‘Think Aloud’- Reflections on a metacognitive strategy

Last week I attended an in-house training session led by a number of my colleagues in the MFL Department including Louise Miller, Dr. Gianfranco Conti and Dylan Viñales.The focus was on developing students’ metacognitive skills. I was particularly taken by an activity which they suggested that they called ‘Think Aloud’.

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.

Screen Shot 2016-01-29 at 12.08.40 PM 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.


“What would a more difficult question on this topic look like?”

Cross-posted to betterQs

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.

MyMaths: Last Refuge of a Scoundrel

Many Maths teachers rely on MyMaths (or indeed MyiMaths) heavily in their teaching. Either as a classroom teaching tool or, more commonly, as a means of setting homework.

I contend that over 90% of the time this is an unacceptable teaching strategy and usually (but not always) stems from laziness.

Perhaps some of you committed MyMaths-ers (and we all know at least one) are reading this now are bristling with indignation and your mouse is being drawn either to the ‘exit’ icon on the top of your browser. Well hang fire, read this and if you disagree post in the comments below.

MyMaths as a Classroom Tool

Many teachers plan entire lessons around MyMaths. I imagine most of us have worked with teachers teachers whose classrooms and you can be confident that a few times a month you will walk past their classrooms and see the familiar MyMaths white on black being projected onto their board.

So what’s the issue here? The resources look nice, feature slick and sometimes helpful animations and are neatly mapped to the curriculum. Why am I kicking up such a fuss?

Firstly using MyMaths as the primary means of delivering a lesson shifts the manner in which teachers think about learning when they are planning. By using a premade resource (which is not even editable unlike those from TES or other sources), MyMaths users are focusing on the activity within the lesson at the expense of what really matters- what students will learn over an extended period of time.

MyMaths increases the likelihood that teachers will use resources unthinkingly and uncritically because it is easy. No more time spent considering how to develop a high quality lesson and the surrounding pedagogical issues because MyMaths have laid on a plate ‘how to do it’. No thought has to go into ensuring that questions match what teachers are hoping to teach as the nice guys at MyMaths have prepared a presentation that takes you through everything and often includes questions for the class to consider. Resources don’t have to be matched to the needs of classes because all topics are ‘helpfully’ matched to a national curriculum level or GCSE grade.

At this point some teachers may say ‘well I always use MyMaths for teaching circle theorems as they have a really nice way of explaining them’ or something similar. Great! Developing a bank of resources is all part of being an effective teacher and if all you are doing is using a particularly nice animation from the website or know that there are a really nice set of questions on a given topic then by all means use them. Indeed, the interactive aspect of MyMaths is one of its (few) positives. However, most don’t stop there and instead proceed onto the mindless clicking through the activities that characterises MyMaths. Is it any wonder then that there has been a call by many in politics to introduce more unqualified teachers into the classroom if this is the sort of lesson delivery that many qualified professionals use as part of their practice?

If you are using MyMaths because you genuinely think it has a passable set of resources and perhaps you find it difficult to navigate TES, please take a look at this website and use it as a jumping off point to start making your lessons your own. There are a great selection of questions, animations and activities that you can incorporate into your lessons which will (hopefully) better allow you to keep ‘learning over time’ rather than ‘keeping them busy this lesson’ at the forefront of your planning.

Ask yourself this question- if you were being observed by a senior colleague, would you do a MyMaths-based lesson? I imagine the answer is a resounding ‘no’. This response should tell you all that you need to know.

MyMaths for Homework

“But it’s a great way of setting homework” I hear some colleagues cry when this issue is raised. “No it isn’t” the more effective teachers shoot back.

