Harkness at Wellington

Now that Torchwood is a distant memory, presumably there won’t be too many readers here by mistake. The ‘Harkness’ in the title refers to the Harkness Method of teaching that developed at Phillips Exeter Academy in the US. To cut a long story short, a benefactor (by the name of Harkness) donated sufficient funds for the college to develop an approach to teaching that he thought ideal. In its purest form, the classroom contains an oval table with a dozen chairs around it. But it’s not so much about the table.

Maths the Harkness Way

To quote from Phillips Exeter’s own page, students taught in this way

are exposed to problem solving in a very student-centered, discussion-based classroom. Students are held accountable for attempting solutions to homework problems and the class as a whole decides on correct solutions.

How does this come about? The homework that students complete comprises a set of problems that are to be tackled in advance of a lesson. Then, during the lesson time itself, they present their methods/solutions and a discussion ensues where misconceptions are tackled, misunderstandings are explained and, for example, the efficiency of different methods is compared. The teacher’s role in this, to grossly oversimplify things, is to guide the discussion, direct the students’ questioning to other students, and to tease out interesting ideas that are worth discussion.

A few people that I’ve talked to about this approach have thought that it’s just “flipped learning” by another name, but I would disagree. Flipped learning, as I understand it, still takes the linear topic-by-topic approach and still explains to students (albeit by video etc) “this is how you do X, Y and Z”. Then class time is spent with the students doing X, Y and Z with support and direction from their teacher.

In contrast, through the Harkness approach, the students are developing their own sense of mathematics through the tackling of problems. Moreover, several themes are developed concurrently.

Teaching Resources

Clearly, a significant factor in the success of such an approach lies in the problem sets themselves. Phillips Exeter have for a long time freely shared their resources, aligned to the American curriculum. At Wellington, Aidan Sproat (@aidansproat) along with colleagues, has worked hard to create appropriate problem sets for the core modules of the UK A level curriculum. Aidan is more than willing to share these resources and has set up a web page where you can access samples, or contact him directly for the full versions.

The Lesson Routine – Beginning

The first thing that struck me is how the students are fully comfortable with the routine and are busy as soon as they enter the classroom. They cooperate in terms of clearing the whiteboards (the classrooms I saw each had a good number of whiteboards – small and large – around the walls) and sharing out pens and board rubbers. Individually, they would elect one of the problems from the set they had worked on prior to the lesson and wrote up their work on a board. I didn’t time the whole process, but I would estimate no more than 10 minutes were taken up of the hour lesson. That 10 minutes is invaluable to the teacher for surveying the work going up on the boards, taking a register, recording which students have contributed solutions etc. Moreover, the class organise themselves well in terms of having notes, pens and calculators out from the very start.

The first time you see one of the problem sets, you will be struck by the variety of topics it covers. The second lesson I observed was a lower sixth Further Maths class. Their problem set comprised 7 questions covering:

1. The chain rule for functions of the form $(ax+b)^n$
2. Creating functions with a given domain and range
3. The chain rule for functions of the form $e^{f(x)}$ (and also including $x^e$ as a function to be differentiated)
4. Exact trig values for an obtuse angle, given $\sin\alpha$
5. Parametric equations that had to be explored, essentially to determine why they were not defined on part of the specified domain
6. The magnitude of 2d vectors (given a definition to work from)
7. A proof of the $\cos(A+B)$ and $\sin(A+B)$ results using matrix multiplication (of matrices representing rotations)

(Note: Aidan explained to me that following the staggered changes to A level subjects, Wellington have opted for no exams in the lower sixth in any subject. This frees up a considerable amount of teaching time! Thus, students in that year will make good progress on A2 study before the summer.)

The Lesson Routine – Middle

Once students were settled and solutions on display, the first order of business was asking the students which problems they wanted to discuss. These were listed on the projector (a Microsoft Surface and a projector with a wireless HDMI adapter makes for a very efficient set up!) and then Aidan added his own request to discuss the second problem. A student had a quick question she wanted to ask about number 4 and that was duly added to the list, too.

