Understanding students’ ideas
I read a really interesting article today entitled “Teachers’ evolving understanding of their students’ mathematical ideas during and after classroom problem solving” by L.B. Warner and R.Y. Schorr. It is a great report that looks at three teacher’s responses to their students’ solutions to a problem, and it discusses in detail how the teachers reflected on them together. It is well worth a read for all maths teachers.
The teachers were middle school maths teacher and they were presented with a problem to solve by the researchers they then presented their classes with the problem and debriefed afterwards. It was clear that the teachers didn’t have the thorough subject knowledge of a high school maths specialist and this lead to them failing to pick up some misconceptions and not allowing students to explore their own methods if they didn’t understand it, rather moving them on to a method that was more familiar to the teacher. The reflections of the teachers are interesting, they all appear to become frustrated with themselves when analysing their responses and are able to reflect on this by offering alternatives. It does show that deeper subject knowledge is important to allow that exploration to take place. The study showed that in this context when the teachers just told students how to fix their mistakes, rather than question students as to why they had made them, this led to student confusion. This suggests that we should be striving to understand our students thinking whenever possible and using that to combat their misconceptions so they don’t fall into similar traps again. This will also allow students to see why they are coming up with these misconceptions.
There are many teachers who, at times, fail to understand the lines of mathematical thinking taken by their students when solving problems. This can lead to not giving the proper amount of credit to valid ideas and it can lead to teachers failing to spot misconceptions. Some students may have a perfectly valid method but as the teacher may not see where they are going they can sometimes block this route off. This has deep links to “Flowery math: a case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency” by K. von Duyke and E Matusov , which I read recently (you can read my reflections here). I feel that it shows that deep subject knowledge is important, as is allowing students time and space to work through the problem on their own. Rather than saying, “No, do it this way” we should, be encouraging students to follow their nose, as it were, and see if they can get anywhere with it. It is always possible to show the students the more concise method when they have arrived at the answer to bui8ld their skill set.
Warner and Schorr believe that subject content, as well as pedagogical content is vitally important to teachers to enable than to know how to proceed when a student is attempting a problem. They look at relevant literature on this and quote Jacobs, Philip and Lamb (2010) who suggest that this is something that can be achieved over time and Schoenfield (2011) who says that teachers tend to be more focussed on students being engaged in mathematics and replicating the solutions of the teacher rather than allowing students to meander their own way through so the teacher scan identify their understanding and misconceptions. The latter would, in my opinion, be a much better way of developing, and I agree with JPL that this is a skill one can develop over time.
Jacobs, V. R., Lamb, L. L. C., and Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41, pp 169–202.
Schoenfeld, A. H. (2011). Toward Professional Development for Teachers Grounded in a Theory of Teachers’ Decision Making. ZDM, The International Journal of Mathematics Education, 43 pp 457–469.
Von Duyke, K. and Matasov, E. (2015). Flowery math: a case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency. Pedagogies: An International Journal. 11:1. pp 1-21
Warner, L.B. and Schorr, R.Y. (2014). Teachers’ evolving understanding of their students’ mathematical ideas during and after classroom problem solvin. Proceedings of the 7th International Conference of Education, Research and Innovation, Seville, Spain, pp 669-677.