The Ethics of “Teaching to Exams”

A recent article in the wonderful Bagehot column of The Economist discussed the so called ‘open v. closed’ divide that some say has come to replace traditional left-right cleavages in politics that has apparently enhanced political polarisation within society. The piece was, as is almost invariably the case with Bagehot, interesting and touched on ideas related to identity politics that are quite intellectually a la mode at the moment given the work of controversial popular intellectuals such as Sam Harris and Jordan Peterson whose work is currently causing quite a stir.

However, what really struck me were the concluding paragraphs which addressed what the columnist referred to as ‘the real divide’ and a key explanation for political polarisation- that of “the gap between exam passers and exam flunkers”.  I think it is worth pasting the entire final two paragraphs in their entirety:

“There is a better explanation of political polarisation than the open-closed split. It is the gap between exam-passers and exam-flunkers. Qualifications grant access to a world that is protected from the downside of globalisation. You can get a job with a superstar company that has constructed moats and drawbridges to protect itself, or with a middle-class guild that provides job security, or with the state bureaucracy. Failing exams casts you down into an unpredictable world of cut-throat competition.

Exam-passers combine a common ability to manage the downside of globalisation with a common outlook—call it narcissistic cosmopolitanism—that binds them together and legitimises their disdain for rival tribes. Exam-flunkers, meanwhile, are united by anger at the elitists who claim to be open as long as their jobs are protected. They are increasingly willing to bring the system crashing down. Talking about open v closed is a double error. It obscures the deeper forces dividing the world, and spares winners by playing down the legitimate concerns of losers.”

Whilst you may disagree with the extent to which not being an ‘exam passer’ can affect people, I think most would agree that there is more than a kernel of truth in this.

At the time of year when exams are very much at the forefront of teachers’ minds, teachers can get bogged and disillusioned at the prospect of ‘teaching to the test’ and of hammering home ‘exam technique’ relentlessly. This is completely understandable. Most teachers love the academic aspect of their subject at least as much as the pedagogical one. Instructing students exactly how to squeeze out a final few marks on a paper at the expense of delving deeply into a topic and seeking out the richness and beauty within a subject is rarely a teacher’s favourite part of the job. I have also heard it argued that this boarders on the unethical and that a culture based around exam passing rather than enriching students’ understanding and their appreciation of the subject deprives them of what education should be doing.

Perhaps this is what education should be doing. But we have to base decisions around teaching and learning within the education system (and indeed, the economic system) that currently exists. The fact is that failing to succeed in exams limits students’ opportunities in life and does not provide them with a ‘shield’ against the more pernicious aspects of modern economic realities (please don’t trot out the straw men of Richard Branson, Alan Sugar et al who failed exams and yet made millions- they are notable because they are the extreme exceptions, rather than the rule). Surely the real ethical failing would be to ensure students are not ready for this and are not equipped with a pocket full of great examinations grades.

Much of the time, it is not a case of either or and we must be careful to avoid creating a false dichotomy. Enriching students beyond the scope of what is specifically defined by the exam specification will often improve their exam results. However, there will come a point where a decision must be made to focus squarely on the exam and perhaps we should be careful of disparaging teachers and schools who decide to do this earlier rather than later (I have heard this happen myself and seen it written on social media).

I say all this as someone who inclined towards academia and instilling within students a love of learning a subject for its own sake, without extrinsic reward. However, like all teachers, I want to give students the very best start in life. John Tomsett has written in a number of places that “I am convinced that the best pastoral care for students from socio-economically deprived backgrounds is a good set of examination results”.

The quote that most resonated from the above article was “Failing exams casts you down into an unpredictable world of cut-throat competition”. (The fact that The Economist even hinted at criticising competition emphasises just how cut throat this world must be!) At some point in the teaching year or in the process of curriculum design, a decision must be made about the trade off between the exam and the richness of the subject as a whole. They aren’t the same thing. Let’s not be too hasty to cast students into this ‘cut throat’ world in order to preserve our selfish view of how a subject ‘should’ be taught.


Oxbridge tips for teachers you can implement right now

Oxbridge application season is upon us. Over the weekend there was some controversy over recent BME statistics released by Oxbridge. I’m not going to comment on that here other than to say irrespective of the role that universities should or should not play in redressing the imbalances these statistics suggest, some schools are significantly better than others at preparing their students for Oxbridge. This is due to a huge range of factors which obviously includes school intake but also encompasses long term decisions about curriculum, stretch and challenge for students and high expectations about the universities students are capable of being accepted to. I imagine that in an average classroom at (say for the sake of argument) Westminster School, it is assumed from the lowest years upwards that a great many students will be applying to Oxford or Cambridge and that this impacts significantly on learning within the school (I’m obviously not saying it is as simple as ‘high expectations’ but it is undoubtedly a factor).

