Culture trumps all

Having recently observed a Harkness lesson outside of my comfort zone, I was blown away by the environment the teacher had created in his classroom. Indeed, part of the feedback conversation went down the route of “sorry, you didn’t see much of me there” and I instantly had to highlight the positivity in that fact. What I did see was a class of students who were so comfortable talking, listening, thinking and writing that the teacher did not need to put much into practice.

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.







Culture trumps all

What do students think of Harkness mathematics?

I write a lot about Harkness and its impact on my teaching and how it makes me think about mathematics but none of that actually matters. What matters is the impact it has on the students and their learning. I thought I would address this with a post based on recent pupil feedback in my year 9 class. The biggest criticism of the Harkness approach is that it won’t work with big groups. I am a staunch opponent of this but I do think things change and educators have to be a little more creative.

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:

  1. What has been most beneficial to your learning in your lessons in this subject?
  2. What would help you to improve your learning experience?

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:

  1. a really good student might not need to discuss very much
  2. Students don’t/can’t stretch discussions

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.







What do students think of Harkness mathematics?

Clever Lands: A first-class education insight


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!



Clever Lands: A first-class education insight

How to Harkness (a maths teacher’s view)

Before you carry on reading this, it is imperative you read Sarah Donarski’s blog about Harkness and how to implement it in the classroom. I aim to discuss the ways in which our Harkness experience is similar and how it is different in the mathematics department. I hope to open eyes to the creativity allowed and debunk the myth that it means the teacher does nothing but turn up, sit down and listen.

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.



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


“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.


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.



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!













How to Harkness (a maths teacher’s view)

`Today I will just observe’

Today was a pretty important day in my approach as a teacher of lower sixth mathematics. It was the first time where I can genuinely say I planned for and observed for learning.

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.




`Today I will just observe’


Edfest Thursday, Edfest Friday morning and then a train up to Leeds for my third straight day of CPD can only mean one thing…it must be time for a maths conference! My third of the year (and ever incidentally) and I still have the same emotions leaving this one as I did after leaving Sheffield in September last year. Getting to Leeds from Crowthorne isn’t the easiest journey but 5 hours after departing, we landed in the hotel, had some food and prepared for the day ahead.

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







Let’s have an argument!

Through recent re-watching of “The History Boys” and “Dead Poet’s Society”, I came to realise that I am far too conservative in my approach to lesson planning and, more importantly, resource writing. The two films, for those that haven’t seen them, centre on eccentric History and English teachers who employ a teaching methodology inspired by argument and free thinking.

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!








Let’s have an argument!