How important is knowledge in mathematics?

The importance of knowledge in any subject is well-known. Indeed, if you are yet to read Sarah’s  (@s_donarski) blog on questioning for knowledge, you can find it here. I was at the same conference as Sarah and Doug’s demonstration of the power of knowledge is something that will stick with me for a long time. Knowing more instantly allowed everyone in the room a lot more freedom to think and allowed us to think more deeply.

My question now is: as a maths teacher, how important is it that students know mathematical facts? How can I adjust and adapt my own practice in order to ensure that these facts are known and easy to access?


Knowing mathematical facts

I listened to Dani Quinn from Michaela (@danicquinn) talk over a year ago now and the massive takeaway I had was:

if we, as teachers, use a shortcut or “just remember” something, why do our students not?

At the time, we were using the example of multiplying by 0.1 and the educators in the room were “just dividing by 10”. A student does not necessarily do that. Why is it that we do? We have had hours and hours of practice across our mathematical life times. We need to get students to the same point as quickly as possible. Dani is a proponent of “drilling” whereby students are asked a lot of one-step questions in a short period of time. Things such as: “will this give a positive or negative answer?”.

I took this idea and ran with it. For example, I’m currently teaching trigonometry to a year 10 group and over the last 5-8 lessons, I think they have answered about 60 questions which are “which rule can we use?”. The students have picked up on the importance of speed in that decision. I’m not rushing them through the process of applying the sine or cosine rule but I am highlighting how experts would act with a year 10 trigonometry question.

Another part of knowing facts within maths is remembering formulae. The sine rule has to be readily accessible in order to answer a difficult, multi-part bearings question. The quadratic formula must be ready to come out at the drop of a hat at the end of a hard pair of simultaneous equations.

This is not a novel idea and comes from cognitive psychology. We can hold 7-9 things in working memory at any one time. We don’t want these things to be formulae. We want these to be relevant to that specific question (the arithmetic required for example).

Take the following GCSE exam question:

A sector with an angle of 144° is cut out of a circle of radius 10 cm. The remainder is turned into a cone by bringing two radii together.

If the volume of the cone is the same as a sphere, find the radius of the sphere, leaving your answer as a cube root.

This question is impossible if you do not have lots of things to hand in your long-term memory. See below.

My thinking:

  • Angle left in circle = 216 degrees
  • Length of arc = circumference of base = 12pi
  • 12pi = 2pi x radius therefore radius = 6 cm
  • vertical height, radius and slant height form right angled triangle so vertical = 8 cm
  • 4/3 pi r^3 = 1/3 pi (36) x 8
  • Solve so r is the cube root of 72


Look at how even the most competent of student will think about this problem:

  • Angle left in circle = 216 degrees
  • Length of arc will be the circumference of the base.
  • Length of arc formula is portion of circle x circumference
  • 2 pi r x 216/360
  • = 12 pi
  • The circumference of the cone base is 2 pi r and that is 12 pi so r = 6
  • How do I get the height from the radius?
  • The height is 10.
  • What’s the volume of a cone?
  • What’s the volume of a sphere?


You can see very quickly how a student has to hold and internalise a lot of thoughts. This comes after having to visualise and conceptualise this whole question – cognitively, this is exhausting! We must help them. These questions are hard even if you can remember all the relevant formulae but they become nigh on impossible if you have to struggle to remember them too.

Organise your students’ knowledge

Knowledge organisers are a hot topic at the moment and lots of people use them in a variety of ways. I must say that I am a complete novice in this area but I am a huge supporter of organising students’ knowledge for them. Every lesson, I will revisit my “things we have to remember” page. I make sure I am the one giving definitions and formulae and I ensure that all students have correctly copied this down.

We test every fortnight and these tests are designed in a way that students should be expecting to get everything correct if they have kept up in class. Part of all of these tests is a memory recall section. Recently, I have decided to be transparent with this, too. Students are given 5 minutes to answer 10-15 minutes of memory recall questions. We mark those ourselves and then they crack on with the “problem solving” questions – where they may be using some of the things in the memory recall section.

I have written tests before where I have asked “use the cosine rule to find the missing length” and I thought I had done enough to bullet-proof the question because I’d told them which rule to use. Unfortunately, I hadn’t foreseen people not remembering, misremembering etc. If I wanted to ask the same question now, I’d put a “write down the cosine rule” in the memory recall section and then the same question later in the problem solving question.


Testing for (and to deepen) knowledge

This really is the crux of the matter in mathematics. I’m not advocating doing a load of past exam papers and you’ll get better at maths. I am advocating short tests on lots of things. Each question should serve an obvious purpose and if a student can’t answer that question, there should only be one reason. Exam practice is a different animal and one for another time. For the purposes of this piece, we are thinking about deepening knowledge and memory – this comes from purposeful assessment and specific feedback.

