Before I go any further, this whole post comes with a huge caveat – this will not do the day justice. You must try and arrange to see the school if time allows.
After meeting Dani Quinn (@danicquinn), a couple of months ago, I was keen to see how maths is taught at Michaela and have a look at their ethos which some may feel is a direct opposite to Wellington. The day started with the students enjoying lunch time and I was immediately struck by the level of interaction of students and staff. Students were happy to converse and allow staff to join conversations freely. At the end of the play time came the first real sign of the expectations set. A member of staff quickly had the students lining up in silence and all were clear on the rules and what was expected of them. This preceded a silent walk into the lunch hall where I sat with a group of students at a university-labelled table (a great way of inspiring children, I think). I was amazed that these year 7 pupils knew about universities and had great ideas about which universities they wanted to go to and which subject they may consider studying. From just twenty minutes at school, I was stunned by the levels expected from the pupils. This expectation can only have a positive effect on the school.
During lunch, a member of staff read out merits and demerits from the last day in a no-nonsense fashion. These students have been rewarded and these students have not met our expectation. After this, we were told the topic for the day and that is what our table was to discuss while eating. Five year 7 and 8 pupils plus myself went on to talk about punctuality and why it was important not to be late and what we can do to stop ourselves. I thoroughly enjoyed the conversation and the students should be applauded on their welcoming and challenging conversational skills. After a donut for dessert, students were invited to show their appreciation for anyone in their life – mainly teachers or others in the school. The duty staff would then offer very specific feedback on how this was done e.g. “that is worthy of reward because you projected very well and it was meaningful” or “if you want a reward next time, make sure to annunciate every word”. Clear, specific feedback throughout the school. Everyone is on message.
Two students greeted us and took us around the school where we were allowed in and out of lessons freely without students or staff batting an eyelid. It was here that I was struck by the quality of teaching at this school. I saw about 5 minutes of a French lesson, some of a Physics lesson, some Art and some maths. In all classrooms, students work was put first. The school may well believe in teacher at front, student at desk but there is also a clear ethos that students must be doing work to learn. The behaviour culture allows for clear boundaries and allows teachers to positively reward great learning habits e.g. sitting up and following a solution very well or tackling a problem straight away. It does also allow for negatives to be enforced quickly e.g. turning around but across my afternoon, there was far more specific, positive praise than negativity.
Each teacher ensured pupil participation was high and there were lots of questions asked across all the classrooms. Teachers involved each student and if something was not quite right, time was taken to correct before moving forward. There is a clear expectation academically as well as behaviourally. I was thoroughly impressed by the level shown by all students. All activities used were activities that students wanted to engage with. Students could see that the teacher wanted them to know what to do so they trust that there is a purpose in what they are doing.
After popping in and out of random lessons, I watched a full lesson with Dani to finish the day. This followed a similar pattern: lots of pupil involvement; great questioning from the teacher; constant assessment of and for learning as well as a chance for lots of success. I did not see Dani using a drill but I got to see a little into her teaching philosophy which will certainly have an impact on my own practice. I found that students did a lot of practice of specific questions and anyone who needed more got that opportunity too. Dani (and the other teachers) kept up the idea of specific feedback on all activities too. In maths, the feedback was even more specific owing quite a lot to the nature of the activities. All of them allowed Dani to quickly work out what the misconception was and then address that. This is certainly something I will be taking into my own lessons – particularly during revision time.
Unfortunately I had to rush off before talking to Dani at the end but I would like to thank her very much for the chat we had and for allowing me to see into her lesson. I had a brilliant afternoon and I very much look forward to going back to see some in the future in the maths department. I also hope to get Dani over to Wellington to have a look at how we do maths and where the similarities may lie even if the surface looks/sounds very different.
Overall, I think that Michaela are doing brilliant things. Students enjoy school, they want to succeed and everybody takes pride in themselves and their school. This is a great achievement by all involved. Expecting the best (“double the minimum”) and reenforcing this message means that students push themselves and have a positive attitude towards this test. I would like to look more into the effect that extrinsically motivating young children can bring about intrinsic motivation as this is something I had an opinion on before but after yesterday, I wonder whether all students can actually generate their own high standards after being held to them by punishments and rewards for a number of years. Certainly, all students I met in an informal setting (lunch, after school) were very polite, interesting and proud. All values that Michaela School has.
I would like to appreciate Michaela for letting me in for the afternoon, for showing me that great teaching relies on a great attitude to teaching and for setting a high expectation of all students.
It is impossible to read the book as an educator without thinking about its impact upon the classroom and the little things we do in order to create an intrinsically motivated culture. I am particularly interested in this to discuss the impact that written and oral feedback can have as well as our questioning in the classroom.
