Previous posts offer a more detailed explanation into what we have tried to create but in summary and itemised this is our progress so far:

By topic (There are 24 topics in Third Level):

- Course plan document written
- Learning intentions agreed
- Resources for teaching developed – especially in topics with pedagogical shifts
- Pupil tasks and worksheets developed and sourced for every learning intention (many individual learning intentions have a variety of resources)
- Rich tasks identified and written in to the course
- Diagnostic assessments for every learning intention
- Summative assessment for each topic created separately and compiled into an extensive ‘phase’ test

- Homework booklet created to support ‘every pupil, every night’ policy
- Summative assessments compiled with a fair balance of each topic to thoroughly assess & marking scheme created to ensure consistency
- Revision booklet created & accompanying solutions booklet for pupil use in their own time
- Pre-test AfL grid sheet for pupils to evaluation and focus their revision
- Post-test feedback sheets
- Post-test remediation work for further practice in order to achieve mastery.

- Members of the department attended MathsConf in Bristol and Sheffield.
- Stirling conference attended by a number of the department two years running
- Department colleagues participation in Glasgow Counts training
- Attendance at Mark McCourt Mastery course in Kendal
- Invitation extended for Bruno Reddy to visit department for a one day workshop on his experiences and ideas
- Reading/Research and sharing ideas at DM’s – a lot of reading on mastery, effective L&T
- More recently thinking has been on applying cognitive science and learning more about effective assessment strategies.
- A lot of conversations and correspondence with PTs in other schools

I have written and spoken at length about the problem we are addressing – a high number of our pupils are leaving without achieving National 5 after 13 years of education. As it stands, we have good S5 Higher results in sets 1 and 2 comparative with other Glasgow schools and satisfactory results for S5 National 5(set 3). However, sets 4 and below are achieving mainly National 4 in S5. The majority of set 4 are failing National 5 and then very few pupils manage to take their National 4 from S5 and achieve National 5 in S6.

Early aims for S5 attainment of the first mastery cohort:

Subjectively we “feel” that it is working, certainly, in terms of culture, things have improved. The homework return rate for our ‘every pupil, every night’ policy has been excellent this year. We have had no in-class revision and instead have issued extensive revision materials to be used by pupils in their own time. Classes have shown the initiative and resilience to study in all classes – they have been taking responsibility for their learning because we expect them to and we enable them to. Not every pupil has engaged fully but far more that would have been the case previously.

We have carried out pupil surveys and the results have been encouraging. I do not place a lot of emphasis on pupil voice as being a key evaluation tool regarding the effectiveness of learning & teaching. The main reason is that enjoyment and engagement are not good proxies for learning. Pupils struggle with this distinction. Nevertheless, I want to hear how the pupils perceive their experience and to understand the patterns that emerge. A few key points that came from this:

- Pupils realise they get homework every night and know why this happens
- Pupils feel that they have used the revision materials
- Pupils overwhelmingly are positive/neutral about their experience of maths (no change on before)
- More pupils in the previous year group (non-mastery) felt we went ‘too slow’. This is the surprising result! These pupils are from a year group who, for sets 3 and below, covered almost twice as much work! This contradiction perhaps confirms why we need to be careful when making judgement based on pupil perception.

Department colleagues also provided feedback. Some key points to emerge:

- Appreciate having less pressure to ‘get through’ the curriculum. Like the idea of covering less but with more depth.
- Lesson planning has been easier in so far as the course is comprehensively resourced with quality materials.
- Time has been required to study new pedagogies before teaching some topics.
- Perception that pupils can achieve more and that pupils are making better progress.
- Good uptake of homework
- While time has been spent talking, both in DMs and informally, colleagues felt they would like even more time to discuss subject specific learning and teaching.

Above are the figures for the third Level Mastery cohort classes.

I feel it is too early to be drawing final conclusions from this data. We have the rest of Phase 5 to complete before we have completed study of Third Level.

Initially my thoughts are that this data is encouraging. If anything the vast majority of pupils are experiencing success. Previously vast swathes of the cohort were failing. The mean in set 1 was 69% last year (albeit on a different assessment). Our assessments are rigorous. I have uploaded to dropbox the decimals section of our phase two test for interest. I feel that we are assessing a comprehensive selection of basic and challenging questions from the breadth of the topic. We also test interleaving of other topics (order of operations, negatives and perimeter in this case). Each summative assessment has 4 or 5 of these topic sections and is taken over two days. For a pupil to achieve even 60% means they need to be able to do quite a lot of mathematics.

I have put the data here in the expectation that those reading will ask questions, challenge us and hopefully offer opinions regarding the validity of our positive feelings regarding the attainment thus far. Please do leave a comment below or send me a tweet!

]]>I feel it is too early to be drawing final conclusions from this data. We have the rest of Phase 5 to complete before we have completed study of Third Level.

Initially my thoughts are that this data is encouraging. If anything the vast majority of pupils are experiencing success. Previously vast swathes of the cohort were failing. The mean in set 1 was 69% last year (albeit on a different assessment). Our assessments are rigorous. I have uploaded to dropbox the decimals section of our phase two test for interest. I feel that we are assessing a comprehensive selection of basic and challenging questions from the breadth of the topic. We also test interleaving of other topics (order of operations, negatives and perimeter in this case). Each summative assessment has 4 or 5 of these topic sections and is taken over two days. For a pupil to achieve even 60% means they need to be able to do quite a lot of mathematics.

I have put the data here in the expectation that those reading will ask questions, challenge us and hopefully offer opinions regarding the validity of our positive feelings regarding the attainment thus far. Please do leave a comment below or send me a tweet!

Drill and Thrill – Dani Quinn

I would encourage all maths teachers, and heads of department in particular, to read Dani’s blog. She is from the (in)famous Michaela school in North West London - the one where a focus on knowledge, hardworking kids and teachers actually teaching is the expectation. Within the first ten seconds of hearing Dani speak it was clear that any of the negative ideas about Michaela being evil oppressors of children were clearly not true. Her enthusiasm was contagious and she smiled and laughed the whole way through the presentation. She had the room in stitches as she told anecdotes about the kids in her classes. How much she really cares about those kids and how well she knew them was evident.

The main focus of the workshop was on the idea of using drill exercises. Drills can be used to improve fluency in several aspects of mathematics:

Drilling is most suitable for those topics which are high leverage (vital/advantageous for later learning and attainment) such as fraction to decimal conversions, rounding, adding and subtracting fractions, multiplying and dividing by powers of 10, adding and subtracting negative numbers etc. Drilling isn’t suitable for topics that are too complex. For example, metric conversions have too many decisions and steps required so don’t lend themselves well to drilling. Things that are too simple and allow pupils to go into autopilot such as multiplying fractions or multiplying 20 values by 0.1 aren’t ideal either. Furthermore, progressively harder questions shouldn’t be used in drilling either – that is normal work! As a guide, anything that will take over 15 seconds to do isn’t good for drilling.

Dani spent significant amount of time explaining the culture in her classroom and how she explains the relevance and importance of the regular drills to her pupils. She also spent time talking about the joy factor that the drills have brought into her class. I really didn’t expect this but as she described the way she has created a culture of kids taking it seriously, aiming to do the drills quickly (but not carelessly rushing) and being slightly competitive while having fun, I could see how it would engage them. Dani uses drills to help make improvement visible to her pupils via ‘repeat, improve’.

Interestingly Dani spent time explaining that they do explain things to the pupils. I’ve heard the “just tell them” soundbite often, but suspected this was more to do with emphasising that the pedagogy isn’t inquiry based. Controversially she suggested that we should just let the kids move the decimal point when multiplying and dividing by powers of ten. We do it ourselves as teachers, so why deny the kids this knowledge? If she’d suggested that we teach kids to ‘change side, change sign’ for solving equations I might have dismissed the idea out of hand, but I now find myself wondering if she has a point. If we’ve explained that the point doesn’t really move, but we can do it as a cheat – will it have any real negative impact later? Dani actually blogged about this yesterday (link above).

Finally, Dani explained the streaming approach at Michaela. An interesting reverse weighting approach is used. The bottom class is for those who have the poorest literacy, while the top class is for those who are the best mathematicians. Pupils stay in these classes for all lessons across the school. Different pupils have different needs – Dani explained that she uses drilling with all classes but less often for fundamentals with the top class, however, with a lower group it is highly effective in developing that procedural fluency in the essentials. Music to my ears again – I don’t like the promotion of the ideology that everyone should go mixed ability. As @mrgraymath pointed out at Stirling last weekend it depends on your context. The evidence also shows this, if you read below the surface.

Kris is somebody I’ve followed closely on Twitter and on his blog over the past couple of years. I was delighted to finally get the chance to hear him speak. I’ve been reading bits and pieces about Direct Instruction over the past year or so, but Kris is the guy who really has his finger on the pulse. Direct instruction can have many meanings, as Greg Ashman pointed out in the latest Mr Barton podcast - however Kris’ focus is on Direct Instruction as explained by Siegfried Engelmann. A disclaimer: Kris said he has found reading this work very hard going and that he didn’t know all of the answers. What I’ve written up here serves two purposes – to share the ideas from yesterday with those who couldn’t be there and also to clarify my own thoughts on what I’ve learned. I may not have this completely accurately and will be happy to correct any misunderstandings.

Kris discussed the use of examples and non-examples. I’d first encountered this idea when Bruno Reddy came to visit our department back in October. Bruno had talked about illustrating what was a difference of two squares and what was not – but showing pupils examples of each. Kris discussed how in some cases comparison is more suitable to example/non example. For instance with gradient, to say that something is a gradient or is not a gradient is a bit of a mute point. However, to compare gradients has merit– this is steep/this is shallow, this is increasing/this is decreasing.

A myth to be exposed was that Direct Instruction is about teacher talk. A message I’ve taken away is that if the board work is good enough, then very little has to be said. Both Kris and Dani were vocal about the the fact we can’t listen and read at the same time. Talking over what we are writing on the board, or speaking as pupils copy notes down is ineffective. Giving pupils time to read before explaining is important. Kris demonstrated a lesson sequence where the only thing said was repetition of the question wording.

Kris explained the difference between a process and a transformation. Multiplying out a single bracket is a transformation while multiplying two binomials is a process. My understanding is that if the vast majority of pupils can infer the knowledge by simply seeing some questions and associated answers (no working) then it is a

I’ll now attempt to do justice to Kris’ explanation.

The format is as follows:

- Initial instruction
- Initial Assessment
- Expansion Sequence

During all of this, there is minimal/no teacher talk. At the end of the exposition the teacher will ask for any questions from the class. Next follows a short assessment sequence of questions, posted to check if pupils have gained the correct inference. If there are errors at this point (which from the sounds of it is actually unlikely) then these can be rectified.

Kris may then extend this knowledge further by posing the following as questions.

Further down the line, having taught pupils the distribution property for two binomials

(a + b)(c + d) = a(c + d) + b(c + d), which is also a transformation and can be taught this way, he may pull it all together by teaching a process. This would then be done as shown in the following slide:

I really liked how the vocabulary is precise. I liked how it pieced together with the previous learning being so explicit. The pupils would be very aware of how they were using the previous knowledge.

Now all of the above may sound a little bit like discovery/inquiry based learning but, as Kris explained to me after the presentation, it really isn’t. The examples have been so carefully chosen that it is very difficult to infer anything else. The difference with discovery-based learning is that it is too wide open in scope. It is possible for pupils for infer all sorts of incorrect ideas, which would then need to be explicitly unpicked in any case.

Kris discussed ‘the best lessons’ he’d ever taught, which were using this method for simultaneous equations. He had very low ability pupils solving by elimination with real fluency. Interestingly he had broken down simultaneous equations by elimination into, I think, 13 different transformations/processes. Kris showed several slides based upon just one of these elements – which was adding two equations together. This was not for the purpose of elimination, but as a skill in its own right. I spoke last week about breaking things down and then breaking them down again. Kris is a master of this. You’ll see how clever the examples below are and how the final examples lead to elimination of a variable:

Kris also showed some examples for calculating area of rectangles and then eventually area of a triangle.