At the heart of the issue with homework is the timeless gripe of Maths teachers- ‘show your workings’. If we want to develop students as mathematicians who appreciate the importance of showing a solution rather than just ‘an answer’, mathematicians who are better prepared for the rigours of A Level study and beyond and mathematicians who are better able to pick up maximum marks in an exam, we have to develop students’ pen and paper mathematics. MyMaths does not allow this

Yes setting MyMaths homework is easy, will check the box that your Head of Department or Leadership team sets and might satisfy the parents of your students, but it isn’t nearly as effective as it could be. I have often thought that while classroom pedagogy seems to be almost continually developing, a corresponding pedagogy surrounding homework does not seem to have emerged. Policies that encourage teachers to focus on just setting homework rather than considering what effective homework actually is have emerged in our schools. School and department leaders make assessments about the quality of classroom teaching, they should also do so about the quality of homework and consider the contribution that it makes to learning over time. MyMaths contribution to learning over time is limited compared to many other options for setting homework.

I can already see the next wave of rebuttals coming my way arguing that the effectiveness of MyMaths lies in the fact that it can be instantly marked and students and teachers can see how well they have done straight away. Consider this:

  1. Effective instant marking is nice, makes the life of a teacher and student easier but isn’t always the best way of doing things;
  2. If you really want instant marking then there are better resources out there, in particular the incredible Diagnostic Questions from Craig Barton that consists of (mostly) high quality questions that focus on misconceptions and encourages students to explain their answers. Teachers can create their own quizzes and questions which obviously makes it far more useful than a premade quiz on MyMaths;
  3. There are ways of setting pen and paper homework that allow for almost instant self marking (give students a mark scheme, jumbled up answers, codebreaker-style activities etc.)

This YouTube video is well worth a watch, though it does contain some rather colourful language so you might want to check there are no students around!

So what now?

Whilst laziness is, in my opinion, the primary reason teachers use MyMaths, there are two other possibilities. The first is ignorance of the other resources that are available and the fact that TES can often seem a bit overwhelming and has resources that vary massively in quality. Try the above links, start following teachers on Twitter and share with colleagues in your department. Secondly, MyMaths can act as a crutch for teachers who are less confident in their planning, delivery or subject knowledge. Whilst this is something of a short-term fix, it certainly isn’t a long term solution. Colleagues and in particular departmental leaders should be supporting those who are less confident in their planning and delivery rather than perpetuating the sub-standard teaching that MyMaths almost always entails. Yes this may be difficult and time-consuming, but isn’t helping colleagues to develop themselves to the best of their ability something that lies at the heart of leadership?

MyMaths subscriptions cost £599+VAT per year for secondary schools. That is not a trivial amount of money. If your subscription is up for renewal anytime soon and you are in a position to influence this decision within your department, consider the above points and whether that £599 per year could be better spent elsewhere in order to enhance pupil learning in the long run. I imagine the answer is yes.

At some point in the not too distant future, I want to make the transition to Head of Department. Getting rid of MyMaths would be one of the first steps I would take to help develop a culture of excellent teaching. I’m not for a second saying such an approach is a quick win, I think these are few and far between in the area of teaching and learning. It would however be a way of ‘setting out my stall’ and showing what I would value in terms of teaching and learning.

"What does the word ‘percentage’ mean?"

Note: Originally posted on Betterqs

I love asking low-ability students or students whose first language is not English questions like this. Unpicking the etymology of words is something that can benefit all students but for low-ability or English as an additional language (EAL) groups this is of the utmost importance as they had anywhere near the same level of exposure to the subtleties, nuance and conventions of the English language. It is the job of educators to increase the level of exposure beyond what they would otherwise encounter.

For young students especially I really ham it up when I reveal that ‘percentage’ means ‘of 100’ and let them know that they are part of a small secretive club that will refuse to use the word ‘percentage’ willy-nilly but will stick to its strict definition.
I usually then go on to tell students that they would now be able to have an educated guess as to the meaning of any word with the word ‘cent’ in and I ask for a number of suggestions. Again for younger students a touch of the theatrics can be useful here (think Sherlock Holmes references).
When I asked this question last week it was with a Year 7 group, most of whom hadn’t encountered percentages yet but who were beginning to gain confidence with fractions of amounts and equivalent fractions. They quickly cottoned onto the idea that 25%=25/100=¼ and were then able to find percentages of amounts.
Focusing on the fact that percentages mean ‘out of 100’ and treating percentages as a special instance of the fractions work they had already encountered was something of a long way round. However, initially and in subsequent lessons students seemed to ‘get it’ and were far better able to explain some of the intuition behind percentages. Additionally, it made subsequent work on converting between fractions, decimals and percentages far easier.
Also posted @NWMaths

