Math to Math Ed
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Resources on Bridging from Mathematics to Other Disciplines
A very useful document discussing the importance of these connections, and the contributions that people in Mathematics Departments could be, and should be, making to the preparation of teachers of mathematics, is the Conference Board of the Mathematical Sciences (CBMS) book:
The Mathematical Preparation of Teachers http://www.cbmsweb.org/MET_Document/index.htm
It can be purchased on line, or downloaded free as a PDF, or viewed as html.
At the MAA winter meeting in San Francisco, January 2010, there were several key resources being discussed. One key discussion was from the Curriculum Revision Across the First Two Years (CRAFTY) discussions with people in a number of disciplines served by Mathematics and Statistics Departments (including Education, Engineering, .... ). The materials are available for download at: http://www.maa.org/cupm/crafty/
A very remarkable observation is that the core, shared recommendations from these sources in other disciplines overlap with both the recommendations of the CBMS book on the Mathematical Preparation of Teachers, and with core recommendations of what the objectives are / should be for all mathematics majors, in terms of refocusing on mathematical processes as in-depth learning of some material, with flexibility around which topics are the focus and which will be local choices or be left to the students to learn if we have prepared students who can read and learn mathematics independently.
This growing consensus does not yet represent what most mathematics program are doing, or what most courses within programs are doing. This is the major challenge.
Programs in Mathematics Departments - for Future Teachers
An interesting example of such a program, at a major research university in the US, is the program at UCLA: http://www.math.ucla.edu/~twg/mathed.html
The York University Program in Mathematics for Education is designed as a Major for either concurrent education students or for students who plan to apply for a consecutive education program: http://wiki.math.yorku.ca/index.php/Mathematics_for_Education_Program
Here are the University Undergraduate Program Objectives for the Math for Education Program.
To support the work of mathematics departments in the mathematical preparation of future teachers of Mathematics, as well as the education of undergraduate mathematics majors, it is important to draw on the findings of researchers in mathematics education.
To support high quality research in mathematics education, it is also important to maintain a dialog between mathematicians and mathematics educators about how mathematicians practice mathematics, as well as mathematics as a discipline in service of other disciplines.
To maintain a strong, interdisciplinary conversation within CMESG, it is essential to renew the participation of Mathematicians in the joint meetings with Mathematics Educators. Such a stimulating mix of mathematicians and mathematics educators has been a key feature of the Canadian community in general, and the CMESG in particular. Such a dialog is also central to the long-term impact of the work in Mathematics Education on the preparation of teachers of mathematics and the development of high quality mathematics education in Canada.
Bridging from Mathematics to Mathematics Education: A Workshop June 2009
A one-day workshop was held Friday June 5, 9-4:00, Department of Mathematics, York University.
This one day program was designed for people working in Mathematics Departments (or studying in Mathematics Programs) who are planning to join the larger CMESG conference, and would like an orientation to the issues, vocabulary, methods, … current in research in Mathematics Education in Canada. We have received funding from the Fields Institute for the Mathematical Sciences to organize this event.
The program was also designed for anyone in a Mathematics Department with an interest in closer collaborations / liaison with Faculties of Education and in making contributions to the preparation of pre-service teachers of mathematics.
Here is the Workshop Program as it was offered. All presenters listed participated.
A group of `experienced mathematical hands’ in these CMESG conversations lead and animated our discussions with some contributions from some experienced people in mathematics education. Detailed planning continues.
In the evening, there was be a public lecture which was also the Opening Plenary for the CMESG conference. All participants were invited to stay on for this, and the BBQ which will take place between the workshop and the Plenary.
Sponsorship of June 2009 Day
This June event received support from the Fields Institute for Mathematical Sciences and from MITACS. With this funding we were able to cover all costs and make the event free of charge.
While not directly organized or ‘sponsored’ by the CMESG executive, this local experimental initiative had their support and a link to this pre-conference program appeared in the materials for the CMESG Conference.
CMESG Working Group on Recruitment, Retention and Attrition in Post-Second Math
This was the topic of a working group at the CMESG meeting in May 2010 at Simon Fraser. The discussion provided a good example of how insights from people in mathematics departments and in faculties of education can combine to provide additional insight into critical problems for teachers of mathematics, for mathematics educators and for people in mathematics and statistics departments.
