Overview: The main goal of this set is to get the students working and seeing that their understandings of long division and irrational numbers are not as deep as they think. It also gives me a chance to have the students doing group work early on. I have them work together in class on the calculator program. Many of the students have never programmed their calculator before and find programming difficult. On the other hand, students also discover that they have never truly thought about the basic operations and lemmas they have learned and how they correspond to the algorithm they know. The other text problems play a similar role. Students have learned the proof that the square root of two is irrational, but usually they have not analyzed the importance that 2 being prime plays in the proof. Giving them other numbers helps reinforce this problem. At the same time, asking them to give other proofs for the primality (like those in the text) also.
Establishing my legitimacy: One problem a mathematics teacher has in teaching future K-12 teachers is that the students "know" that they will never use advanced mathematics in their teaching. (See Arcavi, et al. 1998, for discussion of this in a problem solving course.) This set helps to establish that I do have knowledge they need because it forces them to confront their lack of understanding of long division. Moreover, I can tie their learning of the division algorithm in an advanced algebra course into the teaching of long division. To see that this set creates some legitimacy, I quote Lyn:
Like programming the calculator for division. We know that to tell a calculator that knows nothing we have to go back. For every little thing we did, irrationals, dividing, factoring, we did all that back to basics, so, talking in class was really comfortable and actually learning to see in a different way was really exciting, it's like, "Oh I never saw it that way, or like I totally forgot, like look at it in this perspective." So it like totally helped me out.
Reflecting three months after the class is over, she saw the calculator program as an activity that helped her think about teaching. She then saw that happening over and over again.
A Change in Mathematics: A second goal is to bring forward problems that do not have one simple answer, but rather to show that there are multiple right answers and multiple methods of solutions. Thus, problems asking for them to make conjectures, and gather data are setting the stage for their working on the semester long research projects.
Technology: The students will need to be able to use technology in mathematically useful ways for their research projects and for their teaching. One of my goals in this set is to establish the value of technology for experimentation and discovery in mathematics. Thus the requirement to write a calculator program teaches them to use technology, and the requirement that they observe data and make conjectures (some of which happens in the next set) forces them to use the technology the way I want them to.
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