Lads, Lads, Lads!": The Impact of ‘Laddishness’ on Asian Muslim Males’ Perceptions of Education OR Why you should write a dissertation

I recently finished my MA in Education at the University of Manchester, writing a dissertation entitled “Lads, Lads, Lads!”: The Impact of ‘Laddishness’ on Asian Muslim Males’ Perceptions of Education. My previous school was an all-boys school and the majority of these students were Muslim students of Pakistani or Bangladeshi heritage and researching the dissertation allowed me to engage with a number of these students in a way that I hadn’t before.

The dissertation was something of a labour of love- I was fortunate enough to choose a subject that I was genuinely interested in and passionate about and there was enough literature available to write a rigorous academic piece but there did exist lacunae that enabled me to make some original insights.

This was not a piece of action research and in all honesty, it has only had a very limited impact upon my classroom practice. However, if you are considering embarking upon a course requiring a dissertation or if you given the opportunity to write such a piece of work in the future, I implore you to grab it by both hands. The dissertation was an opportunity to step back from the immediacy of the classroom environment and take a macro-level view of a topic that I am passionate about- education (and in particular the education of underprivileged students). It exposed me to academic debates that I didn’t even know existed yet was relevant to the sorts of behaviour I saw everyday and improved my writing no-end. It’s also very satisfying to know that I have made some original insights into this area of academia, no matter how small.

Those of you not interested in the specific content of my dissertation- stop reading now! For the real nerds however….

The abstract is below:

“Laddishness’ is one explanation often given for the underperformance of males relative to females in secondary education. However, the intersection between ’laddishness’ and ethnicity, particularly Asian Muslim ethnicity, has received a relatively little attention. This dissertation begins by examining the various constructions of ‘laddishness’, the mechanisms through which it operates, and considers how ‘laddishness’ may impact upon academic achievement. Four detailed pen-portraits of Asian Muslim boys at a single-sex school in Manchester are constructed and analysed. The portraits indicate that the influence of ‘laddishness’ permeates the lives of these students to a lesser extent than the literature suggests. In particular the students did not show many of the ‘hypermasculine’ traits one might expect, nor did the students appear to reject many of the characteristics and behaviours required to succeed in education. A range of possible reasons are discussed for this that consider culture, aspirations, the role of the school and a reaction to perceived or actual institutional racism in the higher education and employment markets. “

‘Laddishness’ is a contested concept but broadly, those that claim that ‘laddishness’ is a cause of male underachievement state that there is some, fairly homogenous, concept of masculinity that is adopted by a number of young males that is antithetical to the ethos required to succeed at school. General features of ‘laddishness’ identified in the literature include belonging to a hedonistic peer group, an interest (and even a fixation upon) stereotypically ‘masculine’ pastimes including sport and a focus on ‘having a laugh’ and ‘hardness’. There is also a consensus that ‘laddishness’ includes the rejection of authority and a dislike of anything overtly feminine.

There are two broad schools of thoughts on how exactly ‘laddishness’ actually impacts upon schooling. Sociological views suggest that education is antithetical to the hegemony of ‘laddishness’ many male students are immersed in and thus results in poor class behaviour and low achievement. Socio-psychological theories however suggest that ‘laddish’ behaviour results from students preserving their self-worth. For instance students may misbehave so that if a poor grade is achieved they can attribute it to their misbehaviour rather than a lack of ability. Of course if they do achieve a good grade, they have achieved this ‘effortlessly’.

The overall picture present in my portraits did not reflect what some the popular or academic literature suggested regarding ‘laddishness’ as an example of a crisis in masculinity and as a potential source of underachievement.

The students I worked with were all engaged in their education and recognised its value. They all indicated that, whilst they may participate in ‘laddish’ behaviour at times, this tended to be specific to certain subjects or teachers and was not indicative of a generally negative view of education. These students were not examples of ‘boys in crisis’. Indeed, some made a point of eschewing and condemning the negative behaviours in which some of their peers participated.