Pink proposes there are three things to strive for in order to promote intrinsic motivation:
- AUTONOMY – “the right or condition of self-government”
- MASTERY – “comprehensive knowledge or skill in a particular activity”
- PURPOSE – “the reason for which something is done”
In the classroom, students will be motivated if they focus on these three areas. In particular, teachers can direct student focus to these three things through their feedback and questioning. Sarah Donarski has written a blog relating specifically to feedback and its motivational responses – I will try and take some ideas further, specifically in the context of Pink’s trio.
Autonomy
Autonomy in the classroom can take multiple forms. Pink argues that autonomy will improve engagement and will take over from compliance in the workplace – the same can be said of the classroom. With a truly autonomous student, a teacher can be confident that there is a prominence of engagement and a desire to carry out actions because they want to do them. Jang, Reeve and Ryan found in 2005 that high autonomy was one of the most important characteristics of a “satisfying” learning experience and low autonomy had an even more negative effect on the experience.
The question then remains – how do we promote autonomy in our classrooms? Educators can create situations which require autonomy as much as possible. For example, an activity might require students to make a choice at the start and justify. We also must ensure our tasks are challenging enough that students want to engage with them. As students progress through school and are more skilled at making decisions, we may also set tasks which allow for preference and encourage students to think about why they are making these decisions to choose which activity. We cannot let the classroom become a free-for-all but we can slowly introduce these ideas as students are ready for them. The same can be said for classroom dialogue. Teachers can be flexible and allow a more “free” classroom.
With a feedback hat on, which feedback allows autonomy to grow? Specific feedback with some ideas on how to improve on these specific topics in the classroom give this chance. Darren Carter (@mrcartermaths) has spoken previously about his homework (or lack thereof) “policy” and it strikes me that this is a great way to inspire intrinsic motivation. Of course we know better than the students about what they should improve and how they can go about doing this. This information should still be shared but we are allowing them to decide what to do and explaining why (which encroaches upon number 3). Spend time with students showing them excellent online resources; picking a specific chapter in a specific book or write some feedforward questions which allow immediate improvement. There is no expectancy of completion but all students realise that being active with feedback will result in improvement. Thus an intrinsically motivated action has some extrinsic reward also.
Mastery
This is particularly prevalent at the moment in the mathematics world but in this discussion, we are not talking about deeper knowledge about less subjects – we are interested in the idea that students feel better at a specific discipline. All children want to be really good at stuff – this is not up for debate – whether it is maths, English, sport, dance whatever. Everybody wants to be good. Teachers must tap into this innate part of a student’s make-up. How can we do this through feedback? We must be positive and we must be specific with this praise. See below for a tweet from Ben Ward I saw this week (@mrbenward).
“We remember criticism because it is specific and personal.
Whereas encouragement is general [so it] washes over us.
Aim for ‘precision praise'”
I love the idea of precision praise. It is a big part of sports coaching and every course I have been on in this domain has focused heavily on generic praise and its pitfalls (namely that nobody acts on it and it is wasted energy). Precise praise can mean a student knows they are further along the journey in mastering a topic than they were before. Specific praise on something you have asked them to improve in the past will have the added bonus of showing them that their choice of work has worked and been recognised (their autonomy is improving too). Too much praise can be a negative but using praise in the right scenario in a very specific context will improve student’s internal motivation and reaffirm their belief in themselves.
In Sarah’s blog, she examines the idea of a positivity bias in which students focus on the good things you say or see overly positive messages in circumstances which might not be wholly positive. She proposes that this can be a good thing for students and again specific, precise praise can let a student know that there is positives in what they are doing. This can only be a good thing at all ends of attainment.
Purpose
How many times have we heard “When will I ever use this?” about almost everything taught in the maths classroom? The answer of course is that almost everyone will not use the sine rule nor the area of a trapezium nor differentiating trigonometric functions from first principles outside of their maths lesson. In much the same way that students will not analyse the meter of a poem many times after GCSE English nor testing the pH of something. The point that students aren’t getting is that all of these should be ends in themselves. Carl Hendrick (@c_hendrick) has written a piece recently looking at the idea that education should not be a vehicle to prepare us for what comes after school. At the end he writes,
“Students should study Shakespeare not because of what job it might get them but because it’s an anthropological guidebook that tells them how to live.”
This same sentiment should be held by teachers in all subjects. You are not learning about pi because it will help you in some 21st century job yet to be created, you are learning about pi and its place in history because it shows you something remarkable which was at one point undiscovered. Yes, you will be able to use google maps to tell you how far away something is – that isn’t why we teach triangles but you should have some appreciation of size, number and shape. It is important that teachers highlight this purpose throughout all of their feedback and discussion. Teachers must live the idea that everything taught is purposeful and should not find themselves justifying existence.
Moving forwards, I aim to incorporate these three ideas of internal motivation through all my student interaction. Any feedback given should look to promote at least one of these areas. Remember that feedback should be more work for the receiver than the giver.
The day began as usual with Mark and Andrew Taylor from AQA discussing the day and explaining the processes. It is always inspiring listening to Mark talk about teaching mathematics and he makes it all seem so simple. His message came across very effectively.
