The videos also illustrate how the researcher, in the role of classroom teacher, facilitated interactions among students who worked in small groups, as well as in whole-class discussions, in ways that supported students' articulation of

mathematical arguments and consideration of whether those arguments were convincing as justification for solutions to problem tasks.

Thus, should an author wish to construct a logically correct

mathematical argument he must show that the symbols consistently stand for the same concept.

He covers mathematics, experience, and argument in 19th-century science;

mathematical arguments in The Origin of Species; Mendel, mathematics, and the case for uniform particulate inheritance; rhetorical strategies and Francis Galton's arguments for a mathematical model of inheritance; Karl Pearson and the push for theoretical mathematical biology; and the ethos of biometry.

They follow formal

mathematical arguments for the truth of propositions (for example, "the sum of three consecutive natural numbers is divisible by 3").

In a book-length

mathematical argument that has been percolating through the set theory community for the last few years, Woodin has proved--apart from one missing piece that must still be filled in--that elegant axioms do exist and, crucially, that every elegant axiom would make the continuum hypothesis false.

When Casey explains his thinking to the class, his

mathematical argument becomes part of the ensuing discussion.

So, this paper proposes a pedagogical approach to mathematical statistics education that centres on exploration, discovery, conjecture, hypothesis, thesis, and synthesis such that the experience of doing a

mathematical argument, creating a statistical model, or synthesising ideas is reason enough for the exercise--and the joy of mathematics and statistics is something that needs to be instilled and encouraged in students by having them do proofs, counterexamples, examples, and counter-arguments in a Probability and Mathematical Statistics course (indeed in any course).

So, this paper proposes an approach to mathematics education that centres on exploration, discovery, conjecture, hypothesis, thesis, and synthesis such that the experience of doing a

mathematical argument, creating a mathematical model, or synthesising ideas is reason enough for the exercise - - and the joy of mathematics is something that needs to be instilled and encouraged in students by having them do proofs, counterexamples, examples, (informal) arguments, and counter-arguments in any mathematics course.

So, this paper proposes a pedagogical approach to mathematics education that centres on exploration, discovery, conjecture, hypothesis, thesis, and synthesis such that the experience of doing a

mathematical argument, creating a mathematical model, or synthesising ideas is reason enough for the exercise--and the joy of mathematics is something that needs to be instilled and encouraged in students by having them do proofs, counterexamples, examples, and counter-arguments in a Bridge course to prepare the student for work in advanced mathematics.

Sensing, established by a shared expectation, was used and extended in the form of sociomathematical norms associated with

mathematical argument and authority.