I tried to consider the interplay between ethnicity and masculinity that was at play here and noted that all the students I spoke to valued their strong family structures and had incredibly high aspirations irrespective of the poverty in which many of them lived in. I also suggested that a ‘deficit factor’ could offset the influence of ‘laddishness’. Students from a minority ethnic group face challenges, both real and perceived, that white British students do not. Perhaps this helps offsets some of the desire to indulge in some of the ‘laddish’ pastimes.

Drawing a firm conclusion was difficult as the intersection between race and gender obviously does not exist in a vacuum but is part of an incredibly complex web of factors acting upon students. One such factor is the school which students attended and some credit must be given to the staff and leadership of the school given that students in my research group did not see consider ‘laddishness’ to have a particularly powerful impact upon their education.

I also made the following recommendations:

-That academics, policy-makers, professional bodies, popular commentators and others refrain from indulging in damaging ‘boys in crisis’ rhetoric. ‘Laddishness’ does occur but it is not uniform, is not ubiquitous and isn’t a simple phenomenon.

-That the role schools and individual teachers play in tackling ‘laddishness’ be acknowledged and investigated further.

-That the high aspirations of members of ethnic minority groups, and indeed students generally be cultivated and used to prevent the emergence of ‘laddishness’. Amongst the students I interviewed, there was no ‘poverty of ambition’, only desire to succeed and an acknowledgement of the effort required to do so.

-That authentic first-person accounts of students’ experiences of education be used to inform policy and shape teaching practice. ‘Pupil voice’ is increasingly common within schools and can be useful. However, even when conducted properly (and not as a ‘tick box’ for OFSTED) it usually lacks the depth, insight and sense of advocacy that the ‘pen portrait’ approach that I used can achieve.  

Comments welcome as always or Tweet @NWMaths

Five Traits of Top Teachers

After reading @greg_ashman’s post about the top five traits of best teachers, I have decided to stick my oar in and share my five with the internet. Just like Greg’s original post this is subjective and is based upon the sort of teacher I would like to be, some of the best colleagues I have worked with and the best teachers I was taught by.

So, my five traits in no particular order

  1. Good ‘explainers’ (for want of a better phrase- ‘good at explaining things doesn’t quite seem to do this justice). At its heart teaching is explaining and the best teachers understand and know their subjects well enough to explain things in a way that students just seem to understand. If one explanation doesn’t work, they are comfortable trying others and they are constantly adapting and tweaking these, often on the spot.

  1. Ability to have almost all classes ‘eat out the palm of their hand’. This can come in many forms. It might be the firm but fair disciplinarian, the slightly scatter-brained drama teacher or the quietly spoken teacher who just exudes presence from the front of the classroom. Whatever form it takes, the best teachers are able to develop a relationship with classes that means that students are attentive, will strive to work incredibly hard, will hang off their every word and are motivated to put in maximum effort (almost) all of the time. It goes without saying that the behaviour in classes taught by such teachers is almost perfect.

  1. Know the ‘hooks’. This is linked to point two and point one but deserves its own listing. As well as knowing ways of best explaining a subject, great teachers know the stories, anecdotes and jokes that students remember and engage with for years to come. I remember the teacher at primary school who took us for PE telling us that the tactics we were working on were the same as Kevin Keegan used when England beat Scotland in the 2000 Euro Qualifiers. I still remember being told the Jaffa Cake/VAT court case in year 7 and use it myself whenever I teach percentage increases. My history teacher at A-Level used to tell us gripping stories about areas of history we simply wouldn’t have had any interest in whatsoever if it wasn’t for him (the history of Quebec being one that has really stuck with my over time). In my subject, the history of Maths is a rich area for engaging students- Fermat’s last theorem and the life and beliefs of Pythagoras are two of my favourite things to talk to students about and I’ve been working hard developing my spiel in a few other areas.

  1. Make time. The best teachers are prepared to give their time to students in their classes. Be it staying behind for revision lessons, taking the time to turn around practice exam papers in the run up to GCSEs, taking the time to get to know their class or making the time to prepare really fantastic show-stopping lessons.