All schools have as much a duty to support high-achieving students applying to these universities as they do any student in their school and the best way of doing this is through long-term support. However, what I have tried to do here is to suggest some short term ‘quick fixes’ that may help teachers preparing students for Oxbridge over the coming weeks. These are not gospel by any means, nor are they silver bullets that can replace years of hard work from students and teachers alike. I’m sure that many others have different approaches to this and it would be great to hear about them. I would love for as much discussion, debate and transparency was geared towards Oxbridge preparation as in many other areas of education. Whether or not universities have a role in addressing inequalities in education, schools undoubtedly do. This is my contribution to that debate.

Quick note on my background I studied Philosophy, Politics and Economics at Oxford 2009-2012 and studied for a Masters in Education that was primarily sociology-based. I work closely with Oxbridge applicants at my current school and help oversee the whole application process along with the Higher Education advisor and a range of willing subject mentors. I now teach maths so am able to support (to an extent) both arts and science applicants (although, as I write below, the core of any support should be from subject specialists, or as close as subject specialists, as possible).

I firmly believe in the power of shared terminology so I’ve shared a lot of the names I use for these ‘techniques’ and ideas here. It makes things easier when asking a student “have you prepared your ‘hit them for six’ answers” or telling them “for the first question you did really well to answer your question like an essay” or “why didn’t you use your five sentence summary here”.

1. Encourage students to pause, think and plan before answering.

Students are often reluctant to do this but taking five or perhaps even ten seconds to plan an answer (time that can seem like an eternity to students) is time well spent. This time should be spent gathering one’s thoughts, checking if clarification is needed, identifying any immediate problematic areas that might arise and deciding how the question will be tackled and help give the impression that the student is a thinker rather than an individual with a tendency to rush in and answer without considering the full implications of what they are saying. Of course this can be taken too far- pausing for too long can could also give a negative impression as being prepared to roll up the sleeves and get stuck in is an important part of the interview process. However, in my experience most students tend towards not taking enough time to think rather than taking too much.  (It is also worth pointing out that this does not apply to each and every question- a five second pause isn’t necessary if the student is asked how their journey was or to reconfirmed their name- always gets a laugh when I tell students that…..)

2. Verbalise the thought process- especially when things get difficult.

This is probably my number one tip and is a difficult skill to master. However, for both arts and science studnets it can go a long way both in terms of helping students plan effective answers and cementing the impression that they are a logical thinker. Students find this easy to do when they know what they are doing. Ask a candidate to sketch the graph of y=1/x and doubtless they will talk through the shape of the graph, the location of the asymptotes and why they are at y=0 and x=0 and perhaps talk about whether the function is odd or even. However, when students are asked to sketch the graph of y=e^sin(x) students often go silent as they try to work out what is going on. While some pausing and thinking is good (point 1) interviewers are keen to see what a student is thinking and the only way for a student to do this is talk. For science students carrying out a graph sketching question I encourage my students  to develop a bit of an algorithm they can talk through that will give them a good starting point in most cases:

“First I’m going to think about whether there are any values that the function can’t take [student then talks through this process writing down their findings as well as verbally communicating] now I will need to identify what happens as x approaches infinity and negative infinity….how will the function behave around the maximum and minimum points etc.”

Again this is very easy to do when the question is easy but is difficult to do when the material is more unfamiliar. When I ask students about why they aren’t talking the response is generally “I don’t know what to do” or “this is difficult and I don’t want to say something wrong”. However the process of talking through their thoughts often leads to a “lightbulb moment” and even if it doesn’t it shows the interviewer that they at least have the intellectual confidence to attempt to tackle a difficult problem and makes it easier for them to provide a teaching point about what they need to do which will hopefully set them off in the right direction.

In arts subjects, I often encourage students to ‘answer like a mini essay’: tell them what you are going to say (aim for 3 or maximum 4 points-along with the thinking time this helps form a coherent answer) and define key vocabulary, say it and then tell them what you have said, ideally using the wording of the question asked.

An answer of this form might look like this:

“So in considering how a utilitarian might consider the ethical issue of driving without wearing a seatbelt we could consider three things: firstly, what harm could come to the individual not wearing the seatbelt, secondly what pleasure can be derived from wearing the seatbelt, and thirdly what harm (and if appropriate pleasure) may come to others as a result of this choice. So in terms of the to the individual making the choice [talks through the issues] secondly ……and thirdly…… So on balance the utilitarian could consider these issues when deciding whether or not to wear a seatbelt whilst driving and may reach the following conclusion……”.

