Synthesis of this Course

This course has helped me to realize the importance of incorporating the following ideas into my future mathematics classroom:

-evaluating and being aware of student comprehension as I teach, since comprehension allows students to grasp concepts and relationships found in mathematics;
-teaching and using vocabulary in a way that students can understand and master for themselves;
-encouraging oral discussions as a form of student-to-student teaching and as a way of communicating ideas on mathematics topics;
-using writing as a way to express ideas and understanding, as well as a form of assessment;
-creating a classroom that is conducive to supporting high self-efficacy levels and motivated learning by choosing affective dimensions of literacy (i.e. text selection that is appropriate for diverse audiences);
-becoming a critically literate teacher who values the empowerment of the student and who hopes to teach lifelong learning skills;
-differentiating instruction by providing students with choices and by educating them through different viewpoints, as well as being aware of the needs of English language learners.

I feel, after learning about all of these different areas, that each of these topics change and impact the teaching of an educator for the better. I have been convinced by research and by our class discussions that the implementation of these ideas and teaching styles can help to change the mathematics classroom from a training center to an empowering place of learning, education, and discovery.

As we started and continued through the semester, some of these subjects seemed distant and unconnected for me. However, as we took our oral final and discussed each area in depth, the big picture seemed so clear to me - it all came together. To incorporate a single one of these ideas is not enough because they are all interlaced and overlapping; to get the fullest effect, a teacher has to try to implement them all. I know that I will have a lot of struggles in the beginning of my career with these goals. But I also know that as I become more experienced, that it will be easier for me to notice changes that need to made in order to use and incorporate these ideas more effectively.

Discipline-Specific Writing and Assessment

As I have thought about my experiences in mathematics, I have realized that my experiences with writing have been somewhat limited. I say limited due to the fact that writing in mathematics may be considered different when compared to writing in other disciplines. The main type of writing that I have done in mathematics has been mostly numerical when computing and completing assigned textbook problems, tests, and quizzes. The only assignments that I remember having to write in a "standard English" sense (i.e. a paper or report) are a few reports on a mathematician from history and a project on fractals. So, overall, my math teachers in grade school typically encouraged me to express my understanding through numeric problems.

I feel that my experiences with writing in mathematics changed drastically when I entered college and began to complete upper-level course work. Much of the assignments in my classes required written proof, which was something new to me. However, I have come to appreciate the precision that comes from explaining and comprehending the driving forces behind mathematics because of this shift. I think that my professors have encouraged me to express my understanding by thorough investigation and detailed analysis of my work.

I honestly cannot think of any types of expression that I feel would be considered alternative or creative in presentation. However, this aside, I have been learning a lot this semester about how to incorporate technology related activites into the mathematics classroom. I feel that I have learned a lot of different ways that would display student understanding of mathematics in creative and alternative ways. Some of these technologies include math applets, blogs, and screen-cast videos. I have found that most research supports the use of these technologies in the math classroom, as they help to engage students in higher-level thinking, as well as encourage students to become doers of mathematics and make the discipline their own.

As a whole, I feel that most of the assessments that I took at the grade school level were limiting because they usually did not include questions that explored student understanding and comprehension in writing. Most tests were merely computational. I think that the tests that I have taken in college have been a little bit more diverse in this area, but only for certain classes. I feel that it will be important for me to include assessment prompts that ask students to explain how they would go about completing a problem, rather than always asking students to perform a computation. I think that knowing students' thought processes will help me to adapt my teaching to enhance their learning. I also think that I want students to create either a video or an applet of a proof as another type of assessment. I think that these types of assessments will help to educate students in mathematics rather than just train them in the discipline.