Matthew Martin wrote about the forgetting curve in his blog and that’s perhaps more important in mathematics teaching. Students are very good at compartmentalising topics. This is a sequences lesson. This is geometry. This is calculus. When they are doing calculus, they are not thinking of geometry. Our tests should include things they have forgotten. Effortful retrieval has been shown to be a high-impact solution to deepening knowledge and strengthening memory (Make it Stick).


The key things to think about when thinking about memory in mathematics are:

  1. Think very carefully about what it is you want students to remember.

  2. Test very specifically whether or not they can remember these things.

  3. Test regularly. Explain why you are testing them.

  4. Let them mark these tests – you gain almost nothing from marking it yourself. They gain lots from knowing whether or not they actually remember it. The right/wrong feedback is enough for a memory recall question

  5. Don’t ask students to actively remember and apply within the same question until they are ready for it. This is the main reason past paper practice is a problem.

  6. Deliberate, specific practice of one or two step questions like Dani’s mentioned earlier alleviate this problem.








How important is knowledge in mathematics?

Why do it?

“Those who can’t do”

“It would seem you have no useful skill or talent, have you thought about going into teaching?”

Why would you become a teacher? Are you teaching because there is nothing else to do? Are you teaching because it is fairly safe and there will always be jobs? Are you teaching because you get lots of holiday?

I’ve been bombarded with all of the above and I’m happy to say it is none of those things. Learning fascinates me. How do we pick up the things we pick up? How does it become second nature to do something? I adore the process of learning and I absolutely love watching it happen. I am not teaching for personal satisfaction; I am not doing it because I feel I owe something to someone; I am doing it because I absolutely want to and I really enjoy doing so. The only thing I have in my mind is ‘how can I convey my enthusiasm in a genuine way?’ I would like all of these children to go home and say they enjoy maths (wouldn’t that be nice?!) and I am learning how to be authentic with that every day that goes by.

It would also be silly here not to admit the following…children are brilliant! Every single child is fantastic – I wholly believe that. Every single day they inspire me. This is the essence of teaching and everything mentioned above; it is a two-way process. Everything about teaching and learning should be two-way and teachers should inspire students but students should inspire teachers. Students are far better creative thinkers than we are (in this writer’s opinion) because they are allowed to be.

I would advise everybody to give teaching a go. “I don’t like children though” or statements to that equivalent would soon evaporate. You will understand, very quickly, that you don’t like children who are in a tough spot. It is your job to ease those spots as much as possible.

Secondly, I believe teaching is one of the most professional professions. By this, I mean that I believe all teachers strive to get better. No teacher that I know is doing their job in order to get paid. All teachers are trying to be the best teacher they can be. I’m not sure this is true about other jobs (in my experience).

Hats off to everyone in education. We complain – there are things we would all like to change, naturally – but it is a pretty good system. It is a good system because of you guys. Keep enjoying it – I certainly will!



Why do it?

Teach like a Champion: Ratio

Think Like A Champion: Ratio

Today marked a big day in the future of my teaching and my students’ learning. It was the day in which I think I saw the holy grail of professional development. Maths conferences that I have been on are superbly run but that’s fellow teachers sharing best practice. This, being led by Doug Lemov, Erica and Colleen of UnCommon Schools, was a workshop where almost every 10 minutes, there was something to take away to the classroom. As my neighbour at the table stated, “not one bit has felt like a throwaway comment”.


The day started with an early train over to Waterloo and a trip up to Camden. The venue was perfect for the event and we managed to get there in time for some breakfast and much-needed coffee! Doug started off with some icebreakers which were a bit less awkward than ones I’ve seen before. Pair off and discuss the story behind this…pair off and think how it applies to teaching. Highly engaging and a good, fast-paced start. This is also the first time that I got my first oneline takeaway…


“With common vocabulary, we can build anything”


A key theme throughout the day was a common culture – this was the start of embedding that idea. When talking about teaching, use the same language and others will find it a lot easier to follow and it aids in all areas: feedback, discussion and learning.


We’ve all been in classrooms where we’ve had a great workout – now let’s build classrooms where students do the hard work.”


This is the underpinning message behind the workshop. Ratio relates to a lot of things but at its root, it is doing with the ratio of teacher work: pupil work. We have seen recently that Wiliam thinks “feedback should be more work for the recipient than the donor” and now we are extending this to the classroom. Learning should also be more work. I was hooked.

Thanks @leeDonaghy:

All this stuff about pupils working harder than teachers: it doesn’t mean independent learning. It means RATIO.

That’s what this whole day is about – a proper framework for independent learning.


Participation vs Think


The first session of the morning centred on time students spend participating and time students spend thinking and the balance between the two. Which activities are we setting? Do we know what the purpose of our activities are? Where is the blend between the two which gets us to all participating and all thinking? This workshop led to some great discussion around these areas. We talked about times when actually lots of participating and no thinking might be good and vice versa. One of my key takeaways was:

“if you let a student write straight after a group discussion, you’re testing their listening skills.”