This skill is difficult. It involves planning and speaking almost simultaneously. However, students can and will get better at it with practice and the payoff is high. It both helps students plan effective answers that don’t ramble and, given how difficult the skill is, shows that the candidate has the capacity for ordered, logical thought (for an indication of how difficult but also how useful this skill can be, try it when driving home a point at the next department meeting). This isn’t a call for students to get on their soapbox and speak uninterrupted for five minutes however and they can and should expect to be challenged at any point. However, by having an initial structure for an answer formed, students are able to answer more lucidly and clearly than they otherwise would. Again, this may not be applicable to all questions and overdoing it may sound contrived. However, surgically and subtly deploying this technique throughout an interview can be powerful indeed.

3. Scientists and Economists must practice sketching unfamiliar graphs.

Time and time again these come up in interviews. Your mathematically minded colleagues, or failing that Desmos, are your friends here. Students should  develop an approach to sketching these that is systematic and efficient. Good follow-up questions may include asking students to differentiate the function and using the result to confirm that the turning points are correct, integrating the function or transforming it in some way. Students must be proficient at this skill and fortunately it is something that can be easily practiced.

4. Develop the skill of one and five sentence summaries as ways of responding to textual or graphical stimuli.

Presenting students with an article or graph is a common interview technique but often students get bogged down in the detail of the piece without taking a ‘helicopter view’ first. I encourage, students to attempt to summarise an article and to an extent a graph or table, in one and (roughly) five sentences when first reading it and using that summary as part of their answer.

For instance, a few years ago in an article  discussing how legalising cannabis in a state in America may have national implications , when I asked a student to talk me through the article (an easy opening question I thought) the first thing they said was “Representative Jones from the state legislature said that Cannabis should be legalised because xyz and Representative Evans disagreed on the constitutional lists of this because….”. They got so bogged down in the detail of the argument that they missed the real point of the article. This is something that is not uncommon.

By explicitly encouraging students to form a once sentence summary and or a five sentence summary they are more likely to get a handle on the big picture of the argument which they are then able to use in their answer. This isn’t to downplay the role of close reading and detail (as we know, this is often where the devil is), but students should gain a sense of the big picture and general themes before “drilling down” into the details, as well as of course evaluating the arguments and considering possible implications and drawing parallels when necessary. By forming a one or a five sentence summary however, the process will be considerably easier and allow candidates to see the wood from the trees before they venture in!

5. Insist on high quality verbal communication and make students aware of how they look and sound whilst being interviewed.

Candidates should be assessed on their intellectual ability/potential rather than how they speak. However, for a number of reasons how candidates sound, as well as how they conduct themselves during interviews, will inevitably form part of the decision-making process. Hopefully a whole school approach towards high quality verbal communication has been taken already but you can only work with what you have now. It isn’t about eliminating accents or suppressing individuality but it is about appreciating that there is a standard way of using the English language and that students use it in that way. There are two primary reasons I insist on this, firstly interviewers are well-meaning but they are human. If every fourth word is ‘like’ or if students aren’t using the correct syntax in fairly basic cases this does create a poor impression irrespective of how good their subject knowledge is. The second point relates more to essay based subjects. As my colleagues in English often say “if you can’t say it then you can’t write it”. There is a correlation between quality of speech and quality of writing and given that the interviewer may be reading (or enduring!) an essay a week from the candidate, they want the essays to be easy to read and mark.

Physical tics are less important and whilst they shouldn’t play a role in determining a candidate’s success or not interviewers are not robots and on a subconscious level seemingly small things could make a difference. A few quick fixes encourage students to tie long hair back if they have a tendency to fiddle with it or if it falls across their face, if they are standing up working on a whiteboard and they have a tendency to shuffle slightly or rock from side to side, get them to stand with their feet 15cm wider than they otherwise would (try it yourself it works wonders), if they have a tendency to fiddle with their hands, encourage them to grasp a thumb in the palm of their opposite hand and if they fiddle with their sleeves/cuffs, get them to roll their sleeves up. Whilst these shouldn’t make a difference to a candidate’s success, it would be wrong not to address these issues to help give students the best shot possible.

At this stage in the process it is difficult to make dramatic positive changes, however I often say to students that improving by 15 or 20 percent will make a noticeable positive difference. Encourage students to communicate formally in all lessons (students should ‘practice how they play!’) get in touch with their subject teachers and encourage them to correct students where possible. Film students to help them identify any verbal or physical tics that can be ironed out before the interview. This is often uncomfortable for students but can do wonders in identifying small changes they can make to help them be better presented.

