Does the Harkness Method improve our understanding of Maths?

Elena and Amelia, Y12 Further Mathematicians, explore how the Harkness Method has opened up a new way of thinking about Pure Maths and how it allows them to enhance their mathematical abilities.

For Further Maths A Level, the Maths department has picked a new style of teaching: the Harkness Method. It involves learning by working through problem sets. The problems give clues as to how to get to the answer and this is better than stating the rules and giving examples; we have to work them out ourselves. These problem sets are given for homework, and then we discuss them together during the next lesson by writing the answers on the board and comparing our results with each other.

Elena:

At the beginning of term, I found it quite challenging to complete exercises without knowing what rules I was expected to apply to the problems, as each question seemed to be completely different to the one preceding it. The tasks also require us to use our previous GCSE knowledge and try to extend it ourselves through trial and error and by applying it to different situations and problems. I found it difficult to understand how to apply a method to solve different problems as previously each problem came with a defined method.

Maths diagrams As the lessons progressed, I started enjoying this method of teaching as it allowed me to understand not only how each formula and rule had come to be, but also how to derive them and prove them myself – something which I find incredibly satisfying. I also particularly like the fact that a specific problem set will test me on many topics. This means that I am constantly practising every topic and so am less likely to forget it. Also, if I get stuck, I can easily move on to the next question.

Furthermore, not only do I improve my problem-solving skills with every problem sheet I complete, I also see how the other girls in my class think about each problem and so see how each question can be approached in more than one way to get the same answer – there is no set way of thinking for a problem.

This is what I love about maths: that there are many ways of solving a problem. Overall, I have grown to like and understand how the Harkness Method aims to challenge and extend my maths skills, and how it has made me improve the way I think of maths problems.

Amelia:

When I first started the Harkness approach for Pure Maths in September, I remember feeling rather sceptical about it as it was unlike any method of learning I had encountered before. To begin with, I found it slightly challenging to answer the questions without knowing what topic they were leading to and found confusing how each sheet contained a mixture of topics.

However, I gradually began to like this as it meant I could easily move on and still complete most of the homework, something which you cannot do with the normal method of teaching. Moreover, I found it extremely beneficial to learn the different topics gradually over many lessons as I think that this improved my understanding, for example for differentiation we learnt it from first principles which gave me the opportunity to comprehend how it actually works instead of merely just remembering how to do it.

Furthermore, I think that the best part of the Harkness Method is that you are learning many topics at a time which means that you cannot forget them as compared to in the normal method which I remember finding difficult when it came to revision for GCSEs as I had forgotten the topics I learnt at the beginning of Year 10. I also began to enjoy the sheets more and more because the majority of the questions are more like problem-solving which I have always found very enjoyable and helpful as it means you have to think of what you need to use instead of the question just simply telling you.

Moreover, I very much enjoyed seeing how other people completed the questions as they would often have other methods, which I found far easier than the way I had used. The other benefit of the lesson being in more like a discussion is that it has often felt like having multiple teachers as my fellow class member have all been able to explain the topics to me. I have found this very useful as I am in a small class of only five however, I certainly think that the method would not work as well in larger classes.

Although I have found the Harkness method very good for Pure Maths, I definitely think that it would work far less well for other parts of maths such as statistics. This is because I think that statistics is more about learning rules many of which you cannot learn gradually.

The importance of collaborative learning

How can we encourage collaborative learning? Alex Farrer, STEAM Co-ordinator at Wimbledon High, looks at strategies to encourage creative collaboration in the classroom.

Pupils’ ability to work collaboratively in the classroom cannot just be assumed. Pupils develop high levels of teamwork skills in many areas of school life such as being part of a rowing squad or playing in an ensemble. These strengths are also being harnessed in a variety of subject areas but need to be taught and developed within a coherent framework.  Last week we were very pleased to learn that Wimbledon High was shortlisted for the TES Independent Schools Creativity Award 2019. This recognises the development of STEAM skills such as teamwork, problem solving, creativity and curiosity across the curriculum. Wimbledon High pupils are enjoying tackling intriguing STEAM activities in a variety of subject areas. One important question to ask is what sort of progression should we expect as pupils develop these skills?