Speed dating bought with it some good conversation about raising morale, discussing interleaving and talking about Harkness to anyone that would listen. Please do talk to me about/come and see the teaching philosophy if you are at all interested. A familiar feeling then ensued where I would have liked more time to talk with fellow maths teachers but time meant we moved on to listen to the first speaker of the day: Michaela’s Dani Quinn.
Drill and Thrill (@danicquinn)
Firstly, I must say I really enjoyed listening to Dani talk and I was captivated by her energy, passion and clear desire to be a better educator. Her students are incredibly lucky – I look forward to coming and visiting the department as well as the school.
Dani was talking about “drilling” students with a particular focus on pre-year 9. She explained the processes used and the point behind them with a clear link to the relevant research. The underlying messages were:
I was keen to hear how drilling is incorporated after reading a few of Dani’s blogs. She is certainly not against conversation and engagement and lively classrooms but she was saying that when used at the right time, drilling can improve that instant decision making that we take for granted. A good example was: will this addition give me a positive or negative number? Students may have 20 questions to answer on that. There is usually some competition: quickest one or most correct in a short time. It is purposefully tense and the point is that students should be accessing their long term memory and not using their working memory.
Drilling should be used to:
Dani progressed to discuss implementation and some how to’s and how not to’s. We all make mistakes and Dani was not trying to say that Michaela are perfect by any means and she gave some advice on what a good problem sheet looks like and what a less good one might be. There are risks to drilling and again Dani addressed these with positivity and advice.
I certainly feel like I will go away and think about how deliberate practice can better be implemented in mathematics and in my classroom. I really like the idea of making an individual decision. This idea can be perfected at all levels and I will have a think over the next week or so about the best way to use “drills” in particular with my exam sets. I would like to see happy students who are happy that they “know” the cube roots and square roots for example. I’m not sure I will get them doing a mexican wave though – maybe that’s next year’s task!
The key thing to take: PRACTICE MAKES PERMANENT.
Where Y11’s will go wrong in the maths GCSE…and what you can do about it now
After Dani’s excellent opening, I was treated to one of the more omminently titled workshops I have attended lead by Craig Barton (@mrbartonmaths). One of my all time favourite twitter maths people and his podcast is excellent – well researched and heavily evidenced. I was looking forward to this after attending guess the misconception nigh on a year ago.
Craig’s website has become part of my teaching this year and student feedback is positive – this has reminded me to re-push the students after their mocks to get back and do some maths every day. Craig spoke very well, awesomely confident and as Dani had done, gave off an aura of passion and dedication to the profession. Craig highlighted the top 10 most wrongly answered questions and went into detail on three:
Throughout the three, he referred to what he and his department will do with these findings. His new look website is a staple and that links to a lot of Don Steward’s resrouces which Craig will use to commence purposeful practice.
The key from this talk was that all students benefit from purposeful practice. Purposeful practice can mean some students go from no knowledge to some; some go from some to lots; some go from lots to more. Nobody knows everything about a topic and “overlearning” has many benefits. Craig is clearly evidence-based in his decisions and he has shared his expertise with us all in attendance today. The key things with purposeful practice are:
We must cover key concepts before we can expect students to get better at exam questions.
Craig mentioned this towards the end of the talk and it reminded me of something I heard a few conferences ago: “nobody gets cleverer by being tested”. This is absolutely true (“Minimal guidance doesn’t work”) and we cannot throw past papers at students and expect them to get better. This was great advice at this time of the academic year and has made me refocus my mind on how best to use the remaining teaching time. There is some merit in past papers but don’t just throw them at students and expect results to improve. Learning is not happening.
A fascinating session with Craig and the bar has been set superbly high from the morning. I am now looking forward to lunch and then sessions this afternoon with Andrew Taylor and Kris Boulton.
New GCSE Maths Exams (@aqamaths)
It takes a certain sort of someone to enjoy a session about exams and what constitutes which mark and why questions are how they are. I am that sort of person so I had a great 50 minutes listening to Andrew Taylor discuss the new GCSE exams and why AQA have taken the decisions they have.
Andrew eloquently went through the different types of problem encountered at GCSE and discussed where marks can be gained and what we would expect to see for those marks. There is an emphasis on finding marks not taking them away and rewarding students. This means that some notation mistakes aren’t penalised and examiners are expected to translate without something necessarily written down word for word.
The big decision taken by AQA with the new spec is that there are less marks for doing numeracy or carrying out basic operations. For example, in a problem solving question, the markw ould be for getting to 4x + 1 = 13 and no extra mark for 4x = 12. There will be other questions which tackle this skill.
At Wellington, we teach to the Edexcel board but there is lots to be gained I feel from talking about the final assessment. There are also some awesome questions written by all the different boards and it cannot be a bad thing to see more of these!
The key message from this sessions was that:
Discuss methods with students and ensure they understand what a correct method looks like and why that is so.
One more hit of caffeine and then on to listen to Kris Boulton (@kris_boulton) – another of my favourite tweeters.