  1. Get results. This one should go without saying but in an effort to push back against an overly results-driven education system, the pendulum can sometimes swing too far the other way and this can be overlooked. Terminal exam results are important and to not have that somewhere near the top of one’s priority list is to not do the best for the students being taught. Results are obviously dependent on context- for some classes ‘results’ may be all students getting a D or better whilst for others nothing less than a fistful of A*s is good enough. To me this seems uncontroversial but I’ve encountered a range of articles lately that seem to suggest otherwise. I’m not suggesting that this should be at the expense of student well-being or holistic development (this is a false dichotomy) but getting results is one of the hallmarks of a great teacher and to suggest otherwise is naive.


Asking students is a great way of finding out who the ‘best teachers’ are. However, in my experience, to get an honest answer you should already have in place a good relationship with that student and, I have often found it helps if I add a caveat like “apart from Maths teachers” in case the student feels uncomfortable referring to one of my closer colleagues.
Thoughts and alternative top fives always welcome either on here or @NWMathshttps://twitter.com/NWMaths.

In praise of video self-analysis

I had been delaying it, putting it off, avoiding it and making excuses but the time had finally come…it was time to film myself teaching.

I made a private resolution at the start of the school year that before the half term was out I would film and watch myself teaching. There was only one rule- the filming had to be as ‘no notice’ (or as ‘no-notice as it is possible to be when filming oneself) in order to capture an authentic lesson.

Come the last week of the half term, I still hadn’t taken the plung myself so right before the penultimate lesson on Friday (year 9, four operations with standard form) I grabbed my iPad, propped it up against a set of textbooks and set it filming.

The results were…not as cringe inducing as expected. On the contrary, they were (quite) reassuring. It didn’t set the world ablaze with it’s cutting edge teaching and learning techniques or whizz-bang resources but it was a ‘tight’

lesson with great learning gains for almost all of the students. After the inevitable discomfort that comes with hearing one’s own recorded voice which subsided after the first five minutes I was pretty pleased with what I saw. As someone who has observed a fair few lessons in my time, I was relieved that my ‘bog standard’ lesson without all the out of the ordinary ‘oofle dust’ that (some) people put into their observations was reassuringly ‘solid’.

However, the tape doesn’t lie and did pick up on a few things that I should definitely work on which I don’t think would have been picked up on by an external observer. The inevitable self-consciousness and tendency to be self-critical that inevitably comes with watching oneself worked in my favour. I was that bit ‘harsher’ than some observers might be and thus picked up on things that might have been unchecked by others.

Habits and Body Language- This was the real eye-opener and it is something that no amount of introspection and reflection was likely to identify and something that hasn’t been commented on by those observing my lessons. On more than one occasion when explaining a particularly difficult concept to students I made a strange movement with my hands (picture the ‘whole thing’ gesture from charades but with your elbows touching your sides). Whilst not an issue in itself, I imagine that I would have picked up on this were I was one of the students I teach (and I probably would had a good laugh about it to boot) so it is worth eliminating. I also have a particularly annoying habit of throwing pens up and catch them when circulating the room. This didn’t look particularly professional and I wouldn’t stand for my students doing it. Again this isn’t something that either myself or others have been identified as an aspect of my practice that I should improve.

Choosing Students-I rarely allow hands up when answering questions and instead choose students and differentiate my questioning accordingly. However, I noticed that I seemed to have three or four ‘go to’ students from across the ability range that I called upon more than the others. So much so that when watching the video a week later I was quickly able to predict fairly accurately the students that I was going to call upon after I heard the question I was asking. Whilst an external observer may have noted and perhaps praised the fact that I was moving the questioning around the room to include a range of students, my knowledge of the class and students showed that this wasn’t as inclusive as first appeared.

Video analysis isn’t a complete solution for developing one’s practice and has it’s downsides. Notably, my video analysis was time consuming-I watched a few minutes of video at 10 minute intervals throughout the lesson and made a few notes but this was still a process that took over half as long as the lesson itself. I’d be really interested to hear how others have analysed videos of themselves teaching. However, it was definitely worthwhile and is something I will do again.