The Science National Curriculum for England (D of E gov.uk 2015) outlines the “working scientifically” skills expected of pupils from year 1 upwards. Pupils are expected to answer scientific questions in a range of different ways such as in an investigation where variables can be identified and controlled and a fair test type of enquiry is possible.

However, this is not the only way of “working scientifically”. Pupils also need to use different approaches such as identifying and classifying, pattern seeking, researching and observing over time to answer scientific questions. In the excellent resource “It’s not Fair -or is it?” (Turner, Keogh, Naylor and Lawrence) useful progression grids are provided to help teachers identify the progression that might be expected as pupils develop these skills. For example, when using research skills younger pupils use books and electronic media to find things out and talk about whether an information source is useful. Older pupils can use relevant information from a range of secondary sources and evaluate how well their research has answered their questions.

The skills that are used in our STEAM lessons at Wimbledon High in both the Senior and Junior Schools utilise many of these “working scientifically” skills and skill progression grids can be very useful when planning and pitching lessons. However, our STEAM lessons happen in all subject areas and develop a range of other skills including:

  • problem solving
  • teamwork
  • creativity
  • curiosity

Carefully planned cross-curricular links allow subjects that might at first glance be considered to be very different from each other to complement each other. An example of this is a recent year 10 art lesson where STEAM was injected into the lesson in the form of chemistry knowledge and skills. Pupils greatly benefited from the opportunity to put some chemistry into art and some art into chemistry as they studied the colour blue. Curiosity was piqued and many links were made. Many questions were asked and answered as pupils worked together to learn about Egyptian Blue through the ages and recent developments in the use of the pigment for biomedical imaging.

There are many other examples of how subjects are being combined to enhance both. The physiological responses to listening to different types of music made for an interesting investigation with groups of year 7. In this STEAM Music lesson pupils with emerging teamwork skills simply shared tasks between members of the group. Pupils with more developed teamwork skills organised and negotiated different roles in the group depending on identified skills. They also checked progress and adjusted how the group was working in a supportive manner. A skill that often takes considerable practise for many of us!

Professor Roger Kneebone from Imperial College promotes the benefits of collaborating outside of your own discipline. He recently made the headlines when he discussed the dexterity skills of medical students. He talks about the ways students taking part in an artistic pursuit, playing a musical instrument or a sport develop these skills. He believes that surgeons are better at their job if they have learned those skills that being in an orchestra or a team demand.  High levels of teamwork and communication are essential to success in all of those fields, including surgery!

Ensuring that we give pupils many opportunities to develop these collaborative skills both inside and outside of lessons is key. We must have high expectations of progression in the way that pupils are developing these skills. Regular opportunities to extend and consolidate these important skills is also important. It is essential to make it clear to pupils at the start of the activity what the skill objective is and what the skill success criteria is. It is hard to develop a skill if it is not taught explicitly, so modelling key steps is helpful as is highlighting the following to pupils:

  • Why are we doing this activity?
  • Why is it important?
  • How does it link to the subject area?
  • How does it link to the real life applications?
  • What skills are we building?
  • Why are these skills important?
  • What sort of problems might be encountered?
  • How might we deal with these problems?

Teacher support during the lesson is formative and needs to turn a spotlight on successes, hitches, failures, resilience, problems and solutions. For example, the teacher might interrupt learning briefly to point out that some groups have had a problem but after some frustrations, one pupil’s bright idea changed their fortunes. The other groups are then encouraged to refocus and to try to also find a good way to solve a specific problem. There might be a reason why problems are happening. Some groups may need some scaffolding or targeted questioning to help them think their way through hitches.

STEAM lessons at Wimbledon High are providing extra opportunities for pupils to build their confidence, and to be flexible, creative and collaborative when faced with novel contexts. These skills need to be modelled and developed and progression needs to be planned carefully. STEAM is great fun, but serious fun, as the concentration seen on faces in the STEAM space show!

Twitter: @STEAM_WHS
Blog: http://www.whs-blogs.co.uk/steam-blog/