Most of the things you think are processes probably aren’t
I have read a lot of Kris’ work on Engelmann and his thoughts but have never heard him talk about him and how he uses the theory of instruction in his own teaching. After listening this afternoon, I wish I had! Kris always speaks well and this was certainly no exception. He explained the theory well and it was easy to see why continuous conversion and minimal difference will work in many contexts.
Today’s discussion was centred around a type of concept which Engelmann calls “transformation”. Simply put, this is any concept which takes an input and something happens to create a new output. Kris used examples of the distributive concept of multiplication and finding the area of shapes. At a basic level, the students see your answer to the question you are asking multiple times and then infer a correct answer to a new question based on patterns. I can see how this works in all creation of resource and I don’t think it has to be done in the explicit my turn, your turn way that Kris modelled (although this looks hugely effective). For example, in our Harkness books, we have examples of showing examples and then ask students to complete a similar problem. The next sheet can then be initial assessment. The next sheet would then be the expansion phase and seeing the same concept in different scenarios. The key here is that differences must be minute. As small as changing one number.
Towards the end, Kris mentioned two things which will sit with me forever.
The phrase: Logico Empirico. Loosely this refers to think of something, try it, adapt it, try it, adapt it. If I could sum up my approach to teaching, it would be this.
The sameness and difference principles related to maths problems.
Sameness relates to questions which look markedly different. The cognitive discussion students have is to find the similarities. Difference then is questions which look really similar, students ask what is different and how does this affect the output?
Thank you very much, Kris, for an inspiring finish. Not sure where I will find 180 quid for Engelmann’s book though!
And that was that. The final curtain on another wonderful day. Many thanks to La Salle and Mark McCourt. Thank you to City Academy. Thank you to everyone I listened to and everyone I had a conversation with about teaching maths. Every single moment was a learning opportunity. I will go home now with a head full of ideas and a lot of stuff to read up on.
Well done all!
At the start of this particular lesson, students came up with their ideals for a Harkness discussion and this was put on the board – I am rapidly coming round to Eddie Jones’ idea of ‘showing perfect’ and by leaving these up on board, subconsciously the students are holding themselves to account [a crucial part of a Harkness discussion]. The teacher then listed three questions to look at and gave the students 10 more minutes preparation time to gather thoughts and perhaps rewrite things they had already done before the lesson. The next 20 minutes or so was a full group discussion on torture, the morality of it and what this one source had to say about justifying it etc.
The discussion was chaired by a pupil (something I have previously blogged about) and the teacher sat back and took notes. The impact of the teacher doing this meant when he said something, it was received, students recognised it as important and all were tuned in. Having listened to David Didau discuss findings on feedback and the impact too much can have, this really resonated with me and it was done well in this lesson. One piece of feedback to clear up a misconception (after the students had had time to interject) and a couple to bring the conversation back to the key points of the questions.
The students themselves held themselves accountable to what had been put on the board as a checklist at the start of the lesson. All built on others’ points and agreed/disagreed and they held each other to that too. There were times when quiet students were bought into discussion by their peers – how much more powerful is this than a teacher ‘picking on’ them?
I thoroughly enjoyed the dynamic and it showed me that whether the students have done work before or not; whether the resource is any good; whether you have 3 students or 103; the crucial part to get right is the culture of the classroom. All the students here felt comfortable to express and never under any pressure. There were times a student was asked for their response and a reply would be ‘I have nothing to say now’ only for that student to come back in when they were ready. No judgement from others, just a recognition that each student will contribute when they are happy. Talking to some students at the end, they enjoyed the idea of being challenged by other students as opposed to the teacher. It was encouraging to here this as positive of the lesson – it either reaffirms one’s own belief or it perhaps gives another perspective.
The students found the lesson enjoyable and my current fascination with motivation leads me to believe these students have furthered their intrinsic drive. There was no feeling of them carrying out this discussion for anyone’s benefit but their own. This can only come from the environment created and the time and space given to each student individually.
Before we carry on, these students aren’t professionals – I do not expect them to know what is the greatest impact on their learning but it is illuminating to ask them every now and then how they feel and what is working. It will certainly effect a few changes in their classroom this term. I also know that pupil feedback is limited because of this (a more in-depth blog on this here) but for the reasons above, let’s see what the group of 13-14 year old mathematicians think of this approach.
The students were asked two questions:
To summarise, after ‘coding’ the verbal responses, students think the ‘discussion’ aspect of Harkness lessons is the most important part of their learning experience. They also believe that seeing and completing solutions on the board is beneficial to their learning.
What do students think of talking about maths?