I know that many schools use IRIS Connect and other similar systems which I’m sure are great and even include the teacher being ‘miked up’ and technology to ‘follow’ the teacher around the room. However, I managed with an iPad propped against a stack of textbooks. Given the near-ubiquity of tablets (and indeed textbooks) it shouldn’t be too hard to beg or borrow one. Probably best to avoid stealing however.

Take control of your own development and try filming yourself teaching a lesson. You don’t need to share the results or even tell anyone else about it. If my experience is anything to go by, you will notice things that you might never have, it will improve your practice and it might even be reassuring.

Perhaps, like me, the video will even remind you to stay clean shaven unless you are capable of growing designer stubble or an impressive beard!

Comments welcome either here or @NWMaths.

Micro Plenaries

Just a quick post on something I have been trying to reincorporate into my teaching over the past few weeks after really focusing on it last year- ‘micro plenaries’. I first saw this idea on the impressive ‘but is it on the test?’ blog- worth reading for a detailed account of this and a range of other maths-related teaching and learning ideas. The concept is a scaled down, individualised version of the sort of plenary one might have at the end of a lesson. After each interaction with an individual student, ask them a short plenary question to help ensure that the processes used are explicit and thus their work becomes more meaningful. The most common questions I ask are ‘what was the key step in this question’, ‘what was the way into this question’ and ‘can you summarise what you did to answer this question’. However, there is a huge range of questions that are likely to be just as, if not more, effective.

It takes some practice to remember to incorporate this into part of one’s normal questioning, especially if this takes place whilst circulating the room where the temptation is often to quickly attend to other students or gauge the ‘feel’ of the class after interacting with an individual. However, in my experience it is well worth sticking with and has a real impact in helping the student become more aware of the cognitive processes they are using.

Reflective Journals and Developing Metacognition in Maths

In my school’s dedicated professional learning time on Friday afternoons, I recently read about the techniques that Gianfranco Conti plans on using over the coming year in order to enhance and develop his students’ metacognitive abilities.. The original article is a must read and can be found here but I was particularly taken by two of his strategies which I think have could prove particularly powerful in the Maths classroom.

The first of these is an error log. The idea is simple and one I have used (inconsistently and sporadically) before. When students get back a marked piece of work they read through their feedback and pick out the errors made. They then record these and make a note of their mistake, along with an explanation. When using this with my classes this year, I will not sum up their ‘WWWs’ and ‘EBIs’ for students and will only mark on a question by question basis to ensure that students have to engage in the process of working out their mistakes for themselves. To do otherwise may turn a cognitive exercise into a copying exercise.
The second of these techniques is encouraging students to keep a reflective journal in which students are presented with a stimulus question each week (e.g. ‘which aspects of Maths do you think are causing you the most worry?’) and are then encouraged to respond with a short paragraph.
I have decided to combine these strategies. Students will always be required to produce a written log of errors on each piece of marked work and will be given a stimulus question less frequently (perhaps every other week).
In addition to the metacognitive benefits that Gianfranco mentions, the error log in particular will hopefully have other positive impacts. Specifically it will ensure that students have read in detail the marking and corrections that I have made to their work. Students will therefore gain the benefits of acting upon individualised feedback as well as the metacognitive benefits of reflection.
I plan on introducing this to my Year 10 ‘core’ IGCSE class who I believe will benefit significantly from the increased confidence these metacognitive strategies will have and my Year 12 A-Level group, many of whom achieved a grade B or a low grade A at IGCSE.
All the journals will entries will be made on a Google Document that both myself and the student have access to and upon which I can comment directly. Here is a verbatim entry from two of my year 12 students.
Student 1
“I have to be more careful when multiplying with negatives because one of my first mistakes was that I had multiplied √2 by -√2 and said it equaled to 2 which is wrong. The right answer was negative √2. Moreover, when dealing with surds that are in fractions, and finding the value of P and Q which are numbers in front of the surd, I should not times the denominator by the numerator because the fraction would no longer be equivalent. I should continue to draw a graph when dealing with quadratic inequalities because they helped me a lot in finding which part of the graph is needed. When using difference of squares, I can times back to see if the equation is exactly the same as how it started. If it’s different, then I know that something must have gone wrong which I can then check through the working again.”
This student has clearly given a considerable amount of thought to his journal entry and has picked up on my feedback. I think that by ‘difference of squares’ he means ‘completing the square’ and I added a note to the document highlighting this. The student is using the language that I would expect from an A Level student and this has been a useful way of confirming that the student is responding to my feedback and ensuring the time I spent marking wasn’t wasted!