I will endeavour to tailor this to write-discuss-change your answer.


The necessity for prerequisite knowledge


This was the one for me that I was really excited for today. The workshop after a quick coffee break was about the need for knowledge. We did an eye-opening exercise which encapsulated the point so well, I will use it in every single professional development I ever have the opportunity to run. The message of this session was simple: if you don’t know something, how can you solve problems relating to it?

There was a “criticism” of Bloom’s taxonomy in this period which I liked. The main point was it has become socially embarrassing (within teaching) to ask knowledge questions because they are the bottom of the period but actually knowledge must come before the pyramid at all levels. Knowledge is “Bloom’s delivery service”.



Throughout both sessions (and post-lunch), there was lots of video of great lessons and best practice. I loved this and again is something I will push in more development training. The key feature, whatever we were discussing, was the culture is clear in all these classrooms. “Habits of discussion” (hands down when someone is talking; everyone turns and faces the speaker etc) are consistent throughout. There is a clear argument here for a common, consistent culture in schools.



Maximising ratio through writing


One thing is clear from today – these guys love written text! Whether students are writing it, reading it or discussing it; use written text. After all, most of where

“students will derive facts from is written text.”


The afternoon session followed a more practical model and we were discussing the benefits of showing student work and how you can frame parts of your lesson around this. I enjoyed the practicality of the discussion and this is something where I will be focusing more thought towards in my planning as opposed to my pretty ad hoc approach at the moment.

We looked at group worked examples, comparative judgement and a handful of other techniques and what makes them powerful. The whole day was about being effective and efficient. Again, a plethora of video examples ran throughout and there was plenty of time to discuss.



The run down of each session has been purposefully brief here, out of respect for the organisers but  I would absolutely recommend this course to any teacher. I was sceptical that I wasn’t in enough of a leadership role but there is so much here to get my teeth stuck into. The course was brilliantly run. The three deliverers spoke eloquently; there was lots of time to reflect; there was lots of personal time and collaborative and it was a room full of people all trying to get better. I’m determined to followed a similar model when we run our own development courses moving forwards. Little things that I hadn’t thought about until just now – fluency with material (page numbers etc), boxes to write reflections on everything; clear dynamic of own table, move table etc.


The first day was inspiring and I am looking forward to heading back into tomorrow morning for the second half. Thanks so much, Dough, Erica and Colleen. Genuinely fantastic.

See you in the morn!





Teach like a Champion: Ratio

How can sports coaches help teachers?

Over the last few weeks, as my mind has started thinking about the new school year, I have found myself involved in lots of sports coaching. Specifically, I have found myself involved in multiple conversations about sports coaching. This has highlighted to me just how far ahead sports coaches are in comparison to teachers. I hope to offer some advice on how we can take what they do and know and adapt it to teaching.

Firstly, let’s talk about the shifts in both fields. “Progressive” is banded around a lot in teaching with negative connotations. It’s a shame that such a positive word generates this reaction. If you are progressive (in the twitter sense), you believe in solving problems; creating engagement and motivation breeding success. There is far more to it than this but for now, that’s enough of an overview. The other end of the spectrum – “Traditionalist” means you believe in facts; teachers know best and things are taught not discovered. Again a horridly cut-down version of the actual debate but enough for those who are unaware. The recent shift in education has been to push problem solving. Questions are more “real world” so students have to act as mathematicians do or solve medical problems that doctors face. A similar shift happened in sports coaching. Games were the big thing a few years ago. Spain are the best side at football, they play lots of games therefore we should play lots of games.

Technical vs Tactical

Both of these arguments seem to make sense on the surface level. The problem is that if someone’s skill set is not strong enough, they cannot engage in these activities in any sort of way which will impact long-term memory and therefore learning . For example, 9 year old Spanish kids may be better at passing with both feet than any other country therefore the games they play will work in a different way. In the same way, a student who can recall their multiplication tables will be better at solving multi-step problems involving multiplication than a child who has to work out every multiplication. What do sports coaches do, therefore, to look past this problem?

When games-based coaching first appeared, there was a tendency to go fully in the deep end and let the players play games and expect them to get better. Obviously, this model did not work but it began a very important process in sports coaching. Sports coaches realised they could constrain games however they wanted – numbers of players, size of pitches, type of ball used or certain rules – and this would, in turn, force students to practise the skill necessary. This is taken further with a “whole-part-whole” model whereby a session begins with a “whole” e.g. some form of a game which is used to diagnose flaws and then the “part” comes in. This “part” may look more like a drill or specific practice in which skills are practised. After significant improvement in a more controlled environment, the players are put back into a “whole” in which they play the game and hopefully show improvement in this discipline.