“Good communication between students”
“I like the way that we express our opinions about other people’s work in a constructive way”
“When Mr. McDonald comes over to explain just to me or my table”
“We get a lot more opinions from the class”
Above are a few of the anonymous comments relating to the discussion aspect of a Harkness classroom. I particularly think point 3 is a great benefit of this approach. There is far more time for student-teacher conversations on a more personal level while others are having a discussion. This gives students an immediate feedback and means I can tailor follow-up questions as much as possible. Secondly, the last quotation is the one I was most surprised with. I had envisaged students complaining that there were too many opinions and I wasn’t letting them know which is right or best or whatever. In fact, this student believes the opposite. I love the use of the word opinion as it shows that this student is happy to challenge and question others. Small groups of four students is nothing new but I really try and challenge myself to be economical with my language. Every sentence I say must be impactful. Every sentence that a student says has to help someone, if nothing else to improve their confidence.
Why do they like working on whiteboards?
“I find it a really good way to learn from our mistakes.”
“I think writing up all the answers on the board is a unique and valuable approach…this leads to good discussion”
Above are the two main positives to come from students’ opinions of the work on boards. To reiterate, students put all solutions to the work completed pre-lesson on the board. Spotting mistakes is framed negatively but I think this is meaning when someone else has got a correct answer. I was disappointed that no students commented on feeling more confident to put solutions on boards as this is something I am really striving for in the classroom. The students will challenge each other and some will put up a different solution next to one that is already up. I encourage students never to rub anything out. The second quotation leads to the discussion we have already mentioned and I like that students recognise the point of putting things on the board. It is not a gimmick and it is not to make us look good. We put things up for everyone to see and critique. This allows me to quickly assess for learning too and I am able to prioritise tables and students.
What do they not like?
“I want to do more work in lessons”
“More teaching from the teacher not all Harkness”
“Focus on one topic then move on”
Here we have the main negative themes from the students. Doing more work can come from one of two things in my experience:
The second drawback there is obvious and that is up to me to ensure I have ears all around the room so I can ask high-level questions for groups that have grabbed the concept well while others have. The first point is my biggest issue with the Harkness approach in general. It is really difficult to differentiate for the top-end. I have experimented in the past with student teachers and I will certainly revisit this but I am looking at how we can better do this. I think it is also important that the quality of discussion is monitored and good discussion is championed.
What impact will this have on my delivery?
I will go into this term with a renewed sense of enthusiasm after reading their insight. It certainly highlighted to me how important it is to these students that they have ownership over their learning. I will endeavour to produce more high-end material in class through my own questioning and through big problems related to the current topics. I will also ensure all students are getting involved with discussions and are contributing, whether that is through a solution, a challenge or a question. Harkness can work with a large group of students – it just looks different and the teacher must trust the students to be doing the right thing at the right time.
To put it simply, I cannot think of a person I would not recommend this book to. Educators, parents, leaders should absolutely read it but I will push students and others to read too. Everyone can learn something from Lucy Crehan’s fantastically researched, objective account on the ‘world’s education superpowers’.
Upon opening this book, I was interested in other schools, particularly internationally but now I would go so far as to say I am fascinated. Crehan manages to subtly compare and contrast education systems without putting one above another and without her own opinion coming through. The reader is encouraged to imagine themselves in these different scenarios and imagine whether they would send their own children, or work, there.
Backing up all of Lucy Crehan’s findings is an immense amount of research. Indeed there are more than 250 studies, journals, books referenced – all contributing to a stellar discussion on how students learn all over the world.
To finish, I will not go into details about the book – you must buy it to see that – but I will push the author’s clear fascination with motivations. She is adamant that culture is not the be all and end all but there is a theme throughout that motivation lies at the heart of solid education. This is the take-home message for me and one in which we can all act on. Parents can help. Teachers can help. Leaders can help. Students can help.
All in all, a fantastic book and one I hope to read into more and more in the new year. A great way to finish the year. Here’s to reading and learning more!
What is the motivation for responding to the previous blog written by an English teacher? Having observed a handful of Harkness lessons in different subjects, I can say that Harkness takes different forms depending on the teacher and on the subject. In particular, mathematics through Harkness learning looks and feels remarkedly different to a Harkness lesson on Macbeth or a translation of a text etc. I therefore aim to try and mimic Sarah’s plan to bottom-line Harkness teaching particularly from a maths teacher’s point of view. I encourage you to make the links between the two and see how it can be linked to your own subject, classes and environment.
Sarah highlighted when Harkness can be most effective – I don’t think this changes much regardless of subject. I am absolutely fascinated however by whether or not we need as much teacher-led time as was suggested. With no research to confirm nor deny, I am merely hypothesising but particularly in STEM subjects, I propose that with clever scaffolding of the resource, students can cope without being explicitly led down a path by a teacher’s voice. Instinctively, I feel that this could happen in English, for example, too but would require a different look to Sarah’s outline. In my head, it would take the form of a short worksheet asking students to a) find quotations and then b) find historical context and then c) … etc. I would like to push this subject further and research a little more. Sarah then went on to push the three pr’s of Harkness learning in her subject: preparation, practice and praise.
