Patrick Barmby School of Education & CEM

1. Student teachers’ attitudes towards subject knowledge and teaching in mathematics: The importance of understanding Patrick Barmby School of Education & CEM

2. Context for the research • BA degree in Primary Education • Support for less confident students

3. Background - Teacher Knowledge • Hill et al. (2004; 2005) - teachers’ mathematical knowledge significantly related to student achievement. • Lack of impact of previous qualifications in mathematics on the teaching of the subject (Eisenberg, 1977; Ball, 1990a) • Lack of impact of teacher training courses on subject knowledge and knowledge for teaching (Ball, 1988) • Conceptualisation of teacher knowledge (Ball, 1991)

4. Mathematical knowledge • Shulman (1986) • Subject matter content knowledge • Pedagogical content knowledge • Curricular knowledge

5. Mathematical knowledge • Shulman (1986) in fact emphasised ‘understanding’. “With Aristotle we declare that the ultimate test of understanding rests with the ability to transform one’s knowledge into teaching. Those who can, do. Those who understand, teach.” (p.14) • Also Ball (1991).

6. Mathematical understanding • We draw on a ‘representational-reasoning’ model of understanding (Barmby et al, 2009): • A network model (Kilpatrick, 2009) • The importance of connections • Connectedness of teacher knowledge • The importance of reasoning • Shulman’s (1987) model of teacher reasoning based on the knowledge bases • Research • Case studies • Classroom experience • The difficulty of assessing understanding

7. Aim of our research • To develop ways of highlighting students that may have specific concerns regarding their subject matter content knowledge in mathematics, and also concerns about their teaching of mathematics in the classroom.

8. Methodology • Pragmatic solution - measures of attitude to examine student teachers’ views of their mathematical subject matter knowledge and their teaching of mathematics in the classroom.

9. Methodology • Attitude questionnaire administered to a sample of 144 student teachers (81 first years, 63 second years) at the end of the second term. • Subject audits also administered. • 10 students subsequently chosen for interviews.

10. Results • Exploratory factor analysis revealed two factors • attitude towards studying mathematics • attitude towards teaching mathematics • Reliability analysis – Cronbach α = 0.94 and 0.90 respectively.

11. Individual students

12. Interviews • The interviews with the selected student teachers were based around three questions: • How do you feel about your maths subject knowledge? Why might this be? • How do you feel about your teaching of maths? Why might this be? • How do you think your maths subject knowledge and your teaching of maths are related?

13. Bottom-left quadrant “I know it sounds daft, but I feel like I have cheated my way through. I feel like I don’t understand what I am doing, and then I suddenly get the right answer!” “When it comes to doing good old fashioned addition, subtraction, multiplication … I had every question written down. I had the answers to all my sums on the board, little post-it notes. It went alright but I would never be able to get up there and just explain something” “I don’t know if I will be able to explain how I have done it and I don’t really have a reason for doing it that way”

14. Top-right quadrant “When it came to teaching it, I found it quite difficult to explain what I knew. I accept rules and … I apply it and it works. Trying to explain that to children, I found it at first a bit like, ‘how am I going to break down what I just accept?’ ... But by the end of my placement this year, I felt much more confident in doing that. I would start to go back over what I knew and figure out how I had learnt it and how I had come to the point to be just doing it, which helped when it came to teaching it” “I think I am quite good at teaching as well, because I teach them in a way that I understood it. I know different ways of doing it as well. So no matter which way they give it to me, I can see different ways of doing it as well”

15. Top-left quadrant “The second time round there was a lot more different methods. When I did it the first time round, there was the same way of doing it and if you didn’t get it, you learnt how to. Whereas now, there seems to be three or four methods of doing the one task” “I do not know how I would respond if it was something that I had to revise before I went into the classroom”

16. Bottom-right quadrant “Although I know I have the subject knowledge to be able to teach to the children, I feel as if sometimes, although I have learnt so much on the course, I have not learnt things like, how to teach it, how to do the addition and the subtraction and how they are doing it in schools” “It is not that I would not be able explain it to somebody, because I do know what is going on. But it is how you explain it to them, because when you know, like if you ask a child a question, and you know the answer, if they don’t get one bit of it, you’re like ‘why don’t they get it?’ … I think that is the tricky part”

17. Discussion • Good subject knowledge is sufficient but not necessary!? • What is missing is an examination of understanding • Number of connections • Reasoning/explanation

18. Implications • Williams’ report (2008) and ‘deep subject knowledge’

19. Thank you!