-Various members of the Twitterati.

The Dunning-Kruger Effect is a phenomenon suggesting that people with low ability in a given field are likely to over-estimate their competence. The original paper by David Dunning and Justin Kruger, originally published in 1999, is available here, although, as they themselves acknowledge, the paper builds significantly on prior psychological studies. The authors describe the phenomenon as a “dual burden” for those with limited knowledge in a particular domain- “not only do [low ability individuals] reach mistaken conclusions and make regrettable errors, but their incompetence robs them of the ability to realize it.”

Importantly, Dunning and Kruger suggest two conditions which are likely to be required for the phenomenon to hold. The first condition is when “knowledge about the domain confers competence in the domain”. If one has a significant knowledge of the mathematics then one is necessarily a competent mathematician. Compare this to something like sports where one might possess excellent knowledge of even the most technical aspects of the given sport but not have the physical prowess to execute the skills in the manner they know is required for exceptional performance (this is the position of many of the great sports coaches throughout history). In sports, as in many situations where physical skill is required, knowing what do do does not entail being able to do it.

The second condition that Dunning-Kruger is that *some* threshold knowledge is required of the subject in order for the the effect to hold. I am not likely to overestimate my ability to translate passages of text into Arabic as I simply would not be able to write anything down if asked to perform such a feat and I am all too aware of this fact. If however someone asked me to do the same in French, my *very* limited knowledge of this domain (largely unused since GCSE , much to my discredit), might allow me to generate some correct (or at least, vaguely plausible answers) and thus I might well overestimate my ability. In these cases, a little knowledge can be a dangerous thing.

It seems reasonable to suggest that Mathematics, along with a large number of other school subjects, fulfils both criteria.

Whilst this post focuses on over-confidence and a few possible implications for self-assessment, I am obviously aware that for many students a *lack* of confidence is a major barrier to success in school. This is something I am acutely aware of and it is just as important for teachers to be aware of this as over-confidence. In my experience however, teachers are generally more aware of students lacking confidence than they are of the over-achievers and they take this into account when interacting with and assessing students.

It is reasonable to state that self-assessment is an often used tool by teachers in many schools. RAG, coloured cards, thumbs up, confidence scales from 1-5 etc. are all fairly common sights in secondary classrooms. Whilst unlikely to form the whole of a teacher’s or school’s assessment model, it is often used as a way for teachers to decide the ‘direction’ of a lesson or a sequence of lessons. “You guys are all ‘green for this? Great! Let’s move onto the next topic”. “Ok, lot’s of reds here, perhaps we need to take a while longer looking over this”. However, the Dunning-Kruger effect suggests that this is a poor approach to take as if students lack competence in a particular field, they have a tendency to overestimate their skill. Note that this phenomenon could affect older students as well as younger ones- when students encounter new material they are novices and are unlikely to know what they don’t know in that field.

**A Few Suggestions- both conservative and radical**

**Base any in class assessment as much as possible on data and your own observations and knowledge of the students rather than their perception of their competence.**Rather than asking students if they are Red, Amber or Green for a particular topic, ask them how many questions in that exercise they correctly completed or perhaps give them a test on the subject to get some objective numerical data that you can then use to decide on your course of action. This is simple but it is amazing how often teachers still ask students for something based upon students’ feelings rather than something objective, valid and reliable approach.**Be aware of the Dunning-Kruger effect when discussing learning with students.**This is particularly important around exam periods where students inevitably have to make decisions about the areas on which they need to focus their revision. It might be that students need more guidance than one might think when it comes to supporting them in making decisions about where to focus their efforts. This is particularly the case if students are making decisions on the basis of ‘feel’ (in my experience this is the norm) rather than empirical data about their strengths and weaknesses. Interestingly however, Dunning and Kruger don’t discuss the ability of people to make*relative*self-assessments between similar disciplines. Perhaps a student might overestimate their competence in both vectors and calculus but it might be that they are at least able to make relative comparisons between the two so they can prioritise their efforts. I.e. they think they are better than they are in both subject areas but at least they are accurate in realising that they are generally better at calculus than vectors. Be aware however that some students might incorrectly consider themselves to be so accomplished in particular areas that they do not need to work on these areas at all.**Give students more negative feedback.**This is obviously a more radical approach and isn’t one that I have tried myself. Dunning and Kruger tentatively suggest that a possible explanation for their eponymous effect is that “people seldom receive negative feedback about their skills and abilities from others in everyday life”. I would certainly extend “everyday life” to the classroom. Most teachers I know (and I definitely include myself in this) are reluctant to tell students (and indeed parents) they aren’t very competent in particular areas, especially in such stark terms. Feedback is almost always given a positive spin and it might be that students are not even aware that they are being told that this is an area in which they need to improve. Obviously this is done with good intentions and is designed to preserve students’ self-esteem and confidence. However, perhaps this does more harm than good. Reading Dani Quinn espouse the virtues of competition and public sharing of results in the book*Battle Hymn of the Tiger Teachers*and listening to her interview on the Mr Barton Podcast made me think about this in more depth. Whilst Dani (and presumably other teachers at her school) accompany this with a carefully constructed narrative that focuses on effort rather than ability, this ranking of students means that they are certainly likely to have a better sense of their own ability and are presumably less likely to fall prey to the Dunning-Kruger Effect. Although I know this is something of an anathema for a large number of teachers, it is something worth considering further as there is at least*some*case to be made for this, even if it is not something that one ultimately agrees with.**Just focus on teaching them subject content!**Dunning and Kruger write that “[paradoxically] once [the participants in their experiments] gained the metacognitive skills to recognize their own incompetence, they were no longer incompetent”. If correct, for teachers to develop students’ self-assessment ability then they they could just focus on improving students’ mathematics. As they get better at maths, they will get better at self-assessment. This is not to downplay the importance of self-awareness as a component of expertise, rather it is to say that this awareness will come as one’s knowledge develops. Be patient with the development of this aspect of students’ expertise and don’t rush it.

