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.

 

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!

 

 

 

 

 

 

 

 

 

 

 

 

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How to Harkness (a maths teacher’s view)

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