Tech vs Tact


The difficulty with getting this approach right is selecting which “whole” to start with. If you look above at the graph of technical vs tactical, developed at Oxford Brookes University, we see that as the number of players increases in a game, the emphasis shifts to tactical and less technical. This makes intuitive sense, if 30 people are doing something, it is hard to see each individual techniques but you can see how teams co-adapt and progress in the specific game.

Coaches constantly think about the purpose of their “whole” segment. A 15-a-side rugby match is not the time to judge someone’s passing ability. The match is not designed to test this ability. In the same way that a summative exam is not designed to test a student’s individual techniques. These are designed to differentiate from top to bottom within a cohort. I cannot recommend Daisy Christodolou’s book “Making Good Progress?” enough for more on summative vs formative assessment. We must realise what we are asking students to do and the purpose of it.

The equivalent to whole-part-whole in a traditional classroom would be to sit an exam; highlight flaws; practise flaws; sit an equivalent exam. Repeat. The problem here is that an exam is far more on the “tactical” end of the spectrum if we think of the same diagram above. Exams are designed summatively and need to be able to put students into labelled boundaries. Therefore, these exams will test multi-step problems and questions will take on multiple techniques at once. As a question increases in marks, this becomes more true. In this sense, we can think of more players = more tactical/less technical for sport and more marks = more tactical/less technical for teachers.

What do we do then to ensure we maximise the effectiveness of this approach? Sports coaches are very creative with what constitutes a “whole” and very specific in what they take from the performances. A “whole” does not need to be a full game, it can be small-sided games or it can be a very variable practice focusing on one or two techniques. The point of the part is to “screen” the players. After players have received very specific feedback, they are asked to play either the same whole or a slightly different version but coaches offer precise feedback on the one or two techniques discussed. We can certainly take that approach into our classrooms and ensure that students are specific in what they are trying to improve. For this to happen, we must be creative when asking students to complete diagnostic assessments. This doesn’t have to always look like the final exam. What do we want to test for? How can we constrain an activity so we can effectively and efficiently test for it? What feedback will we offer to ensure the students know where they are and how improve?

The idea here is not necessarily always to get the right answer/win the game. Sometimes, you will want students to improve technically in one specific skill. Other time, you will ask to get the right answer. This perhaps fits more into the tactical side – in sport, you’d be asking players to win and in class, you’re asking students to get it all right.


Degrees of Freedom

The next part of sports coaching which I think can be taken into account is the idea of removing degrees of freedom. Simply, this means removing variability from a skill. For example, when teaching a player to throw, we might ask the player to sit on the floor and just use their wrist and elbow. This allows a chance for the coach to break down a massive group of concepts into parts. From here, we challenge the players to do different things – throw further, throw higher, throw left, make it bounce etc. This allows players to “feel” each part of a movement. They can quickly see how each part fits together and what happens to their process if multiple steps are required.

This can happen, too, when teaching in a classroom. Kris Boulton (@Kris_boulton) has done some great work, inspired by Engelmann, in breaking down big topics into concepts. Kris took “simultaneous equations” and said there are thirteen parts; let’s go through them. This is a similar idea to removing degrees of freedom. There is so much potential for errors in big topics that we have no chance of spotting where the actual mistake is and students have no idea where their own flaws are.

For example, if we look at a simple one step algebra solution at the end of a long, simultaneous equations equation:

2x = 6

x = 4


What is the mistake? Does this student think 6 divided by 2 is 4? Do they think 2x means 2 + x? Have they just made a silly error through fatigue? All of this could be true – this question is not designed to test this ability. The more we break down concepts and assess students abilities to solve those problems, we can be happy/unhappy that students can or cannot grasp this specific concept involved in simultaneous equations.

If we break things down, concept-by-concept, we can see specifically where students are struggling. This also has an added benefit of instant fixes and a chance of immediate success depending on the difficulty of the concept. The teacher’s role is to ensure each concept is a small enough jump that success can still occur but large enough that it feels challenging.


Removal of Anxiety

Finally, the last thing I would love to see in classrooms is a removal of anxiety. Sports pitches should be the least anxious of all environments. You kick a ball and it doesn’t go to your teammate, oh well, let’s go and get it back. The attitude here is a “what’s next” mentality instead of “what’s happened”. I think this probably comes from matches (without scaffolding) being played an awful lot – in some sports, players will play multiple times a week – whereas students might only sit assessments (without scaffolding) once a half term or even less frequently.

More low stakes quizzes/tests/assessments may contribute to a forward-thinking attitude as opposed to a backward-looking one. What we do with these assessments is also important. I saw a great tweet a few months ago saying “you wouldn’t write a letter to each player after a football match” to give them feedback so why are we doing something equivalent after these assessments? There can be some feedback for the whole class and something active to ensure it is more feedforward e.g. targeted questions, active reflection. This frees teachers to have the opportunity to have individual conversations with students. Feedback is far more likely to be acted upon if we have a short, specific conversation explaining what we noticed in their work which could be improved.