**Wrap up**

I am well aware of the irony of me, a Maths Teacher with no background in psychology, writing an article on novices overestimating their ability. Throughout I have tried to couch my thoughts in the language of uncertainty given that this is not my area of expertise. However, I am confident that Dunning and Kruger are experts in this field so if this has made you do nothing more than read their papers and relate them to your own classroom practice then for me that is a job well done.

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The case for the prosecution has been made in many cases including here, here and here. Here are a few of the salient points:

- Students are likely to remember the ‘fun’ activity rather than the learning itself;
- ‘Fun’ lessons are a way of trying to ‘trick’ students into enjoying the learning, rather than encouraging students to appreciate and value learning for its own sake.
- Student engagement is a poor proxy for student learning;
- David Didau quotes John Hattie in saying that the hard work of learning “is not always pleasurable and easy; it requires over-learning at certain points, spiralling up and down the knowledge continuum, building a working relationship with others in grappling with challenging tasks… this is the power of deliberate practice and concentration.”

It is also worth addressing the straw man in the room- that rejecting this idea of ‘fun’ does not entail planning dour and boring lessons, rather, it stresses that learning, and an engagement with the subject for its own sake rather than for any extraneous gimmicks, should be at the heart of one’s classroom practice. This relates tangentially to another of my pet peeves- contrived ‘real life’ maths questions in which the context serve no purpose (not the sort that encourage students to identify relevant information as part of the question- that is a huge part of a good mathematical education….another blog post for another time).

A discussion last summer with a friend of my father’s about cycling (obviously- it’s the subject du jour for men of his age) shed a different light on the concept of ‘fun’. He made the distinction between what he calls ‘type 1 fun’ and ‘type 2 fun’. I contend that we as teachers should almost always be striving for type 2 fun whereas type 1 is what leads to many of the pitfalls listed above.

**Type 1** fun stems things that feel great when we are doing it. It is things that make us want to laugh and smile. Depending on one’s preferences this may include laughing with friends, dancing, sex or reading a good page turner. As one of the articles on the subject that I read put it, Type 1 fun is “fun to do, fun to remember”.

**Type 2** fun occurs with things that aren’t that fun at the time but bring a sense of pleasure when one looks back and reflects upon them-‘. My dad’s friend used the example of a hilly cycle. From what I understand, cycling has it’s share of ‘type 1’ fun- the sensation of speed for instance or the views that might be experienced on the course of a particularly scenic ride or the sense of solitude one might have cycling on a quiet road in rural France. The essence of the enjoyment of these activities is based in the moment itself. However, much of cycling’s enjoyment is ‘type 2’. The sense of accomplishment afterwards, reliving the challenges of what was at the time a quadricep-busting lactic-acid inducing climb that wasn’t enjoyable in any way whilst one with your companions *after.* In other words ‘not fun to do, fun to remember’.

**Type 3** fun is also interesting- ‘not fun to do, not fun to remember’ BUT makes a great story while sitting in the pub or round a campfire! Often these are life or death situations and make great films (Apollo 13, Touching the Void, Everest etc.). We should probably avoid type 3 fun in the classroom (I don’t think this is what is meant when SLT talk about ‘taking risks….’).

