Today I managed to find a few minutes to browse the latest issue of Maths Teaching, the ATM journal. One article that caught my eye was the “from the archive” section, where Danny Brown (@dannytybrown) introduced an article that was first published in 1957. The article was written by Ruben Schramm and is entitled “The student’s passive attitude towards mathematics and his activities.”
The article discusses mathematics teaching, particularly the nature of students who often, for whatever reason try to find an algorithmic method to follow to solve a problem, looking to recognise the problem and answer it in a similar way to how they have answered questions before. This is a problem that was obviously prevalent in the 1950s, as evidenced in the paper, but it is still prevalent now, and I feel the nature of our exam system must at least hold a portion of the blame. The questions on exams tend to be very similar and students will learn methods to answer them whether the teachers like it or not. This is one issue I hope will be dampened a little with the upcoming changes to the exams.
Schaum suggests that this passivity in maths, this tendency to look for algorithms, is in part down to how students see mathematics. He suggests that when they see teachers solve problems on the board by delivering a slick, scripted solution they can get a feeling that it is via “witchcraft” and see the whole process of uncoordinated steps, rather than a series of interconnected mathematical ideas. The latter would encourage the students to drive the mathematics from their internal ideas, and this would lead to them being more able to apply their knowledge in new contexts. If we can develop this at all levels then I feel we really would be educating mathematicians – ie giving students the skills to be able to apply their knowledge in new contexts, rather than teaching them to follow a recipe to answer a question.
Schaum goes on to discuss authority, the infallible authority that students see in their teachers and in the mathematical theorems and formulae. It is suggested that students see these theorems as infallible, and as such they reach out for them in their memories and try to apply them to problems. This can mean that the problem they are applying them too is only vaguely similar to the problem the theorem or method is actually there to solve. Schaum calls these “analogy mistakes”, and suggests that it is down to how comfortable with the content students feel that mean they revert to them. I feel that this is true in part, but that also the pressure of exams can lead students to confuse things in their head if they have opted to learn algorithms rather than looking to develop a deeper understanding.
I’ve had a couple of examples of these “analogy mistakes” in lessons and exams recently. A year 12 student came to an afterschool elective as she was trying to solve some coordinate geometry problems involving tangents. She had gotten herself really confused because in her notes she had written tangent gradient is perpendicular (when discussing circles) but she didn’t think it should be perpendicular because a tangent at a point should have the same gradient as the curve. I spend a little time discussing where her misconception had come from (her notes should have said “perpendicular to the radius”) and discussed how she could remember this more easily if she has thought about the graphs and sketched them.
Another example was in a recent exam one of my students had answered part of a question on alternative from incorrectly, she had done the alternative form bit well and the answer was 25 Sin(x + a), but it then asked her for the maximum she had written -25. When I questioned her about this after it seems she had fallen victim to an “analogy mistake”, she had remembered that “maximum is positive” when discussing second derivatives and in the pressure of the exam this memory had taken over, rather than the rational thought process that should have flagged up that the maximum or the function would be 25, which is definitely bigger than -25.
In his preface Danny Brown suggested that one way to counteract this would be by questioning and discussion, if we remove the authority from the discussion and don’t validate the answers by issuing statements saying they are correct or incorrect, but rather open them as conjecture to the class who then can discuss this, then we can allow students to develop their own mathematical ideas. Lampert (2001) also discussed this idea and suggests that as teachers we need to be striking the right balance between allowing students to discuss and conjecture and ensuring they understand what is important and aren’t making mistakes. This is something I strive for in my own classroom, and something I am currently working on trying to improve.
This post was cross posted on Cavmaths here.