The hope here is that this approach creates a culture of seeking feedback after an assessment and not a score. This is very much mirrored in the best sports teams – players are concerned with how they did rather than how many points/baskets/goals they scored. This culture comes from the coach and I believe can be created by the teacher.

Before finishing, the final thought is about success. Sports coaches are also trying to convince players to come back a lot of the time. They use success to breed this motivation. Instead of using motivation to breed success. This is the last idea I think more teachers can latch on. All students want to be good. All parents what their students to be good. Let them be good. The more success that students feel, the more motivated they will be in your subject. Yes, failing and making mistakes is important, but don’t forget how important it is to feel good, too. All of us are more likely to stick with things if we think we can do it.









How can sports coaches help teachers?

Today I would like to appreciate Michaela…

…for allowing me to come and see a splendid school in all of its glory.

Before I go any further, this whole post comes with a huge caveat – this will not do the day justice. You must try and arrange to see the school if time allows.

After meeting Dani Quinn (@danicquinn), a couple of months ago, I was keen to see how maths is taught at Michaela and have a look at their ethos which some may feel is a direct opposite to Wellington. The day started with the students enjoying lunch time and I was immediately struck by the level of interaction of students and staff. Students were happy to converse and allow staff to join conversations freely. At the end of the play time came the first real sign of the expectations set. A member of staff quickly had the students lining up in silence and all were clear on the rules and what was expected of them. This preceded a silent walk into the lunch hall where I sat with a group of students at a university-labelled table (a great way of inspiring children, I think). I was amazed that these year 7 pupils knew about universities and had great ideas about which universities they wanted to go to and which subject they may consider studying. From just twenty minutes at school, I was stunned by the levels expected from the pupils. This expectation can only have  a positive effect on the school.

During lunch, a member of staff read out merits and demerits from the last day in a no-nonsense fashion. These students have been rewarded and these students have not met our expectation. After this, we were told the topic for the day and that is what our table was to discuss while eating. Five year 7 and 8 pupils plus myself went on to talk about punctuality and why it was important not to be late and what we can do to stop ourselves. I thoroughly enjoyed the conversation and the students should be applauded on their welcoming and challenging conversational skills. After a donut for dessert, students were invited to show their appreciation for anyone in their life – mainly teachers or others in the school. The duty staff would then offer very specific feedback on how this was done e.g. “that is worthy of reward because you projected very well and it was meaningful” or  “if you want a reward next time, make sure to annunciate every word”. Clear, specific feedback throughout the school. Everyone is on message.

Two students greeted us and took us around the school where we were allowed in and out of lessons freely without students or staff batting an eyelid. It was here that I was struck by the quality of teaching at this school. I saw about 5 minutes of a French lesson, some of a Physics lesson, some Art and some maths. In all classrooms, students work was put first. The school may well believe in teacher at front, student at desk but there is also a clear ethos that students must be doing work to learn. The behaviour culture allows for clear boundaries and allows teachers to positively reward great learning habits e.g. sitting up and following a solution very well or tackling a problem straight away. It does also allow for negatives to be enforced quickly e.g. turning around but across my afternoon, there was far more specific, positive praise than negativity.

Each teacher ensured pupil participation was high and there were lots of questions asked across all the classrooms. Teachers involved each student and if something was not quite right, time was taken to correct before moving forward. There is a clear expectation academically as well as behaviourally. I was thoroughly impressed by the level shown by all students. All activities used were activities that students wanted to engage with. Students could see that the teacher wanted them to know what to do so they trust that there is a purpose in what they are doing.

After popping in and out of random lessons, I watched a full lesson with Dani to finish the day. This followed a similar pattern: lots of pupil involvement; great questioning from the teacher; constant assessment of and for learning as well as a chance for lots of success. I did not see Dani using a drill but I got to see a little into her teaching philosophy which will certainly have an impact on my own practice. I found that students did a lot of practice of specific questions and anyone who needed more got that opportunity too. Dani (and the other teachers) kept up the idea of specific feedback on all activities too. In maths, the feedback was even more specific owing quite a lot to the nature of the activities. All of them allowed Dani to quickly work out what the misconception was and then address that. This is certainly something I will be taking into my own lessons – particularly during revision time.

Unfortunately I had to rush off before talking to Dani at the end but I would like to thank her very much for the chat we had and for allowing me to see into her lesson. I had a brilliant afternoon and I very much look forward to going back to see some in the future in the maths department. I also hope to get Dani over to Wellington to have a look at how we do maths and where the similarities may lie even if the surface looks/sounds very different.