6. Use subject specialist colleagues whether they have experience of Oxbridge or not.

Oxbridge interviews are academic and subject-specific. Whilst developing written and verbal communication can be done by a non-specialist, any benefits that might be gained from developing students’ subject knowledge. In an ideal world subject specific enrichment will have been ongoing for some time. Whether it has or not however, there is still time to ensure students get some exposure to subject-specific academic material that will benefit them in their application. Ideally this would probably take the form of a number (yes a number the schools with impressive Oxbridge success rates don’t rely on one or two mock interviews in late November) of subject specific interviews. I am incredibly grateful for the willingness and calibre of the staff at my current school in this regard. If your school does not have teachers with the confidence or experience of doing this then arranging a half hour conversation about an area of the subject about which the teacher is interested can still be very beneficial for the student, especially if preceded by some reading. A conversation about a dissertation they wrote, something the teacher themselves found challenging or regarding at university or a subject-related book the teacher is currently reading can expose the student to new ideas as well as get them used to formal academic discussion about their subject.

Of course if you have teachers that are both willing and able to give subject specific mock interviews then this is gold dust and is something that can really underpin a successful application (most scientists will have a maths component to their interview so your maths colleagues are an incredible resource here). The main take home from this is that subject specialists are a vital part of supporting a student in a successful application. Mock interviews and mini tutorials are brilliant but even just allowing students the chance to talk in an academic context about their subject in a way that goes beyond the A Level syllabus can be very beneficial. Furthermore, often staff involved find the process challenging and rewarding and it is great professional development for them.

7. Don’t start a sentence with ‘because’.

This has been one I’ve really pushed with students this year (the fact that they are beginning to groan whenever I make the point means I’m doing something right). Although to the best of my knowledge it is not grammatically incorrect, beginning an answer to a question with ‘because’ often leads to closed, short answers that lack depth. It also means that students are less likely to go through the process of verbalising their thought process discussed in point 2 and in my experience means that they are less likely or able to view all sides of the  argument as they are honing straight in on one particular point. Again this won’t just happen and students should be encouraged to do this across all lessons.

8. Ignore, or at least handle with care, lists of interview questions released by Oxbridge.

In an effort to increase transparency, Oxford and Cambridge periodically release lists of interview questions. The latest batch are here. In the best case scenario, these are accompanied by a short commentary on the question and ways that a candidate might tackle the answer and things that interviewers may be looking for. Often, when reported second-hand, this commentary is ignored. Context is king and in both these cases, to lesser or greater extents, context is lacking. In the link above, the question presented for a Modern Languages applicant is “what makes novel or play ‘political'”. Despite the short paragraph of commentary provided, this is still an intimidating question and the lists do not give due acknowledgement to the fact that these questions will almost certainly be part of a discussion that has come before. Perhaps the interviewer led into a question referencing something an applicant made in their personal statement or had given the student a piece of text about political works of fiction to analyse beforehand. What is unlikely to have happened is for this question to be asked straight ‘off the bat’. Schools with staff with significant experience of Oxbridge are probably aware of this and are able to use these questions to their advantage in their mock interviews as make students aware of the possible drawbacks of viewing these questions out of context. For schools with less experience in this area, the questions can be unfairly intimidating or off-putting to students in a way which they would not be if the proper context is given.

There is a better way. A quick search on YouTube reveals a significant number of mock interview videos (of varying quality). These are more useful to get a sense of the style and format of interviews and get a sense of how an interview develops over the fairly short space of time in which it takes place.

9. Ensure students have their five or six ‘hit them for six’ answers ready, although they probably won’t use them.

Trying to guess Oxbridge interview questions in advance is folly. Upon being offered and interview students often scour college websites looking for their interviewers research interests and areas of teaching but ultimately this is likely to be a waste of time. The level at which the interviews are pitched and the academic calibre of the interviewer means that the range of topics covered is often incredibly broad and not necessarily linked directly to their research interests.

However, I do emphasise for students that there are certain questions they should prepare for. Many of these are unlikely to be asked other than as a bit of a ‘settler’ as they aren’t particularly academic in nature (though it might lead down a particular academic avenue) but if they are asked these questions, they should be prepared to ‘knock the ball out of the park’ as they are the only questions to which they can plan their answers in advance. They include:

  • Why this course?
  • Why Oxford (very unlikely but again it would be foolish not to prepare an answer)
  • What have you covered recently at school that has interested you and why (this one does come up and is often used as a jumping off point for discussion)
  • ANYTHING relating to their personal statement. What do you mean by…..what did you think of …… etc.