Like most Maths teachers, I enjoy geeking out on a particularly good problem that is pitched just right for my level of expertise. But my feeling *at the time I am doing the **problem* isn’t one of enjoyment. It is a feeling of being challenged, of curiosity and more often than not, frustration. But I persist because I know that the reward I get if I complete it and reflect on what I have done and look at different strategies will be huge! (As an aside I would be interested to know whether anyone feels that solving difficult problems is for then a ‘type 1’ rather than a ‘type 2’ activity). I suggest that we should be instilling and developing this feeling in our students.

‘Type 2’ fun in the classroom goes hand in hand with encouraging a lifelong love of learning. If enjoying and relishing the challenge of learning for its own sake is something we are seeking in our learners then perhaps dismissing ‘fun’ out of hand is too short sighted. Aim for type 2 fun, avoid type 1 fun (and definitely avoid type 3 fun).

If you are after an excellent selection of puzzles that you can use for school competitions each and every week that will help develop students’ appreciation of type 2 fun- head over to my colleague Andrew Sharpe’s excellent Puzzle of the Week Website http://www.puzzleoftheweek.com/

(Credit to the following-the ‘Three and a half types of fun’ from Teton Gravity Research and ‘3 types of fun’ from the Pebbleshoo blog).

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For those not aware, marginal gains is an approach popularised by Sir David Brailsford, a cycling coach who was in charge of the gold-medal machine that is Team GB’s cycling team and is still manager of Team Sky who won the Tour de France in 2012, 2013, 2015 and 2016. In Brailsford’s words:

“The whole principle [of marginal gains] came from the idea that if you broke down everything you could think of that goes into riding a bike, and then improved it by 1%, you will get a significant increase when you put them all together”.

By making lots of small improvements, tweaks and changes in a range of previously unconventional areas, Team Sky and Team GB were able to achieve an edge over their competitors. For instance, athletes were shown how to wash their hands correctly by a surgeon in order to minimise the chances of illness. Hotel rooms were scrubbed down before athletes arrived in order to reduce the possibility of an athlete contracting an illness. Famously, the same type of mattress and pillow that athletes would use when at home were taken with them when travelling in order to help ensure a good night’s sleep for athletes. Such changes alone may be insignificant but together they have been credited with helping to produce a golden era of British cycling. For the reasons outlined above, the principle has been championed by those outside cycling as a way of improving performance in a range of fields including teaching. However, this thinking is flawed.

Constantly striving for improvement and considering innovative and creative ways of improving performance is to be commended. Indeed, a key component of professionalism in the teaching profession is seeking to reflect on and improve what we do. However, we are not elite cyclists. A marginal gains approach is not right for us.

There are over one billion bikes in the world. Even dividing this number by 10 and assuming there are 100 million cyclists, Team Sky cyclists make up the top 0.00001% or so of cyclists in the world. Even the very best teachers in a given school are almost certainly not in a comparably elite category simply on the basis of probabilities. The cyclists Brailsford oversaw were at the top of their field, as were their competition. Their nutrition was already very very good. Their training protocols were world-class. Their technique was exceptional. They had been living the life of an athlete and dedicated thousands of hours of practice to their sport from a young age and were incredibly genetically gifted. They literally had no other way of improving their performance and gaining an edge over their rivals other than to go down these non-conventional routes. The marginal gains approach made a difference because all of the athletes from all of the teams were doing everything else right. No amount of hand washing will make up for even a slightly sub-par nutrition, recovery and training schedule. Marginal gains worked because substantial gains had already been made.

Another area in which I am keenly interested is diet and nutrition. People focus on meal timings, no carbs after seven, paleo, organic, skip breakfast, don’t skip breakfast, high fat, low fat, high carb, low carb or consider buying the latest thermogenic fat-loss supplements. These things may make a difference, but only once the basics are in place and have been adhered to for a substantial period of time. If people seek to alter their body composition they should control how many calories they eat as their priority. After that they should control how much protein they take in. For 99% of people just doing these two things, combined with a sensible exercise programme, will see them making far more progress than if they ever would by worrying about balancing the carbs in their evening meal. If everything else is in place and being successfully adhered to and has been for a long time then the timing of breakfast might make a small difference to a person’s body composition, but securing the substantial gains first has to be the priority and will be enough for 99% of the population. If calorie control is not in place, no (legal) thermogenic supplement in the world will make a jot of difference to how a person looks.

Back to the classroom….we are not elite cyclists and there are significant substantial gains we should all make as classroom teachers before we start even thinking about gains at the margins. For Maths teachers, I strongly believe we have two areas in which as individuals could all make substantial gains. Firstly, thinking about and literally rehearsing how we explain concepts to students *will* lead to dramatic improvements in learning if done consistently lesson in, lesson out for an extended period of time. Mathematical explanation is a skill and takes time and effort to practice and perfect and is sadly underrated, especially given that it lies at the heart of what teaching actually is! Secondly, actively taking the time to really consider students’ misconceptions for each and every concept in each and every lesson will have a similarly large impact (hat tip to Craig Barton for the superb training session he delivered on this in Kuala Lumpur last month).