Kris also showed some examples for calculating area of rectangles and then eventually area of a triangle.

A final important point to make is that, contrary to the beliefs about Direct Instruction being about the teacher lecturing and pupils being passive, the duty is on the pupil to be engaged and thinking. The cognitive demands are the on the pupil too, however, because the lessons are broken down so much there is no risk of cognitive overload. The teacher isn’t going to explain a process/transformation, until the pupil has considered it.

This all correlates with the idea of learning as being the residue of thought. As pupils really are being active then there is much more ‘active learning’ than when copying down notes and having a teacher do all the explaining and talking.

From a personal perspective I intend to reflect about where this pedagogy can appropriately be incorporated into my own teaching. I also need to be sure how it will fit with the existing pedagogies we are trailing such as visual representations for algebra. Rigorous lesson planning will be required. This isn’t something you can do in two minutes before the end of lunchbreak. Kris suggested that index laws, adding fractions with like denominators, multiplying fractions, simplifying by collecting like terms, angles on a straight line, adding and subtracting negatives etc could be taught this way.

Once again I found the conference to be thought provoking and challenging. I would encourage all of my colleagues in Scotland to support this event when it makes its Scottish debut in Dunfermline in August. I think there is a lot to be gained from cross-fertilising ideas from both sides of the border. At under £30 it is a bargain and a great addition to our conference calendar. Details can be found here: https://completemaths.com/events/scotland

I think context is important here. My previous role was at a very high attaining school, in an affluent commuter town near Edinburgh, where my experience of curriculum design had a different focus. I was heavily involved in writing out ‘blue course’. For over half of our learners five years was more than adequate to allow them to get to Higher. We had the luxury of being able to give them regular experience of the subject outside of the prescribed curriculum. 55-60% of the cohort arrived with a very solid basis (in old money, mastery of substantial amounts of Level E). For instance, with these pupils we felt we had the time for a large S3 statistics unit, which went way beyond the scope of the curriculum. It was fun to teach and the kids found it interesting. It didn’t add value to our SQA results except perhaps encouraging pupils to consider taking Statistics at Advanced Higher. Certainly, these learners didn’t seem to suffer with this time spent elsewhere outside of the Es&Os as results were continually impressive.

Fast-forward to August 2015 and my arrival at my current inner-city comprehensive with a much more diverse clientele in terms of SIMD, ASN, LAC and EAL. Unusually for the city, we present over 30% of cohort at Higher in S5. However, the reality is that even our most able pupils arrive and are already 3 to 6 months behind their peers in one of the near-by private schools. For the majority of our learners the story, on arrival, things are much bleaker than this. We struggle to get over 40% of the cohort anywhere an N5 pass by exit. My task was not to write a course, which met the needs of just those pupils arriving with a good basis, but instead, a course that met the needs of every pupil.

It is within this context that our vision has developed. While our main focus has been slowing down to achieve more, as per the Mastery cycle, this has not meant that we have been willing to sacrifice precious time on a topic where the outcomes on later attainment are negligible. We have had to be pragmatic at times. For all of our pupils, but especially those outwith our top 30% we don’t have a moment to spare on any activity which fails to add tangible attainment value either in skills or meta-skills (problem solving practice, strategy selection etc).

At the outset of planning our Third Level we had the aim that our curriculum would equip learners with all of the necessary skills and knowledge required for a successful engagement with the subsequent levels. We wanted them to know how to find a fraction of an amount, we wanted them to be able to use angle facts to solve problems, and we wanted them to have numerical fluency and so on. We also explicitly wanted to plan for problem solving and for the development of conceptual/relational understanding as well as the development of the hardworking, studious culture that I described in a previous blog post.

As I’ve written before, we have no timelines. We move on once the majority of the class can demonstrate mastery. I’m often quizzed by colleagues from other schools about how boring it must be for our pupils and how disengaged they must be. My reply is always the same. Evidence shows that the single biggest impact on motivation and engagement with mathematics is previous success. If we fall into the trap of using extrinsic motivation ‘look at this fun worksheet/PowerPoint/maths game’ then we are doomed to failure. Also aiming to find pseudo-realistic situations to motivate the learners is normally contrived and questionable. Occasionally there is something very relevant and useful, but often not. As much as I admire Dan Meyer’s work, the likes of Greg Ashman have torn many of his nice ideas to pieces by discussing the lack of evidence for a positive impact on learning. Instead of trying to get their attention in these ways, we decided to throw out most of the Tarzia puzzles and calculated colouring sheets and focus on quality learning experiences with maximum mental engagement and time on-task. We want to develop intrinsic motivation. I am delighted with the engagement of our S1 cohort. Our learners are beginning to see that what we are doing is working for them – the fifth class of seven sets almost all passed the whole numbers unit. Last year a tiny number of pupils in that class achieved the same. The homework policy and revision policy play a part but I think most importantly is the fact that we’ve been very precise in our planning and allowed learners the time to achieve mastery.

I’ve always been of the opinion that if we teach mathematics well then learners will pass the exams as a consequence of that. I’ve never much favoured the tail wagging the dog with passing the exam the focus. That approach all too often leads to rote learning with no relational/conceptual understanding. It can also result in pupils with weak problem solving/application skills. Despite this, I decided to make a quick analysis of the last National 5 mathematics exam. Basic number permeated everything as one would expect. I have listed, below, the percentage of N5 exam questions in which each of the following Third Level skills was required. Of course, many other Third Level skills appear, but for brevity I have listed only those appearing in over 10% of the questions.

I don’t think this tells us anything most teachers didn’t already suspect. Third Level algebra is vital as is number. Shape less so, but it has a role to play. I plan to do a similar exercise for Higher and expect similar results.

I’ve had many discussions about course order with colleagues from both outside and within the department. We knew that number had to come first as it is the corner stone. After this we felt that we had a lot of flexibility. I know some people have looked at our order and thought it to be ridiculous. ‘Why would you teach powers before you taught integers?’ or ‘why are you leaving equations so late?’ To be honest I don’t have clear answers on every single aspect of this. Shockingly I’ll admit, some of it is arbitrary, as we didn’t feel there was a strict order in which some of it had to be taught. I’m still open to being told that topic X should come before topic Y if there is a good reason for it. Even Bruno Reddy looked at our course and was surprised by some of it, but acknowledged that with our approach to interleaving the order is less important.

We haven’t cut a lot of content, save the poster about a Scottish mathematician, but what we have changed is emphasis. For instance we felt an awareness of rotational symmetry to be adequate. We don’t need learners to know about the order of rotational symmetry; we don’t need them to be able to complete pictures so that they have rotational symmetry. What purpose does it serve? I’ve certainly used this knowledge rarely in later maths. Odd/Even functions at AH come to mind as one of the few points it appears.

For other topics we drill much deeper. For order of operations we need fluency in a variety of contexts. Leaners need to be able to handle brackets within brackets with integers and fractions. For this reason we’ve really worked at breaking this topic down and creating more learning opportunities for it. Coincidentally, we’ve banned the term BODMAS as at best it is confusing and at worst just wrong! Using simple concrete models the learners understand intuitively why we multiply before adding. I thin BODMAS belongs in the trashcan with FOIL, change side, change sign and all those other nice but naff ideas.

As mentioned, whole numbers comes first as it is the single most important topic. By this we meant: place value, four operations, order of operations, multiples, powers and roots. Doing this first gives us a working toolkit upon which to build (and some nice ideas we could interleave throughout). Shape seems to dominate the rest of phase one. Symmetry lasts only a couple of lessons and has to come early due to an IDL project. We decided to do properties of shapes before area or length as this gives us a solid footing upon which to explore these topics. Again I must emphasise the learning experience is what makes the difference, not the course order. For properties of shapes we ditched the dire exercises in the normal texts and instead put together a short department booklet full of tasks, which developed spatial reasoning and application of the key facts about the regular polygons/quadrilaterals. One such task is this classic from Don Steward: http://2.bp.blogspot.com/-f24JVscUKkQ/URgJEnge9mI/AAAAAAAAIvg/kQt74bnwQjc/s1600/Picture1.png

It was decided to do angles early as it is a topic our older pupils seem to have a poor grasp of. Angles is a topic which can be easily interleaved into many other topics and as such gain repeated exposure, which would held with retention after mastery was attained first teaching.

Coordinates also give us an essential platform, which the learning of other topics can be built upon. There are no coordinate picture sheets in the suggested resources for the new course. On reflection: a pupil can plot and read points or he/she cannot. If he/she can do it then this is simply a time consuming drill, with little learning value, where one small error leads to a deformed picture. If the pupil cannot plot points then we end up with spaghetti like masterpiece! So either the pupils need practice at basic point plotting or is ready for deeper thinking and application of their knowledge. Again, Don Steward has an excellent task we used, which led to some excellent discussion and helped to set the scene for some important ideas for later. http://2.bp.blogspot.com/-ukHyU4o-3YU/VXc-CHfYGiI/AAAAAAAAQqQ/Dc9ZPuj0y2M/s1600/Picture1.png

Integers (1) is adding and subtracting negatives. This is very comprehensive and takes is well beyond any existing Scottish text in terms of the complexity of calculations the learners are expected to be able to perform. Integers(2) is separate as we don’t want to confuse learners by having too many similar ideas near each other. Decimals come before fractions and percentages as the research I’ve read says it should.

Only once fractions, decimals and negatives have been mastered do we consider looking at equations. I don’t see many questions such as -5 = -x + 2 in the normal Scottish S1 books. In our view, there is no point in engaging with equations if you have to “hide” certain questions. We want to be able to take the shackles off and allow pupils the freedom to use whatever operation they want, so long as they do it to both sides. Comparing solutions is one of Swan’s principles of effective mathematics teaching. Equation solving is a great area for this, if learners have the required skills.

Later topics give more scope for application of skills and knowledge learned in the earlier phases – money and speed distance time etc.

It’s difficult, even within the blog to explain the rational for every single decision taken. However, I would emphasise that what is more important than the order is how we interleave the topics. For instance in decimals we interleaved integers(1) skills. Pupils were working on -3.2 – (-2.8), questions, which allowed decimals practice but also developed their integers knowledge (particularly magnitude). This particular example is subtle but important. Try the question with your own S1 – for many this is not intuitive. While doing these questions is nothing groundbreaking, our course is littered with these little opportunities for thought. In integers (2) we have pupils finding cube roots of negative numbers. Again, utilizing prior knowledge, but generating deeper thinking about the current topic. Today our two fastest classes were clarifying knowledge of integers, decimals, order of operations and powers all at once, with questions such as (-3+0.7)-(0.3 + 2.90.4 - 0.4 ).

There is much I could say about the curriculum design. I could laboriously discuss which learning intentions we planned for each topic and those we omitted, but I doubt anyone would want to read it! If you really do want to know more about a specific topic then let me know or say hi at Stirling in a few weeks.

**How long did the process of designing your curriculum take? **

At times I wish it was all over – it’s overtaken everything else for the past 18 months! Any spare moment that has been available between trying to run a dept and teach a fairly heavy timetable has been spent on this. Between the colleagues involved several hundred hours of work on this is not a modest estimate. Two of us have really taken the lead on the development, with others beginning to contribute more as they’ve learned about the ideas. The first 3 or 4 months were simply on reading and research. We read a lot. I’ve got at least 50 papers on mastery, memory and evidence based pedagogy stashed away in a folder. There have been countless hours of discussion in the department – not just in planned meetings or DMs but on a day to day basis. I’m sure @shivmckenna would agree that, on average, we talk about curriculum development for at least a couple of hours a week, even now. It was way more when we were getting started. Each topic takes hours of work consisting of: breaking down learning intentions, refining them further, investigating evidence based approaches to learning and teaching, finding resources which meet our aims, creating resources where none exist that we know of (very often), developing teaching slides, writing summative assessments, writing diagnostics (one for every learning intention), writing the associated pages for the homework packs, writing the associated questions for the revision packs. At present we have finished phase one and two (for now) and are almost done with phase 3. Our fastest classes won’t get to phase 4 until August, so this buys us some time. Despite hours of writing back in the Spring of 2016 we found that evaluating some topics a few months later made us realise that something was missing, needed enhancing, needed broken down more or that there were opportunities for interleaving of problem solving missed. In the case of equations, only this week we decided that the whole thing was a piece of junk and started again. For some topics we’ve created entire booklets, from scratch, as teaching resources.

