Many of your assignments will involve exploring a mathematical situation and then communicating about that exploration. An effective write-up will usually contain the following:

- A setup of the situation you explored
- A discussion of the various avenues and dimensions of your exploration
- The making, exploring, and proving of conjectures
- A pedagogical paragraph, wherein you discuss the mathematics that
can be learned from this exploration and issues relating to learning
and teaching mathematics with this exploration (or some variation of it)

You will find that you usually do not get a 10/10 on your first attempt at these write-ups. It is most common to get an initial grade ranging from 6-8. As the assignments you are working on are usually very open-ended, the work necessary to improve from this initial grade to a 10/10 will differ from person to person. Although I have outlined below the general rubric I use in evaluating your work, I want to be able to give you individualized feedback on your own mathematical understanding. I want you to feel free to be creative and innovative in the work that you do and I will try to play off of that in my feedback.

General Grading Rubric

Any evaluation of your write-ups will necessarily be somewhat subjective on my part. The following rubric will give you some idea of what I will be taking into account when I assess your work:

A: The task is explained by thoroughly developed mathematical ideas
and is enhanced by other mathematical ideas. A thoroughly
developed
plan using pictures, charts, words, graphs, and/or symbols is used to
correctly
solve the task. The task is verified and/or defended possibly by
solving the problem using a new strategy. The path connecting the
concepts and strategies to the identified answer is very clear and
enhanced
by graphics and examples. The essence of the problem, the strategies
used,
and the reasons and justifications for using those strategies are
communicated
explicitly in a clear and organized manner.

B: The task is explained by mathematical ideas that work. A
complete
plan, using pictures, charts, words, graphs, and/or symbols is used to
solve the task (all work is shown). Some parts but not all of the
work is checked. The path through the work to the identified
answer
is complete. The work may have a small mistake, but the important
parts of the work are fine. The essence of the problem, the strategies
used, and the reasons and justifications for using those strategies are
communicated explicitly in an organized manner.

C: Parts of the task are explained by mathematical ideas that
work.
The plan could solve parts of the task or the work is only partly
shown.
The check is underdeveloped (only a small section of the work is
checked.)
The path through the work is partly shown. The work has a small
mistake,
but the important parts of the work are fine. The essence of the
problem,
the strategies used, and the reasons and justifications for using those
strategies are somewhat evident, although this is somewhat implied.

D: The plan is underdeveloped (many missing sections) or the plan
included
some strategies that cannot work. The check is ineffective for
the
task, is only minimal, or no identifiable check is shown. The
path
is not clear or is underdeveloped showing few connections within the
work.
The answer is mostly incorrect, not finished, or does not match the
work.
The essence of the problem, the strategies used, and the reasons and
justifications
for using those strategies are not communicated clearly.

F: Inappropriate or minimal concepts are used or no ideas are
shown.
The plan is ineffective, the work is minimal, the work conflicts with
the
answer given, or no work is shown. The check is ineffective for
the
task, is only minimal, or no identifiable check is shown. The
path
is ineffective, minimal, or is not shown at all. The answer is
not
correct, not finished, or does not match the work. The essence of the
problem,
the strategies used, and the reasons and justifications for using those
strategies are not communicated.

Writing To Learn Mathematics Education

A writing component accompanies most of the learning-to-teach activities of this class. Some wonder why there would be so much writing in a course about learning to teach mathematics, sharing protestations such as, “This is about mathematics, not English” or “I chose to go into mathematics to get away from writing”. Because you may have had thoughts along these lines, below are three primary reasons why writing is so prominent in this experience:

- Teachers are communicators, and mathematics teachers are communicators of mathematics. You will develop the facility to communicate about mathematics, its learning and teaching as you write about your learning-to-teach experiences. This ability to communicate will be an invaluable asset as you teach and as you reflect on that teaching.
- We build knowledge by sharing it. The writing process will require you to seek not only for connections among the various concepts you are learning, but also for ways to articulate those connections. In turn, efforts to synthesize and organize your ideas will lead you to explore yet deeper and broader connections. This recursive process is the essence of learning to teach.
- Writing is a
lens on understanding. Throughout your career as a teacher,
those you work with (students, parents, colleagues and administrators)
will judge your competence, at least in part, on the quality of your
writing. They will take what you write as evidence of what you know and
how you know it. The same is true during your student teaching
experience. You should interpret all writing assignments as an
opportunity to make a case for your own understanding.

Article Reflections

As article reflections differ a bit from the general write-ups you will be doing, I will give you somewhat more specific guidelines for these assignments:

For each article reflection your initial assignment is to read the article and come to class prepared to participate in a class discussion. You should be prepared to pose questions to the class and to make comments on specific aspects of the reading. The last part of the assignment is to write a reflection paper on the article as well as the class discussion and post this reflection to your webpage. This reflection should consist of three substantive (as opposed to whimpy or shallow) paragraphs of the following format:

Paragraph 1: Summarize both the article and
the class discussion. Heading "Summary".

Paragraph 2: Provide a critique of both the article and the class
discussion. What aspects were the most convincing (or not convincing)
to you and why (don't forget the why). Heading "Critique".

Paragraph 3: Describe what the message of the article and the class
discussion mean to you personally. You could consider how the article
and the class discussion relate to our past class activities or
discussions, to your beliefs about teaching and learning mathematics,
to your own experience as a mathematics student or as a user of
technology, to your pedagogical plans as a future teacher... well, you
get the idea. Heading "Connections".