When reading the article, I kept in mind that although there is a big difference in Type 1 and Type 2 lessons, there is no “right answer” in choosing a lesson type to model after. Type 1 lessons are narrower and don't elicit as much critical thinking and decision making, but they are more structured and make a classroom more organized. Type 1 lessons may be necessary depending what kind of students I will teach. If I am teaching students that are “behind the curve” and may need extra assistance with math, Type 1 lessons may be required. On the other hand, Type 2 lessons may be more beneficial for gifted students who can handle a more independent learning environment.

As I read the section describing Type 2, a lot of questions popped up in my mind: If the Type 2 lesson allows students to be independent and explore on their own, how do the goals of the lesson (and curriculum) fit in? How do they learn what they're expected to learn? What happens to classroom management if students get off track? What is my role as a teacher if technology allows students to learn on their own? Is my purpose solely that of a ‘marshall’—I'm just there to make sure students are doing what they're supposed to be doing? These questions have significant purpose not only tonight but throughout all the readings. I want to be able to use technology to guide students' learning, not to replace my teaching. My purpose as teacher is to help students investigate and understand relationships among objects, and to complexify

this investigation so that the students can simplify

(p. 309) it. Rather than making decisions about what and how to investigate, my job is to guide students to make those decisions, and reaffirm that they're making the right (or wrong) ones.

A miscellaneous topic: I noticed that teachers and students often throw around the words ‘lesser,’ ‘greater,’ ‘bigger,’ and ‘smaller’ a lot without realizing their meanings. ‘Bigger’ and ‘smaller’ can be quite confusing in the domain of negative numbers. (Positive numbers are easy to understand in this regard). As teachers, we should be careful when choosing these words to describe relationships and ordering of real numbers. We should use the terms ‘lesser’ and ‘greater’ to make explicit the ordering of numbers. For example, negative five is definitely less than negative three. When we ask, “Is negative five bigger or smaller than negative three?”, students will most likely get confused because they will think of absolute value. Negative three is closer to zero (also known as ‘nothing’) than negative five is, therefore it must be ‘smaller.’

One last point: Tonight's reading enforced my belief that the main goals of any lesson should be **investigation and discovery**. When I first transferred to the Math Ed program here at Tech, I was excited to teach in a math classroom and lecture to the students, much like many college professors do today. However over time I realized that in a high school environment, students do *not* want to be lectured to. They want to investigate and discover things on their own. So rather than giving out all the answers, I hope to guide students to create and find their results and own them, so that way they will learn better, remember easier, and maybe even be enthusiastic about math.

References

- McGraw, R., & Grant, M. (). Investigating mathematics with technology: Lesson structures that encourage a range of solutions. In W. J. Masalski & P. C. Elliott (Eds.), Technology-supported mathematics learning environments (pp. 303–317). Reston, VA: NCTM.

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