The other aspect of the curriculum planning has been and will be the macro-curricular picture. By that I mean looking at the longer-term pathways for learners. We have a rough idea of what our fourth level course will look like. Moving forward we’ll need to plan this in a more coherent manner. We also need to look at how this links with National 4. There will be no distinct N4 course in the department. I fully expect it to take some of our weaker learners five years to do all of third and fourth level properly. Similarly our Level 5 course will need to map with N5 for some learners, but not for our Higher candidates as they only need the Level 5 knowledge which is required for Higher. We don’t present pupils at all in S4, for anything. It gives us tremendous flexibility to plan long term learning pathways. We view this as very long term project. The level of planning and preparation that has gone into Third level will in time be rolled out all of the way to Advanced Higher. It’s hard work, but all of the teachers delivering this new curriculum have been thrilled by pupil progress. For that, it is worth it.

**How have primaries reacted to mastery?**

Our local primaries are all participating in the excellent Glasgow Counts programme. I understand that this is about evidence-based pedagogy, influenced by work in Singapore, Shanghai and Hackney. The aim is to increase the quality of subject specific learning and teaching and to develop primary teacher’s subject knowledge in mathematics. In this respect, we are not a lone voice in the wilderness. We have built links with colleagues who run Glasgow Counts and hope to reach out to primary colleagues in a more coherent manner in coming months. So far we haven’t got to a deep level of pedagogical discussion. I believe that Glasgow primaries are moving to a common planner for maths so this should help to develop a shared understanding, coupled with the similar pedagogical approaches being promoted by Glasgow Counts.

]]>For other topics we drill much deeper. For order of operations we need fluency in a variety of contexts. Leaners need to be able to handle brackets within brackets with integers and fractions. For this reason we’ve really worked at breaking this topic down and creating more learning opportunities for it. Coincidentally, we’ve banned the term BODMAS as at best it is confusing and at worst just wrong! Using simple concrete models the learners understand intuitively why we multiply before adding. I thin BODMAS belongs in the trashcan with FOIL, change side, change sign and all those other nice but naff ideas.

As mentioned, whole numbers comes first as it is the single most important topic. By this we meant: place value, four operations, order of operations, multiples, powers and roots. Doing this first gives us a working toolkit upon which to build (and some nice ideas we could interleave throughout). Shape seems to dominate the rest of phase one. Symmetry lasts only a couple of lessons and has to come early due to an IDL project. We decided to do properties of shapes before area or length as this gives us a solid footing upon which to explore these topics. Again I must emphasise the learning experience is what makes the difference, not the course order. For properties of shapes we ditched the dire exercises in the normal texts and instead put together a short department booklet full of tasks, which developed spatial reasoning and application of the key facts about the regular polygons/quadrilaterals. One such task is this classic from Don Steward: http://2.bp.blogspot.com/-f24JVscUKkQ/URgJEnge9mI/AAAAAAAAIvg/kQt74bnwQjc/s1600/Picture1.png

It was decided to do angles early as it is a topic our older pupils seem to have a poor grasp of. Angles is a topic which can be easily interleaved into many other topics and as such gain repeated exposure, which would held with retention after mastery was attained first teaching.

Coordinates also give us an essential platform, which the learning of other topics can be built upon. There are no coordinate picture sheets in the suggested resources for the new course. On reflection: a pupil can plot and read points or he/she cannot. If he/she can do it then this is simply a time consuming drill, with little learning value, where one small error leads to a deformed picture. If the pupil cannot plot points then we end up with spaghetti like masterpiece! So either the pupils need practice at basic point plotting or is ready for deeper thinking and application of their knowledge. Again, Don Steward has an excellent task we used, which led to some excellent discussion and helped to set the scene for some important ideas for later. http://2.bp.blogspot.com/-ukHyU4o-3YU/VXc-CHfYGiI/AAAAAAAAQqQ/Dc9ZPuj0y2M/s1600/Picture1.png

Integers (1) is adding and subtracting negatives. This is very comprehensive and takes is well beyond any existing Scottish text in terms of the complexity of calculations the learners are expected to be able to perform. Integers(2) is separate as we don’t want to confuse learners by having too many similar ideas near each other. Decimals come before fractions and percentages as the research I’ve read says it should.

Only once fractions, decimals and negatives have been mastered do we consider looking at equations. I don’t see many questions such as -5 = -x + 2 in the normal Scottish S1 books. In our view, there is no point in engaging with equations if you have to “hide” certain questions. We want to be able to take the shackles off and allow pupils the freedom to use whatever operation they want, so long as they do it to both sides. Comparing solutions is one of Swan’s principles of effective mathematics teaching. Equation solving is a great area for this, if learners have the required skills.

Later topics give more scope for application of skills and knowledge learned in the earlier phases – money and speed distance time etc.

It’s difficult, even within the blog to explain the rational for every single decision taken. However, I would emphasise that what is more important than the order is how we interleave the topics. For instance in decimals we interleaved integers(1) skills. Pupils were working on -3.2 – (-2.8), questions, which allowed decimals practice but also developed their integers knowledge (particularly magnitude). This particular example is subtle but important. Try the question with your own S1 – for many this is not intuitive. While doing these questions is nothing groundbreaking, our course is littered with these little opportunities for thought. In integers (2) we have pupils finding cube roots of negative numbers. Again, utilizing prior knowledge, but generating deeper thinking about the current topic. Today our two fastest classes were clarifying knowledge of integers, decimals, order of operations and powers all at once, with questions such as (-3+0.7)-(0.3 + 2.90.4 - 0.4 ).

There is much I could say about the curriculum design. I could laboriously discuss which learning intentions we planned for each topic and those we omitted, but I doubt anyone would want to read it! If you really do want to know more about a specific topic then let me know or say hi at Stirling in a few weeks.

At times I wish it was all over – it’s overtaken everything else for the past 18 months! Any spare moment that has been available between trying to run a dept and teach a fairly heavy timetable has been spent on this. Between the colleagues involved several hundred hours of work on this is not a modest estimate. Two of us have really taken the lead on the development, with others beginning to contribute more as they’ve learned about the ideas. The first 3 or 4 months were simply on reading and research. We read a lot. I’ve got at least 50 papers on mastery, memory and evidence based pedagogy stashed away in a folder. There have been countless hours of discussion in the department – not just in planned meetings or DMs but on a day to day basis. I’m sure @shivmckenna would agree that, on average, we talk about curriculum development for at least a couple of hours a week, even now. It was way more when we were getting started. Each topic takes hours of work consisting of: breaking down learning intentions, refining them further, investigating evidence based approaches to learning and teaching, finding resources which meet our aims, creating resources where none exist that we know of (very often), developing teaching slides, writing summative assessments, writing diagnostics (one for every learning intention), writing the associated pages for the homework packs, writing the associated questions for the revision packs. At present we have finished phase one and two (for now) and are almost done with phase 3. Our fastest classes won’t get to phase 4 until August, so this buys us some time. Despite hours of writing back in the Spring of 2016 we found that evaluating some topics a few months later made us realise that something was missing, needed enhancing, needed broken down more or that there were opportunities for interleaving of problem solving missed. In the case of equations, only this week we decided that the whole thing was a piece of junk and started again. For some topics we’ve created entire booklets, from scratch, as teaching resources.

The other aspect of the curriculum planning has been and will be the macro-curricular picture. By that I mean looking at the longer-term pathways for learners. We have a rough idea of what our fourth level course will look like. Moving forward we’ll need to plan this in a more coherent manner. We also need to look at how this links with National 4. There will be no distinct N4 course in the department. I fully expect it to take some of our weaker learners five years to do all of third and fourth level properly. Similarly our Level 5 course will need to map with N5 for some learners, but not for our Higher candidates as they only need the Level 5 knowledge which is required for Higher. We don’t present pupils at all in S4, for anything. It gives us tremendous flexibility to plan long term learning pathways. We view this as very long term project. The level of planning and preparation that has gone into Third level will in time be rolled out all of the way to Advanced Higher. It’s hard work, but all of the teachers delivering this new curriculum have been thrilled by pupil progress. For that, it is worth it.

Our local primaries are all participating in the excellent Glasgow Counts programme. I understand that this is about evidence-based pedagogy, influenced by work in Singapore, Shanghai and Hackney. The aim is to increase the quality of subject specific learning and teaching and to develop primary teacher’s subject knowledge in mathematics. In this respect, we are not a lone voice in the wilderness. We have built links with colleagues who run Glasgow Counts and hope to reach out to primary colleagues in a more coherent manner in coming months. So far we haven’t got to a deep level of pedagogical discussion. I believe that Glasgow primaries are moving to a common planner for maths so this should help to develop a shared understanding, coupled with the similar pedagogical approaches being promoted by Glasgow Counts.

**How have the primaries responded to this?****Are these assessments designed to assess current knowledge or potential?****Do you have any other plans for this transition in the future?**

Ultimately, realistically, I want all of set 1 to 5 to be able to get H or N5 by exit, so they all need to learn the material eventually. In last year’s S1 (and this year’s S2) we’d see scores of 90% in set 1 and then a slide as we looked at each set below, to the point that, say, set 5 had a mean score of 50% (or less). Because we are going to be going much slower, and into more depth with classes, as they require it, then this phenomenon should become a relic of the past. Summative assessment should be once learners are in a position to succeed.

I think the analogy of learning to drive is applicable here. Some people do a crash course of 5 lessons and pass first time, which others take over a year! Ultimately though, nobody would be put forward for their test unless their was a realistic chance of passing. Regarding our approach, I will post the scores of the five classes following the course once all have completed the assessment.

**This is one of the biggest challenges that we face. I’m fed up of kids doing the same 5 Calculated Colouring exercises and playing on Cool Maths Games for three years before we “get them” their National 3 in S4 and National 4 in S5 before finding them completely out of their depth in a National 5 class in S6 (if they are lucky!).**

**One of my main goals this year has been to pick up the pace across the department and ensure more accountability - I feel that I’m doing the opposite of what you are doing in this regard. Your logic is hard to argue with but I am not sure that I am in a position to do this yet.**

**Every PT that I meet is tearing their hair out about this****You mentioned:**

**I would also add:**

Agree with all you’ve said there. We did anonymous surveys of S2 classes who recently were given a test and provided with some revision materials. From set one 87% said they did their pre-test revision as advised, only 23% of set 5 said the same. I’d like to think that after a year of the new curriculum and interventions we are putting in place that the culture in next year’s S2 will be better than at present. I agree with thinking Habits. I’m current reading “Thinking Mathematically” By John Mason, recommend it. So much of what we do as maths/science/engineering/computing/whatever graduates is second nature and internalised. We sometimes can’t externalize and codify this stuff for learners, as we don’t really know what we do either, in terms of solving problems. We just have a knack, or feel or intuition. This has been learned though. I’m not talking explicit skills, but more our range of approaches to questions and mind-set when tackling something new. This is why regular engagement with problem solving and rich tasks is vital in developing mathematicians.

