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Participants' testimonials (A)
Selection from methods course participants' reflections
| The following excerpts come from the final "What have I learned" paper that each participant was asked to write at the end of the methods course. These excerpts were selected to illustrate the range of benefits that participants identified; therefore, you are invited to read the whole set, which comprises the following items: |
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"What have I learned" paper #1 (2 pages) -- elementary pre-service teacher discussing how the course changed her view of mathematics and helped her overcome her math anxiety.
Consistent with the inquiry based approach to mathematics which encourages numerous formats to demonstrate our understanding of mathematical concepts, I have chosen to write you a letter to express "What I Have Learned in Your Class."
First, as I am sure you have noted in my journals and the size of the smile on my face, I have had a RADICAL change in feeling about mathematics in the past the three months.
As you may remember, I called you very upset the day after our card trick experiment. I really felt like I was in the wrong class and wondered how I could ENDURE eleven or twelve more sessions if I was going to go home feeling like I did that night. [...]You told me that this course was designed for people like me and not to give up. Wow! I hung up the phone, measurably cheered, but I must admit, I still didn't believe that I was the kind of student that would most benefit from this class. [...]
Because I knew that for our final project I would be writing "What I learned about Mathematics," I kept a critical eye open to trace the evolution of my thinking from day one. I wanted to know exactly what I was learning. Ironically, I expected to learn a lot of mathematical "techniques" in this class. I expected a lot of skills "instruction" (via transmission model) and to focus on concepts the ways I had learned 25 years ago. After all, in my original definition of mathematics, there was only one answer. It would logically follow then that there could only be one way to teach math, right? Wrong! What I experienced in your class seemed to be related to mathematics only because there were number symbols involved. I encountered none of the feelings of isolation experienced as a result of individualized learning, silence, worksheets, or an insensitive instructor. When I was sitting in class on Wednesday night [during our last class], I though, "I hope I don't get so overwhelmed by my first year of teaching that I forget how much I loved math and learning about math in this class." What I experienced as "mathematics" in your class was so enriching, enjoyable and enlightening. In sum, it didn't resemble what I learned as "math" growing up at all.
[...] When I "revisited" my Math Survey in preparation for this paper, I re-read my answer to question #3:
"Math is so abstract. Because it was taught without any "real world application it only existed in my class in books or on the paper in front of me. It was right or wrong and I always had to force myself to is down and open my math books."
The math I experienced in your class was tangible, not abstract, had definite real world applications, it was learned with many laughs, songs, skits, posters, delicious food and scaffolding from you and my classmates. Whereas I never talked about math outside of math class before, I talked about the "radically new math that exists now" with my family and friends. I read my "Math Their Way" book as bedside reading, I devoured "counting" books at the library and eagerly scanned the math catalogs for manipulatives when they arrived in my mailbox as discussed in my last journal. Furthermore, everything we experienced in class was structured around the goals of the NCTM standards. Additionally, I learned that math is colorful, and involves manipulatives, literature, and games. It is vehicle for teaching people about many facets of life and understanding. Whereas I learned about tessellations, area and other mathematical concepts, in those lessons, I learned more about problem solving, critical thinking, empowering students, reflective thinking, the benefits of cooperative grouping and learning as a social practice. Because all opinions in our class were respected and valued, this created a huge sense of trust amongst us. Often, our class never reached a consensus about the "right" answer but that was encouraged. Very often there were many solutions to the problems, certainly a contrast to the "right" and "wrong" manner of my early schooling.
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"What have I learned" paper #2 -- (3 pages) secondary math pre-service teacher identifying what he learned with respect to other people's views of mathematics, an inquiry approach, and the emotional aspects of learning mathematics.
One of the most important things which I have learned from this course is the many different ways in which people view mathematics. [...] Ever since I began to study mathematics seriously, I have viewed the discipline as much more than a dormant set of facts which are just waiting on the pages of a text somewhere to be memorized. Similarly, I have come to view mathematical knowledge not as the ability to recall these theorems, axioms, and definitions verbatim from class lecture notes or a text, but as the ability to "make sense" of these theorems, axioms, and definitions and apply them to new and unique problem situations. [...]
However, as was emphasized during the group sharing portion of the "What Is Math?" activity and the many different metaphors for math, from the "Math Survey" (Homework #1), this is by no means the only view of math. This was also expressed by Raffaella Borasi in The Invisible Hand Operating in Mathematics Instruction: Student's Conceptions and Expectations, "many students assume that mathematics consists of a predetermined set of rules and procedures 'passed on' by teachers to the next generation" (page 175). A direct consequence of this is that these students do not consider "thinking on their own" (Borasi, page 175) as a viable approach to solving mathematical problems, and they come to see mathematical theorems, axioms, etc. as nothing more than the words, sentences, and examples which are used in the texts to describe them.
