This follows from my previous post, in which I described a plan for teaching students how to solve problems about circles using this prompt:
This post contains the student responses to the prompt, particularly in relation to finding the centre and radius of the circle.
The lesson - initial discussion
I started by asking the students to work in pairs to discuss these initial questions:
I then opened up the discussion to the whole class. Students started by stating the obvious, such as 'there is a point is on the edge of the circle.' This gave me the opportunity to explore terminology and notation.
Some students conjectured that P was the centre of the circle, and that M was a midpoint of the line segment AB. We discussed how we could not know this information unless we were told.
Some students conjectured that the line L bisects and is perpendicular to the chord AB, and we discussed that this would be the case if and only if M was indeed a midpoint.
Some students proceeded to work out the length of AM using Pythagoras' Theorem. I drew the students attention to this important technique and drew in the all important right-angled triangle:
When it become apparent that we could not proceed much further without some more information, I presented the students with my question:
Solving the problem
Students started work on this question in pairs or small groups. This is where the pedagogical patterns came in: I directed students towards each other if I felt it would be helpful. I listened to their conversations carefully, being careful not to loiter, and rarely intervening unless asked to. Many of the students referred to their books, or textbooks, instead of asking me for help, which is something I am encouraging them to do.
After a while, I decided to draw the class back together to discuss what we had found. Here are some of their responses:
- A couple of groups of students found the gradient of AM, but only one group of students then used this to find the equation of the line L. I made this good idea public, and suggested these groups might like to work together. Although they found the equation of the line correctly, they realised they didn't have enough information to use it to find the point P.
- A couple of groups of students had used the length of AM and applied this to PM, but also realised this would not be useful for finding P.
- One of these pairs realised they could then use Pythagoras' theorem again to find AP - although interestingly they hadn't realised this was the radius until I accidentally told them they had solved part of the problem!
- One pair guessed two possible coordinates for P - see below*.
- A couple of groups of students had found the 'midpoint formula' in the textbook and were trying to apply that to various lines; although this was a 'dead-end', it had value in that they familiarised themselves with midpoints.
- One group of students found B by using the fact that AM = MB, counting along 2 and up 3 from M. Although finding B is not helpful in solving this problem, the idea of counting squares in this way will lead to the solution.
What interests me here is that I did not anticipate most of these approaches when thinking through the lesson, but they were all valuable.
(Unfortunately) No students managed to connect these ideas together, so I then reverted back to the MANY RESPONSES pattern and decided to write up the guessed (*) coordinates on the board, which were (5,-1.5) and (6,-0.5). I asked students to test whether these answers were true or false, which some of them did using Pythagoras' Theorem.
Following this, I then acted in-the-moment to create a version of the problem in geogebra, the rationale being that I wanted students to consider/visualise the technique of 'counting squares':
Many students at this point conjectured that the (correct) coordinates for P was (6,-1). I didn't know whether they had reached this conclusion through guesswork or through calculation, but it had reached the end of the lesson. I asked them to explain how they knew this was correct as homework, and gave them a couple of similar problems to attempt, to be discussed next lesson.
And in case you are interested, here is my solution to the problem:
I wrote these posts as an exploration of the planning (and teaching) process using one lesson as an example. I have suggested three main planning aspects (prompt, what will students attend to, pedagogical patterns) but this structure may only be applicable to this lesson.
Is it possible to create a structure for planning that will be (a) applicable to all lessons, and (b) useful for others?
This is what I am trying to explore with these posts. One problem I have always found with lesson plans is that they aren't flexible enough to capture the various types of lessons we teach, or the types of teachers we are, or the various students that we encounter. Also, they don't capture more fundamental issues in our classrooms, such as student independence, resilience and status.
Perhaps we can create higher level pedagogical patterns for this too, for example:
PROBLEMS FIRST: Problem A is used as a prompt, forming the basis for the teaching in that lesson or unit. Practice is subordinated therein, rather than building up to problems through practice. The rationale here is that students learn how to solve these kinds of problems by doing these kinds of problems. Students should be encouraged to think mathematically, to explore dead-ends, to use their intuition.
SELF STUDY: Students are given a problem that requires use of specific methods and theorems. They should try to find these for themselves, without the input of the teacher. They should then apply these methods to the given problem, gaining a deep familiarity with theorems and techniques, and when they might be applicable, through trying to apply them. The teacher should ensure disparate ideas are brought together for consolidation.
PARTIAL SOLUTIONS: Students present whatever partial solutions to problems that they can. They must try something rather than leave a question blank. There should be evidence that they have tried a number of approaches and failed, to have thought about a particular problem for a long time. The teacher wants to see their failed attempts, and work with them to develop their methods and techniques. This is scaffolding but in reverse, perhaps more like an apprenticeship.
STUDENTS AS SENSE-MAKERS: All student responses are valued, and discussed. Students should be given time to explore ideas and discover what makes sense, and what doesn't, with the guidance of the teacher and others.
These patterns formed the rationale for the lesson I described in these posts.
Do you think this idea could be useful, or is it perhaps too general? Any comments would be welcome.