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.

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‘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.

"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.

IMG_7112
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

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