Conversations with friends and classmates (which usually occur directly after they find out that I am math major) have long since helped me to understand that this latter view of math, and not the view which I share, is the one held by many, if not the majority of, students. What this course has helped me to understand is the multiple benefits Which can come from helping students to see math as a more humanistic subject. One of the primary benefits, in my opinion, of a more humanistic understanding of mathematics, is that students will begin to shift their view of math from something which simply exists and is all discovered, to something which people do, end they will also hopefully learn to emphasize the process by which they come to a solution, rather than the product of this process. Further, as Borasi points out in "Major assumptions and implications of an inquiry approach to mathematics instruction:"
The recognition of limitations and "human" elements in the discipline could make it more attractive to people who have so far been intimidated by the absolute and authoritarian image of mathematics currently presented in schools ­ especially women and ethnic minorities. (page 4)
One way for mathematics educators to facilitate this is by not placing an emphasis, and students evaluations, on answers, but instead concentrating on how the students went about obtaining these answers. This is supported in the NCTM s discussion of their goal for mathematics students to learn to reason mathematically: "A demonstration of good reasoning should be rewarded even more than students ability to find correct answers" (NCTM Curriculum and Evaluation Standards, page 6). [...]
Dealing with this issue of process over product will also help to clear up the widely held misconception that nothing can be learned from mistakes. On the contrary, I have come to believe the mistakes we make are very powerful tools in our learning and understanding mathematics. [...]
Learning to communicate mathematically and gaining an understanding of the symbols and notation will also help students in assimilating new information into their overall understanding of mathematics. As I stated earlier, since I began to study mathematics and think about what it means to know mathematics, I have conceptualized mathematical knowledge as an ability to form an understanding of, or "make sense" of, mathematical concepts. My introduction to constructivism and the inquiry paradigm has helped me to not only clarify this view of mathematical knowledge, but to also understand its importance in my future teaching:
The influence of the constructivist position on mathematics education can be seen in the call for creating instructional environments that encourage children to make rather than receive knowledge from the teacher or the text. In such classrooms, the positions of teacher and student are reversed as students give up the job of listening and take responsibility for their own learning while teachers give up the job of telling and begin to listen to students' thinking so as to help them reflect on their personal constructions. (Borasi, "Reading, Writing and Mathematics: Rethinking the 'Basics' and their Relationship," page 6)
The inquiry approach to teaching mathematics is grounded in the idea that mathematics is a humanistic discipline, and that mathematical concepts are socially constructed within a community of learners. Another fundamental assumption of this approach is that learning should be seen as generative process of meaning­making from the part of each students (Borasi, "Major assumptions and . . ., page 4). This process of "meaning­making" is quite often initiated as a result of students' desires to better understand new concepts and problems. In contrast to many traditional classrooms, in which it would be the responsibility of the teacher to "clear things up," in an inquiry classroom, the teacher functions as a facilitator as the students collectively attempt to "make-sense" of the situation themselves. Along with acting as a facilitator in the classroom, teachers utilizing an inquiry approach also have the responsibility of developing activities which are not only thought­provoking, but which also raise questions and "engage the students actively in inquiry and meaning­making" (Borasi, Major assumptions . . ., page 6). Although at this point, I am unsure what role the inquiry approach will play in my teaching, implementing some inquiry learning in my classroom will help me to achieve my goals of emphasizing process over product and presenting my students with an opportunity to engage in open­ended problems and problems which may take several days to solve.
The class activities which were integral in my understanding of the inquiry approach were the guided inquiry activities on area, the mid­term paper, and our own inquiry project. The inquiry on area was an important first step for me because it served to not only introduce inquiry, but it also demonstrated what this may look like in the classroom. The mid­term paper, in which we analyzed a classroom experience in light of the NCTM Standards and the principles of an inquiry approach, was helpful in that it allowed me to evaluate which characteristics of an inquiry approach I felt were important and pertinent to the goals of the NCTM Standards (especially those standards which I believe are important, such as helping students to become mathematical problem solvers and creating a classroom with comfortable, and fairly open discourse between students and teacher). Finally, my own inquiry allowed me to participate in a "genuine" inquiry, which gave me a better understanding of how an inquiry develops; from developing a topic and raising" some questions, to refining these questions and deciding just where you want to begin).
Another major thing which I have learned this semester is the emotions which many students tie into math. As I commented in my Follow Up on the Card Trick (Homework #2):
As a student of mathematics, I often found that math made sense for me, and that I was more often than not able to understand certain aspects which others became frustrated and confused with. I know that since math has never frustrated me as I know it has, and still does, many people, I will need to . . . make sure that I remember how I felt in instances such as learning the card trick [which was initially a frustrating experience for me], and learning topics in non­math courses, which seemed at times almost foreign to me.
