1. Naomi Ingram, University of Otago From Brussels Sprouts …

2. Naomi Ingram, University of Otago … to Summer Strawberries

3. Naomi Ingram, University of Otago Engagement …involvement in the activity of the classroom. mathematical?

4. Naomi Ingram, University of Otago Relationship with mathematics   Views of maths   Feelings about maths   Perception of ability   Mathematical knowledge   Habits of engagement

5. Naomi Ingram, University of Otago Engagement in mathematics is important because of the nature of the subject. There’s so much in maths. If I get behind in maths, it would probably be the worst subject to try and catch up on (Debbie). I am aware that I need to try my hardest. I am more attentive than in other classes. I know we need to know this (Sean).

6. Naomi Ingram, University of Otago Tensions … I didn’t work to the depth required because I was bored with it (Jason). I work hard in other subjects. I talk the most in maths. In other subjects I’m quiet the whole time because I’m doing it (Tracey) It’s easier to talk than making my brain hurt (Bridget).

7. Naomi Ingram, University of Otago Students develop habits of engagement. Homework Getting ready to learn How much they socialise How they ask questions How they listen How they engage in each mathematical task How much they get involved in class discussions

8. Naomi Ingram, University of Otago A mathematics problem: A machine prints four books in 10 minutes. How many will it print in two hours? How many hours would it take to print 390 books? 48 16 and ¼ hours.. or is it? g

9. Naomi Ingram, University of Otago Reflect on your feelings and engagement How did you feel before you saw the problem? How did you feel after you saw the problem? How aware were you of others doing the problem? Did these feelings affecting your doing of the problem?

10. Naomi Ingram, University of Otago Reflect on your feelings and engagement Imagine the problem was given to a class of mixed-achievement Year 10 students as a starter. What range of behaviours would you observe?

11. Naomi Ingram, University of Otago Relationship with mathematics 5. Habits of engagement Context of the moment Engagement in a mathematical task Unique learning experience and outcomes

12. Naomi Ingram, University of Otago Context of the moment When students engage in a mathematical task, they are each situated in a unique context of the moment.

13. Naomi Ingram, University of Otago Teacher classroom hot classroom cold too bright social awareness seating arrangements Who I sit next to totally affects me (Moira) All Blacks romantic relationships issues with friends after-school jobs parents’ divorcing other subjects lack of vegetables family illness Relationship with maths Engagement in a mathematical task

14. Students have habitual pathways of engagement when doing a mathematics task Avoidance Full engagement Superficial engagement

15. Naomi Ingram, University of Otago Pathway of Engagement - Avoidance Some things I just don’t attempt (Debbie). Look at what I did today. No maths (Mark). Going to the toilet, ruling up pages in their exercise book, organising their equipment, programming their calculator, eating, daydreaming, writing notes to each other, writing poetry, doodling, talking socially, writing out the problem as neatly as possible …

16. Naomi Ingram, University of Otago Pathway of Engagement - Superficial I try and figure the problem out. After a wee while if I really can’t do it at all, I just give up (Bridget). I kind of get frustrated when I can’t do something. I just give up (Paul) Unprepared to think deeply about a problem.

17. Naomi Ingram, University of Otago Pathway of Engagement – Full engagement I’ll read it a couple of times and think about it logically and sort of see what it actually means. I talk to some other people about it … see what their methods are (Katrina). When fully engaged, students think about the processes of solving problems, they persevere, use a variety of strategies and discuss mathematics with others.

18. Naomi Ingram, University of Otago Asking for help I need the teacher to go over answers I get wrong. The way I learn is by the teacher explaining it to me and helping me understand on my level. Is this an indication that a student is engaged? I get frustrated. Can’t you just tell me what it is? Maths is like a silent movie because you either understand the storyline by following the action and another person explaining it, or you sit back, bewildered.

19. Students have a set of engagement skills that operate at different levels of effectiveness.

20. Naomi Ingram, University of Otago Perseverance Integrity Intimacy Independence Concentration Utilisation of feelings Cooperation Reflection

21. Naomi Ingram, University of Otago The skill of continuing to do a mathematical task, despite experiencing difficulty. Perseverance

22. Naomi Ingram, University of Otago Perseverance Integrity Commitment to searching for mathematical truth and understanding

23. Naomi Ingram, University of Otago Perseverance Integrity Intimacy Emotional engagement with mathematics

24. Naomi Ingram, University of Otago Perseverance Integrity Intimacy Independence The skill of solving problems autonomously.

25. Naomi Ingram, University of Otago Perseverance Integrity Intimacy Independence Concentration The skill of remaining focussed, despite disruption.

26. Naomi Ingram, University of Otago Perseverance Integrity Intimacy Independence Concentration Utilisation of feelings The skill of being resilient to negative feelings felt during a task, and instead using them as a signal to persevere or change strategy.

27. Naomi Ingram, University of Otago Perseverance Integrity Intimacy Independence Concentration Utilisation of feelings Cooperation The skill of discussing mathematics, solving the tasks cooperatively, and to ask for help as a strategy, rather than as a form of disengagement or dependence.

28. Naomi Ingram, University of Otago Keep going Integrity Intimacy Independence Concentration Utilisation of feelings Cooperation Reflection Being self-aware. Reflecting on own and others’ engagement.

29. Naomi Ingram, University of Otago What can we do to enhance engagement? Help students to recognise and act on feelings of confusion and worry when working on mathematical tasks. Encourage students to develop effective engagement skills. Hmmmmmmm …….

30. Naomi Ingram, University of Otago Some ideas … 1. Be explicit about the importance of doing mathematics to learn. 2. Get to know students’ engagement. 3. Wait until everyone is ready to learn before revealing the problem. 4. Give students a variety of problems of different contexts, difficulty, depth, and length. 5. Make sure ALL students have the opportunity to experience confusion. 6. Talk about your own engagement and/or the engagement of a tame mathematician. Normalise and EMBRACE confusion. 7. Ask students to try a problem by themselves for a time before talking to a friend or asking for help. 8. Ask students to reflect on how they solve problems. 9. Report formally on engagement skills to school and parents.

31. Naomi Ingram, University of Otago Try the problem yourself before asking for help. Don’t give up when you are stuck. Seek understanding as well as an answer. Expect confusion Concentrate. Maths is thinky. Remember that mistakes are windows into your thinking. Cooperate with others. Be proud of your progress. Reflect on your doing. When doing maths …

32. Naomi Ingram, University of Otago Naomi Ingram University of Otago, College of Education naomi.ingram@otago.ac.nz