Of course, all of this is built upon knowledge. Unless pupils have first acquired the relevant knowledge and understanding of a topic then problem solving is very difficult. That’s why in the old days of credit you rarely saw a pupil score better on the reasoning component than on the knowledge component. The planning of the development of problem solving skills has to be explicit. As part of this the ideas and understanding of resilience and perseverance can be teased out over time too. While I often disagree with Jo Boaler’s comments (especially on memorisation of timestables), she states that

**Again I find myself nodding along here. We should absolutely be holding all of our learning to the highest standards. For me, this extends to the private sector who we have a huge amount to learn from. I’ve tweeted a bit on my Off Piste account about the standard of their entry exams. Although I find this horribly elitist, I feel that there is no reason that kids from Glasgow can’t be held to the same standard of those at Eton.**

**My ideal scenario is for classes to be investigative, challenging and creative and homework to be the essential “boring”, repetitive drills for retention. However, my problem is that kids are not doing their homework meaning that these exercises have to have a place in class. Trying hard to crack this one but finding it tough! I’m going to look at your policy at our DM and see what the response is.****That said I’d like to read into the current trend and see what the research says about homework.**

Agree – I like to do the harder stuff in class, where support is available to the kids. I’ve had battles with older kids across the dept regarding homework in various folks classes, but I hope if we get S1 on board doing it from day 1, it won’t be a shock when we ask them to do a bit more in N5. At present, I’ve had to speak to less than a handful of pupils in S1 about not returning homework. The kids now know to expect homework every lesson.

**Totally on board with this. This will make a huge difference when coupled with the supported study.****Have you created the revision pack?****How “leading” are the questions in the pack?**

**On what basis do they disagree?****I don’t think that anyone can argue with having high expectations for all learners. I could maybe understand the argument that we are putting learners under too much pressure at a young age. Saying that, perhaps it’s better that we give them that responsibility earlier - I’ve seen too many students crumble in S4 when the expectation is suddenly heightened.**

**Can I please please see these pathways?****What planning went into these and what did you use for guidance?****My planning principles are different from what you’ve described but, again, I find it very difficult to find fault in your thought-process. Would be really keen to compare notes on this.**

**Do all staff use the same resources?****This is my biggest dilemma (as you will have read in my tweets). On one hand, I want high-quality learning experiences to be common place in all classrooms. On the other hand, I want resources to be reactive to the class being taught. I feel that teachers can become resource reliant. Often the “turn to page…” approach is no worse that the “here’s a Tarsia” approach. Really keen to hear what your thoughts are on this. I’m trying to implement the Swan tasks as a compromise.****On that note, since being introduced to the Swan tasks at your presentation in Stirling, my teaching is the strongest that it’s been so thanks for that. I feel that these tasks offer a framework that almost guarantees learners to be creative as well deepening understanding.**

predicted_timings_for_fastest_classes.xlsx |

Our new Mastery Curriculum is now running with first year classes. We began the year by very broadly setting classes based upon two diagnostics – one pre-summer and one post-summer. Five of the classes are embarking upon our Third level course and the other two are looking at a Second level version with a view to moving onto Third Level at a later point. I think we have to be realistic about the starting point of our learners. It’s an uncomfortable truth that 2/7 of the cohort arrive at secondary not in a position to meaningfully engage with Third level, lacking knowledge of times tables and other key ideas, but that is where we are. It wouldn’t be doing these learners any favours to course them into a third level course, when there are so many gaps in their knowledge of Second and, in some cases, First level. We need to get these pupils equipped with the basic skills to meaningfully engage with the Third level course. I’ve seen too many pupils coursed at a level where they are missing substantial pre-requisite knowledge. Advocates of mixed ability may criticise the decision to set at all, however, we are ambitious for all of these learners. We want them to catch up – I have no intention of having ghetto classes of forgotten pupils who are destined to failure. I am delighted to say that the colleagues teaching these pupils share this mindset and I am hopeful, that we can make a difference, with careful planning.

With regard to the other five classes, they may be set, but over time this setting will be by pace of learning – not by ability and not by expected depth of coverage. The bar that learners have to clear in order to achieve what we consider security at Third level is the same for our of our leaners. (Don’t mention Benchmarks!) The rate at which learners progress towards this and achieve this standard will vary. Classes will be flexible and will reflect this. There are no timelines dictating that we spend 2 weeks on fractions etc. We move on when mastery has been achieved by a majority of the class (see previous posts for more details). Learners working at similar paces will be in the same class but there will be relatively frequent movements as and when required. I have the second fastest class – note the use of language- not second top set. We have spent EIGHT weeks on whole numbers. This thinking, on spending time on the essentials, is very much influenced by Bruno Reddy’s work at King Solomon and some of the amazing stuff happening at Michaela. Until learners are secure at the four basic operations then why would we consider moving on?

On Monday two of the classes will be sitting a summative assessment on this topic. Later assessments will be based upon whole phases of the curriculum, encompassing four or more topics, however, for an early indication of where we are and how things are going I felt that an assessment at this point was necessary.

This is a phrase we often hear – but what does it mean? I believe that our curriculum changes are about culture as much as they are mathematics. Our S4/5/6 middle/lower classes are disengaged and underachieving for two many reasons. 1. They lack the pre-requisite knowledge and understanding to engage with the level now being attempted and 2. They have not developed the study habit.

I regularly see kids from oversees arrive at the school and make up two years of progress in one, because they work hard. I want to help all of our leaners to be like this. We are doing a few things to support this in our new curriculum, which will be followed on into S2-6 in each of the subsequent few years.

We have produced booklets for each phase of our curriculum, covering almost every learning intention, with enough short homework tasks to keep our pledge of every pupil every night. Most exercises are single topic based – usually practice and drill. However, we have included some at the back of the booklet for teachers to drop in at appropriate times, which are based upon various topics covered up to that point in the curriculum. This comes from the idea of interleaving and spaced practice. In fact, 5 weeks after doing order of operations I may drop in one of the order of operations exercises for a nightly homework. Thinking about retention is important. Vitally, the homework policy is to ensure that learners are in the study habit. Four pieces of maths homework every week (not counting the formal end of topic exercises or revision materials issued) ensures that every pupil is spending time on maths outside of class. We have supported this with a very transparent homework policy. If a pupil ends up on the fourth occasion not doing homework then a letter goes home (there are sanctions for one, two and three occasions too). I haven’t had to send a single letter home for any S1 pupil after the 36 homework opportunities so far. There are always concerns about pupils copying, but the insistence on showing working and our regular in-class diagnostic testing should almost entirely eradicate this. For example, if a pupil scores 10/10 on an adding integers homework yet scores only 5/10 on a diagnostic in class on the same material then suspicions are obviously raised! We intend to roll the Every Pupil-Every Night policy all the way through the curriculum from Third Level to National 5/Higher in coming years. This culture of doing homework every night – if sustained would make a significant difference to our senior pupils. An extra 30 mins per week of maths, 37 weeks of the year, for 5 years means a total of 92.5 hours extra mathematics work. The equivalent of roughly 2/3 of a school year. Ultimately, if learners are used to doing maths homework then it will take something radical to

This four-pronged approach to pupil ownership is, to recap, about equity of expectation and equity of opportunity for every pupil, regardless of home circumstances.

It’s taken eight weeks to ensure that my class is at a point where I am confident in their handling of the above. Some of the other classes are perhaps two or three weeks away from this point. Fine. These are building block topics that are vital to everything else we will do. Colleagues have been, at times, a little concerened about the slow pace, but our clearly planned pathways all the way to S5 show there is nothing to be worried about in terms of pace relative to making H/N5, as appropriate, by end of S5.

Notice above that multiples are included but no factors or primes. One of our design approaches is to keep minimally separate, often confused topics apart initially. The next topic is negative numbers (1) – which is just adding and subtracting. No multiplication or division until a good while later. Other topics later in the course such as area and perimeter are also kept separate initially.

The results of the whole numbers assessments will be interesting over the coming weeks as classes get to the point of readiness. I am very optimistic that there will be good scores across the board. Pupils will hopefully get used to succeeding in maths, and teachers too, will really begin to believe that all of the learners can succeed.

That is, explicit learning intentions, resources for delivery and pupil practice (high quality stuff as previously discussed in other posts), teaching advice if applicable, diagnostic questions and diagnostic assessments, rich tasks and then additional homework tasks – to supplement the booklet mentioned above. The Whole numbers scheme covers six pages of A3. This, perhaps, shows you the scale of development and resourcing we are currently still working on.

The resources are vital. If the schemes of work have good resources, which encourage sound evidence based effective pedagogy, then it makes it easier for teachers to be more effective. The longer I’ve been teaching the lesson planning seems to become more complex- maybe due to my own understanding growing. The examples we use with pupils, the practice that pupils get on each of the specific common cases of a problem and the way pupils get to apply and reason their knowledge all needs to be carefully planned. I’m really interested in the work at Michaela who are creating their own textbook. Perhaps in four or five years time, our compendium of electronic resources could also be made into a department book. Most of our exercises and tasks are sourced from others: take a bow Corbett Maths, Resourceaholic, Median Don Steward, Nrich and Malcolm Swan amongst others. In other cases we have to design our own where we are using, as yet, nationally uncommon approaches (such as algebra tiles for the teaching of integers). We are mindful of Swan’s effective mathematics tasks while also aiming to incorporate the philosophy summarised here, when creating resources and tasks from scratch.

One of the key components of our new curriculum has been the development of teacher capacity – including my own. Masses of reading, research, a lot of trial and error of various approaches in my own classes are being filtered into a scheme of professional development which is running through every department meeting we have this year. Forthcoming sessions I’ll be delivering to colleagues include multiple representation approaches to integers, development of algebraic understanding and fluency, using the bar model effectively for the teaching of fractions and percentages. This follows on from last year where we looked at ideas such as abandoning other forms of solving equations except balancing. My main question is, as ever, how can we help pupils to have more procedural and conceptual fluency with the key topics? The excellent mathagogy blog suggests that just because our conventional methods work for the more able kids, it doesn’t mean the methods are good. It’s a point to consider. I’ve been able to develop fraction skills in many pupils over the years, but also failed miserably to build sustained fraction knowledge in many others. There’s no point being able to teach something to a class who won’t retain it. The learning needs to be deep enough and meaningful enough such that it**IS** learning, not just instantaneous performance. We need new pedagogy which is evidence based which works for the majority, not just some. So, as well as development of the curriculum and resources, there is a lot of professional learning going on.

]]>The resources are vital. If the schemes of work have good resources, which encourage sound evidence based effective pedagogy, then it makes it easier for teachers to be more effective. The longer I’ve been teaching the lesson planning seems to become more complex- maybe due to my own understanding growing. The examples we use with pupils, the practice that pupils get on each of the specific common cases of a problem and the way pupils get to apply and reason their knowledge all needs to be carefully planned. I’m really interested in the work at Michaela who are creating their own textbook. Perhaps in four or five years time, our compendium of electronic resources could also be made into a department book. Most of our exercises and tasks are sourced from others: take a bow Corbett Maths, Resourceaholic, Median Don Steward, Nrich and Malcolm Swan amongst others. In other cases we have to design our own where we are using, as yet, nationally uncommon approaches (such as algebra tiles for the teaching of integers). We are mindful of Swan’s effective mathematics tasks while also aiming to incorporate the philosophy summarised here, when creating resources and tasks from scratch.

One of the key components of our new curriculum has been the development of teacher capacity – including my own. Masses of reading, research, a lot of trial and error of various approaches in my own classes are being filtered into a scheme of professional development which is running through every department meeting we have this year. Forthcoming sessions I’ll be delivering to colleagues include multiple representation approaches to integers, development of algebraic understanding and fluency, using the bar model effectively for the teaching of fractions and percentages. This follows on from last year where we looked at ideas such as abandoning other forms of solving equations except balancing. My main question is, as ever, how can we help pupils to have more procedural and conceptual fluency with the key topics? The excellent mathagogy blog suggests that just because our conventional methods work for the more able kids, it doesn’t mean the methods are good. It’s a point to consider. I’ve been able to develop fraction skills in many pupils over the years, but also failed miserably to build sustained fraction knowledge in many others. There’s no point being able to teach something to a class who won’t retain it. The learning needs to be deep enough and meaningful enough such that it

Raising attainment in mathematics is a complex problem. Developing independent and resilient learners is also a complex problem. There are no quick fixes. Offering extra supported study sessions (or Higher revision weekends like the one I am on just now) can help a pupil approaching a final exam. However, I compare this to treating the symptom while avoiding the root cause. Of course, supported study and exam revision sessions are important and add value for learners. However, I offer the opinion that a bigger impact upon senior phase attainment will be given by, not only the initiatives put in place for S4/5/6, but also what goes on in the broad general education.

