I teach mathematics whilst sat in a circle. I sit at the same level as the students, they come to class with 45 minutes of work done and they talk all lesson. We don’t do a lesson on integration followed by a lesson on the trapezium rule; we don’t write our answers down and we advocate mistakes.
Above is a link to a blog I wrote a few months back about the “new” style of teaching that we are implementing at Wellington College. I write “new” because I don’t think it is a million miles away from an ideal that is based on general good practice. I won’t go into full detail here but we embrace the Harkness philosophy in the Wellington College maths department and it is something that I am 100% of the opinion is the best way to teach mathematics; no matter what level. I likened it to playing sport and this is an analogy I have seen extend even further now. We want our students to go and play. We want to create an environment whereby students feel no step is embarrassing and not doing anything is worse than doing the wrong thing.
Our Harkness course takes all the subtopics of the A-Level and pupils explore these all together. We do not have a week teaching calculus then a week teaching series then a week teaching graph transformations; we have two terms teaching maths. For example, if I open our C1 and C2 book to a random page, I will see Q1 about the binomial expansion, Q3 about coordinate geometry and Q5 relating to the discriminant. On the next page, the students will revisit similar ideas and improve their knowledge in each of these areas. Topics within mathematics are not discrete and as educators, it is not our job to make boundaries between them. Let’s see how coordinate geometry and finding the area under the curve are linked; let’s not call that Chapter 5 and Chapter 7 from the textbook.
Learning new things just means extending prior knowledge a little bit more. Without prior knowledge, we can’t learn anything. At this point, I cannot answer why World War 1 finished when it did but if someone built it up slowly while looking at other wars, I’d be confident that eventually I could coherently form a conjecture about its culmination. This is what we are trying to achieve, we know how to find the area of a triangle, so let’s use that fact to find the compound angle formula for sine. Why not? We know how to use similar triangles, heck let’s use that knowledge to find out about the secant and cotangent functions!
Last Summer I was lucky enough to be a part of a course delivered by the EMI (http://www.exeter.edu/summer_programs/7327.aspx) which opened my eyes to this style of teaching. There were 60 maths teachers from all over England sitting in classrooms tackling problems and talking about how students would go about it or what questions we can draw out of these problems. It was a fantastic 4 days and a course I would recommend to anyone either as subject knowledge enhancement or as teaching CPD. The course itself is delivered by 4 supremely talented teachers from America and I cannot speak highly enough of them. They certainly created an environment in which no question was silly and no answer was a waste of time. By the end of the week, almost everybody felt like they would be welcomed when answering a question. An absolutely invaluable course for myself and much like the maths conferences run by La Salle Education, any atmosphere where maths teachers are talking about teaching has to be positive for maths education.
We are always happy to invite people in to come and see this pedagogy in practice and/or indeed talk about how we are teaching our A-Level course. Please do get in touch with me on twitter (@jk_mcd) if there is anything here you would like to talk about.