These are my potential substantial gains. Whilst I try and do both of these, I do not do them with the consistency and frequency with which I could and I know of no colleague that does (especially the former). This is where I should focus my energy rather than trying to implement a number of small changes that will aggregate to an improvement in learning far smaller than I could achieve by spending more time carefully considering my explanations in lessons. I could spend time adopting a triple-colour marking approach that, along with a number of other approaches, *might* lead to a small improvement in learning. Instead I will focus on better content in lessons that supports weaker students whilst still stretching stronger students because there are still substantial gains I can make here and I suspect that there are similar substantial gains staring most teachers right in the face; certainly I don’t know of any colleague who makes a point of really considering in fine detail every explanation for every single concept in every single lesson.

Putting my armchair psychologist hat on for a moment, I am also inclined to think that from a behavioural perspective we are less likely to be successful if trying to change a large number of small things in our classroom practice *a la* the marginal gains approach rather than one or two larger things due to being better able to form habits when only focusing on a small number of things.

The real lesson from Team GB’s and Team Sky’s success is the same lesson that can be drawn from any successful team or person in any field. Success requires practice and perseverance over time and doing things consistently well. Whilst the marginal gains approach is a far more appealing whole school INSET than a story about someone having the discipline and determination to eat, sleep and train cycling for 20 years, it is this that we should be focusing upon as teachers. Once you have achieved truly elite status, then start worrying about the marginal gains but until then, consistency is king. Don’t look substantial gains in the mouth by worrying about small things at the margins. Get brilliant at the basics and keep getting better at them over time. We are not elite cyclists- we are (hopefully) decent teachers trying to get better.

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Sound familiar?

My knowledge of primary education is far too limited to even begin to comment on any of the above issues. However, what is indisputable is that a fairly large number of students, for whatever reason, arrive at secondary school ill-prepared for the demands of the mathematics curriculum.

Rather than trying to unpick why this is the case, I want to outline some of the considerations I find to be particularly important for these ‘low ability’ Mathematics students. Note that this term is imperfect and has far more negative connotations than I would like. To me it implies an inability to improve, something that is the polar opposite of what we as teachers work towards. However, I will use this term a. for lack of a better one and b. because I assume that most at least understand what the term means irrespective of whether or not they agree with my view that it has negative connotations.

**Don’t teach them what you ‘should’ be teaching them, teach what they need to know to know now**

If students are lacking basic skills and if they do not have certain facts committed to memory, they will not be able to make progress. Fact. Much of the philosophy behind the mastery approach to teaching is predicated upon this idea. Nail the basics so that they do not take up space in students’ working memory, thus allowing them to apply these concepts to more complex problems.

In Mathematics, I would go so far as to say that if the four operations and an understanding of place value and the decimal system is not 100% secure then that is what you as a teacher should be almost exclusively focusing. Trying to teach students to multiply two decimals together, to calculate area and perimeter or to carry out any sort of task involving algebra (as well as most other tasks on a typical Year 7 scheme of work) is likely to be to no avail without these in place first. The added cognitive strain of carrying out these operations whilst applying them to a new context will make committing any new processes to memory nigh on impossible.

And by basic I mean basic. Number bonds to 10, 50, 100. Place value. Chanted multiplication learnt by rote, basic inverse operations and not much else (For more on this, see Bruno Reddy’s account of how King Solomon Academy in London designed a ‘Mastery Curriculum’).

Many Year 7 schemes of work do not start with these ‘basics’. At the previous two schools I have worked in, ‘factors, multiples and primes’ has been the first topic covered in Year 7. Finding the factors of 48 without having multiplication tables committed to memory is an almost impossible task.

If you identify with this and your students are not **100%** confident with the ‘building blocks’ of Mathematics, implore the powers that be to let you focus on that at the expense of all else. They ** will** catch up later.

Complaining ‘they should know this by now’ is pointless. Make it your job to teach them this. Right now. And don’t rush. embedding these concepts takes time and once embedded must be periodically recapped to help commit them to long-term memory and to allow them to be effortlessly deployed as and when they are needed. This is a process that takes months and even years, not weeks.

**Get them enthused**

Even amongst the most disaffected students, it is rare to find a student who does not exhibit at least *some* excitement in starting secondary school. Harness this. For the love of God grab onto this and don’t let go. Keeping this level of excitement and enthusiasm for the entire year is a tough and seemingly impossible task but if you are constantly stoking that enthusiasm and creating a positive mindset, students will *hopefully* overcome many of the negative perceptions that they associate with Mathematics. New year, new you etc!

