am most interested in evolving.

Phases of LearningConnect/Start

Based on Teaching for Mastery’s evidence of the more effective schools having on average 49 minutes as the main part of their lesson I know I need to make every second at the start of my lesson count. Step one is the students will come and sort their equipment and sheets whilst I register and setup.

Then I plan to use 4 retrieval questions on the board which will be testing prior knowledge from previously taught content. Answers will be pre-prepared on the board with students raising their hands to show if they have got it right. This year I will then only cover 1 question in depth so the main learning aim(s) progresses and is given the majority of the time. 5 minutes maximum given to complete the questions.

These retrieval questions will have a mixed of problem solving and non standard questions so students develop more resilience and strategies for these type of questions. 1/2 questions will be checking prior knowledge retention ready for the next topic area so I am clear if any corrective work is needed before that topic so selected students can be supported prior to topic delivery.

This part of the lesson must be short, focussed, sharp, timed and with the key aim of checking retention and diagnosing prior learning.

The Teaching bit

I am certainly going to be a student myself this year learning to use manipulatives to show how new topics build on prior knowledge/learning. Think this will be the biggest habit change for me as I think I have not explicitly explained some of these connections to students before when introducing a new idea. I have to force myself on creating my own method bias by immediately using abstract ideas.

The manipulatives on mathsbot.com seem excellent and very flexible. The key for me will be that I will have to accept initially that this phase which has become almost automatic will take practice and preparation time to be confident enough to deliver well in the classroom.

Each new idea/topic will likely start with a problem that students will have a short period to discuss. I’ll then discuss initial thoughts/methods and dispel any concepts which aren’t mathematically sound so the students have clarity. Using examples/non examples and Frayer models will support initial concepts too.

My general style this year will be using directed instruction with purposely planned periods of time for students to discover and find certain areas of maths for themselves. I’m going to be clearly aiming to get that optimum balance right as detailed by @emathsuk in his book.

Activate/Guided Learning

For me structures are important here because reflecting on last year I have moved too quickly in lessons from teaching into independent practice at times. An idea that really resonated with me was from @teacherheads book emphasising the need to get roughly 80% success rates in learning. This was very powerful as I can identify independent practice phases where i have moved on too quickly to this and then had a boat load of corrective work to do with students. Key structures I will use this year are:

  • Backward Fading (removing more and more steps as problems progress)
  • Lots and lots of whiteboard use repeatedly getting and reacting to the performance in front of me
  • Use Example-Problem Pairs
  • Use the routines of problem , example, problem, example…
  • Clearly verbalise my own thought process and why I choose a certain strategy where support is required

Demonstrate/Independent Learning

This is still the area I am still developing within my own practice. Interesting that a tweet went out asking about problem solving in Maths and there was a general concensus that there are no real standardised ways to improve this critical area for students.

The key principles behind this phase is that I need to focus on what students are giving their attention to and carefully consider the activities that they undertake which allows them to transfer near to the topic but also challenge them to apply this also a range of mathematical areas.

The students will need to work for longer sustained periods with a variety of AO1/2/3 areas to consider. Some collaborative learning structures will help here when looking at techniques for AO2/3 questions.

Consolidate/Review

When I was reading about the Study, Test, Test, Test principle being the most effective for learning it got me reflecting about the end of lessons and how I use them and other opportunities for assessment. There was also the stark realisation that its suggested that mathematical ideas can take 2 years of maturation to then be applied well in contexts.

Over the last few years i have heard and read much about the difference between performance and learning. Students immediate performance does need to be assessed in the end of unit as this can show you immediately the areas which have not been taught well or need a different method.

There is also a good opportunity at the natural end points of ideas and topics to produce clear notes which reinforce the steps taken to solve a problem and allow students to refer back to in revision and recall.

Strategically I need to plan out the retrieval topics at the start of lessons and the purpose of the mini quizzes ensuring there is also good reinforcement on topic concepts through note taking. Both of these can be excellent vehicles to assess learning and prior knowledge.

This blog has been a bit of a thought vomit but hope its given a few people ideas to apply in lessons, would love feedback/thoughts via twitter @blastmaths.

Week 1 Update – 08.09.19

Its always a challenge getting back into the routines of teaching after the summer off but after the reading over the summer I was looking forward to trying out some new techniques. Hopefully each week I am going to blog a short piece reviewing my findings.

I have put a renewed focus on speed of my starters in lessons and it is already paying dividends. As the week has gone on I have become more and more precise and demanding over that first couple of minutes in lessons embedding routines (like books out, whiteboard etc…) and getting the students on the 4 retention questions as quickly as possible. My general aim is to have everything complete and one of the questions covered (if required) within no more than ten minutes of the lesson. Managed seven minutes in one lesson this week. The pace of this really sets the tone for the lesson and gets the students ready for the main teaching.

Another key focus this week has been the use of manipulatives to explain concepts using prior knowledge. Its been fascinating watching students reactions when I have modelled a decimal multiplication using bars, initially quite a few thought ‘this is primary sir’ but when I proved that visually maths can just be as clear they were won over, I have a feeling they will retain the information more as its not just purely abstract. Looking forward to connecting others areas using the same models over the coming months. I have to say though I have had my fair share of mini-mishaps in the lessons as I need to ensure I spend the time modelling my explanation myself before delivery.