Overall, I think that Michaela are doing brilliant things. Students enjoy school, they want to succeed and everybody takes pride in themselves and their school. This is a great achievement by all involved. Expecting the best (“double the minimum”) and reenforcing this message means that students push themselves and have a positive attitude towards this test. I would like to look more into the effect that extrinsically motivating young children can bring about intrinsic motivation as this is something I had an opinion on before but after yesterday, I wonder whether all students can actually generate their own high standards after being held to them by punishments and rewards for a number of years. Certainly, all students I met in an informal setting (lunch, after school) were very polite, interesting and proud. All values that Michaela School has.

I would like to appreciate Michaela for letting me in for the afternoon, for showing me that great teaching relies on a great attitude to teaching and for setting a high expectation of all students.








Today I would like to appreciate Michaela…

The three keys of intrinsic motivation

At the end of 2009, Daniel H. Pink wrote a book entitled “Drive: The Surprising Truth About What Motivates Us”. You can find him talking about his vision here. Within this book, Pink argues that businesses are not acting upon what we know about the human brain and what psychologists have known for a long time about motivation. The key message being that rewards and extrinsic motivators only increase productivity for straight-forward tasks. In other situations, these motivators may actually inhibit creativity.

It is impossible to read the book as an educator without thinking about its impact upon the classroom and the little things we do in order to create an intrinsically motivated culture. I am particularly interested in this to discuss the impact that written and oral feedback can have as well as our questioning in the classroom.

Pink proposes there are three things to strive for in order to promote intrinsic motivation:

  1. AUTONOMY – “the right or condition of self-government”
  2. MASTERY – “comprehensive knowledge or skill in a particular activity”
  3. PURPOSE – “the reason for which something is done”

In the classroom, students will be motivated if they focus on these three areas. In particular, teachers can direct student focus to these three things through their feedback and questioning. Sarah Donarski has written a blog relating specifically to feedback and its motivational responses – I will try and take some ideas further, specifically in the context of Pink’s trio.


Autonomy in the classroom can take multiple forms. Pink argues that autonomy will improve engagement and will take over from compliance in the workplace – the same can be said of the classroom. With a truly autonomous student, a teacher can be confident that there is a prominence of engagement and a desire to carry out actions because they want to do them. Jang, Reeve and Ryan found in 2005 that high autonomy was one of the most important characteristics of a “satisfying” learning experience and low autonomy had an even more negative effect on the experience.

The question then remains – how do we promote autonomy in our classrooms? Educators can create situations which require autonomy as much as possible. For example, an activity might require students to make a choice at the start and justify. We also must ensure our tasks are challenging enough that students want to engage with them. As students progress through school and are more skilled at making decisions, we may also set tasks which allow for preference and encourage students to think about why they are making these decisions to choose which activity. We cannot let the classroom become a free-for-all but we can slowly introduce these ideas as students are ready for them. The same can be said for classroom dialogue. Teachers can be flexible and allow a more “free” classroom.

With a feedback hat on, which feedback allows autonomy to grow? Specific feedback with some ideas on how to improve on these specific topics in the classroom give this chance. Darren Carter (@mrcartermaths) has spoken previously about his homework (or lack thereof) “policy” and it strikes me that this is a great way to inspire intrinsic motivation. Of course we know better than the students about what they should improve and how they can go about doing this. This information should still be shared but we are allowing them to decide what to do and explaining why (which encroaches upon number 3). Spend time with students showing them excellent online resources; picking a specific chapter in a specific book or write some feedforward questions which allow immediate improvement. There is no expectancy of completion but all students realise that being active with feedback will result in improvement. Thus an intrinsically motivated action has some extrinsic reward also.



This is particularly prevalent at the moment in the mathematics world but in this discussion, we are not talking about deeper knowledge about less subjects – we are interested in the idea that students feel better at a specific discipline. All children want to be really good at stuff – this is not up for debate – whether it is maths, English, sport, dance whatever. Everybody wants to be good. Teachers must tap into this innate part of a student’s make-up. How can we do this through feedback? We must be positive and we must be specific with this praise. See below for a tweet from Ben Ward I saw this week (@mrbenward).

“We remember criticism because it is specific and personal.

Whereas encouragement is general [so it] washes over us.

Aim for ‘precision praise'”

I love the idea of precision praise. It is a big part of sports coaching and every course I have been on in this domain has focused heavily on generic praise and its pitfalls (namely that nobody acts on it and it is wasted energy). Precise praise can mean a student knows they are further along the journey in mastering a topic than they were before. Specific praise on something you have asked them to improve in the past will have the added bonus of showing them that their choice of work has worked and been recognised (their autonomy is improving too). Too much praise can be a negative but using praise in the right scenario in a very specific context will improve student’s internal motivation and reaffirm their belief in themselves.