How to do this? Well that is the million dollar question. I only have one real guideline here- *get students excited about getting better*. Students like being good at things. They like this more than almost anything else. If you can *show* pupils how they are progressing and getting better (track quiz scores, table tests etc.) and if you can give them real, genuine, praise about their progress, they are more likely to ‘buy into’ your maths lessons. A generally positive learning environment with great relationships underpins this obviously, but you must get students excited about doing and progressing in maths, not about making posters, using technology or anything else.

**Planning….or more accurately thinking and rehearsing**

Obviously all lessons should be carefully and thoughtfully planned but when teaching lower ability students, meticulous planning is more important than ever. By planning I don’t mean full ‘Ofsted-style’ lesson plans. Rather, I mean taking the time to think through possible misconceptions, requisite prior knowledge and to carefully rehearse explanations, focusing in particular on the language that you are using and the expectations for how the work should look on the page.

Given that the concepts being taught to these students are at the more ‘basic’ end of the spectrum, there is a tendency to almost ignore the explanation and to try to explain it off the cuff in class whereas when teaching (say) a difficult A Level topic, many teachers spend more time considering this aspect of their practice. Given that these concepts are going to underpin the rest of the students’ Mathematical understanding and the fact that these students have almost certainly been taught these topics before but haven’t managed to retain the information, these explanations are *the* most important you will give as a maths teacher. Take time to get them right.

One area that I am very keen to develop in my own practice is the use of concrete resources to aid explanations. This is something I have never felt confident using myself having received a fairly half-baked 45 minute training session on concrete resources during my ITT year and very little else since other than being told in my NQT year to “just use some Deans Blocks with them” (I imagine this is not uncommon among secondary maths teachers). From what I gather however, if these resources are used correctly, the impact on student understanding can be significant.

**Feedback**

There is no shortage of research on feedback and its importance. There is also no shortage of information telling you exactly how to give feedback. It must, of course be written (for parents, senior leadership, Ofsted etc), must give the opportunity for the student to give a written response and should involve peers…..

Whilst this might be applicable in some cases, my personal experience suggests that feedback for lower ability students should, by and large be immediate and is most effective when given verbally. Students who struggle in maths tend not to be able to effectively link the written feedback on a page with answers and solutions that they wrote down a day or two previously. Immediate verbal feedback given while circulating the classroom and mini whiteboard work combined with regular low-stakes quizzes to assess learning are far more effective than written comments in books for these students in particular.

Additionally, I find that, whilst students can be ‘trained up’ to be fairly effective peer markers, the opportunity cost of doing so (using time that could be spent *doing* maths) more than negates the relatively small benefits of being able to review another piece of work and almost all of the benefits derived from peer marking can be given through teacher produced model answers.

**Scaffold the transition**

One thing I am often struck by when entering a primary classroom (something I do not do enough of, especially given I work in a very large ‘all through’ school) is how different it is to secondary classrooms and in particular, my ‘unfashionable’ classroom layout with students seated in rows facing the board.

Whilst I am convinced that seating students like this has a positive impact on student learning, I am also of the view that for Year 7 students in particular, this sort of classroom environment can be something of a shock to the system. Similarly, having students work on questions from the board rather than on worksheets or listen in their seats rather than in carpet spaces can be very difficult for students who haven’t experienced it before. While the end goal for me is to get students to a position where they are able to work effectively in a traditional secondary setting, secondary teachers must acknowledge these challenges and try to wean students away from the primary mentality but must do so gradually.

If you have a space to do some work ‘on the carpet do it. Use worksheets. Sit down with students and work through problems together. Occasionally group tables together and move students around. Interleave ‘primary’ methods with ‘secondary’ methods and over time begin to move away from the former and towards the latter. Even if you are able to keep up this ‘primary-style’ approach, for better or for worse it is likely that your colleagues will not and students need to be prepared for the realities of secondary education.

**Wrap Up**

This obviously isn’t a ‘how to’ guide. It is just a set of reflections from my experience teaching so-called ‘low ability’ students in mathematics. I’d love to hear the experiences of others and what works and doesn’t work for you.

Teaching these classes can be challenging but, at the risk of sounding sanctamonious, can be *the* most rewarding classes to teach. The planning and delivery of these lessons requires patience and takes time to get right but, when the students start making steady and sustained progress, the sense of personal satisfaction is right up there with some of the best feelings in teaching.