My final area in the blog this week is looking at the guided practice part of my lessons. I have fallen back in love with mini-whiteboards, the snapshot it gives you of performance in maths at that moment in time cannot be beaten. Have developed a couple of new routines this week, using a visualiser to show students live misconceptions and also writing all the answers I see in front of me on the board and asking them to put their boards down when I have written down theirs. This second routine has really allowed me to see some fascinating misconceptions which I may not have discussed as much. Have enjoyed using Example-Problem pair where I model on the left hand side of the board and then the students complete a similar example on the right hand side. Find this particular use where new learning is requiring multiple steps.

Actually another thing worth mentioning is an idea taken from Mark McCourt’s book which is ‘reprogramming’ students brains when they are seeing new learning for the first time. I put a new topic question on the board and ask for ideas on how to solve it, I politely correct them and state that it is not the correct method but it seems to do two things for me. One it builds curiosity for the methods for the rest of the lesson and it also allows we to stop and rework incorrect prior learning to give the lessons some solid foundations.

Week 2 Update – 15.09.19

The starts of my lessons are now becoming more routined with the first 3 students who enter the room giving out books, I am not one to line up the kids outside i have found that less issues happen if I can get the first 5-10 students in the lesson when they arrived helping me out. This year have also labelled each of my eight tables with a number and the first student that gets to their table goes into the resource cupboard and picks out plastic holder which has 4 pens, 4 whiteboards and 4 board rubbers. This again cuts down lesson transition time when I want to get the whiteboards out. Also developed a new routine with getting them out of the packet this week by the first students pulls them all out instead of what was previously happening was it was taking a minute or so each student picking out what they needed out of the bag individually. Some of my groups in the connect (starting) phase of the lessons are now completing more than one question in each of the four blocks on the board. They have around 5 minutes to complete the activity below with answers then revealed and one area discussed.

Part of a bit of training we had last week was tracking the students work as you walk around the room. I think my own default style has been to get around the room to ensure students are on task and working now I am ‘tracking’ the students work in their book as I walk and stopping any areas where they can’t get to the next step. As I spot things I have to be very disciplined and not help the student straight away instead after a quick walk of the room I will start writing prompts on the board for the questions above or do the first step. For example changing the mixed numbers to improper on the top right example, students now are then using that board for the scaffolding they need. We are integrating prior learning checks each lesson to aid future planning and focussing on Algebra and Geometry based on historical exam performance as areas to further develop.

I am still adoring mini-whiteboards but particularly the discussions we have when I displayed a student’s mistake using the visualiser. The students love seeing what others have put down and then unpicking why their brain made then do it in that way. This is all connected to Mark McCourt’s (@emathsuk) section on correcting those misconceptions in prior learning and its having impact as I think previously some students were trying to add to their knowledge/skills building on very shaky foundations.

I am still learning about the right phased learning approach based on the topic. I know I default to whiteboards too much because I like the snapshot of the room. I am starting equations soon and looking forward to trying out backward fading which is as the examples progress you take out more and more of the steps which the students then fill in @mrmathsgordon is doing some great work in this area. As I am writing this blog I am going tweak me use of whiteboards on problems which have more steps – instead of getting this all down on a whiteboard I am going to ask the students to do the work in their book and then share just the answer on the whiteboard. Always seems a shame that multiple steps are wiped off in an instant, this will allow them to be stored in their book and then they can correct over the top If they have made an errors.

My use of visual/concrete work is developing and the more I practice the more I am naturally connecting areas together and live modelling concepts. What I am finding is I need to be very careful at the ‘size’ of numbers I use in the initial examples so I don’t spend too long showing to concept on the board and focus on the explanation @mrmattock book is a constant source of reference when I am thinking about explanations. I just have to be careful not to get too into this in lesson as I am making connections to other areas which I didn’t realise where there before and I need to give me classes a bit longer time so they share the same level of connection as they have not been shown that other area yet. When I am getting this right I think my modelling/explanations take around 20% longer in lesson but students are then able to work for a longer period of time of a variety of question applications.

Week 3 Update – 22.09.19

Been doing a bit more guidance fading this week but not in the guided learning part of the lesson have used it in the retrieval activity at the start of my lessons. On occasion I can see that i have put up a questions that requires prior learning which they have forgotten or it was built on shaking ground in the first place. As I have been aiming to get my connect/starter done in around 7/8 minutes I have started to put prompts on the board as I am observing which questions the students aren’t completing or have decided to leave till last.

I created the modelling on the right hand whiteboard on the wall as the students were working through the questions. Some needed the additional support to complete the questions this then supported my aim of around 80% success rate in the classroom which built momentum into the main teaching of the lesson. I feel myself getting more and more efficient at the start of lessons and getting the students doing maths sooner which seems to result in a more productive lesson. When the connects/starters drag you can feel it for the rest of the lesson.