In mathematics, Harkness learning looks slightly different than that outlined in the blog. In lesson one, a student takes home a booklet of problem sets and gets asked to spend [insert time period here – 45 mins for me] on the first problem set before coming back to the next lesson. In the next lesson, we spend 15 minutes or so with students putting solutions on the board before talking through these and asking questions about where mistakes occurred or why things work. This would then continue for 4 problem sets. After this, the students have a short test on the material covered so far which is marked and given back to the students. This process is then repeated. For larger groups, we are slightly more creative – less time spent on problems to bring to class and some class time dedicated to problems involving the skills being learnt at the time.
Preparation
The skill from the educator’s point of view, then, comes from creating a resource which means the students can tackle problem sheet 2 after discussing problem sheet 1 and seeing correct solutions to those. This has been the toughest challenge for our department. The head of department has written most of the problems and the resource (the book of problem sets) is constantly evolving. An example for sheet 1 to sheet 2 would be:
“Compare three quadratics in how difficult they are to solve” in sheet 1
to
“Explain why [quadratic with imaginary roots] can’t be solved with the quadratic formula” in sheet 2.
We do not discuss the word ‘discriminant’ until sheet 5 but by then students are very comfortable with the idea that sometimes a quadratic equation will not have real solutions. Each problem set has multiple topics so students will look at quadratic equations, trigonometry, sequences, areas of a circle and much more in almost every sheet.
This is where the ‘preparation’ aspect is not aimed at the students but at the educators. Teachers must be prepared – indeed the resource has taken our department a long time to put together and it is changing all the time – and they must look forward to where each question leads. My advice for any maths teacher using this philosophy is to plan for concepts not for lessons. For example, I would like all of my students to understand a sequence which increases by a constant amount each term can be summed by finding an average and multiplying by the end of the first 4 sheets. When that happens does not matter but as long as it happens in that time frame, I am happy. This means teachers can probe individuals who may not be there yet and can let others lead who grab the ideas really quickly.
Praise
I have purposely re-ordered Sarah’s three because this is an area of mathematics learning which has to be at the top of every teacher’s list. Success breeds success and students are more likely to engage in the premise if they feel successful. Our job as the teacher in this environment is to ensure every student is successful and understands when they have been. Success in a maths discussion takes many different forms. It may be that a student has a correct solution, another student may have a more efficient solution and another student may ask a really good question about that solution. All three of these should be heralded by the teacher. The third student may have walked in without a solution to offer but by asking that question and being praised on it, they may well have enough knowledge now to have a go at the next sheet.
Success also takes the form of giving things a go. ‘Maths anxiety’ is very real and particularly in lower school the fear of getting something wrong will hold some students back from putting things on the board for all to see. Teachers must create a culture whereby everything that is up on the board is a chance for us to learn from. If it is not correct, let this be a chance for someone else to feel success by finding where the solution breaks down. It is then important, in my opinion, to let the original student have another go at trying to get the right solution with that mistake pointed out. You then have two people feeling successful and learning has taken place.
Praise in the Harkness classroom works very much in the same way as any maths classroom but you can create a genuine team-orientated classroom when praise comes at the right moment.
Practice
This happens less in mathematics. The practice element is more about refining their abilities to talk through solutions, which questions to ask, when to ask questions etc. This practice comes through each individual lesson. If we get through a sheet particularly quickly, I would actively reflect with the group on the lessons and talk about when learning takes place in the most effective way. Usually, students will bring up ideas of ‘good explanations’ or ‘clear diagrams’ and we will talk about the importance of these. Students are then held accountable and I would use these phrases with students as they are explaining or showing a solution.
Just give it a go!
Finally, I would encourage everyone to try this idea at some point. If you are trying to get students talking about maths, actively engage with challenging problems and have the chance to ask some real open-ended questions about mathematics, this gives you a fantastic springboard for that.
Harkness teaching relies on empowering students and letting them be in charge. Teachers can throw in probing questions when the time is right but trust the students and trust your worksheets to bring out the skills required. Don’t be afraid to let them run with a discussion and see where it goes. What’s the worst that can happen in one maths lesson?
The resource created at Wellington College is available. Please tweet me @jk_mcd or @AidanSproat [head of department and creator] for details. As always, if anyone wants to learn more, come and see it in action!
I have blogged previously about our approach at Wellington to Sixth Form Maths – following the Harkness philosophy where students come in with questions done and we talk about their solutions in class.
Usually these lessons involve me chairing a discussion and letting student A talk through his/her solution before the odd pupil chimes in. I then proceed to ask some question to try and touch on the conceptual ideas. “Why is it important to write the equation in this way?” “What does this mean we can now do?” “Why do we care how many roots this quadratic has?” etc.
This morning, however, I decided to remove myself from the lesson and watch. I told the class this was my intention and I would only intervene when they had confirmed they were happy with the discussion that took place and I had spotted some mistake.
