RELATIONSHIP BETWEEN PROXIES FOR LEARNING AND MATHEMATICALLY RELATED BELIEFS Peter Liljedahl

MAVI 12 – preservice elementary teachers

MAVI 12 – preservice elementary teachers

MAVI 12 – preservice elementary teachers TOOLBOX SYSTEM PROCESS

MAVI 12 – preservice elementary teachers BELIEFS ABOUT MATHEMATICS BELIEFS ABOUT THE LEARNING OF MATHEMATICS BELIEFS ABOUT THE TEACHING OF MATHEMATICS

MAVI 22 – inservice secondary teachers BELIEFS ABOUT MATHEMATICS BELIEFS ABOUT THE LEARNING OF MATHEMATICS BELIEFS ABOUT THE TEACHING OF MATHEMATICS PROXIES FOR LEARNING

Three Views of Mathematics Toolbox View – mathematics is seen as a set of rules, formulae, skills, and procedures System View – mathematics is characterized by logic, rigorous proofs, exact definitions, and a precise mathematical language Process View – mathematics is considered as a constructive process where relations between different notions and sentences play an important role (Dionne, 1984; Ernest, 1991; Törner & Grigutsch, 1994)

Three Types of Belief Systems Quasi-logical relationship – beliefs can be either primary or derivative Psychological strength – beliefs can be held either centrally or peripherally Isolated clusters – beliefs are held in clusters in isolation from other clusters (Green, 1971; Nespor, 1987; Rokeach, 1968)

Three Types of Belief Systems Quasi-logical relationship – beliefs can be either primary or derivative Psychological strength – beliefs can be held either centrally or peripherally Isolated clusters – beliefs are held in clusters in isolation from other clusters (Chapman, 2006; Liljedahl, Rösken, & Rolka, 2006; Wilson & Cooney, 2006)

Proxy for Learning something serving to replace or substitute for another thing (Merriam-Webster Dictionary, 2016) metaphors, gestures, and manipulative proxy measurement is as old as measurement itself (Scott, 1995) proxies for engagement (Liljedahl, 2016) proxies for learning (Liljedahl & Allen, 2013a, 2013b) faking stalling mimicking

Proxies for Learning TEACHER PRACTICE STUDENT BEHAVIOUR

Methodology 12 inservice secondary mathematics teachers (1-20 years experience) Master’s of Secondary Education Educ 847 – Teaching and Learning Mathematics Journal entries Week 1: What is mathematics? What does it mean to learn mathematics? What does it mean to teach mathematics? Week 2: Consider some of the proxies for learning that you witness in your own students. How are you enabling these within your own teaching? Week 13: What is mathematics? What does it mean to learn mathematics? What does it mean to teach mathematics? Think back on your practice before this course. What will you not do anymore?

Analysis Beliefs and Belief Clusters → three views of mathematics (Dionne, etc.) Proxies for Learning → analytic induction (Patton, 2002) studenting (Liljedahl & Allen, 2013a, 2013b) constant comparative method Belief Clusters + Proxies → case by case analysis comparison between cases

Beliefs

Problem Solving Beliefs Art ThinkingI have, in the last four years, viewed math as solving problems. I’ve maybe viewed it as solving puzzles. But now I think of it more as an art. (David, week 13) … but my belief that mathematics is an art has been reaffirmed after this class. (Nancy, week 13) Thinking Mathematics is a way of thinking about patterns, shapes, numbers and concepts through discovery, invention, and problem solving. In order to learn mathematics, one needs to be engaged by personally thinking through a given problem. (Ellen, week 13) Discovery of relationships through problem solving. I still believe that the best way to teach is through guided problem solving where the teacher provides interesting material and hints along the way, but the students are responsible for the discovery and learning. (Larry, week 13) Developing tools Mathematics is the tools we develop through problem-solving. (Barb, week 13)

Belief Clusters Mathematics is the study of developing mathematical thinking skills. Learning mathematics allows students to be able to connect mathematical ideas with new and existing ones. Teaching mathematics is to aid students to think mathematically by helping them conceptually understand mathematics. (Alison, week 13). 10

Belief Clusters Art – making connections Nancy Evan

Proxies for Learning Faking, stalling, mimicking and slacking are all evident on a daily basis. [..] I certainly have a role in enabling these proxies for learning; however I am starting to become more aware of them and taking some action to give my students some skills to move away from them. Certainly the ‘you try one’ style problems lend themselves to these actions and it makes me more aware of student actions. By spending the majority of time speaking to students I am not giving them time to think for themselves and construct their own relationships to the material. This teaches them that I will likely continue to offer the information or ‘answers’ and that they can wait for me to continue to do so. A learned behaviour of waiting for the result then occurs. By always offering answers or strategies students never truly take the ownership of the learning that needs to take place. We can then wonder whether or not they are truly learning at all. (Barb, week 2) 9

2 start 5 stop 2 mixed Proxies for Learning Faking, stalling, mimicking and slacking are all evident on a daily basis. [..] I certainly have a role in enabling these proxies for learning; however I am starting to become more aware of them and taking some action to give my students some skills to move away from them. Certainly the ‘you try one’ style problems lend themselves to these actions and it makes me more aware of student actions. By spending the majority of time speaking to students I am not giving them time to think for themselves and construct their own relationships to the material. This teaches them that I will likely continue to offer the information or ‘answers’ and that they can wait for me to continue to do so. A learned behaviour of waiting for the result then occurs. By always offering answers or strategies students never truly take the ownership of the learning that needs to take place. We can then wonder whether or not they are truly learning at all. (Barb, week 2) 2 start 5 stop 2 mixed

Proxies for Learning … there are numerous students who engage in mimicking behaviour, a proxy for learning. These students are good students, but I feel that their confidence in mathematics is lacking as they find it difficult to stray from the “notes” and create understanding on their own. If I changed the “you try” question to an application question these students struggled and then demonstrated “stalling” behaviour. I believe this is partially my fault as I always encourage them in my teaching to refer back to the notes when they are not sure what to do. (Nancy, week 2)

Proxies for Learning Students tend to chat with others which usually results in a seat change. However, they keep trying to engage in other meaningless activities while taking notes including doodling and constantly checking time. I am enabling these proxies because my lesson is mostly teaching with notes on the board—students are sitting back listening to the lesson and taking notes down. (Abby, week 2) 2 Jessy And Abby – saw they are responsible, but saw math as forcing this.

Proxies  Belief Clusters

Proxies  Belief Clusters

Proxies  Belief Clusters

Proxies  Belief Clusters

Conclusions and Implications inservice secondary teachers ≠ preservice elementary teachers (mathematics-teaching-learning) cluster is common proxies are mostly seen as the domain of the teacher process + (mathematics-learning)  proxies are the teacher’s domain extends Liljedahl, Rösken, & Rolka (2006) introduces proxies for learning implicates proxies for learning as a associate of (mathematics- teaching-learning) belief cluster

THANK YOU! liljedahl@sfu.ca www.peterliljedahl.com/presentations @pgliljedahl