Some Follow Up: December 2009 CMS Adrien Pouliot Talk
On December 5 2009, Walter Whiteley gave an invited 45 minute 'Adrien Pouliot Award' talk at the Canadian Mathematics Society Winter Meeting on the same theme of Bridging Mathematics and Mathematics Education. In a number of ways, this reflected on the themes, and the discussions from the June workshop.
Here are the slides from the talk: Bridging Mathematics to Mathematics Education (Windsor 2009). I would be interested in feedback, discussion, debate, ... on these themes.
Big Ideas as a Bridging Theme
We have heard that the next round of curriculum revisions in Ontario will be focused on 'Big Ideas'. This is an interesting shift from detailed lists (which are pedagogically destructive), so we have started work on this. (We being the people in the four year old 'Three Year Curriculum Project!).
There was a one-day session at the Fields Institute in October 2010, and this will be the topic of the Canadian Mathematics Society Mathematics Education Session in December 2011, in Toronto. Some reports from this are at the site:
Uncertainty, chance, statistical inference
One big idea identified is the cluster of ideas around chance, uncertainty, statistical inference ... . In New Zealand one of the three strands of the entire mathematics and statistics curriculum k-12 is in this area. Chris Wild from New Zealand will be a Plenary Speaker on this theme.
A second big idea is focusing on the mathematical processes, with less attention to details of which content.
The process expectations of the Ontario curriculum are:
Problem Solving problem solving, and selecting appropriate problem solving techniques 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 …
For comparison, a version of the Undergraduate Degree Level Expectations for the Mathematics for Education program at York University are:
integrate relevant knowledge to pose questions, solve problems and learn new mathematical concepts, methods and tools… construct, analyze, and interpret mathematical models and understand the value and limitations of mathematical models employ technology effectively, including computer software and algorithms (numerical, graphical, simulation), to investigate, conjecture, solve … analyze data and make inferences using appropriate concepts and techniques from statistics and mathematics and recognize the limits of inferences. take a core mathematical concept and ‘unpack / repack’ the cognitive network compressed in the concept communicate mathematical and statistical concepts, models, reasoning, explanations, interpretations and solutions clearly, to multiple audiences in multiple forms … recognize the limits of their knowledge (and of mathematics and statistics) identify and describe some of the current issues and challenges (professional, ethical, … )
Do these still encode big ideas (e.g. limitations of knowledge, modeling) or have they become 'small'?
Transformations, symmetry and Invariance
A third big idea we are investigating involves Transformations, symmetry and invariance. Inside this theme are the modern (post 1800) approaches to geometry, to much of the physical sciences, etc. While young children have a perceptual sense of bilateral symmetry, and of rotation, translation of an object, the modern approach represents a paradigm shift (starting with Legendre in 1794, and going on through the development of group theory from multiple sources, Klein's Erlanger Program, Lie, Cure's and Poincare's reworking of physics in terms of symmetry, invariance, conservation laws .... ).
Here are some further reflections on this theme:
An interesting book, available electronically from some university libraries, is
Giora Hon and Bernard R. Goldstein: From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept
This makes the case that a paper by Legendre in 1794 marks a shift in paradigm towards the modern development of 'symmetry' connected to transformations, groups and invariants.
Possible Graduate Diploma in Teaching Post-Secondary Mathematics
Under construction - April 2011
It is now clear that a substantial fraction of Ph.D. graduates will seek and find work in positions that are primarily teaching post-secondary mathematics (as well as courses at post-secondary institutions which cover secondary mathematics material). What program can accomplish this?
Here are some thoughts, stimulated by a discussion in April at the University of Kentucky:
A course in the mathematics graduate program that introduces some basic themes and issues in post-secondary mathematics teaching (including grading, classroom management, running a discussion, ... )
A broader course in University Teaching and Learning,such as the UTAL course at York:
As well as the more complete University Teaching Practicum:
Some supervised teaching, with observations, reflections;
Another set of resources at York are the courses in the Graduate Program in Education Graduate Diploma in Post Secondary Education:
Some Links and References
A more general commentary on how to improve undergraduate education, with similar recommendations, is the Boyer Report from the Carnegie Foundation: http://www.as.wvu.edu/~lbrady/boyer-report.html
For further information, contact Walter Whiteley: mailto://email@example.com, 416-736-2100 ext 22598.