The BGE should not be the after-thought that it so often is. It is the fundamental underpinning. It is the basis upon which all future attainment is built. For a learner to reach a Higher or National 5 class with a chance of success in maths, the BGE experience has to be very effective. It must equip the learner with all of the fundamental explicit and implicit skills that are required to study the senior qualifications.

This complex problem does, indeed, require a complex solution. In this post I will discuss our curriculum design principles, discuss some work which exemplifies the quality of resources we will require to support our curriculum and then, finally, set out our rationale on ability setting.

In my previous post I stated that the Mastery cycle was the defining characteristic of a Mastery curriculum. Implementing this alone, according to some research can have an effect size of 0.5. This is unsurprising, given the amount of formative assessment and feedback that is built in to the cycle.

Successful implementations of Mastery curricula not only involve the cycle, but also demonstrate similar pedagogical approaches, best encapsulated by two words: conceptual understanding.

With this in mind I propose the following as the guiding principles of our curriculum. These principles are a refection of various reading on effective mathematics teaching approaches and curricula.

The Mastery cycle, automatically considers pupil prior learning through the use of pre-teaching diagnostic assessments and corrective teaching. These diagnostics should also seek to expose any of the common misconceptions pupils have on topics.

There is a significant body of evidence which suggests that effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems. The development of conceptual understanding will promoted through teaching advice in our schemes of work and supported by high quality resources. A time investment will be required for professional dialogue and learning, to increase our own understanding of how best to aid the development of conceptual understanding.

To support teachers, our schemes of work will offer suggested question stems for each topic. A source of inspiration for this is the Kangaroo Maths schemes of work (freely available online)which support a Mastery approach to the English curriculum. Much of the questioning support material from these schemes can be incorporated into our own. Many of the question stems in Kangaroo are from the work of John Mason and Anne Watson – another couple of names unfamiliar in Scotland, who are outstanding academics in the world of mathematics education.

Dylan Williams mentions that formative assessment is an activity of the short term, medium term and long term. In my experience, many teachers are good at short-term formative assessment – it is almost instinctive. In our course planes, this will be supported by the incorporation of hinge questions and the question stems from Kangaroo in addition to other formative assessment tasks. Our medium term formative assessment will be supported by the diagnostic tasks that are built into the Mastery cycle. Longer term formative assessment will be supported by both learning logs and post assessment feedback sheets.

While a Mathematics exam may be an individual pursuit the classroom is an environment where productive collaboration between learners can be very effective for learning and teaching. The use of open ended and or challenging questions are often very suited to such situations. These sort of tasks tackled in pairs, particularly, can have a significant impact on learner understanding. I tend to agree with literature that takes a constructivist viewpoint. A significant part of learning is based upon learners constructing their own meanings and interpretations. Learners will do this whether they have learned through a rich collaborative task, direct instruction or any other approach. A quality task, with proper scaffolding, used in a situation where on-task discussion is allowed can allow learners to develop their own meanings and interpretations. The role of the teacher is to ensure that the meanings do not include miscomprehensions. My personal preference is to have learners tackle problems individually initially and then ‘pair and share’ – this can ensure that both learners have engaged properly with the material.

This is a fundamental in mathematics education. We already do it in class. “You must show all of your working” is the most common used phrase in maths classrooms. In secondary mathematics some errors are unsatisfactory e.g. basic arithmetic, however, as we advance further then the contrasting of approaches becomes the most interesting aspect. A simple example from my own teaching, inspired by the typical Shanghai lesson, is that when we are engaging with new material I will set a short initial task and will invite two pupils to do the task on the boards at the front while the rest of the class work. I will then write up my own solutions. This simple recent addition to my teaching armoury has had an incredible impact. This can result in the most terrific learning conversations. We have real pupil misconceptions on display. We sometimes have really unusual and alternative pupil approaches on the board. Very often teaching points I never even considered making are considered. Particularly with solving equations I find it useful to discuss two strategies, which have resulted in a correct answer – but open it up for discussion with pupils to decide which they think is more elegant and desirable. Of course there is the issue of classroom culture to consider. This approach might not be able to launched with an S4 class midway through the term. However, I have S1 and S3 classes of very different levels of ability who have really enjoyed and benefited from this approach after recent adoption.

There is a range of evidence, which suggests effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving. High quality tasks, such as those mentioned below, can facilitate this. Our curriculum will be built with the idea of making connections between representations built into schemes of work. For instance, in addition to the allocated separate fractions, decimals and percentages teaching times there will be an additional part of the course plan where several lessons are spent looking at how they are different representations of the same thing. See below.

Effective teaching of mathematics consistently provides learners with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships. Learners must also develop the ability to persevere and accept “stuckness” as a natural part of doing mathematics. Appropriate problem solving activities for all learners from the first moment of first year have to be encouraged. Learners need to be able to think for themselves and have the courage to attempt problems. I am sure you have heard “we don’t have an example like that in our notes” on many occasions. The reality is – we cannot, through exemplification alone, prepare learners for every possible situation that will arise in an examination. We can however, equip learners with experiences where they have to apply their knowledge to unusual situations. This can only serve to develop their confidence, resilience and enjoyment of questions where reasoning is required.

We need to take account of the research on how learning and memory operate (more of in a later blog post). The interleaving of topics will be a fundamental aspect of our curriculum. Simply put, when teaching new content the application of previously learned material will be a key component of the learning. Where possible pupils will encounter problems that develop and consolidate their algebra, negative numbers, fractions, decimals and percentages within the context of the current topic. For example: algebraic expressions in angle problems or area problems with fractional lengths. The key skills must be encountered in as many contexts as possible, not just when being taught explicitly. This will ensure that the key skills are kept fresh. Memory and recall will also be addressed through our homework policy and scheduled revision within the course plans.

The report Mathematics Matters from the NCETM states that ‘

Malcolm Swan is relatively unknown in Scotland – at least judging by the small show of hands when I asked delegates at the Stirling conference if they had heard of him. This is unfortunate as, without exaggerating, his work is world class. This section will summarise a few of his ideas about mathematics teaching. His work has provided many points for consideration and much inspiration for effective learning and teaching. His work will provide a point of reference in the development of our Mastery curriculum.

Swan discusses two broad views of mathematics teaching: A transmission culture or a collaborative, challenging culture.

Mathematics is seen as

- a body of knowledge and procedures to be ‘covered’

- an individual activity based on listening and imitating

- structuring a linear curriculum for the learner
- giving explanations and checking these have been understood through practice questions
- ‘correcting’ misunderstandings when students fail to

‘grasp’ what is taught

Mathematics is seen as

- a network of ideas which teacher and students construct together (the teacher is seen as a facilitator of pupil learning)

- a social activity in which students are challenged and arrive at understanding through discussion

- non-linear dialogue in which meanings and connections are explored
- recognising misunderstandings, making them explicit and learning from them

I think, in reality, these perspectives can be placed on a spectrum, with many teachers somewhere between the two. It is my opinion, after a lot of reading and a decade of teaching that the collaborative, challenging culture is the one that should be the aspiration for the learning and teaching within my own department. This fits comfortably our principles for curriculum design.

Much academic evidence supports my opinion. Boaler(1997) states that Learners who had engaged in collaborative work develop relational forms of knowledge that are more useful in a range of different situations (including traditional examination questions). Boaler also states:

- Textbook exercises or discussion of mathematical ideas are not merely vehicles for developing knowledge, they shape the forms of knowledge produced.

- Learners, who work through textbook exercises, find it difficult to use mathematics in applied or discussion-based situations.

In reality learners need a period of practice to hone and perfect routine skills. Fluency only comes through practice. I doubt there are many maths teachers who would disagree with that. Textbook exercises themselves are not intrinsically a bad thing. They can be extremely useful for learning and teaching. The danger, however, is that the textbook can become the course. In a number of our prominent Scottish texts problem solving, tasks that aid conceptual development, and high quality non-repetitive tasks are a rarity. The second bullet point, in particular, should be a point of reflection for many of us in Scottish mathematics education. No book alone should be the basis of our curriculum. The same can be said for any set of worksheets. A wide range of resources is required to support the fulfillment of the ambitions we have.

**Evaluating****mathematical statements**

Swan suggests asking learners to decide if a given statement is Always, Sometimes or Never true? Having used his examples and similar of my own making I am convinced that these are of a high value. Swan states that learners are encouraged to develop ‘rigorous mathematical arguments and justifications and examples and counterexamples to defend their reasoning.’

**Classifying****mathematical objects**

**Interpreting****multiple representations**

**Creating****and solving problems**

The examples above a small snapshot of a wide range of Malcolm Swan's work. John Mason and others have produced similar standard of materials. These will in some cases be directly useful for us, and in other cases will serve as a basis for the development of our own tasks.

The examples above a small snapshot of a wide range of Malcolm Swans work. John Mason, Ann Watson and others have produced similar standard of work – be it tasks, questioning advice or pedagogical approaches etc. These will in some cases be directly useful for us, and in other cases will serve as a basis for the development of our own tasks.

I must also mention Don Steward and his outstanding blog ‘Median’. This is a source of excellent materials that will be of major use in the development of our schemes of work.

The example below will work nicely in our topic on the multiple representations of fractions, decimals and percentages.

Many people have asked, since my first blog post on Mastery, how we will be able to slow down but meet time pressures etc. As I stated previously, a lot of the pressure is put on ourselves. We have 5 blocks/phases of material in Third Level. Within that there are 4/5 topics in each. The key-differentiating factor is pupil pace of progress not expectation. I have included a PDF with our pathways. I am happy to take questions on these – but can assure you that these are a slowed down version of what has been happening previously. It should be noted that we do not present anyone for an S4 exam in our school, except S4 leavers. Unlike the west coast school you may have heard about in the news that tried this approach, the impact has been very positive to date. This model being in place in the school offers the maths department greater flexibility in panning long-term pupil pathways.

third_level_mastery_curriculum.pdf |

I have a confession to make. A large number of the Mastery implementations I have read about are firmly run with mixed-ability classes. We will, against that notion, continue to set the classes. A number of considerations have led to this decision.

In the existing curricular model, the department has a “common course” which runs for the first few months of S1. That is all of the pupils, unset, do the same work – with everyone sitting the same test. I must admit that I didn’t enjoy my experience of it. I didn’t know how to resource it, manage the lessons effectively or give all of the pupils what they required. After ten years of teaching rigidly set classes, my own skillset was inadequate. Looking at pupil attainment it was clear that I likely wasn’t the only one of my colleagues who struggled to make it work. It was also a horribly negative starting point for a number of our learners.

All of the above is not to say mixed ability couldn’t work here, however, having pupils on First, Second and Third level in one room is something which most maths teachers in Scotland are unaccustomed to. In order to attempt that level of mixed ability teaching we would need to spend a lot of time developing new strategies and skills. Given that we will be investing so much time in developing our skills in Mastery and conceptual development, I feel this is simply too much to ask of colleagues at this point. Further, if implemented badly, any positive effect, which can come from mixed ability, would undoubtedly be wiped out. As I said in my previous post, there is no

In future as we develop our skillset at operating multi-activity lessons, which will be required by our Mastery cycle (some pupils on rich tasks, some on re-teaching) then moving to true mixed ability might seem a smaller step to take.

I must emphasise: the decision not to do mixed ability classes is due to the fact that in the short term, I do not think it will make a positive impact in our context. Put simply: is it better to have a skilled teacher delivering lessons to a rigidly set class or a teacher who is unskilled working with a mixed ability class? I suggest that in this case it is best for the learners that the classes stay rigidly set. I am very open to how we arrange the classes but feel that, in our context, this is for the medium term.