The majority of the class time was then spent with a very active discussion on the part of the students: whoever had written the solution explained their approach, and then questions were directed between students. The teacher’s role in this is a delicate one: orchestrating the discussion without jumping in too quickly; taking care to ensure the mathematical accuracy of points discussed (but again through questioning more than telling).

One of the most interesting discussions (from my point of view) was the exploration of the parametric equations. I’m not doing the original question justice here, but its mathematical essence was:

Explain why $x(t)=2t+3,\, y(t)=\sqrt{(2t-1)(t-1)}$ is not defined on the whole interval $t\in[0,5]$.

Points that came up included: is it possible to square root 0? Can we form a Cartesian equation for this curve? I tried every number from 0 to 5 but didn’t find any problem (by which they mean they tried every integer – a nice aspect of the problem); It came to light nicely that the values of $t$ for which it was not defined corresponded with the values of $x$ for which the Cartesian equation was not defined. Moreover, the skill of sketching $y=\sqrt{f(x)}$ from the graph of $y=f(x)$ also arose naturally.

Question 3 was also fascinating for me. The way $e^x$ is introduced to the students is a whole blog post in itself. But by the time they reached this problem set, the students were piecing together knowledge developed over previous weeks so that they could combine their understanding of the chain rule with the fact that $\frac{d}{dx}e^x=e^x$ to differentiate functions such as $y=e^{x^2+3}$ without any explicit instruction.

The Lesson Routine – Ending

In the remainder of the lesson time, the students were invited to complete a few problems from the ‘review’ section of their problem sets. These are designed to give them extra practice: to reinforce what they have already covered and to prepare for a test which takes place once a fortnight. During this time, the teacher dealt one-to-one with the student’s question about problem 4 but, again, through a careful process of questioning and relating the problem to earlier work that had been tackled.

So, does it work?

Desperate as we are to simply ask ‘does it work?’, that’s not really a well-defined question. Even to ask if it is a successful approach to teaching the subject, you need to define some success criteria. Mathematics is renowned for being a ‘chalk and talk’ subject: “Here’s a topic, here’s more or less how it works, here are several examples of all the main variations that can come up. Now you try a few.” In my teaching, I try hard to include ‘warm up’ activities that prompt students to retrieve prior knowledge and push them to try and tackle a new problem before the spoiler of the main part of the lesson. Going all out for a full Harkness approach is a brave move and I cannot imagine how much time and effort has gone in to the planning of the problem sets. As I alluded to above, for example, the development of $e^x$ as a function is incredibly well-crafted and it takes place through individual problems over the span of many sets.

What this approach certainly does achieve, and what I particularly like about it, is the community of students discuss mathematics together and they are able to explain their ideas (and indeed their questions) very coherently. Questions came up occasionally such as “do we need to learn this?” (formula for $\cos(A+B)$) or “should we copy this?” (a student’s protracted and not completely correct working on a problem) but, in fact, the students demonstrated very good independent and discerning study skills.

What about the student that doesn’t do the homework? To be honest, that’s not a mathematics teaching issue. That’s a pastoral concern.  If anything, this Harkness approach makes it very clear to students that to study mathematics, they must come to class having made the best attempts they can on the full problem set. To prepare for their fortnightly tests, they must ensure they are confident with all the problems and they can use the review sections of problems to practise. As in every department, additional support is available outside of lesson time.

Am I a Convert?

As I already mentioned, I like to think I have a significant amount of problem-solving in my teaching already. However, I am fascinated by this approach and I think many of my students would get a lot out of being taught in this way. It’s early days for me: after observing it ‘in action’, I have been able to answer many questions I initially had but only for them to be replaced with new thoughts and questions.

In July, Wellington are holding a training course led by instructors from the Exeter Mathematics Institute. I was initially surprised by the fact that this runs over four days, but with so much involved in the development of the material and so many subtle details in how to lead the classroom, I think it will be an intensive but fascinating experience. I can’t wait to learn more!