In this talk, I will outline my attempts to highlight the processes with my students, particularly the process expectation {\it reflection}. Examples of evaluation items and student responses will be shared.

In each of our examples the discussion of a case-invariant solution has a slightly different emphasis. In the first example we demonstrate the importance of consideration of special cases: the key contraction that was found in a special case of our problem suggested the solution to the original problem taken in full generality. In the second example we illustrate the possibility to notice some additional geometrical facts useful for proving other statements while looking at different cases of a theorem’s proof. Another example shows that trying to unify various cases of a problem using a DGE may allow one to deepen their understanding of certain geometrical notions (such as area) and make connections with other branches of mathematical knowledge.

A pre-test/ post-test scheme is used to collect the evidence. In the first week of classes in September, students are administered an unannounced 50-minute survey, which gives an initial assessment of their general math knowledge and skills. After students complete the survey, no aspect of it is addressed in lectures. Eight months later, at the end of the school year, students are given the same survey, again unannounced. What is the purpose? None of the survey questions are explicitly discussed in lectures. However, throughout the first-year instruction in iSci students are exposed to a number of activities (such as problem-solving, critical thinking, creating precise scientific arguments, and so on), which can help them answer test questions better than at the start. The purpose of this approach is to determine whether students did learn math in the sense of being able to apply it to situations that were not explicitly addressed in lectures.

In my talk I will present the data, discuss interesting and relevant findings, and comment on the implications for other disciplines, within as well as beyond the iSci program.

Like mathematical reasoning, statistical reasoning requires clear, logical thought. But in contrast to the certainty prevalent in mathematical arguments, statistical reasoning requires the use of rational thought to make sense of uncertainty. We will explore issues relating to integrating stochastic thinking—both process and content—into the mathematics curriculum across all grades, from K–14. Questions might be based on the relative importance of intuitive, experimental, and theoretical probability; the difference between causality and correlation; understanding randomness; or teacher preparedness in terms of understanding and embracing uncertainty and how to best teach the concepts and processes.

The ‘tangram’ has been originally referred to as the 7-pieces dissection (or tangram problem) consisting of seven flat shapes forming together a square shape (five triangles: two identical large, two identical small and one medium triangles; a small square and a parallelogram). It was originally invented in China at some unknown year in history, and then carried over to the world by trading ships in the early 19th century (Wang and Hsiung, 1942; Read, 1965). In particular, assuming that the small square has an area of one unit square then each large triangle has an area of two unit square and each small triangle of an area of half unit square and medium triangle of an area of one unit square and a parallelogram have an area of one unit square too.

In collaboration with Radcliffe Siddo, Lakehead University (rsiddo@lakeheadu.ca)

Although we could, from a strictly academic perspective, develop a set of big ideas to encompass K-12 mathematics, the real issue is whether and/or how this is of value in the classroom setting.

• Is it the teacher who is assisted by this focus, the students, or both?

Is the main value of focusing on big ideas to provide rich enough connections for students that new content related to those big ideas becomes more accessible? Or is the main value to streamline the curriculum? Teachers are often concerned that there is too much to teach and they see a focus on big ideas as a way to have less to teach.

• How would classroom practice be affected?

Would it really change the way in which instruction is offered to students? Is it more about the problems that are posed or the questions that are asked about those problems? Or would it simply be a change in optics?

• Should big ideas be the "content equivalent" of processes, i.e. those content ideas that are revealed repeatedly in different guises throughout the curriculum or should the big ideas be the processes themselves? Is it enough for the big ideas to be just the processes?

In physics, picking your coordinate frame and starting time do not change the solution of the problem. The laws of physics are unchanged by where the origin is, what the 'start time is' or what direction you call 'x'. Choosing your frame of reference is a transformation of the equations for the laws of physics, and the answer transforms in the same way - it is invariant. In physics, this 'symmetry in the laws' is recognized as a big idea and connects to conservation laws, through Noether's Theorem.

How do these connect to how children learn - and how the curriculum can be structured? Are there other examples back in arithmetic, in algebra, in statistics ... ?