Bridging Mathematics and Mathematics Education

Bridging Mathematics and
Mathematics Education
Walter Whiteley
Mathematics and Statistics
Graduate Programs in:
Math, Education, Computer Science,
Interdisciplinary Studies
York University
Bridging Math and Math Ed
Why (for mathematics)?
Why (for education)
What within Mathematics Programs
Possible Impact on other programs
How - within Mathematics (some ideas)
Obstacles and Opportunities
• Thanks for continuing recognition of value of
engaging with mathematics education
• a research mathematician and a mathematics
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• I asked some graduating students in a capstone course:
should they be evaluated on these Degree Level
• 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
• 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
• What beliefs do the future teachers develop about
• 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
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
• Alternate ‘official calendar’ for courses - based on
• 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
• to engage,
• to try (and to make mistakes),
• to question
• to expect the mathematics and the connections to make
• 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
• Interest among graduate students in both
• 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
• 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!
Link under Conferences: Bridging Mathematics
to Mathematics Education