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Eventually we started using the term ‘card sort’ as a term of derision used to describe a particular type of teacher who insists on using ‘card sorts’ (and any other activity where the input didn’t seem to justify the output). “Oh that Mr Jones, he’s a bit of a card sort him” or “Ms Khan has got six different coloured worksheets out again, what a card sort”. Furthermore, as a teacher, nothing irks me more and immediately switches me off than having to undertake a ‘card sort’ in an inset session as part of an attempt to disseminate and drip-feed staff ‘good’ classroom practice and I always imagined many students think similar things when faced with one in the classroom.

And that was that. No card sorts for me ever. Or so I thought….

I was recently given a ‘card sort’ activity by a colleague on classifying data and I threw caution to the wind and gave it a go thinking of it as a substitute for a large number of textbook style questions which I didn’t have to hand on this particular topic.

The activity was, to misquote Obi-Wan Kenobi, (it was Star Wars Day this week after all) “more powerful than I could have possibly imagined” and far more effective than textbook style questions could have been. The activity allowed students to appreciate that not all categories of data are mutually exclusive and that data can for instance be both primary *and* continuous.

Perhaps this is an obvious point and but too often I have seen card sorts used, and indeed been ‘subject’ to, card sort activities which just get students to consider and debate whether something belongs to one of a number of categories or just as a substitute for getting students to write things down in oder. A quick search on TES yielded the following card sorts:

- A Geography card sort-‘Reasons for population control vs. reasons against population control’;
- A card sort in which students had to order chronologically the events and personalities leading to the discovery of the structure of the atom;
- A card sort of the chronology of the events leading to the outbreak of the Second World War;
- A card sort on the positives and negatives of nuclear power.

In such cases I just think teachers can use their and their students’ time far more efficiently than preparing and doing such an activity. If you want students to learn and put things in chronological order, just get them to write the damn things in chronological order. There is only one answer here- moving cards around on a page won’t change that. You want students to debate the positives and negatives of nuclear power? Great- get them to draw out a table (a skill in itself), or better yet, get them to write about the issue. If you want to encourage students to consider all sides of the argument or decide between the importance of various factors and are worried writing things down will lead them to keep one opinion rather than consider changing it, then perhaps a card sort does have more value. Although I still think there ways of doing this that won’t require the preparation, cutting and sorting that could instead be used planning, teaching, learning and thinking (mini whiteboards that can quickly be erased spring to mind here).

However, in cases such as the one above, where students have to consider non-mutually exclusive overlapping categories, I found that it genuinely contributed to learning in a way that I don’t think could have been done by other means.

Card sorts shouldn’t be used to order things chronologically. They shouldn’t be used to sort things into simple, exclusive categories ‘chemical change vs. physical change’ for instance. They probably shouldn’t be used to encourage students to debate issues where there is no right answer, although there is perhaps some merit to this. Where card sorts do have a use is in encouraging students to appreciate that things can fit into multiple categories at once (sorting shapes is another area that could lend itself to this sort of activity). I wonder how many of the card sorts used at INSET sessions fulfil this criterion!

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I teach in a traditional and consistent style. Most of my lessons look the same- a starter or quiz lasting 10-15 minutes recapping a range of previous topics that have been covered so far in the year, an introduction or reintroduction as appropriate of the concepts being learnt and practised in the lesson, some modelling, independent practise (often in silence) with me circulating and discussing answers with students and identifying any common misconceptions, more modelling if required, and possibly a final question to check what students can do at the end of the lesson (note I write what they can *do*, not what they have *learnt*). The lessons feature lots of mini whiteboard work so that students can’t ‘opt-out of thinking’ and lots of questioning. I don’t use many of the ‘gimmicks’ or ‘hooks’ that some teachers choose to and I believe in the value of didatic teaching methods. I am unashamedly strict with my classes whilst simultaneously being warm with them. I rely on my personality and ‘humour’ (very much of the cringeworthy/dad variety) and try to inject a lot of energy and life into most of my lessons.

I am by no means perfect at any of this and, some days I am certainly not as good as I want to be. However, this is what I strive towards each lesson of each day.

I am not claiming this is a better or worse style of teaching than other methods (in terms of content learnt over time). My aim is to argue that this fairly traditional style of teaching isn’t ‘boring’ or ‘unenjoyable’ for students (though, even if it was I would still be inclined to teach this way as I do consider it to be the optimal method).

