@canning_mrmaths recently asked me a few questions about designing our curriculum on twitter. I tried to reply via a tweet, but after a lot of meandering thoughts decided the best way to answer would be by a blog post. I apologise for this being the second consecutive post that doesn’t read like a proper article. Despite the poor writing style and messy structure I hope to stimulate some thought/debate.
How did you decide what to keep and lose?
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.
Equations 28.5%
Substitution 28.5%
Expressions 25%
Integers 21%
Fractions 17%
Co-ordinates 11%
Area of basic shapes 11%
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.
How did you decide what to keep and lose?
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.
Equations 28.5%
Substitution 28.5%
Expressions 25%
Integers 21%
Fractions 17%
Co-ordinates 11%
Area of basic shapes 11%
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.
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.