- How is knowledge relevant to math or science constructed? How is it possibly generated in these networked communities? Provide examples to illustrate your points.

Lampert (1990) provided an illuminating account of how she conducted her lesson on exponents to a grade 5 class in order to change the meaning of knowing, and learning Mathematics in her classroom to adhere much more closely to how Mathematicians would argue, and establish Mathematical facts. I believe that her method more closely resembles how knowledge in math is constructed.

Lampert argues that there is a sharp contrast between Mathematical practice and how the subject is perceived in popular culture and in the classroom. In the scientific community, Mathematical ideas are often questioned, with the assumptions frequently evaluated and foundations tested, and as such the subject is open to discussion and the possibility of uncertainty. This is in contrast to how mathematics is discussed in the classroom, as the teacher, and the textbook is believed to hold all the facts and are rarely questioned. It was also believed that the concept of “proof” and the challenging of assumptions are rarely brought into classroom practice.

Below are some of the practices that Lampert used in her class:

- In starting a new unit, Lampert gave students wide open problems that encouraged participation, and discussion. For example, students were asked to find a way to determine the last digit in the expressions 6^4 and 7^4 without multiplying. Problems like these could be solve through a variety of strategies, and was open to student hypothesis and discussion. Lampert was not only looking for the strategy to solve the problem, but also the actual solution. This trains students in the act of forming hypotheses, and discussing their ideas. I also believe that this closely resembles how Mathematical knowledge is created: Mathematicians observe a problem in the real world, and conjectures are made about how to determine the solution to the problem.
- Lampert wrote down student solutions on the board, along with their names. When students asserted that certain answers needed to be removed from the board because they were incorrect, the students were asked to provide reasoning as to why the answer is incorrect, and why the person who gave the answer thought the way they did. I thought this practice resembled academic discourse, as Mathematical proofs are often placed in the public eye, argued, and agreed upon before acceptance as fact.
- Lampert followed, and engaged in the mathematical argument with the students in order to show students what it means to know Mathematics. Lampert made explicit the knowledge that she carried with her, and how she used that knowledge to carry an argument about the legitimacy of their proofs. The analogy Lampert used was one of navigating “cross country mathematics”. The teacher uses their knowledge to move along the path traveled by students on the mathematical terrain, and to help students move along. Instead of directing students along a carefully laid out path, Lampert suggests that teachers should show what it means to have mathematical expertise, and that it is more than being able to navigate down a straight and clean path, but rather the ability to navigate through sometimes rugged terrain.

Networked communities help generate mathematical knowledge by allowing information to be collected from a large population at once. One feature of the GLOBE library for example, is for students to contribute data to scientific studies (Penuel, 2004). Numerical data can be collected from a variety of sources, and analysed for patterns and possible relationships. I believe networked communities are a big frontier in mathematics. Many self driving car trials and experiments for example, is a result of data collected by a a large fleet of semi-autonomous vehicles.

Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. *American educational research journal*, *27*(1), 29-63.

Penuel, W. R., & Means, B. (2004). Implementation variation and fidelity in an inquiry science program: Analysis of GLOBE data reporting patterns. *Journal of Research in Science Teaching*, *41*(3), 294-315.

Hi Gary,

I really like the parallel you draw between Globe and the self driving car trials. . . I never thought of it that day. I’m finding that medicine is really similar. We have a networked community of scientists through a site called Researchgate (https://www.researchgate.net/about). It makes finding out about new articles in our respective fields easier. It’s also a great way to establish relationships with people you have never met and collaborate on research projects, ask researchers questions regarding their publication etc. I would argue that communities like this or Globe allows us to progress in our fields at speeds unthought of in previous decades.

Mo