1. Bridging Mathematics and Mathematics Education
Walter Whiteley
Mathematics and Statistics
Graduate Programs in:
Math, Education, Computer Science, Interdisciplinary Studies
York University

2. 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

3. 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, …

4. 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;

5. 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.

6. 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.

7. 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!

8. 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.

9. What in Mathematics Programs for Teachers (cont)
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 …

10. What in Mathematics Programs for Teachers (cont)
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

11. 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, … )

12. 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!

13. 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, … ?

14. 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’

15. 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?

16. 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.

17. 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, … .

18. 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.

19. 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.

20. 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

21. 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?

22. 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.

23. 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.

24. 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.

25. 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!

26. Thanks Questions? whiteley@yorku.ca wiki.math.yorku.ca/
Link under Conferences: Bridging Mathematics to Mathematics Education