What does it mean to be a ‘good’ maths student? [ AND WHERE DO THESE PERCEPTIONS COME FROM ] Glenda Anthony Oct 2013 Using findings from Learners’ Perspective Study (NZ) with colleagues Tim Burgess, Anne Lawrence, Peter Rawlins, and Margaret Walshaw

In your class what do students need to do to be regarded as ‘good’ learners?

Good learning ↔ Good teaching What constitutes ‘good’ learning and ‘good’ teaching are complex and contestable issues.

Linking mathematical identity to the teaching/learning nexus Can teacher attributes and practices most valued by students offer insight into students’ understanding of what it means to do mathematics?

Construct of Identity: Obligations (Cobb, Gresalfi, & Hodge, 2009) Normative identity: obligations that students have to fulfil in order to be effective and successful in the classroom. General obligations concern the distribution of authority and the ways that students are able to exercise agency. Specifically mathematical obligations concerns what counts as being mathematical competent within the classroom.

Communicate Explain your thinking Seek help to understand the mathematics. Struggle with some problems Learn from errors Make connections Listen & take everything in Talk at appropriate times Be cooperative Be responsible when you are putting you hand up Try your hardest one hundred percent Pay attention Give them respect Do the work Take notes Careful listening, watching Be able to explain how one worked out a problem Seeing where the steps of a problem come from Being resourced and completing homework

Data from the L PS Study: Year 9 classes with ‘good’ teachers NZ1: Low achievers from a low decile school. The lesson sequence involved decimals. NZ2: High achievers from a high decile school. The lesson content was fractions. NZ3: Banded group of high achievers from a mid decile school. The lesson sequence involved linear equations.

Generating and analysing data 1.What attributes of the teacher were valued? 2.How did the students’ perception of the ‘good’ teacher mediate their developing mathematical identity (and thus their perceived obligations as a mathematics learner)? “Tell me about this class.” “What are the things that you do to help you learn? “What are the things the teachers does to help you learn” “Was it a good lesson for you and why?”

Good Teachers/Teaching: NZ1 He cares and he makes mathematics accessible Sense of belonging, and being valued – not being embarrassed, encouraged to put answers on the board, knows his students, and freedom to learn in various ways: We don’t have a seating plan and we can sit next to anybody we want and most of my other classes we aren’t allowed to. We have got the coolest class ever...I like to have my friends around me...I can’t work by myself, I can’t think straight. We are allowed to speak out and in the other classes you have to put your hand up and that would take forever. In out class we can say it and he won’t get angry.

Good Teachers/Teaching: NZ2 “She is really helpful and she teaches me new things and she explains it really well.” Helpful included an ethic of care for students’ mathematical participation, progress, and confidence: We are all very friendly towards each other. Everybody else gets a chance to explain what they are doing. She won’t like just tell you something about it and then walk off like other teachers. Explaining was associated with “knowing what she is talking about”, “being clear”, “knowing how you got the answer.”

Good Teacher/Teaching: NZ3 The teacher is really good at explaining stuff to you and helping you if you don’t understand what is going on…AND he helps us to communicate. A sense of belonging in this class related to being able to “get on with our work, we don’t just do our own thing we get on with the work.” Caring related to students’ mathematical progress – expectation to struggle, building on errors. He “helps you to extend what you already know.”

Developing mathematical identities NZ1 General classroom obligations: Volunteering solutions to review questions and being prepared to write these on the board. Allowing time to make responses and respecting other students’ contribution. Listening to and taking notes of the solution methods to computational problems demonstrated by the teacher. Asking the teacher for help when stuck. Specifically mathematical obligations Being able to specify the calculational steps involved in a computation.

Their role as mathematics learners related more to social obligations listen and take everything in and talk at the appropriate times; be cooperative; be responsible when you are putting your hand up to talk; try your hardest one hundred percent; pay attention; give them respect. Teacher efforts were directed to making students feel valued and comfortable in the mathematics classroom’ Creating a climate that fostered students’ confidence in themselves as mathematics learners was enacted by lowering the cognitive challenge of tasks.

Developing mathematical identities NZ2 General classroom obligations: Being resourced - equipment and homework completed. Solving problems linked to prior knowledge. Being prepared to explain how one worked the problem out to the teacher. Careful listening, watching attending to peers (respect) Taking notes. Asking the teacher clarifying questions in order to understand the demonstrated methods.

Mathematical obligations Seeing where the calculational steps come from with reference to concrete models (e.g., overlapping arrays to explain multiplication of fractions). Being able to specify the calculational steps involved in a computation using a preferred strategy. Awareness of more than one way to ‘do’ or ‘solve’ a problem

Developing mathematical identities NZ3 General obligations : Solving problems - explain and justify how one worked the problem out to peers and teacher. Asking clarifying questions in order to understand the student or teacher demonstrated methods. Working collaboratively with peers.

Mathematical obligations : Understanding reasoning for solving problems, both the how and why a particular strategy works. Indicating and giving reasons for disagreement with other students’ mathematical arguments. Making connections.

Caring NZ1: Social care, lowering mathematical demands: Wanting to participate, but expecting to be followers. NZ2: Confident mathematically, development of them as individuals: ‘Good’ student works hard, attends, does what the teacher wants, learns from the teacher. NZ3: Caring about the mathematical understanding: Developed strong conceptual agency.

Explainer NZ1: explanations make things clear and simple: ‘Good’ students need to remember the steps the teacher showed them. NZ2: Explains clearly; the focus is on how to do problems, link with prior knowledge, and multiple representations: ‘Good’ students need to seek help from the teacher, they need to ‘see’ themselves how to ‘do’ the problems NZ3: Explanations deepen understanding: ‘Good’ students need to communicate and question and actively struggle to understand.

Conclusion 1.Perception of a ‘good’ teacher is influenced by the socio- political realities of the classroom. The nature of the classroom tasks and the quality of supportive discourse is of critical importance for the roles that students choose to take on. 2.Clear link between the students’ perception of their teacher as ‘good’ and the co-constructed normative identities of each classroom. 3.In all classrooms students identified with the required mathematical practices; however, what it means to do maths within each of these classrooms was distinctly different - behaviours that are constructed as competent in one setting may be drastically different from another.

What norms/expectations operate in your classroom? How are the obligations to behave as learners of mathematics negotiated? What opportunities do these obligations offer students to engage in mathematical thinking, develop mathematical agency, and productive mathematical identities?