A final point on setting is that, even though classes will be set, the differentiation is through pace of progress. Not through content or expectation of Mastery. Time is the variable, not the standard expected. Mark McCourt uses the excellent analogy of the driving test to describe this. The standard required to pass a driving test remains static. Some people achieve a pass after 10 lessons while others take years. However, each individual has passed the test, the passes are considered of equal value with both being awarded driving licenses.

Yesterday I attended the Scottish Maths Council annual conference at Stirling. This was, in my opinion, a pivotal day for mathematics education in Scotland as Professor Ruth Merttens made the first significant mention of Mastery learning in a Scottish context, in her keynote address. Professor Merttens’ speech was entertaining and passionately delivered. She is clearly an expert in the teaching of early years mathematics. However, having read her SMC journal article prior to attendance I knew that her reaction to Mastery, Shanghai and Singapore would be largely skeptical. I felt that a lot of this skepticism was, correctly, placed at the way these ideas have been politically hijacked in England. However, in the address there was no detailed explanation of what Mastery learning really is, what benefits it could have or any reference to its’ successes in a UK context. This was unfortunate given that Mastery is, at best, a peripheral idea in Scotland, and the majority of the audience most likely hadn’t read much into it or heard of it at all.

I am passionate about Mastery learning. My own presentation yesterday wasn’t about Mastery, per se, but more of a collection of thoughts on what makes effective mathematics teaching. (Slides available on request). The sort of approaches I discussed, as part of a well-implemented Mastery curriculum could have an incredibly positive impact upon pupil experience and final outcomes. I am currently working with some colleagues in the planning and development of a Mastery curriculum for CfE Third Level. I believe this to be only the second attempt at this in Scotland. I have spent the past seven months attending conferences in England, reading countless books, blogs and online articles, corresponding with experts in the field and reflecting upon Mastery learning and other aspects of evidence based approaches to mathematics teaching. Over the course of this and subsequent posts I will attempt to lay out the rationale behind my decision to proceed with a Mastery curriculum and discuss exactly how we are adapting this, and other notions, to fit with our own context.

Let us not confuse Shanghai and Singapore with Mastery. Mastery, as Professor Merttens correctly identified, is an Anglo-American concept, first prominently discussed by Bloom (of taxonomy fame). Shanghai and Singapore are cities that have mathematics education curricula, perform well in PISA studies. We can learn from the practice in Shanghai and Singapore but it would be incorrect to attempt to take their curricula transplant it to the UK as Professor Merttens correctly stated. We can consider Mastery without even thinking about East Asia. It isn’t all that relevant to my arguments for Mastery.

Mastery learning is the belief that students should

We can say a concept has been mastered if:

- The student can demonstrate or explain the concept orally, concretely, visually and abstractly.
- The student can apply the concept automatically, so that it is not dominating their working memory.

This idea of Mastery is very important because of its effect on working memory. Students who have mastered previous skills have their working memory freed to learn new ones, while students who haven’t get bogged down in the basics and don’t have the working memory space to learn something new. A student cannot afford to expend a high cognitive load on, for instance, multiplication facts, during a more complicated problem. The problem being studied should be the challenge. Not the underlying pre-requisite.

Bruno Reddy (@mrreddymaths) who is well known south of the border for his work in implementing a very successful Mastery curriculum at King Solomon Academy, London raises some flawed assumptions that teachers often have.

- ‘There is so much to learn that we have to keep moving on’
- ‘They might not get it now, but we’ll recap it again next year and *then* they’ll get it’
- Because pupils can follow the algorithm they have learned the concept.

I agree entirely with Bruno that this sort of thinking is a mistake.

In terms of the volume of work there isn’t actually as much as we would sometimes think. In my own department if a pupil embarks upon Third level at arrival in S1 they will have to learn 12 blocks of material in order to reach National 5 standard. For many pupils, in 5 or 6 years, they do not master 12 blocks. The progress required is only 2 to 2.4 blocks per year, for the median student aiming to leave school with National 5. In simple terms, 8 topics per year. How many Scottish school maths curricula have as little content as this for a year? We often rush on through a topic, but then end up having to repeat it the following year!

The idea of repeating the topic a year later is based upon a false assumption. Quite simply the chances are the student won’t improve the following year as there will be recap of the previous year and then a whole new set of concepts delivered.

If a student doesn’t master a topic first time around there must be a deficit in underlying skill, which is stopping the student accessing/succeeding with the topic. This is the start of a vicious cycle. The student fails to master the Third level content, due to Second level miscomprehension. A year or two later the same student finds him/herself studying the topic at Fourth level, with a shortfall in understanding which goes back two levels. This is then further exacerbated when said student is coached through National 4 and ends up in a National 5 class. By this point, all hope of achieving proficiency in the topic is gone as the work in question is now, often, three levels above the level the pupil is actually working at in that topic.

If you think this sounds far fetched then try a National 4 Mathematics or even National 3 Lifeksills paper, without preparation and coaching, with some of your senior N5 pupils who are on track to fail. Many of them will struggle to pass the levels below. Simply, because they haven’t mastered it.

Bruno’s final point is the crux of the matter in mathematics education for me. Just because a student can copy a series of steps, you have shown them, and can do this with practice questions with the only difference being the numbers doesn’t mean that they understand! What is required is the development of real understanding and conceptual development. Coupled with this, there is a requirement for teachers to utilise tasks which allow pupils to develop and demonstrate that real understanding. The work of Malcolm Swan showcases a variety of tasks, which encapsulate these principles.

Essentially:

How we teach really matters.

The tasks we assign really matter – maybe even more so!

The Mastery cycle is what distinguishes a Mastery curriculum. Professor Merttens overly simplified diagram and explanation wasn’t a fair representation of what a well-implemented Mastery cycle looks like. Mark McCourt @EmathsUK (check out his excellent La Salle education website) has collated 100 years of research into the diagram above.

Check out https://docs.google.com/drawings/d/1jxN60kmTjluFnKY7VTfLkadXt3U0OprCuKdwru0CT-w/edit?pref=2&pli=1 for the original version with comments (I have used some of these comments as stems for the following).

Some key elements of the Mastery learning cycle:

- Very precise learning objectives are part of curriculum design and shared with colleagues and pupils alike. In Scotland most of us have recently been through the idea of unpacking outcomes from CfE. In order to develop a Mastery curriculum those outcomes need to be decomposed even further.
- Diagnostic Pre-Assessment with pre-teaching. This part of the cycle is one of the key elements of a successful Mastery curriculum. The pre-teaching diagnostic is vital. We must ensure the initial conditions for success are set, before teaching of new material begins! Leyton, 1983 found that students who have reviewed missing prerequisite concepts and skills were far more likely to achieve mastery on new content. This is one of the big differences from conventional teaching. In a Mastery curriculum you do not start teaching new content unless students can demonstrate Mastery with the pre-requisites.
- High quality – whole class, initial instruction. The teacher is the single biggest resource in any classroom. A quality Mastery curriculum will encourage a variety of effective pedagogical approaches. The curriculum will support teachers in its abundance of excellent resources. The aim in our department is to develop this vision into reality. Every department meeting next year will have Mastery teaching on the agenda. This is new to all of us and we will need to share ideas, reflect upon and discuss curriculum content, resources and theory and how we are implementing all of this in our classes. Nobody wants an overly prescriptive curriculum. However, a helping hand with a substantial bank of excellent materials, which build upon our vision, surely cannot be a bad thing? Teacher autonomy is not in question. A starting point and teaching advice in a course plan does not eradicate this.
- Progress monitoring through regular formative assessment: Diagnostic assessments. Students must demonstrate a high level of performance AND understanding before proceeding. We have settled on a figure of 90% of the class demonstrating 80% Mastery before moving on. Mark makes the point that the threshold of 80% is only as useful as the design of the questions / tasks / problems. Designed badly, 80% (or any other score) can be utterly meaningless. This is such a key statement; it is easy to fool oneself into believing that the students have mastered a topic by setting an inadequate test and preparing students for it!
- High quality corrective instruction/interventions - this is not the same as re-teaching! It is easy to repeat the same words louder and call this corrective instruction, but it is not. A variety of approaches are required for pupils who have failed to master an outcome. Interventions include: additional teaching via alternative approaches/models and representations, peer support, small group discussions, homework (encourage parents), online resources – online lessons etc. Should be attainable in class time.
- Enrichment/extension activities. This poses a challenge as we design our curriculum. We must ensure that students are engaged in valuable learning experiences. However we do not proceed to the next topic until we have hit the 90% at 80% figure. Having students simply bide their time, doing more, harder problems or completing busy work while others are engaged in corrective instructions would be highly inappropriate. The enrichment tasks must provide opportunities for these students to extend their understanding and broaden their learning experiences.
- Retention – needs to be continually monitored and addressed. Adequate planning is required in curriculum design on how learned knowledge will be retained. (more of in a later post)

The notion of pace is something, which, in my opinion, is dangerous in the wrong hands. Depth and understanding matters most of all. As I said earlier we really should look at slowing down. 8 topics per year for the middle of the road kids is all we need to achieve. Yet the course outline will have as many as 15-20. A “teach once – deeply” approach means we have to alter how we look at our approaches to scheduling.

Mastery aims to avoid unnecessary repetition across years by regularly assessing knowledge and skills. Extra time is obviously required for pre-teaching diagnostics, development of deep pupil understanding, ccorrective instruction and remediation, problem solving and enrichment and formative assessment diagnostics. Academic and anecdotal evidence indicates that this shouldn’t concern us. Because pupils embark upon subsequent learning with mastery of pre-requisites, initial instruction in later topics can proceed more rapidly. This means that remediation time spent by students and teachers significantly decreases as the student reaches higher instructional units. That is, in essence, start slow and then accelerate later. Subsequent learning will be “easier” as pupils will have fewer gaps in pre-requisite knowledge. This then leads us to be able to be really ambitious with our final outcomes.

I will post our predicted curriculum time pathways in a later post, demonstrating how by slowing down to begin with, we can still arrive at the same end point. The arrival at the end point, contrary to now, should have been after a journey, which has fully equipped learners with the required skills for success.

I’d like to address two points raised by Professor Merttens yesterday. The claim was that meta-skills – ie those things which are hard to capture in a learning intention, don’t sit well within Mastery. Our model will address this by a variety of teaching approaches, interleaving of topics (which is a key part of the curriculum design) and a lot of rich tasks and problem solving. For me, the meta-skills argument is something which shouldn’t be an issue in a well-implemented Mastery curriculum. Professor Merttens talked about “instruction”. While it is true that much of the literature on Mastery uses this word, it can be readily changed to teaching. It is only instruction if you say that is all it is. What I have laid out above, and will do so in greater depth in future posts, is about outstanding teaching. Not lecturing! However, let us not pretend that direct instruction won’t be part of the mathematics classroom even within a Mastery curriculum. (John Hattie does show that is has a positive effect size after all.) It is just one of the many approaches that an effective teacher would utilise.

We must be clear that there is no

In subsequent posts I intend to look more at the sort of lessons we want to include in our curriculum and the types of resources, which will be required to support this. I will also discuss our planned approaches to assessment (both formative and summative) as well as homework policy. Further posts will focus on academic supporting evidence, the development of a mastery culture within the department and the specific vision for what we are trying to achieve in our department. I will also look at how we will build in retention from topic to topic.

Finally I must thank the following, each of whom have, unknowingly in some cases, helped to develop my understanding of effective mathematics teaching, effective tasks, textbook-less curricula, curriculum design and culture. @MrReddyMaths @Maths_Master @mrbartonmaths @EmathsUK, @KrisBoulton, @BodilUK, Malcolm swan, @JohnMOxford, @MichaelOllerton and many others who I forget at this time.

I would love to hear from anyone who reads this. I am still at an early stage of my development of this curriculum and would welcome any comments and criticisms. I know I’ve a lot more to learn. However, I felt it imperative that Mastery had somebody arguing the case for it in Scotland.

After the excitement, came the guilt. I’ve been in post since early August and, at times, feel like I’ve not made any tangible difference. Would an external observer see any positive difference in the way the department operates or, most importantly, in the learner’s experience? All those big ideas I had in the summer seem to have been lost under the day-to-day management and administration of the department.

