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Bridging Mathematics and Mathematics Education Walter Whiteley Mathematics and Statistics Graduate Programs in: Math, Education, Computer Science, Interdisciplinary Studies York University Outline Bridging Math and Math Ed • • • • • • • Introduction Why (for mathematics)? Why (for education) What within Mathematics Programs Possible Impact on other programs How - within Mathematics (some ideas) Obstacles and Opportunities Introduction • Thanks for continuing recognition of value of engaging with mathematics education • a research mathematician and a mathematics educator • two way collaboration, listening, learning • Canadian math education success (PISA) • Settings for collaboration: CMS, Canadian Mathematics Education Study Group • Some Limited Funding: NSERC, SSHRC, MITACS, Fields, PIMS, … Why for Mathematics Programs? • Hard Times - cut backs: programs being shifted, scaled back (cut). • Programs measured by recruitment and retention (graduation) • Substantial number of our math majors plan to be teachers. • Future teachers have different motivations / different sources of engagement. • We value better teaching prior to university: improved, relevant preparation of teachers; Why for Mathematics Programs (cont/) • Goal of increased engagement in mathematics programs • Want a pump not a filter (more students). • Want graduates who see themselves as (young) mathematicians, and mathematics educators. • These goals are already part of the goals in primary / junior education. • Similar needs for our graduate student /Post Docs • Mathematicians have much to learn from Mathematics Educators about how to achieve these outcomes. Why for Education? • Mathematics Educators want mathematically wellprepared teacher candidates - with broad, pedagogically relevant, mathematical knowledge; • Mathematics as Processes, • Big Ideas in design of the next curriculum; • Students with the capacity, and the confidence, to apply the knowledge to new situations, in the classroom. • Mathematics Educators want support to provide better preparation (more time with students) • Currently need to spend time on pedagogically relevant knowledge of mathematics. Why for Education (cont)? • Math (and science) education are generally secondary (or lower) in admissions, in structure of education programs; • Compare ‘literacy’, essays for admission, language expectation, with gap around math. • Faculties of Education in Ontario are turning away qualified applicants (B+ students). • No / limited math or science requirements for Primary /Junior teachers. • Mathematicians can provide support in these larger discussions! What in Mathematics Programs for Teachers? • Programs, not courses, are the level of design. • What mathematicians do, what students should be prepared for, what teachers need to believe in and communicate, practice. • Present Mathematics as Big Ideas and Processes. • Mathematics as reasoning and sense-making; • Focus on processes: mix of embedded mastery and explorations / reflections builds these. What in Mathematics Programs for Teachers (cont)? Processes (Ontario Version) • Problem Solving problem solving, and selecting appropriate problem solving techniques • Reasoning and Proving: • Reflecting and monitoring their processes • Selecting Tools and Computational Strategies • Connecting … • Representing and modelling mathematical ideas in multiple forms: concrete, graphical, numerical, algebraic, and with technology • Communicating … What in Mathematics Programs for Teachers (cont)? UUDLES -University Undergrad Degree Level Expectations • integrate relevant knowledge and pose questions … • apply a range of techniques effectively to solve problems … • construct, analyze, and interpret mathematical models • use computer programs and algorithms: both numerical and graphical, • collect, organize, analyze, interpret and present conjectures and results … • analyze data using appropriate concepts and techniques from statistics and mathematics What in Mathematics Programs for Teachers (cont)? UUDLES (cont) • employ technology effectively, including computer software, to investigate … • learn new mathematical concepts, methods and tools … • take a core mathematical concept and ‘unpack’ the concept • communicate mathematical and statistical concepts, models, reasoning, explanation, interpretation and solutions clearly… • identify and describe some of the current issues and challenges (professional, ethical, … ) What - in Mathematics Programs for Teachers (cont)? • Mathematics as Big Ideas and Processes: Need capstone course(s) for teachers to draw these out. • This does not happen for most students in current programs. • I asked some graduating students in a capstone course: should they be evaluated on these Degree Level Expectations? • Their answer: • not until our previous courses and our instructors are evaluated on them! What - in Mathematics Programs for Teachers (cont)? A sample investigation • f(x+y) = f(x) + f(y) tell me about the function f . • Investigate and present your answer(s) using at least four different representations of functions. • Can you predict how fourth year math majors approach this? • Can they, working a group, understand the reasoning of their peers? • What questions do they ask? • How do their approaches line up with the historical evolution of the concept of function? (A handout on that - by Israel Kleiner) • What do their difficulties and approaches say about what they were thinking through three courses in calculus, through linear algebra, …? What in Mathematics Programs for Teachers (cont)? • Breadth in math - recommended areas: Geometry, History, Modeling, Statistics & Probability, Proofs, Calculus, Linear