In Sarah’s blog, she examines the idea of a positivity bias in which students focus on the good things you say or see overly positive messages in circumstances which might not be wholly positive.  She proposes that this can be a good thing for students and again specific, precise praise can let a student know that there is positives in what they are doing. This can only be a good thing at all ends of attainment.



How many times have we heard “When will I ever use this?” about almost everything taught in the maths classroom? The answer of course is that almost everyone will not use the sine rule nor the area of a trapezium nor differentiating trigonometric functions from first principles outside of their maths lesson. In much the same way that students will not analyse the meter of a poem many times after GCSE English nor testing the pH of something. The point that students aren’t getting is that all of these should be ends in themselves. Carl Hendrick (@c_hendrick) has written a piece recently looking at the idea that education should not be a vehicle to prepare us for what comes after school. At the end he writes,

“Students should study Shakespeare not because of what job it might get them but because it’s an anthropological guidebook that tells them how to live.”

This same sentiment should be held by teachers in all subjects. You are not learning about pi because it will help you in some 21st century job yet to be created, you are learning about pi and its place in history because it shows you something remarkable which was at one point undiscovered. Yes, you will be able to use google maps to tell you how far away something is – that isn’t why we teach triangles but you should have some appreciation of size, number and shape. It is important that teachers highlight this purpose throughout all of their feedback and discussion. Teachers must live the idea that everything taught is purposeful and should not find themselves justifying existence. 

Moving forwards, I aim to incorporate these three ideas of internal motivation through all my student interaction. Any feedback given should look to promote at least one of these areas. Remember that feedback should be more work for the receiver than the giver. 









The three keys of intrinsic motivation


Maths Conf 9 bought with it my first ever trip to the city of Bristol. The conference was hosted at City Academy school which was a great setting as expected.

The day began as usual with Mark and Andrew Taylor from AQA discussing the day and explaining the processes. It is always inspiring listening to Mark talk about teaching mathematics and he makes it all seem so simple. His message came across very effectively.

Speed dating bought with it some good conversation about raising morale, discussing interleaving and talking about Harkness to anyone that would listen. Please do talk to me about/come and see the teaching philosophy if you are at all interested. A familiar feeling then ensued where I would have liked more time to talk with fellow maths teachers but time meant we moved on to listen to the first speaker of the day: Michaela’s Dani Quinn.

Drill and Thrill (@danicquinn)

Firstly, I must say I really enjoyed listening to Dani talk and I was captivated by her energy, passion and clear desire to be a better educator. Her students are incredibly lucky – I look forward to coming and visiting the department as well as the school.

Dani was talking about “drilling” students with a particular focus on pre-year 9. She explained the processes used and the point behind them with a clear link to the relevant research. The underlying messages were:

  • If teachers don’t have to think about it, students shouldn’t have to
  • Children enjoy getting stuff right
  • A speedy mathematician isn’t a good one but good mathematicians are speedy (in certain facets eg. index laws)


I was keen to hear how drilling is incorporated after reading a few of Dani’s blogs. She is certainly not against conversation and engagement and lively classrooms but she was saying that when used at the right time, drilling can improve that instant decision making that we take for granted. A good example was: will this addition give me a positive or negative number? Students may have 20 questions to answer on that. There is usually some competition: quickest one or most correct in a short time. It is purposefully tense and the point is that students should be accessing their long term memory and not using their working memory.

Drilling should be used to:

  • Practice an isolated decision
  • override system 1 type misconceptions (I think I should add the numerators and denominators when adding fractions)
  • consolidate on material already taught


Dani progressed to discuss implementation and some how to’s and how not to’s. We all make mistakes and Dani was not trying to say that Michaela are perfect by any means and she gave some advice on what a good problem sheet looks like and what a less good one might be. There are risks to drilling and again Dani addressed these with positivity and advice.

I certainly feel like I will go away and think about how deliberate practice can better be implemented in mathematics and in my classroom. I really like the idea of making an individual decision. This idea can be perfected at all levels and I will have a think over the next week or so about the best way to use “drills” in particular with my exam sets. I would like to see happy students who are happy that they “know” the cube roots and square roots for example. I’m not sure I will get them doing a mexican wave though – maybe that’s next year’s task!

The key thing to take: PRACTICE MAKES PERMANENT.



Where Y11’s will go wrong in the maths GCSE…and what you can do about it now

After Dani’s excellent opening, I was treated to one of the more omminently titled workshops I have attended lead by Craig Barton (@mrbartonmaths). One of my all time favourite twitter maths people and his podcast is excellent – well researched and heavily evidenced. I was looking forward to this after attending guess the misconception nigh on a year ago.

