Bridging Mathematics and Mathematics Education

Walter Whiteley

Mathematics and StatisticsGraduate Programs in:

Math, Education, Computer Science, Interdisciplinary Studies

York University

OutlineBridging 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 well-

prepared 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, … )

• 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)?

• 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)?

A sample investigation

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