I’ve spent some time thinking about this. 12-14 hours days have been frequent occurrences recently. I doubt that effort is the issue. Perhaps I could have made better use of some of the time. I’m a big fan of the 80/20 principle – the concept that 20% of your effort generates 80% of your desired outcomes. It will take me time to fully develop an appreciation of my core 20% - although I would hazard a guess that it is directly related to teacher capacity and curriculum.

So, what have I achieved? Over a couple of days, I added to the list below as I remembered some of the small steps I’ve taken. The length of the list has surprised me. This list is there to remind me that I have made small steps forward. Without many of these small steps, the bigger changes I have planned for the future wouldn't be able to proceed.

I am currently/am well under way with…

- Building relationships with colleagues and pupils
- Learning how things work in the department and the school, from the photocopier to the curriculum pathways.
- Implementing coherent monitoring and tracking and data gathering systems in the department
- Making learning and teaching the central focus of our department meetings. Admin had a place early after my arrival, but now is reduced to a few minutes.
- Supporting colleagues with learning and teaching – professional dialogue of approaches and ideas.
- Carrying out an in-depth curriculum review
- Making tentative steps at trialing a “Mastery approach” in some of my lessons and designing a new Level 3 curriculum based upon this approach for launch next August
- Thinking and reading about strategies for raising attainment for all
- Encouraging staff to engage with basic number skills in S1 classes - strategies shared and discussed at DM
- Encouraging staff to let the pupils “think” in order to develop problem-solving skills - strategies discussed at DM
- Appointing new staff – involved in hefty process of reading many applications, and a 3 day visit and interview process.
- Ensuring SQA courses admin is correct
- Working on class lists and setting and answering the inevitable parental queries
- Running the maths segment of the P7 parent's open evening
- Discipline support within dept
- Verification procedures for SQA qualifications - updated policy and paperwork and ensured all dept have engaged with at least one level
- Learning and teaching of my own classes - this is a big one! I'm more passionate than ever to do the best for the kids. Too many horror stories from PTs telling m their teaching isn't up to par as they have too much to do. I genuinely think I've actually upped my game from last year.
- My own CPD on leadership, management, mathematics education etc
- Results Analysis - massive time investment here. SIMD v maths attainment, UPS v average in other depts etc
- Improvement Plan and implementation - I only put on 9 items. 49 would be closer to reality if it reflected what I was really aiming for over the coming few years.
- Liaising with learning support PTs and pastoral care PTs
- Overseeing L&T - observations, reflective conversations etc
- PRD process - review meetings with all colleagues
- Ordering of stock, jotters, stationery - trying to figure out how to spend the money effectively in an authority where ICT isn't high on the list of priorities. No wi-fi. Would have loved to have bought a class set of Netbooks. So much potential we can't tap into.
- Supporting student teacher (including sitting in on the crit, writing report etc), supporting probationer
- Taking over higher class from most popular and successful teacher in schools history, to see them achieve less than spectacular prelim results, six weeks into my care.
- Trying to push on with an S4 set 3 and get some of them towards higher, against conventional wisdom in the dept. Put together a proposal document for SMT on pathways within maths
- Wrote an S1 Level 2 skills diagnostic multi-choice, used this in conjunction with S1 common course test (which I'd changed) to decided upon s1 setting.
- Updating our approaches to assessments to reflect SALs Grouping Es&Os.
- Made a re-organistation of the shared file system
- Put in place accurate record keeping for Senior Phase UASPs
- Organised the S5 prelims – producing the higher paper and marking-scheme and collating the other levels
- Attended two days of CPD in England. Maths Conference in Sheffield and Mastery course in Kendal.
- Begin the spadework for the schools' first ever Higher Maths revision weekend.
- Working with local authoitry on National 5

Most of all, I've spent a lot of time thinking and reflecting upon things and planning a path forward for both the coming six months until summer and the following academic year. I should also add in that I've made mistakes. I've got a lot to learn before I feel totally comfortable in the post. Despite the fact I don't really have a clue if i'm doing it well or not at the moment I have to say that these have been some of the best months of my teaching career to date. They have also been the hardest few months, by far, but doing this job has revitalised my career to such an extent that I can't remember ever feeling as enthusiastic!]]>

Having decided to start blogging to help focus my professional reflection I suppose that I should articulate a little bit about what my teaching actually is. I’ve spoken in person to people about various parts of my perspective on teaching but have never gathered it all together in one place. So here goes…

My ambition is for the following outcomes for every pupil in my classes:

1. Able to apply algorithms to solve standard problems

2. Have confidence to attempt to apply knowledge to non-routine problems

3. Understand the concepts underpinning the topic

4. Have some sort of idea of why we are studying the topic

5. Enjoy the experience

I make no claims that all 5 outcomes are met, even most of the time for most of the pupils. I do, however, feel that what I do has a positive impact. My exam results have shown a positive trend, across various levels for a number of years. Pupil and parent feedback has been positive. Some might ask if all of the above is important. I don’t know if there is an absolute truth to it all. To me the above is important - that is why I aspire for it.

I don’t believe that all lessons should follow the widely accepted template of starter, middle, plenary. I am more interested in the progression over a number of lessons. In my opinion successful Maths teaching is far too complicated to fit into an identikit template. For that reason I can’t describe what a ‘typical’ lesson would look like in my class. It can vary a lot. Sometimes there is a starter task, sometimes not. There is rarely a mention of a learning intention(a future blog post). There is sometimes a textbook although, increasingly frequently not.

There is nothing revolutionary about my approach to teaching. I simply want the students to do most of the thinking and most of the work. I really do view myself as being there to nudge and guide them in the correct direction. I’m not an a strong advocate of any particular approach. Usually I will introduce a topic through:

E.g. sketching straight lines from a table of values and developing their own comprehension of y = mx + c. This task might be done in pairs or individual. Similar approaches for transformation of trig, quadratic and log graphs etc.

E.g. For teaching area of a sector I would ask pupils to find area of a circle, semi-circle and circles with angles of 90, 45,60 and 120 degrees indicated. The key question in the task would be the next question. Where the angle is, say, 17 degrees. Pupil discussion and then whole class discussion would pull together and solidify what the pupils where meant to conclude.

E.g. I introduce exponential graphs through various practical ideas. One of which is to ask pupils to count the number of sections created for each fold of a piece of paper. Pupils often come up with y = 2^x and are able to graph it themselves with minimal input from me. I might prompt that we create a table of value and suggest testing when x = 0, 0.5, -1 etc.

E.g. I will discuss a real life scenario and demonstrate a new skill which ties in. Dan Meyer has a nice idea on Standard Deviation which I adapted and often use. It starts with a game on guessing ages of celebrities. It leads to a teacher led conversation (with a lot of planned questioning) which results in the concept of Standard Deviation. http://blog.mrmeyer.com/2007/how-old-is-tiger-woods/

The Ronseal approach - does what it says on the tin. ‘Here is some new material we have to learn’. Even with this approach I would try to bring it to life using Desmos/Autograph or Smartboard software.

Of course, a lot of the above applies when introducing a new idea. A lot of lessons are follow up lessons where we are consolidating or adding a slight extension. In essence though I do find that kids ‘get it’ more when they have participated through some form of directed activity rather than just been lectured to.

I’m not scared to take a little bit of time in class developing an idea and letting a concept secure itself at the expense of routine practice. Of course routine practice is important. For example for core skills such as adding fractions I will ask pupils to add a lot of fractions together! More interesting, though, is non routine practice.

Lower down the school, I often push the boundaries of what a pupil is capable of achieving with the knowledge they have at that point. I’ve always felt if I pushed a pupil slightly past what was required for ‘just now’ then the current content would seem more trivial to them. Eg: sub x = -1/2 into y = x^2 - 2x - 12 This is harder than a typical second year test question, but is a combination of second year skills. Later in the school, once a pupil has a fairly solid footing with a topic, I like to twist it and manipulate the questions as much as possible to ensure the pupil can not only handle the unexpected but also to ensure that the pupil actually understands the concepts underneath.

The preceding paragraph does seem like it is focused on more able pupils, however, I apply these same approaches, at different levels, with all pupils. I would never just teach National 4 to a class who were looking at Nat 5 in S5! That sounds like a future blog post in the making.

I’m not anti-textbook, but sometimes I’ve felt that they can cover up for a lack of planning on my own behalf. My new department has pretty much no class sets of textbooks, they use a lot of these legendary (at least in Scottish Maths teacher circles) worksheets: http://www.knightswoodsecondary.org.uk/personal/Resources/Hillhead/Resources_hillhead.htm. As they were written by one of our colleagues. I know of good exercises in various books that I can rely on, but a lot of the time there is nothing

I am by no means into ‘trendy’ educational ideas. True, I do engage with the likes of John Hattie and the work he has done on the relative effect sizes of various L&T approaches. However, I am very traditional in the sense that I like my senior classes to copy down notes from the board - where I model (what I hope are) mathematically rigorous examples of how I would like them to attempt their own workings.

I have a real dislike of ‘tricks’. I don't think that we should be trying to make maths easier, as such. Instead I think that we should helping the kids become equipped with the skills required to engage fully with the subject. For example, I never teach ‘change side, change sign’ for solving linear equations. This isn’t for any pretentious reason such as it not being mathematically rigorous (which it isn’t) but because it doesn’t fully equip pupils for later and, in fact, can lead to a lot of misconceptions.

x = 2/-4

I would rarely, if ever see this from a kid taught the balancing method for equations. Where as, from my experience of marking with SQA, tutoring kids from other schools etc, I see this is a common sort of mistake. The culprit is almost always that they have this idea of solving equations. Why not just persevere in S1/2 with balancing. ‘Whatever you to do one side you do the other’, ‘ what is the inverse of addition/multiplication?’ etc. Pupils do find balancing harder initially, no doubt. But by giving up on them and going with ‘change side, change sign’ I feel that a dis-service is being done. The same applies to cross multiplication - whatever the hell that is! Consider solving an exponential equation. To eradicate the exponential we take logs of both sides. Everything becomes seamless and fluid. If you start with a rule such as ‘change side, change sign’ then you are building on an unstable base to begin with and the further you go, the harder it is to say that the rule always works. Because it doesn’t. I know this might be controversial - but that’s where I’m at!

I can make similar arguments for novelty ideas of adding fractions etc. None of it makes any real sense and we aren’t doing the kids any favours, particularly if they are going to take their maths forward. These tricks and shortcuts work for very specific cases but aren’t sustainable. So why waste time on them?

I like to do one large homework task per week. Occasionally a question or two to do that evening, but not much more. Pupils have a lot of other subjects to focus on too. Every senior class (in my place that is really S3 up) get an ‘ink exercise’ every week. I will mark two out of every three of them. The one I don’t mark will be marked by the pupil from a solutions sheet I issue in class. I issue forms which pupils complete indicating how many marks they got for each question. I feed the info from these sheets into a spreadsheet. Even though I know the marking isn’t always accurate I still get a picture from the class of where the difficulties lay and can then plan lesson starters/future homework questions accordingly. The homework the pupil self mark is never new content. It is always a revision exercise of work previously covered. I’d love to mark for every class every week, but it’s infeasible. Being PT, having a heavy timetable, being a dad of twin toddlers and generally wanting to have a wee bit of a life I find that two sets of jotters is what I can manage, particularly if Higher/Advanced Higher is one of those sets. In first/second year I rely much more heavily on peer/self marking from a set of solutions. I run round the class to check homework is done and have and talk to pupils about questions they have got wrong or are unsure about. I try to mark a handful of formal exercises for each S1/2 during a term. However, I just can’t seem to manage to incorporate it into my work schedule with any sort of regularity. To this end, S1/2 exercises are often drill exercises designed to reinforce key skills. A lot of my feedback and knowledge of S1/2 pupils comes form AiFl approaches in class such as using mini-whiteboards etc. I do think this is an area where I need to improve. This leads nicely to…

I’d say that my strength lies with Higher and Advanced Higher. I have rarely had many/any fails at either of those levels and tend to help get the kids better grades in maths than they get elsewhere. I’m don't feel that I add as much value for the kids who struggle most with maths. I’m aiming to improve on this over the next couple of years. I’ll be focusing a lot of time on lower ability S1/2 classes. Because I’m now PT I’m investigating ideas such as the mastery curriculum models which seem to be having some success in England. I’m looking to improve the quality of my explanations for these pupils too and also find better resources than I’ve used previously. I feel I’ve done OK with these classes in the past, but could do better.