What then followed was some brilliant student leadership – a couple instantly took it upon themselves to dictate pace and timings of the lesson. They decided to focus on 4 questions and the atmosphere was far less judgmental than it feels when I am involved. The students seemed far happier to interject and offer slight changes to solutions than when I am leading the discussion. I think this was probably because they are waiting for me to jump in sometimes.
I only had to step in on two occasions. One was to be slightly more rigorous with a proof and the other when a conversation was going round and round in circles. The second in particular was a valuable learning point for all about how to bring yourself back to basics when finding a problem with a mathematical question.
The problems with this lesson were that the conversations were very procedural. “I did this and then I did this” “But how have you got from that line to that line?” “Yes I got that answer too” “Did you do that on the calculator?” “Is my answer the same?” and so my challenge for them was that next time I want to listen to some conceptual conversation.
This is something I will almost certainly try with all my classes in the near future. I felt like an empowerment happened and the students surprised me in the quality of the discussion. I have since learnt that “this won’t work in maths – the teacher is too important” was an egotistical feeling more than a fact.
The day itself began with the usual coffee and chats with a couple of stands and I was particularly impressed with the number and quality of the stands present at this conference. A very good representation from all exam boards and some new exhibits that I hadn’t seen before – I am particularly intrigued by how Brix can benefit our students. We arrived quite late so there wasn’t much time before Mark McCourt opened up.
As a semi-experienced #MathsConf goer, I knew the routine of Mark introducing the keynote and then speed dating before starting the first session but this was slightly different in that the keynote speaker, Professor Mike Askew (@mikeaskew26), really captured my attention. I would listen to him speak all day.
Research into problem solving techniques
I will be as brief as possible but in short, Mike was discussing the current research into how students solve problems, what does and doesn’t work, ways in which we can incorporate this into our teaching and the journey between abstract and concrete and back again.
Some superb things came out of this talk. He emphasised the idea of testing students on topics they have learnt previously (not necessarily the ones being studied right now) before discussing, with evidence, the ideas behind ‘sleeping on’ a thought and remembering it better. His final introductory point looked at the impact of asking ‘deep explanatory questions’ and their impact on student learning. This was profound and if I take anything away from today, it will be that “we are a problem solving species”. It is in our nature!
Mike had us thinking about the questions we ask and the tasks that we set against an “indirect” criteria: is it improving fluency; is it improving problem solving or is it improving mathematical reasoning? What a brilliant checklist for every question you ask of a student. If you can answer yes to one of those, I believe it is worthwhile. If not, why are you asking it?
Shortly after, we look at Kahnemann and his ideas behind quick and slow thinking. There is research proving we are a little sad when we don’t get something quickly, is this why students push back? How can we fight this in the classroom? Teachers must think about the culture in their classroom and how they can aid their students in coping with this sadness that comes.
Mike finished talking about the ABC’s of problems (Burkhardt) and highlighted a key point. “The real skill of an educator is when it comes to facilitating the discussion of solutions”. I cannot agree more with a statement and his idea of private vs. public talk is one that anyone should talk to him about if you get a chance. Mike offered some really simple advice to further understanding when students are speaking in a public domain – repeat, rephrase, revoice, agree. In that order.
A great way to start the conference before a little speed dating!
Speed Dating and Mark McCourt
The speed dating is one of the highlights of the conference, always, and it did not disappoint this year. I picked up a couple of valuable teaching ideas and hopefully shared our Harkness resource with others to see how we are going about teaching A Level maths at Wellington. The excitement is palpable between all when this happens.
Mark (@emathsuk) then spoke himself. This was the first time I have heard Mark speak but after the reviews of his researchEd chat, I was not missing this. He always talks so passionately when hosting and the same love was evident this morning. He discussed the amount of hours teachers spend not teaching and ineffective ‘working’ hours. I could almost copy and paste his transcript but the major messages that hit home were that “every lesson has been taught before” and “if something isn’t working, stop it!”. This actually resonated a lot with me after listening to Clive Woodward say that we must put enjoyment at the top of our priorities and Mark is trying to encourage the same message.
We listened to Mark talk about the Singapore approach of teaching less and discussing teaching more. This strikes me as an ideal we will not quite reach in England but it does leave me thinking, how often do I sit and talk about teaching? What worked and what didn’t? Was it effective? How could it have been more effective? Mark showed us how Complete Mathematics is trying to help address the balance and regain some of that enjoyment which was exciting and showed that maths teaching does not need to be reinvented every year.
The overriding question Mark has burnt into my brain is “Am I being impactful?” This can be transferred to a whole spectrum of teaching. Marking. Resource design. Questioning. Testing. Is what I am planning going to impact the students’ learning? If not, why am I doing it?
16 Things about the new A-Levels
Straight after Mark, we went off to listen to Andrew Taylor (@aqamaths), Gary Wing from Hills Road VIth Form and Christine Andrews from AQA talk about the new A Level. This was fascinating and I always find it interesting to get inside the exam board’s head. We were given answers to the most common questions asked of AQA and were advised on topics such as large data sets, calculators and which papers to enter. All in all, a very helpful and important session to attend.