Some thoughts about why this is the case:

**Progress**– students enjoy making progress. They enjoy being able to do things they couldn’t previously do and gaining confidence as they begin to master their subject.*Genuine*and precise praise is clearly important for this, as is reminding them of how far they have come.**Relationships-**these are one of*the*keys to to any teacher’s success. Building up strong relationships, getting ‘buy in’ and gaining the trust from students, whatever the teaching methods used will lead to students enjoying lessons, irrespective of the style of teaching used, whether ‘traditional’, ‘progressive’ or anything in between.**Enthusiasm**– same as above- if a teacher is suitably enthusiastic, the style of teaching and the structure of lessons fades into insignificance. Students enjoy having enthusiastic teachers, however they teach.**Structure**-having taught in both a ‘challenging’ inner city school in the UK and an extremely high performing international school, I believe that despite protestations that some students might make, almost all students prefer structure and clear boundaries (in my experience this is especially the case with boys but this is both a generalisation and purely anecdotal). A ‘traditional’ teaching style goes hand in hand with such an approach and provides students with an environment in which they can thrive.

Again, I am not arguing that jazzy lessons filled with group work, technology and discovery are worse than the approach to teaching I personally favour. Indeed, some traditional lessons certainly are boring- I know that sometimes mine are (to my discredit). Rather, I am claiming that ‘traditional’ style teaching can be just as *enjoyable* for students as any other method of teaching and ultimately, might even be more likely to lead to the most desirable form of enjoyment- the intrinsic enjoyment that comes with the challenge and mastery of a subject.

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A perennial problem for Maths teachers is developing students’ ability to ‘see’ their route into the question when it is not immediately apparent which mathematical techniques they need to apply. This week I began teaching the first lesson of the Edexcel C2 Trigonometric Identities and Equations chapter to my Year 12 Class. For those unfamiliar with this chapter, it requires knowledge and application of the following identities:

Normally when selecting the examples I am going to model with the class, I take a bit of time to carefully select the question I am going to use to ensure that the example contains an appropriate level of challenge whilst still keeping the concept that I am trying to teach at its heart. However, this week when modelling for the class I made a point of (somewhat theatrically) picking a question that I had not reviewed beforehand and talking students through my thought process as I looked at the question. By chance the question I had chosen was a great one from Integral Maths:

Obviously I knew that the question would involve the use of one or both of the above identities, however, the students also have this knowledge. What students don’t have is:

- A full mastery of the two identities including the ways in which the identities can be rearranged;
- Exposure to a sufficient number of questions that allows them to spot the ways into the question that examiners tend to use

In order to help expedite the process of spotting some of the commonly used methods required to solve these problems, I channelled my inner James Joyce and gave students a stream of conciousness style account of how I would look at and complete the question.

I started off by pointing out that the first thing I noticed was that the LHS was a difference of two squares, although not in the form students would be used to seeing given that this was their first foray into trigonometric equations. I then said that I looked at the RHS and spotted that there was no cosine squared on the right hand side. I then said that I know that the sine squared theta + cosine squared theta identity can be rearranged to give cosine squared theta in terms of sine squared theta and so a substitution would likely be the way forward with this question once the brackets had been expanded. I then continued to talk students through exactly what I was thinking as I solved the question. In the second example, students had to do a bit more of the ‘leg work’, spotting some of the interesting features of the question and doing more of the talking than in the first question.

Students really appreciated being explicitly told about how I went about looking at this question, especially given that I was doing it ‘live’ and so was, essentially in the same boat as them. Allowing students an insight into how an ‘expert’ thinks is one of the most valuable things any teacher can do and is one reason why didactic teaching is not the evil many seem to have argued. Discovery-based, project-based or inquiry-led learning just isn’t an efficient or effective way to develop students into competent Mathematicians. They need to ‘see’ how an ‘expert’ might see a problem, at least when developing their understanding of a topic.

When modelling all teachers do some work talking through how they ‘see’ a question, however, I contend that most of the time we can do more to ‘make the implicit, explicit’ (to borrow a phrase from David Didau) and really talk students through exactly how we, as Maths teachers, view and solve a the problems presented to us.

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Recently I have moved away from Zs, Fs and Cs completely. I have banned any mention of Fs, Zs and Cs from my classroom and focused more on the correct Mathematical words, their meaning and a slightly more conceptual understanding of why the angles are equal/supplementary. I have stressed that ‘corresponding’ means ‘in the same position’, ‘co-interior’ means ‘inside together’ and alternate means ‘one way and then the other’. I have also tried to encourage students to get a feel for the intuition as to *why* these angle facts exist- noticing for instance that because corresponding angles are in the same position on two parallel lines they *have* to be equal- they just couldn’t be anything else. Or that co-interior angles are supplementary because of the fact they combine the fact they know about corresponding angles with the fact that angles on a straight line sum to 180 degrees. Similarly for alternate angles- a combination of corresponding angles and opposite angles.

This is a long term approach. It isn’t quick and easy. It takes time, some well-thought out and clear explanations and a focus on long term improvement rather than rapid progress. However, I find that ultimately it improves student understanding and ability to identify and use the angle rules as well as providing students with an added layer of mathematical rigour.

