Representations of mathematics teaching and their use in teacher education: What do we need in a pedagogy for the 21st century?

The introductory content of this paper is taken from the first author’s Pattishall Award Lecture at the UM School of Education in May 2010.First discussion document for PMENA’s RMT Working Group:“Facilitating sessions where teachers interact with and discuss representations of teaching”The work of project ThEMaT has been done with the support of NSF grants ESI-0353285 and DRL- 0918425. All opinions are those of the authors and do not necessarily represent the views of the foundation.http://deepblue.lib.umich.edu/bitstream/2027.42/78158/1/WG-RMT-Final.pd

Integral Basis of Pure Cubic Fields

Algebraic number theory is essentially the study of number fields, which are subfields of the complex numbers that are finite extensions of the field of rational numbers. A pure cubic number field Q[³√m], m cubefree, consists of polynomials with rational coefficients evaluated at [³√m]. An algebraic integer is a complex number that is a root of a monic polynomial with integer coefficients, and the set of algebraic integers in Q[³√m] forms the ring of integers of Q[³√m]. We can define a basis of this ring of integers, called an integral basis of Q[³√m], and any linear combination of elements from this basis will produce an algebraic integer. For m=hk², where h and k are relatively prime and squarefree and for α = ³√m, we prove that there exists an integral basis for Q[³√m] of the form: 1, α, α²/k if m ±1 (mod 9) 1, α, α² ± k²α + k² /3k if m ≡ ±1 (mod 9

Proof: Examples and beyond.

Asking middle school students to verify the math they do requires them to think about proof. By doing so, students construct arguments in the middle school and are more ready for proof in high school

REPRESENTATIONS OF MATHEMATICS TEACHING AND THEIR USE IN TRANSFORMING TEACHER EDUCATION: THE ROLE OF APPROXIMATIONS OF PRACTICE

This is the discussion document for the 2012 meeting of the Working Group on Representations of Practice and their use in Teacher Education at PMENAhttp://deepblue.lib.umich.edu/bitstream/2027.42/91280/1/WGPMENA_RMT_2012_finalweb.pdf-

REPRESENTATIONS OF MATHEMATICS TEACHING AND THEIR USE IN TRANSFORMING TEACHER EDUCATION: CONTRIBUTIONS TO A PEDAGOGICAL FRAMEWORK

teacher education, technology, representations of practice, teaching, mathematics, comics, cartoon, animation, pedagogy, casesThe use of representations of mathematics teaching, particularly those that are maintained in a digital form, calls for specialized pedagogical practices from teacher developers. They also open new areas for investigation of how future professionals learn to practice and the role that various technologies play in scaffolding that learning. In the discussion paper for the PMENA working group on representations of mathematics teaching, Herbst, Bieda, Chazan, and González (2010) reviewed literature on the use of video records and written cases in teacher education and noted that classroom scenarios sketched as cartoon animations have begun to be utilized for those purposes, arguing that they have affordances that are distinct from those of video and written cases. That document also noted existing literature on the use of written and video cases in teacher education and cited examples that concern mostly face-to-face facilitation and argued that the increased capabilities of information technologies for creating, manipulating, and collaborating over multimedia point to a promising future for teacher development assisted by representations of practice. In this document we complement the previous year’s review by briefly accounting for three areas of emerging scholarship: (1) information technologies that support teachers’ learning from representations of practice; (2) the particular challenge of helping prospective teachers understand students’ thinking; and (3) research and theory about what is important or possible to achieve in having prospective teachers look at or work with representations of teaching. We also describe present developments in the articulation of a pedagogical framework.Some of this material has been produced with the support of NSF grants ESI-0353285 and DRL- 0918425 to Herbst and Chazan.http://deepblue.lib.umich.edu/bitstream/2027.42/86657/1/PMENA2011_RMT_Framework.pd