Student 2
“I feel that the the effort grade you gave me was a fair one as I actually spent quite a bit of time on this homework. Overall I felt that most of my homework was well done by me as I got the marks I felt I deserved however about the marks I didn’t get, I understand how I did not get those marks. I need to work on the discriminant questions that include inequalities in them as well as having to remember to draw the graph when working with inequalities.”

This student has commented on the effort grade I have given him as well as mentioning some of the topic areas he has lost marks on and how he can improve (drawing a graph to help solve quadratic inequalities). However, he doesn’t seem to have engaged with the reflective process in as much depth as he is capable of and the quality of his written English could be much better. Given this was his first entry I responded very positively and thanked him for his reflections so as not to put him off the reflective process. In future I will encourage him to be more reflective than he has been on this occasion, but that is a job for another day.
Key to the whole process will be consistency on my part in keeping this up over a long period of time, as well as ensuring that students really ‘buy into’ the idea of a reflective journal. Initially I expect the journal to be focused on the errors made rather than student perceptions of their Mathematical education. However, with some well-placed stimulus questions, I envisage this becoming will become more and more reflexive over time. I have also made a point of periodically checking the documents and as soon as I have read a completed entry emailed the student thanking them for their contributions and giving personalised feedback on their entry (perhaps encouraging meta-meta-cognition?!).

I really look forward to reading the reflections of my students, especially my low-confidence year 10 group. As always, comments welcome either below or @NWMaths

Direct instruction and inquiry…thoughts on starting at a new school

Starting teaching at a new school (inevitably) has its challenges. New systems, new colleagues, new parents and a different context all combine to produce an intense and challenging first couple of weeks.
My previous role was teaching mathematics in a state school in inner-city Manchester- a world away from my new position at a prestigious fee-paying international school in Kuala Lumpur. One particular challenge that I have encountered during the first few weeks is in adjusting my classroom ‘style’.
The students I taught in Manchester were, by and large, incredibly positive, motivated and hardworking. However, due to a range of factors, I found that the students responded best to a fairly traditional and heavily disciplinarian style of teaching. I have always been keen to embrace best practice from colleagues both in my school and across social media in terms of pedagogy and assessment, but this has until now been coupled with a fairly ‘old-school’ approach to behaviour and discipline. Students would line up in silence before entering the class, would wait silently behind their chair before being allowed to sit down and would be expected to work for periods of time each lesson completely individually in silence.
A range of strategies to encourage collaboration and communication between students were of course used (the latter being particularly important due to the exceptionally high number of EAL students at the school). In particular cooperative approaches were used in problem-solving activities which I increasingly used with Key Stage 3 classes in order to best prepare students for the new style GCSEs. Students were also never spoonfed and were encouraged to persevere on difficult problems, find their own solutions and make their own mistakes. However, the metaphorical leash was firmly kept on students at all times. In order to maximise mathematical learning gains, I created and sustained an environment in which me as the teacher exercised considerable influence and control over all aspects of the class at all times and students had relatively limited control over the macro-direction of their learning.

John Hattie’s oft-quoted work Visible Learning appears at first glance to vindicate this approach. High quality ‘direct instruction’ (NB this is distinct from didacticism) in which the teacher specifies the learning outcomes, engages students, models, checks for understanding and provides opportunities for both guided and independent practice has an effect size of 0.59. This was very much the aim back in Manchester and was predicated on my complete control of the classroom environment at all times. Inquiry-based teaching, in which students pose their own problems, ask their own questions, observe phenomena, form and test hypotheses and and have far more influence over the direction of their learning than under direct instruction has a much lower effect size- 0.31. Without going into the nuts and bolts of Hattie’s meta-analysis it is worth noting anything above 0.4 is considered effective whilst a negative effect size has a regressive impact upon a student’s learning.

