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Area inquiry experience--Overview (D2.2.0)
Sample of written reflections on the experience
Writing prompt: Once again, I am asking you to use your [journal] entry, at least in part, to reflect on the experiences done in class, specifically the activity of reflecting on the "math of area" (though, once again, feel free to add to this whatever else you wanted to share to the rest of the class this week if you wish to).
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| Secondary pre-service teachers: Lina , Steven |
| Elementary pre-service teachers: Monica, Kara, Carol , Kirstin |
Lina Maine (secondary math / Masters student)
"To give more students access to mathematics as a powerful way of making sense of the world, it is essential that an emphasis on reasoning pervade all mathematical activity" (Curriculum and Evaluation Standards, NCTM). This quote summarizes what the area activity meant to me. Questioning what area is and how we measure it, really forces one to develop a line of reasoning and thought that perhaps was always "unspoken" or in plain terms: "assumed". The area questions that we were trying to tackle surfaced many thoughts and feelings about mathematics that we may/may not be accustomed to. I myself never thought to question what area was or meant. However, now that I have, I understand how this form of inquiry exercises our ability to critically think-a method of reasoning that is applicable to everything we do. In our discussion of question 1, I found it interesting to hear how others described their solution. There were a lot of answers and perspectives that were very thought provoking: i.e. the quarter technique and why it would, or would not work. When realizing why we can get the exact area of circle, I couldn't help but feel that my knowledge base had become a bit "shooken". Everything that previously made sense all of a sudden, didn't. We know that making sense of things around us is not always an easy task. This is part of mathematical inquiry-it helps us make sense of the world. Inquiry in mathematical exercises, such as the questions we were solving, helps one construct, visualize, draw, measure, compare and contrast ones thoughts and arguments with what is already known. Throughtout mathematics we can see how to use inquiry by asking questions such as Why? When? How? I also learned that formulas and perhaps even some mathematical definitions really do not explain waht a topic such as area is all about. The questions that were brought up in class made me want to revisit the definitions I had formed earlier on in my head-especially forming an argument of how one can on cannot use circles to measure area.
Steven (secondary math / undergraduate)
I genuinely enjoyed the opportunity to reflect on the math of area. In a standard high school lesson, one learns formulas for area and strategies for breaking down more complex figures into simpler shapes but ratrely as much depth of consideration of the fundamentals as we put forth. That I was intrigued by the questions posed is evidenced by the fact that I wrote up a paragraph response to each even thought only two write-ups were called for officially by the assignment, and I was pleased to note that others had also thought as extensively. In most cases, the justifications I was able to come up with seemed fully logical and adequately compelling, but the experience would not have been complete without the hearing of other people's perspectives. True, we often failed to reach a concrete completely satisfying answer; nontheless, I want to pull away from any implication that this made the activity any less rewarding. It is not new for me, as an amateur philosopher, to value critical thinking even when it yields no resolution, a point that is highly de-emphasized in mathematics taught as the systems for acquiring single correct answers. Reflecting also in relation to further reading from the NCTM standards, I do believe such experiences as this area one we've worked at should have a strong place in our math curriculum.
I must say I particularly enjoyed the deriving area formulas activity and can sympathize with Lina's notion of its potential addictiveness; imposing your own sense of order, even on a simple geometric figure, is engaging work. I was intent on uncovering a formula that didn't depend on angles or too many variable. Hence, I started very simply by extending the diamond concept to concave figures as well as convex. I then added parallel lines and was pleasantly surprised to discover one formula would hol for wide variations of the shape. Simplifying my formula revealed an unexpected similarity to trapezoids which I only observed afterward in the figure themselves. The more I thought, the more I wanted to add to my poster.
Monica Warnsman (elementary)
Last week's class was interesting to hear and see the different figures people worked on at home. I think doing the area problem at home was beneficial. I experienced how difficult it can be to develop a figure, as well as a definition, and justification. More importantly, I experienced what it means to question and explore in mathematics. As a student, I learned how much more fulfilling it can be to really learn through inquiry. Instead of just solving problems by a certain formula, it was much more challengin and interesting to explore different questions and options. For example, at the end of last week's class, it was fun to go ahead and explore these different questions that have developed throughout the lesson. What made it so interesting was that we were able to choose the question we wanted to pursue. Throughout the whole unit, I felt that I, as the student, had a lot of choice of what I wanted to do (choosing the shape, question to explore). I was having trouble seeing how rectangles that have the same perimeters could have different areas. After I drew some figures and discussed this with others I understood. However, that question led to further questioning. By the end of the experience, I learned how the exact size of a square and a circle can be determined by only one dimension. That fact can lead to further questions, regarding formulas, etc.