Craig’s website has become part of my teaching this year and student feedback is positive – this has reminded me to re-push the students after their mocks to get back and do some maths every day. Craig spoke very well, awesomely confident and as Dani had done, gave off an aura of passion and dedication to the profession. Craig highlighted the top 10 most wrongly answered questions and went into detail on three:

  • y = mx + c
  • Ratios
  • Enlargements

Throughout the three, he referred to what he and his department will do with these findings. His new look website is a staple and that links to a lot of Don Steward’s resrouces which Craig will use to commence purposeful practice.

The key from this talk was that all students benefit from purposeful practice. Purposeful practice can mean some students go from no knowledge to some; some go from some to lots; some go from lots to more. Nobody knows everything about a topic and “overlearning” has many benefits. Craig is clearly evidence-based in his decisions and he has shared his expertise with us all in attendance today. The key things with purposeful practice are:

  • All students can do the same material [better students engage with it in a different way – ask why/spot connections]
  • Key skills can be practised by those that need it.
  • Methods can be shared by more able students


We must cover key concepts before we can expect students to get better at exam questions. 


Craig mentioned this towards the end of the talk and it reminded me of something I heard a few conferences ago: “nobody gets cleverer by being tested”. This is absolutely true (“Minimal guidance doesn’t work”) and we cannot throw past papers at students and expect them to get better. This was great advice at this time of the academic year and has made me refocus my mind on how best to use the remaining teaching time. There is some merit in past papers but don’t just throw them at students and expect results to improve. Learning is not happening.

A fascinating session with Craig and the bar has been set superbly high from the morning. I am now looking forward to lunch and then sessions this afternoon with Andrew Taylor and Kris Boulton.


New GCSE Maths Exams (@aqamaths)

It takes a certain sort of someone to enjoy a session about exams and what constitutes which mark and why questions are how they are. I am that sort of person so I had a great 50 minutes listening to Andrew Taylor discuss the new GCSE exams and why AQA have taken the decisions they have.

Andrew eloquently went through the different types of problem encountered at GCSE and discussed where marks can be gained and what we would expect to see for those marks. There is an emphasis on finding marks not taking them away and rewarding students. This means that some notation mistakes aren’t penalised and examiners are expected to translate without something necessarily written down word for word.

The big decision taken by AQA with the new spec is that there are less marks for doing numeracy or carrying out basic operations. For example, in a problem solving question, the markw ould be for getting to 4x + 1 = 13 and no extra mark for  4x = 12. There will be other questions which tackle this skill.

At Wellington, we teach to the Edexcel board but there is lots to be gained I feel from talking about the final assessment. There are also some awesome questions written by all the different boards and it cannot be a bad thing to see more of these!

The key message from this sessions was that:

  • Marks are accessible throughout the GCSE papers. The final questions are harder but marks can be gained
  • Words of explanation during methods will help both the student and the examiner
  • Most marks in the paper are for using the correct method.


Discuss methods with students and ensure they understand what a correct method looks like and why that is so.


One more hit of caffeine and then on to listen to Kris Boulton (@kris_boulton) – another of my favourite tweeters.


Most of the things you think are processes probably aren’t

I have read a lot of Kris’ work on Engelmann and his thoughts but have never heard him talk about him and how he uses the theory of instruction in his own teaching. After listening this afternoon, I wish I had! Kris always speaks well and this was certainly no exception. He explained the theory well and it was easy to see why continuous conversion and minimal difference will work in many contexts.

Today’s discussion was centred around a type of concept which Engelmann calls “transformation”. Simply put, this is any concept which takes an input and something happens to create a new output. Kris used examples of the distributive concept of multiplication and finding the area of shapes. At a basic level, the students see your answer to the question you are asking multiple times and then infer a correct answer to a new question based on patterns. I can see how this works in all creation of resource and I don’t think it has to be done in the explicit my turn, your turn way that Kris modelled (although this looks hugely effective). For example, in our Harkness books, we have examples of showing examples and then ask students to complete a similar problem. The next sheet can then be initial assessment. The next sheet would then be the expansion phase and seeing the same concept in different scenarios. The key here is that differences must be minute. As small as changing one number.

Towards the end, Kris mentioned two things which will sit with me forever.

  • The phrase: Logico Empirico. Loosely this refers to think of something, try it, adapt it, try it, adapt it. If I could sum up my approach to teaching, it would be this.

  • The sameness and difference principles related to maths problems.

Sameness relates to questions which look markedly different. The cognitive discussion students have is to find the similarities. Difference then is questions which look really similar, students ask what is different and how does this affect the output?

Thank you very much, Kris, for an inspiring finish. Not sure where I will find 180 quid for Engelmann’s book though!


And that was that. The final curtain on another wonderful day. Many thanks to La Salle and Mark McCourt. Thank you to City Academy. Thank you to everyone I listened to and everyone I had a conversation with about teaching maths. Every single moment was a learning opportunity. I will go home now with a head full of ideas and a lot of stuff to read up on.

Well done all!