My initial impressions of my new workplace is that ‘inquiry’ within the classroom plays more of a role than I have yet encountered in my (fairly short) career. That is not to say that in each and every lesson students are completely independent inquiriers, exercising ultimate control over the direction that their learning takes. Far from it. Rather, it is just the case that there is more inquiry-based learning taking place and more discussion of inquiry than I have been exposed to until now.
Two questions about this have been occupying my reflections on this. Firstly, what are the merits of an inquiry-based approach to learning given that Hattie’s meta-analysis suggests its impact is limited? Secondly, how can I adapt my current practice to allow students to benefit from the positive aspects of inquiry-based learning?

A few initial thoughts about Hattie’s findings regarding inquiry (a full critique of inquiry-based teaching is far too broad a topic for a Friday afternoon):

  • Most teacher training courses that I am aware of tend to develop a ‘direct-instruction’ approach to teaching. Coupled with the fact that it is reasonable to assume that there exists a pedagogy specific to inquiry-based teaching, direct instruction is likely to have a greater impact than inquiry-based approaches simply because more teachers are better at it, rather than it being an inherently better approach.
  • Some topics and subjects are better taught in different ways. Indeed Hattie suggested that pre-teaching core content in order that inquiry can be focused more on process will enhance the effectiveness of any inquiry. Learning times-tables through an inquiry-based approach for instance would not perhaps be the best way for students to learn.
  • Inquiry may or may not lead to subject-based learning taking place at as quick a rate but it is reasonable to assume that the transferrable skills developed are greater than those developed through an approach focused primarily on developing mathematical learning.

So to what extent will my classroom practice be influenced by a more inquiry-based (and some would say progressive) approach to teaching?

Firstly, it will be a useful personal reminder for me to avoid one of my own flaws as a teacher: the tendency to talk too much. ‘Chalk and talk’ isn’t what Hattie meant by direct instruction. Knowing the premium that many around me place upon inquiry-based education and that the students I am teaching are used to this approach will help remind me to reduce this aspect of my teaching.

Secondly, it will encourage me to leave my comfort zone, something which can only be a good thing. I frequently remind my students that they should find the work ‘slightly too difficult’ as that means that they are learning. I see no reason as to why this shouldn’t apply to my teaching. Having been trained and indeed immersed in a system in which ‘direct instruction’ is at a premium, any initial foray into inquiry based approaches will likely be met with limited success. By persevering and developing this part of my teaching, at worst I will have another ‘string to my bow’, whilst at best I will have found a new and exciting way of engaging students in Mathematics and encouraging them to excel.

Thirdly, it will encourage me to think about what I want the students that I teach to be good at (examinations aside). I want students who are being taught by me to develop into good mathematicians and good people. The inquiry approach can certainly play a role in the latter through the development of excellent social and interpersonal skills and it seems reasonable to suggest that there is more scope for doing this through inquiry than direct instruction. But what about the former?

I tentatively (and perhaps rather boringly) contend that a combination of both approaches is likely to result in the best mathematicians. A good grasp of mathematical methods is vital for a mathematician and instinctively I feel that amongst most students that I teach, this is best achieved by direct instruction. However, effective mathematicians look for their own patterns, ask their own questions, and pursue their own routes of learning. This is something that is not just developed by inquiry, this is inquiry.

I currently believe that the more able students that I teach will be more able to benefit from a greater number of inquiry-based lessons than the less able ones. Maybe this is because of I have less confidence in my ability to provide scaffolding and support for inquiry-based approaches at the lower end or perhaps because some of the students that I teach lack the requisite skills to gain as much from an inquiry lesson as from direct instruction.

One thing is for certain- I won’t be stopping old fashioned regular ‘rolling numbers’ style practise of multiplication tables anytime soon!

Any comments, thoughts or criticisms welcome either below or on Twitter @NWMaths