Completed my first Year 11 intervention session after school this week and highlighted that all the students had not accessed the solving a quadratic question. So built a corrective resource around this. The aim of this resource is to allow students to study hard through the stages of the resource and look at steps in method and then apply them themselves, it also has a prior learning check so they completed new learning. Most students actually liked completing the full sheet to get that sense of completeness. Using this resource changed the dynamic in the room with the students continually working for the 45-60 minutes and then I was on hand to explain if they did not grasp any of it. The blog is here for anyone wanting further details. Aiming to hopefully produce one each week for each session I complete.

The key aim for this week was to improve my feedback question to students after I have marked an assessment. Last year I was finding a lot of students were correcting much of their work with help of a resource but not really unpicking the reason why they made the mistake in the first place. Looking at their final GCSE assessments proved this to me as even a month before the exam I could see a student correcting a very similar question only to underperform in the final exam.

This is still a big development area for me but i’ll try to outline my stages in the feedback lesson and things I am going to tweak.

Stage 1 – Do it – I work through the assessment myself typing up an abridged version of the used mark scheme (aim for a page!). Have also changed my routine this year of marking the assessment the same night and doing the feedback lesson straight away after. I previously left the marking to the weekend but then I found the quality of my lesson planning suffered.

I wrote a blog a while ago on ‘lean working’ here which explains the process of timing myself going through the regular tasks of my job and then reflecting on any ways I could improve to make myself to be as time efficient as possible.

Stage 2 – Mark it – Print the one page mark scheme off and then mark an assessment at a time. Circle the mark if they are correct, a single line if incorrect and then a single line and the mark above if they have gained some marks. I give them a numerical effort grade from 1 to 4 based on how hard I feel they have worked.

Stage 3 – QLA it – This may surprise that you that I QLA every assessment. Whilst it does add a couple of minutes on (not as much I you would think). Using a numpad the grid can be produced very quickly. It then gives my greater incite into the questions which have scored the lowest overall. I then write a percentage on the front of their assessment and double check I have not been to harsh on the effort grading. If you are at the bottom of my marking pile I know I start getting tougher!

Stage 4 – Rank it – Once I have done the QLA I do a formula at the bottom of each column which works out on average how many marks to students where off full marks and then use a RANK formula to tell me in order which was the worst. I used to just visually look but this method really shows up the big marker questions.

The ranking formula then in order of questions that I cover in class. So Question 7 was the first one I re-modelled explaining their misconceptions then Q8, then Q 2c etc….

Stage 5 – Impact Questions – These are fortunately centrally provided by the department so I snip each question I what into the format below:

The question on the right hand side of this is a new question similar to the one the students did not answer correctly.

Lesson Phase 1 – Student Reflection – The lesson starts with the student immediately have their assessment in front of them. Because I want this to be as instance as possible the first 8 students on entry to the room get about 4 assessments to give out. Ever had that lesson where you’ve given 2 students the worksheets to give out and it feels like an age before they are all out? They then get their purple pen and start adding up their own scores for each question or groups of questions and then highlight to areas that went well and two areas to improve. This should take no more than five minutes maximum (my first attempt took an age!!!!)

Lesson Phase 2 & 3 – Self Improvement – As they are working through this I drop the one page mark scheme on the desk one between two (want to encourage collaboration and save printing costs!) and then display on the board the QLA for the class. I do not show the overall percentages for each student as my classroom does not shame any student I aim to have those conversations individually if needed, but shows each question mark breakdown. They then have around 10-15 minutes (dependent on their performance in the assessment) to use the mark scheme and go and speak to students in the classroom who got that nasty Question 7 or 8 right. I emphasise that I want them to write over their assessment their corrections and move through the questions at pace as I let them know I will be modelling some questions to them after that phase. From next week phase 2 and 3 are going to just a single phase with students decided whether to use the mark scheme or QLA to improve.

Lesson Phase 4 – Teacher Led improvement – As i have marked the assessment the night before I find it much easier to recall the misconceptions I have seen. I now model on the board the original Question 8 and then talk about the misconceptions. Very rarely do I insist on silence but this part needs it, as I am explaining I am clear that I want to see students correcting this on their assessment and then I give them a period to complete the two boxes below.

As I know the majority got Question 7 wrong i will then give them a further period of time to work through the new question above. The key difference for me in this process (compared to previous practice) is the use of the ‘What mistake did you make’ and ‘What do you need to remember’ boxes and ensuring that the students complete these as they are then hopefully reprograming their brain. This process is then repeated for other questions on the assessment. The new question solutions are then shared at the end of the lesson so they can see what improvements they have made.

I have now completed this with three groups and the biggest challenge I find with feedback phases is that each student is unique and all will immediately want support on the one question they are working on. There is only usually one teacher so its important that structures are put in to support their own self improvement. I know that I have to be very disciplined in these lessons and encourage students to come to me at the front for support so I do not get to engrossed in a one to one conversation and not survey the rooms engagement.

I am a bit of a control freak in the classroom and know that I need to be a bit more flexible in phase 4 and let students work through at their own pace once I have modelled and that’s something I aim to develop further this year.

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