All of the above is, of course, generalisation. It is how I think today. It wasn’t how I thought five or ten years ago and I assume my opinions will continue to change over time. I like to think I can be adaptive and if that means compromising my principles occasionally if I genuinely think it will be best for a particular child/group of children then I will do that. ]]>

I had the pleasure of making the trip from Glasgow down to Sheffield for the @LaSalleEd Complete Mathematics conference. I had never heard of the conference but after looking into it I quickly spotted that some of the workshops seemed to be talking about things which have been on my mind a lot recently. Some of the speakers are people whose blogs and Twitter contributions have at times motivated, challenged and inspired me.

I don't have time to write about everything I experienced at this excellent conference, and I see that other blogs have provided good summaries of the various workshops. Instead I will focus on where my own thoughts have travelled after some of the workshops. In this first post I'll focus on the excellent leadership session offered by @mrbenward.

**"Lead in your sphere of influence. Manage what is outside."**

I didn't realise that this was something I've been doing already. Hearing Ben say the words drew my attention to a skill I didn't know I had! I could allow myself to become frustrated by antiquated ICT provision, by the lack of free time for colleagues to work collaboratively on development or by the bureaucracy involved in gathering and tracking internal assessment evidence for SQA courses. However, I have to decide to manage these things as best as I can as all of these issues are outwith my sphere of influence. I have to implement low(er) tech compromises, employing the ICT we do have to it's greatest potential. I have to plan collegiate time wisely. I have to develop departmental systems for gathering assessment evidence in a rigorous, organised, but simple way which isn't onerous or time consuming for colleagues or myself. These are management tasks.

Having the remit of providing leadership within my sphere of the mathematics department is the exciting part of the job. The management scenarios discussed above are simple in comparison to leadership. In those cases, concrete, well defined problems exist and as such it is relatively straightforward to put in place solutions. Leadership, on the other hand, is about trying to understand complexity. Complex issues do not have a simple solution. Ben used the following quote:

*'Leadership is not mobilising others to solve problems we already know how to solve, but to help them confront problems that have yet to be successfully addressed' Heifetz*

Nobody has education cracked. I want to work with my team to confront the complex problems. The curriculum, the quality of learning and teaching, the learners experience, the quality of resources, whether we are meeting learners needs, the professional culture and professional learning - all of this is within my sphere of influence! I.e. all of the important stuff! As a head of department I have an enormous opportunity/responsibility on my shoulders to do the best for every pupil who studies with us.

My conclusion: there are a few minor hindrances. There always will be. None of this is an excuse for not doing the key stuff well.

**"Know your staff, know your data, know your key docs"**

Ben made the point that a head of department should know the strengths and weaknesses of his department. Ben asked the question that "as a head of department are you able to list a major strength and an area for development for any member of your team?" I've only been in my current department for about 7 weeks so will excuse myself for not being able to answer this. I've obviously been building up a picture based upon data, observations and views. I plan to track data closely across the department - this will highlight teacher capacity in certain areas. I aim to do formal lesson observations over the coming month for all staff members. This will also improve my picture of the individuals on my team. I have been listening to the views aired by all stakeholders. I listen to my teachers. I am beginning to know them as people and as colleagues. All of this, in combination, will help me to understand my team's capacity better. However, I am not sure if it is enough. I feel that this is perhaps an area where I could improve as I don't know that even in another 7 weeks that I'll be able to answer Ben's question.

I love data - no surprise for a maths teacher. I am working hard on building up a COMPLETE picture of our department. Not just the obvious such as % A-C grades at Higher. I am looking deeper down than this. What percentage of our EAL pupils make it into the top sets and how do they progress relative to their peers? What is the impact of socio-economic indicators such as SIMD against attainment? Do our pupils with additional support needs progress as well as other pupils who arrive at us with a similar knowledge/skill base? Data is only as good as the questions which you are asking/answering with it. For me it boils down to two fundamentals. Are we helping every pupil achieve his/her potential? Are barriers to learning holding pupils back from attaining in our department? I want every pupil to be included, engaged and attaining. I am not a slave to data but feel that it is my duty to know the data and to try to understand what it means. It should help us to make a case for improvement and identify exactly where it is needed.

In terms of key documentation , I feel well informed of the current state of key curriculum documentation and what it implies. I need to maintain my level of knowledge of what is going on at a national level. Ben made the point that if the head of department doesn't know what is in the latest policy document then how can he expect his department colleagues to know, let alone implement the recommendations? The latest instalment of 'How good is our school?' was released last week. I have it favourited on Twitter to remind me to have a proper look at it soon. Ironically, where I feel less secure is with some of the new qualifications in maths. Higher, Nat 5 and Advanced Higher I am confident with as I have worked hard in the implementation of these courses. I have little or know knowledge of the requirements of the National 4 maths or National 5 life skills courses, for example. I don't have these on my timetable, but can't afford to wait before becoming familiar with them. This is a very short term development goal for me. I should understand the specification and assessment requirements of every course offered in our department.

**"Teachers don't develop in isolation"**

I agree with this 100%. A lot of my values and opinions about mathematics education have been shaped by others either through direct conversation, through reading, conferences or social media. I have developed my reflective capacity as a result of reading thought provoking blog articles from others. I have improved my knowledge by reading Chris Smith's @aap123 excellent newsletter. I have developed understanding of standards by going to marking meetings. Most of all I have developed by having long and winding conversations with excellent teachers, usually over a cup of tea at the end of a manic day. I want my department to be a place where learning and teaching is central to the agenda. I aim this year to dedicate half of every 2nd department meeting to L&T (no admin). This is an increase on what a lot of depts do. However, longer term, I want less and less admin on the agenda. Ben had the wonderful idea of putting out a regular sheet with admin tasks required of staff. This freed up his very limited department time to talk about their core business of L&T. I want a culture where we are open and supportive but also able to challenge each other.

Overall, I really enjoyed Ben's workshop. He talked a lot about Michael Fullan @MichaelFullan1 and his often referenced publication "Leading in a culture of change". I have read this cover to cover and am quite famliar with the content. It was interesting to hear how another colleague had interpreted Fullan's words.

I'd like to finish on a quote from Fullan which I recently posted on Twitter.*'Focus on fundamentals: curriculum, instruction, assessment and professional culture.'* This is, for me, the key.

]]>I don't have time to write about everything I experienced at this excellent conference, and I see that other blogs have provided good summaries of the various workshops. Instead I will focus on where my own thoughts have travelled after some of the workshops. In this first post I'll focus on the excellent leadership session offered by @mrbenward.

I didn't realise that this was something I've been doing already. Hearing Ben say the words drew my attention to a skill I didn't know I had! I could allow myself to become frustrated by antiquated ICT provision, by the lack of free time for colleagues to work collaboratively on development or by the bureaucracy involved in gathering and tracking internal assessment evidence for SQA courses. However, I have to decide to manage these things as best as I can as all of these issues are outwith my sphere of influence. I have to implement low(er) tech compromises, employing the ICT we do have to it's greatest potential. I have to plan collegiate time wisely. I have to develop departmental systems for gathering assessment evidence in a rigorous, organised, but simple way which isn't onerous or time consuming for colleagues or myself. These are management tasks.

Having the remit of providing leadership within my sphere of the mathematics department is the exciting part of the job. The management scenarios discussed above are simple in comparison to leadership. In those cases, concrete, well defined problems exist and as such it is relatively straightforward to put in place solutions. Leadership, on the other hand, is about trying to understand complexity. Complex issues do not have a simple solution. Ben used the following quote:

Nobody has education cracked. I want to work with my team to confront the complex problems. The curriculum, the quality of learning and teaching, the learners experience, the quality of resources, whether we are meeting learners needs, the professional culture and professional learning - all of this is within my sphere of influence! I.e. all of the important stuff! As a head of department I have an enormous opportunity/responsibility on my shoulders to do the best for every pupil who studies with us.

My conclusion: there are a few minor hindrances. There always will be. None of this is an excuse for not doing the key stuff well.

Ben made the point that a head of department should know the strengths and weaknesses of his department. Ben asked the question that "as a head of department are you able to list a major strength and an area for development for any member of your team?" I've only been in my current department for about 7 weeks so will excuse myself for not being able to answer this. I've obviously been building up a picture based upon data, observations and views. I plan to track data closely across the department - this will highlight teacher capacity in certain areas. I aim to do formal lesson observations over the coming month for all staff members. This will also improve my picture of the individuals on my team. I have been listening to the views aired by all stakeholders. I listen to my teachers. I am beginning to know them as people and as colleagues. All of this, in combination, will help me to understand my team's capacity better. However, I am not sure if it is enough. I feel that this is perhaps an area where I could improve as I don't know that even in another 7 weeks that I'll be able to answer Ben's question.

I love data - no surprise for a maths teacher. I am working hard on building up a COMPLETE picture of our department. Not just the obvious such as % A-C grades at Higher. I am looking deeper down than this. What percentage of our EAL pupils make it into the top sets and how do they progress relative to their peers? What is the impact of socio-economic indicators such as SIMD against attainment? Do our pupils with additional support needs progress as well as other pupils who arrive at us with a similar knowledge/skill base? Data is only as good as the questions which you are asking/answering with it. For me it boils down to two fundamentals. Are we helping every pupil achieve his/her potential? Are barriers to learning holding pupils back from attaining in our department? I want every pupil to be included, engaged and attaining. I am not a slave to data but feel that it is my duty to know the data and to try to understand what it means. It should help us to make a case for improvement and identify exactly where it is needed.

In terms of key documentation , I feel well informed of the current state of key curriculum documentation and what it implies. I need to maintain my level of knowledge of what is going on at a national level. Ben made the point that if the head of department doesn't know what is in the latest policy document then how can he expect his department colleagues to know, let alone implement the recommendations? The latest instalment of 'How good is our school?' was released last week. I have it favourited on Twitter to remind me to have a proper look at it soon. Ironically, where I feel less secure is with some of the new qualifications in maths. Higher, Nat 5 and Advanced Higher I am confident with as I have worked hard in the implementation of these courses. I have little or know knowledge of the requirements of the National 4 maths or National 5 life skills courses, for example. I don't have these on my timetable, but can't afford to wait before becoming familiar with them. This is a very short term development goal for me. I should understand the specification and assessment requirements of every course offered in our department.

I agree with this 100%. A lot of my values and opinions about mathematics education have been shaped by others either through direct conversation, through reading, conferences or social media. I have developed my reflective capacity as a result of reading thought provoking blog articles from others. I have improved my knowledge by reading Chris Smith's @aap123 excellent newsletter. I have developed understanding of standards by going to marking meetings. Most of all I have developed by having long and winding conversations with excellent teachers, usually over a cup of tea at the end of a manic day. I want my department to be a place where learning and teaching is central to the agenda. I aim this year to dedicate half of every 2nd department meeting to L&T (no admin). This is an increase on what a lot of depts do. However, longer term, I want less and less admin on the agenda. Ben had the wonderful idea of putting out a regular sheet with admin tasks required of staff. This freed up his very limited department time to talk about their core business of L&T. I want a culture where we are open and supportive but also able to challenge each other.

Overall, I really enjoyed Ben's workshop. He talked a lot about Michael Fullan @MichaelFullan1 and his often referenced publication "Leading in a culture of change". I have read this cover to cover and am quite famliar with the content. It was interesting to hear how another colleague had interpreted Fullan's words.

I'd like to finish on a quote from Fullan which I recently posted on Twitter.