These lessons and exercices were great ways to teach some of the goals of area according to NCTM standards. The NCTM standards require students to understand that the area of a figure does not change if it is partitioned or rearranged. The fish exercise was a great example of a way to teach that. Students will use different strategies, including rearranging and partitioning, and that came out in that exercise. The other exercises we participated in also addressed many of the requirements of the NCTM on area such as, the formula of an area is not magic, but rather tells how many units cover the figure. Also, the exercises demonstrated the association between multiplication and determining the area.
Kara Jackson (elementary)
My experience with reflecting on the "Math of Area" ran the entire gamete, from feeling sheer stupidity to gaining a greater confidence and appreciation for a topic that was previously quite hazy. When I first began working on the questions assigned last week, I could not find two that I had any clue how to answer. I spoke with another member of the class, and with a handful of teachers at Fairbanks Road Elementary School, and came away with little more information than when I began {NOTE: From the people at FRS I got answers like "That's just the way it is, Kara!"}. I felt frustrated, stupid, and negative about the entire experience. I did nonetheless, write up 'answers' and ideas for three of the questions, and went into class vowing to not utter a word and avoid embarrassment.
When class began, I began my facade of being enthusiastic about breaking into small groups to discuss the area questions. My group consisted of three, and from the onset I felt like a complete loser because my answers were blatently wrong, and my reasoning almost non-existent. When my stupidity reached its height I snipped at a group member and vented my frustrations because of his criticism about my reasoning. At this point, another group member stepped in and explained the questions in very basic terms that almost immediately made some kind of sense to me. Once my initial frustration began to subside, the group worked very effectively with me, and helped me to gain a greater understanding about area.
What strikes me as most powerful, was the simplicity that surrounded some of my problems [and confusions] with area, that I made out ot be so difficult. It was this opportunity to rethink my past reasoning that helped me the most. The connections that I made about patterns, area, and math in general were also helpful in turning this experience into a more positive one. I think that I very rarely thought about how I came to solve a problem, especially those problems that came with a formul that need only be applied. I can see and sympathize with those students who are now balking at the idea of justifying their answers and responding to why they think that their answer or their process is "good." Obviously, like me, these students have had limited opportunities to believe that they "have the power to do mathematics [or figure out what's behind mathematics] and that they have the [power to] control their own success or failure" [p. 29, Standards].
Carol Fruehan (elementary)
As we are coming to the end of our unit on Area as Inquiry, I feel I have a much better understanding of area and its attributes. This was a totally new experience for me, but I am much more confident with my new understanding of area. One of my major learnings from this whole experience is that how a problem is solved is as important as its answer.
During the first class on area inquiry, I shared Kate's feelings about wanting more time to complete the problems. The homework assignment of developing an area formula for our own figure helped me, as it reinforced what we did in class, and I felt I could finish the problems at my own pace (and my partner's). During last week's class it was interesting to look deeper into some of our thoughts and questions about area and related topics. The variety of questions that came up in class amazed me! I think it important that students are given opportunities to explore and explain their thoughts and questions. Most importantly these experiences have caused to think about area and its connections to other math topics and issues.
Kirstin Prior (elementary)
I found the first class on area interesting. The fish + diamond were learning experiences -- they got me thinking about different approaches and also got me thinking sort of "like a mathematician." They also led to some interesting questions. The activity that involved defining our own shapes was good because it exposed me to the issues involved in defining a figure.
In spite of all of this, I must admit that I just couldn't get into "area" last week. Steve's questions about rethinking the whole concept of area as covering a surface was interesting, but that's about it. I tried to analyze my reaction -- it wasn't that it was over my head, or that it was stuff I'd done before or anything like that -- I just coundn't get engaged. My mind wasn't in an inquiry state.
I do think this was a valuable experience as a whole -- the inquiry approach can lead to new thoughts and questions and to real understanding -- because the students are actively constructing meaning.
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