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On paper this was identical to a Kagan ‘Rally Coach’ activity that I have seen elsewhere numerous times before. Students work in pairs and talk through a particular question while their partner quizzes them on what they are doing. This always felt like something of a gimmick in the past- a case of getting students to talk for its own sake or perhaps because it checks a box for an observation. At best I have heard some generic comment about how somehow, *as if by magic*, student learning is enhanced by working cooperatively with no explanation of the mechanism behind this .

It was refreshing to hear some research and rationale justifying this strategy (that it improves metacognition) and some advice as to how best to implement it. Modelling a **constant stream** of discussion worked well, but what was most useful was asking students to model the sorts of ‘coaching questions’ they could ask as I made deliberate mistakes on the board. Initial comments such as ‘you’ve got it wrong there sir’ or ‘you need to square root at the end’ were corrected and improvements such as ‘have you considered what is happening between lines three and four?’ or ‘what have you found here, does it make sense?’, ‘could a different diagram help you here’ quickly came to the fore.

Students then worked through a fairly difficulty Pythagoras’ Theorem problem set that I had put together for them. I circulated listening for good coaching questions and periodically asking students to share with the class good questions they had been asked.

Questions courtesy of http://corbettmaths.com/

Problems with the questions included students ‘getting lazy’ and neglecting to talk their way through the problem (vigilance is key here) and students not speaking when they become unsure of how to proceed (convince them that just saying exactly what they are thinking is a good strategy and often leads to a ‘lightbulb moment’ of how to proceed.)

Did the students find completing the problem set any easier using these strategies? Probably not. Were the students stretched in other ways and becoming more aware of the ‘voice in their head’- probably. Were the students being forced to think like a mathematician to frame questions that supported students but didn’t involve just telling them the answers? Definitely. Indeed it was refreshing, if a little spooky, to notice that some of the questions students asked each other sounded remarkably like something I would ask.

The key with any metacognitive improvement strategy is progressing slowly over time. Rome wasn’t built in a day and effective metacognition is not developed in one lesson (or even one year- it’s a lifetime skill). Crucially a proportion of the training session focused on a neglected aspect of metacognitive strategies- that of developing automation. In Maths terms, this should include students reviewing their own solutions, drawing diagrams and ‘sense checking’ answers **without prompting **(though I think I could soon end up out of a job were this to become the norm!).

To recap:

- Model, model and model again. This was the most important part of the process.
- Encourage students to share good examples periodically.
- Be aware of pitfalls, particularly student reluctance to speak when struggling- nip these in the bud by modelling what to do in these situations.
- Don’t expect improvements to happen straight away, progress in any aspect of learning is not predictable or linear (or even monotonic) and indeed
**should not be rapid**despite what many observers and school leaders might have you believe. Metacognition is no different

In some cases students may find completing the problems *more* difficult using these strategies. Good. Difficulty in learning is beneficial for students and is an indicator that they are being stretched.

One further point- a good proportion of the work I do at my school is with students to students applying to Oxbridge. Oxbridge interviews require students to talk *a lot*. Indeed, any interview for any mathematics or science interview will almost certainly involve students having to talk their way through a number of problems. Many impressive mathematics students lack the ability to do this. Like anything, practice is crucial and activities aimed at developing metacognition like this one are an excellent way of developing students for these intellectual challenges.

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A classic ‘extension’ activity that Maths teachers often use is to ask students to create a question on a topic when they have finished their work. It’s an easy win for teachers; they keep students busy whilst supposedly ‘stretching and challenging’ them by encouraging them to work on the so-called higher order skills required to engage in the creative process.

Creating questions is usually a more difficult skill to master than answering them, particularly when you want a ‘nice’ answer to emerge. Think for instance about the knowledge and understanding required for creating a trigonometry question giving an integer answer compared with merely answering such a question.

However, I prefer to ask certain questions and give particular prompts in order to refine this process and move it away from a ‘keep them busy’ or box-checking activity and move it towards a learning activity. For instance: “what would an easy question on this topic look like?”, “why is question a harder/easier than question b?”, “what would you expect to see in a more difficult question?”. Students can then use these prompts to create easier, medium and harder questions. They are forced to engage with the material and considering the different difficulty involved in each question really develops their metacognitive skills.

Here are some examples of the work that my year 10 class carried out on rearranging formulae:

I was especially pleased with the ‘hard’ example on the far right hand side- putting the intended subject as the denominator was a subtle but important difficulty this student grasped.

The question “can you create an easy, a medium and a hard question on this topic?” is a useful and powerful way of refining the process of students creating questions.

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