Factoring in California

I am always curious about different styles of teaching especially across cultures and I have had the great privilege of visiting a local high school here in California. I have been able to meet the maths teachers and watch maths lessons and attend maths team training. It has been a great experience.

I thought that I might just jot down some thoughts about my experiences. Obviously the course structure is quite different to the UK, the external assessments are different and so on but I am personally interested in what happens in the classroom between the teacher and the students.

One class I attended was an advanced maths course titled Theory and Proof. The content includes complex variables, linear algebra, combinatorics, mathematical induction, and number theory. I don't think that this is a regular high school course, I believe a dedicated and enthusiastic teacher has set it up as an extra. But, I love the fact that students can get credit for this rather than just the curriculum set by exam boards or governments.

The lesson itself was really interesting. I would have liked to see how it may have been set up at the beginning of the year as the students seemed very ok with 'having a go'. Other than a bit of test admin the lesson started with a 4 part questions on the board with lots of room around and under each one:

And that was pretty much the set up for the lesson.

So what happened. The students were sat in groups and most groups did discuss the questions. Some students worked on their own for a bit but there was a healthy buzz in the room. 

I positioned myself on a table with two boys to observe. I watched as they thought around the problem. They knew what the expansion of (a)(b)(c) was and they were discussing how to get rid of the x and x-squared terms. One of the practices I have been working on is watching, listening and not jumping in. (Sometimes this is easier when they aren't your students!) So I did this for a bit. The discussion wandered through some vague knowledge of cubics, and through a bit about complex numbers. I was really fascinated about the way they made these connections.

Not quite being sure what their prior knowledge was I eventually suggested that perhaps we weren't looking for the complex factors and maybe we could start by looking for a linear and a quadratic factor. 

Around about this point the teacher called the class together. He called on a few students to share their thought process for the first part. A few had noticed that = 2 was a root of the polynomial and had decided that (x - 2) must be a factor. It was clear that the students had not been taught division of polynomials by any method so from this point it was pure problem solving. The teacher let the students get back to work and came over to the table I was working with. He checked that they had understood the roots part and then suggested that they 'try something'. I wondered whether this was a bit of a mantra he used. They each chose some quadratic expressions to multiply by the (x - 2) to see what came out. Through trial and improvement I could see they were getting towards an answer.

The teacher called the class together again and had a student write up the solution for the first part. He then called on a selection of students who had taken different approaches and finally he quickly demonstrated a tool he thought they might like to use, which was basically the area method of factorising. This visual tool appealed to the boys I was working on. 

We recreated the first example together and had a go at the others. With the area tool they were away. They picked up the strategy quickly and other than a few sign errors they completed the rest of the questions. 



I then set them an extension. All the examples they had been given had odd powers, what would happen if we had even powers? This led to some great conjecturing and an 'Oooh!' moment!

Towards the end of the lesson the teacher asked several students to present their work. The first 3 parts were done with the area tool and then the last one was presented as an extension of the pattern. One student asked about how this one had been done and there was a little discussion about what patterns had been noticed and when it could be used. 

There was definitely an aspect of headache/asprin in this lesson. I don't know whether this was a frequently used style of lesson though I got the impression that the students were familiar with struggling on their own, thinking back to previous work and working in groups. 

I really enjoyed the lesson, there was a great vibe and the teacher managed a balance of giving the students time to explore and giving them some tools they could use. In some classes I have seen students refuse to engage initially but wait until the teacher gives a hint or a technique. Persuading them that having a go is worthwhile and might give insight is hard but I think the connections made with other topics was really worthwhile. I also liked that the tool shown was conceptual, versatile and not given as the prescribed way, just an alternate way that no one else had come up with.  


Comments