Algebra, … • Capstone integrative courses. • courses designed to use multiple representations, multiple approaches to solve problems. • support reflective learners, learners who can listen to other approaches, present, explore in peer work. • introduction to research in Mathematics Education - become life long learners. • ‘How’ can be more important than ‘What’ What in Mathematics Programs for Teachers (cont)? • Are our future teachers engaged as ‘young mathematicians’? • What beliefs do the future teachers develop about mathematics? • What beliefs do they develop about how they learn, how others learn? • Does our assessment value these processes? • Do we structure first year so that we primarily value processes, and assess them (transition)? • Do we reinforce the key skills from the High School Curriculum? Possible Impact on other programs • The goals (UUDLES) of all our mathematics programs! • What mathematicians do, what students are prepared for, what they believe in and communicate. • Overall Mathematics as Big Ideas and Processes; • Mathematics as reasoning and sense-making; • Focus on processes: mix of embedded mastery and explorations / reflections builds these. • First and second year courses will be shared classes with mix of Mathematics Majors. Possible Impact on other Mathematics Programs (cont)? Applied Mathematics Program Learning Objectives (York) • ability to construct, analyze, and interpret mathematical models … • ability to use computer programs and algorithms: numerical and graphical, to obtain useful approximate solutions to difficult mathematical problems … • ability to learn new mathematical concepts, methods and tools and to apply them appropriately. • ability to communicate mathematical concepts, models, reasoning, explanation, interpretation and solutions clearly and effectively in multiple ways: orally, written reports, visual displays, … . What impact in Mathematics Programs (cont)? Less is More? • If a sequence of courses focuses on these goals, and processes, evidence is that: • In the first course, less material is covered and learning is different. • By the end of a sequence of courses (four plus) like this, more material is mastered; • Broader objectives can be achieved. • Pedagogy of courses is more important than what content. • Alternate ‘official calendar’ for courses - based on pedagogies. • Different ‘course mandated ’ given instructors. What impact on Mathematics Programs (cont)? • Courses which are best for future teachers can be better for all mathematics majors. • Develop their self-efficacy - the confidence and capacity: • to engage, • to try (and to make mistakes), • to question • to expect the mathematics and the connections to make sense. • These would be interesting, engaging classes to teach! • Spending energy convincing the students they do not have the capacity is too common - and too destructive. How to Build Bridges? • Collect evidence of numbers of future teachers in classes designed ‘for math majors’; • Collaborations - find allies: • inside department, among students, across faculties. • Interest among graduate students in both programs. • Collect resources / literature / evidence. • Groups: Canadian Mathematics Education Study Group, Fields Math Ed Forum, MAA, RUME How to build bridging programs (cont)? • • • • • • Experiment with engaged pedagogies; With group work - study groups, projects. Appropriate integration of technology Hands on materials, extended investigations. Modeling what we do in mathematical practice. Work at understanding how students think: • Needed to be effective in any teaching except ‘filtering’ out those who ‘are not like us’. • Possibility of visiting across classrooms; • Lesson Study in University Teaching? How to build bridging programs (cont)? • This is what we want high school teachers to do and we should model / give them the opportunity to see that it supports good learning. • Can learn a lot from classroom teachers, even primary teachers (Fields Math Ed Forum, OAME, …) • about engaging students, • about differentiated instruction and assessment, • about using multiple approaches, • about threading material on big ideas. Obstacles and Opportunities? • Difficult to get financial support • exclusion from basic NSERC funding • difficult to break into SSHRC funding • Hard for Mathematicians to evaluate quality of Math Education Contributions • Low status in Mathematics T&P processes • Hard work to learn results of math education research (and how to evaluate the quality) • Even harder to become a quality mathematics education researcher. Obstacles and Opportunities (cont)? • Hard work to teach in these ways (extra time) • Extra preparation time, extra marking time. • We are not trained to teach writing, to lead discussions, to coach presentations, … • requires appropriate rooms / materials / computer access, • Limitation on class sizes. • My surprise experience: further proposals progressed from the department to the faculty to the VP Academic, the stronger the support. Obstacles and Opportunities (cont)? • Very similar issues in Science Education • Opportunities for allies within and across science, and among science educators; • Respectful engagement with classroom teachers and their organizations gives support. • Curiosity / excitement among students. • Outreach - recognized within departmental priorities, MITACS priorities. • Small network of people in Mathematics Departments working on bridging; • Ask for support from others working on bridging. • High tolerance for ambiguity - a survival skill and necessary for collaborations! Thanks Questions? whiteley@yorku.ca wiki.math.yorku.ca/ Link under Conferences: Bridging Mathematics to Mathematics Education