10. Remember that interviews are just a part of the process.

Interviews are a vital part of a successful Oxbridge application but they are just a piece of the puzzle. Pre-interview tests, references, predicted grades and personal statements all help tutors develop a picture of the student as a whole and can both help tutors sift out students before the interview stage or make up for a slightly below par interview performance. Pre-interview tests in particular have over the past 9 or so years increasingly become an important part of the evaluation of candidates (a fascinating fly on the wall account of the admissions process at Oxford can be found here).

This has two implications. Firstly, don’t focus solely on interview work. All of the tests can and should be prepared for and specific knowledge is needed to succeed on them. This is despite the Admissions Testing Service’s Website making the utterly false assertion that the Thinking Skills Assessment, used for a wide range of arts courses at Oxford and similar in style to a number of the Cambridge pre-admissions tests, “is a test of skills and aptitudes that students already possess”. This is a statement that damages the prospects students applying from schools less familiar with the Oxbridge application process. For instance, off the top of my head, I know that amongst other things, the TSA requires knowledge of:

  • The definition of a conclusion
  • The definition of a premise
  • The definition of an assumption
  • The definition of

Those more in the know (rightly) prepare students specifically for the tests.

Bonus: Encourage any arts student to subscribe to The Economist or at least try to ensure your school’s library offers it. They offer both a print and a digital subscription (with a pretty nice app for the smartphone generation). This really is an investment. The quality of the articles is high, the breadth of reporting consistently amazes me and the analysis is generally very sound and covers both technical economics and finance as well as politics and current affairs. Like any publication it has a political and economic bias (free market capitalism, pro free trade and globalisation and generally fairly social liberal) which it is worth making students aware of but in terms of how much students can gain from reading it, I’ve yet to find a better alternative (pair with a quality daily paper for maximum efficacy). I’m not on commission honest!

As stated above, these are short-term ideas and quick wins that can support students applying. The real work takes place in the medium and the long-terms. However, with interviews being around 6 weeks away, hopefully a few of these ideas could be useful to you if you are supporting students applying to Oxbridge. Any further suggestions, comments or ideas get in touch!

The Dunning-Kruger Effect and Self-Assessment

“The first rule of the Dunning-Kruger Club is that you don’t know you are in the Dunning-Kruger Club”

-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

  1. 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.
  2. 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.
  3. 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.
  4. 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.

Type 1 and Type 2 Fun

Railing against ‘fun’ has been done.

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

(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).




We are not elite cyclists

Marginal gains. The stuff from which whole school INSET dreams are made. Interesting story? Check. Intuitively makes sense? Check. Seemingly applicable to the classroom? Check. Celebrating British success and ingenuity? Check (though the recent TUE controversy does throw some shade on things).

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.

Thoughts on teaching so-called ‘low ability’ Year 7

By now most teachers will have spent a week with their new classes. Inevitably some of you reading this have been throwing your hands up in despair about your new Year 7 class, complaining to colleagues that they are “the weakest you’ve ever taught/taught in years” and lamenting the fact that your primary counterparts have clearly fiddled the data and that the students must have been coached to death through their SATs exams without actually learning anything.

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.


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.


Sort of Rethinking Card Sorts

During my ITT and NQT year, my colleague (a science teacher) and I just couldn’t get our heads around the idea of ‘card sorts’ which were pushed on us by our university tutors in particular. Why use them when, even then, we realised there are more time efficient and yet equally effective ways of encouraging students to think about and sort information.

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!

Traditional doesn’t equal boring

Making my lessons enjoyable isn’t my number one priority when teaching in the classroom. Or my number two. I’m not even sure it is third on the list. However, despite teaching in a fairly traditional and ‘boring’ way, I think it would be fair to say that most students I teach do enjoy the majority of their maths lessons.

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.

More on modelling

I’ve seen a few blog posts on modelling in Maths this past few weeks so I thought I’d add my thoughts on this absolutely vital skill for teachers…

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:

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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:

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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:

  1. A full mastery of the two identities including the ways in which the identities can be rearranged;
  2. 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.

For F’s Sake….why I don’t teach F, Z and C angles

Throughout my teaching career I have only ever known teachers to teach alternate, corresponding and co-interior angles with reference to Zs, Fs and Cs. Obviously all teachers have stressed that they need to learn the mathematically correct names but invariably students have found this hard to do which hampers them in the long run. The letters, whilst intended as a scaffold have actually become part of the supporting structure. When they are removed the house falls down. I myself have done this time and time again and found myself frustrated by student’s inability to remember that ‘Z angles are alternate’.

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.