All 3 spoke excitedly about the new reforms and showed the positive side to the new specifications and how they will benefit students. I enjoyed listening to this side and I can certainly see the benefit myself now. Well done, all.
We then scooted off for lunch and enjoyed a more relaxed hour or so before the last two sessions with Danny Brown and Craig Barton.
Be aware of being aware
Danny Brown (@dannytybrown) really impressed me. I have followed him on twitter for a while now but listening to him speak showed me that this is an educator who cares. His students can count themselves very lucky.
This was one of the few workshops I have been at where not one person is tweeting about it or taking notes/photos. The whole message was about being present and recognising whether or not you are present when listening/talking/reading aloud. Danny is obviously well informed and he was a pleasure to listen to and discuss mathematics teaching with – I would certainly like to do so again some day. I took a lot from his session and I certainly want to implement a values-based teaching philosophy of my own moving forwards. What do I care about? How does that affect my decision making as a teacher?
Incidentally here, a discussion came about into how 1-2 year olds count and whether or not they know they are counting. Why do kids say “one, seven, two, six, three” before they know that these are numbers and why do they eventually work out the order in which these go? Truly fascinating and I look forward to reading more!
Guess the misconception
Craig Barton(@mrbartonmaths) closed the day for me with a fabulously engaging talk about students’ misconceptions. Craig’s website is fantastic and offers students a chance to give an explanation for why they selected the answer they chose. This offers tremendous insight to us as teachers and should be something that is seen more in classrooms across the country. Very rarely are students just plucking a number out; they have some sort of thought process. It is our job to find this out and act on it.
Craig showed us some data analysis from around the world – the Americans can’t do lower or upper bounds – and then looked at our own children’s issues. We were shown 5 questions that students struggle with (in the easier parts of the syllabus) and discussed why we thought these students had misconceptions.
This was a great, light hearted way to finish the day and it left me thinking about how often we delve into wrong answers and how often do I really probe and discuss why a student made a mistake. I should be asking more about what it is that made a student give a particular answer.
Overall, this was a superbly impactful day and, as always, left me excited about being part of the greatest profession in the world. This is a profession, we all care and we can all help each other. I will keep going to these events as long as time and means allow and I will always recommend them to others. Everything went smoothly and La Salle should be complimented on their part in this as I am sure it is not easy to organise something on such a scale while keeping the quality that high.
Well done all on a great day. See you all at #MathsConf8
I wonder how often we, as maths educators, write a resource or ask a question with the answer we want in mind vs. not having anything but the beauty of the subject in mind. What happens in both of these films is that the teacher conveys a complete love of the subject – their faces light up when a pupil starts arguing about Hardy’s use of adjective, for example. I wonder whether that happens enough in our mathematics classrooms. Do the educators truly convey a passion for the subject? Do all your pupils really know that you love maths? This shouldn’t just come up when you talk about irrational numbers or proofs or the real top end of your interest but it should come up when talking about similar shapes and scatter graphs and everything else.
It is with this in mind that I have gone about writing my latest set of resources for next year’s Year 9. With Year 11 and two upper sixth form sets gone, I have some time this Summer term to really knuckle down and concentrate on creating a quality resource. I have taken our Scheme of Work and gone through the relevant things I have to cover before half term. This gives about 10 different areas to go through. The general pattern I am following is to have all the topics combined on each sheet and teach them all incrementally. To carry out this process, we write 10 different problem streams and then combine the questions as we see fit ensuring that things are not introduced too quickly nor too early.
What does all that have to do with arguing and promoting passion? Well, when putting together the sheets, I have tried to include at least one question where all the students in the class will have an opinion. This is not an easy goal and has truly had me thinking about the “low threshold, high ceiling” idea that Craig Barton (his rich maths tasks do this fantastically) first instilled in me. The brilliant thing about this will be that when the students are having their discussions from these questions, I can embrace it and I will be a part of it. I will just be another part of the discussion, hopefully a truly passionate part. With the questions included (for example, should we include 0 as a counting number?), I don’t know any more than a 14 year old, we can have a live discussion.
I also hope to achieve the dream goal of getting away from “is this right?” or “will this be in an exam?” by asking questions on each worksheet which are so obviously not examinable that the students can just try. Robin Williams in “Dead Poet’s Society” has a wonderful phrase whenever a students answers of “Thanks for playing!” and rings an imaginary bell. It doesn’t matter what the answer was, the positive feedback is for answering the question. This is the sort of thing that I hope will push the students to truly love the subject and it is nothing to do with the methods they learn or the numbers they meet. It is do with the subject itself and the history of maths and all the things that are constantly changing. For example, how would Deep Blue the Computer write “100” in comparison to Tutankhamun…which is more intuitive? Which is easier to understand? What problems are there with these number systems?
I encourage all to argue more and let your students discuss stuff. It might be complete rubbish but there will be one statement from one student that may just remind you why you love the subject so much. Here’s hoping!