Fortunately, I was able to move beyond the initial frustration of the card trick and focus on the experience as a problem solving situation; a situation I am far more comfortable and confident with. I recognize that it was confidence in my abilities which allowed me to be persistent in attempting to solve the Card trick problem" and eventually reach a solution. As the NCTM Curriculum and Evaluation Standards points out, it is only through "numerous and varied experiences" with mathematics and problem solving that students can begin to gain such a level of confidence that they become comfortable with their own mathematical abilities (page 6). The diversity in mathematical abilities, which the "joint sessions" of this course provided, was crucial for me in understanding just how much confidence and other affective issues impact mathematical behaviors, such as problem solving.
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"What have I learned" paper #3 (2 pages) -- elementary pre-service teacher discussing his new insights about what constitutes a worthwhile math task and how it affected his student teaching experience.
The methods used to teach this math methods class reflected the instructor's sincere belief in what she was teaching. All too often, I'll take a course on education in which a professor sings the praises of a changing traditional methods of teaching, only to use traditional methods throughout the course. I found the instruction to mirror the methods that were being advocated and very much appreciated the experience. Not only that, having been able to experience an inquiry approach of teaching mathematics from a teacher and a learner made the course doubly effective.
Among the major things that I learned this semester was what constituted a worthwhile task. Before taking this course, I simply thought that by making math more fun that it would facilitate greater learning for students. Worthwhile tasks involve much more than that. I see worthwhile tasks in two overlapping categories. The first are authentic, 'real life' experiences involving situations that transcend the classroom. [...]
The "cost analysis of a pet" activity is an example of such an activity. Experiences like these are important as they validate students' thinking, and display the subjective and humanistic side to mathematics. Secondly, tasks can be mathematically worthwhile, yet seemingly have no direct real life application. Certain activities have merit in that they inspire the discovery of mathematical principles. The area activity in which we created formulas to find the area of stars would be in this category of worthwhile tasks.
As I mentioned changing how mathematics is taught doesn't entail just making math fun. On the contrary, learning math can (and sometimes should) be uncomfortable at times. I learned this lesson painfully during the "card trick" activity. My journal entry gives proof that I found this experience to be very humbling. As with all quests, learning takes place during the trials and tribulations of the journey, not upon reaching the destination. So it is with learning math.
Developing a sense of community is vital to the success of teaching. students need to feel safe to explore and learn from their efforts. During my student teaching experience this semester, I was able to experience teaching mathematics using both traditional and inquiry based methods. [...] The student presentations that were a major part of my area versus perimeter lesson offered [a new] look at the potential for creating a safe community in the classroom. The students were given an opportunity to deal with math concepts by working with concrete materials, through writing, drawing pictures, and orally explaining their ideas. This helped to create an successful learning environment by first giving the students several, varied ways with which to explore mathematical concepts. Secondly, students shared their ideas and talked among themselves to create a shared understanding. Even if a student had an example or answer that didn't work, it simply wasn't wrong. As a class we could pinpoint and talk about what made sense and what did not. The emphasis in this activity was on the process, not on the product. This helped to validate student's thinking and language, thereby improving their confidence and allowing them to internalize the concepts they were working with.
I have grown to appreciate the concept of a process oriented method of assessment. Ongoing assessment of student's learning is necessary to modify instruction and give feedback to better facilitate learning. Observation is a key tool to be used as students perform worthwhile tasks. This of qualitative assessment focuses on a student's strengths as opposed to rewarding them for being able to apply computations and algorithms on a test.
[...] I feel that a comment from a student I was working with while student teaching best sums up what I learned about math this semester. After completing a two day lesson on the difference between area and perimeter I asked the class as a whole what they had learned. One student replied "Someone just giving you the answer doesn't necessarily help you learn. It's better for you to come up with the answer yourself." At no time during the lesson can I remember saying anything to this effect, yet she seemed to have insight to the premise of inquiry approach of teaching mathematics. To use a cliche' proverb, "Give a man a fish and he will eat for a day. Teach a man to fish and he will eat for a lifetime." By the same token, if we teach our students to learn through discovery, they will be mathematical thinkers for the rest of their lives.
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"What have I learned" paper #4 (2 pages) -- in-service middle school mathematics teacher discussing how the course helped her rethink her teaching practices and the value of implemented a scaffolded inquiry experience in her class.
This course has presented mathematics in a new light for me. [...]
Through my own experiences with mathematics, I learned to view mathematics as neat and predictable. I was quick to catch on to procedures and could use them very well. I excelled in what Schoenfeld (1985) calls exercises. However, I felt intimidated by real problem solving experiences. I never gave much thought to the fact that mathematics we alive and changing. Math was math; period. I can't say that I ever was excited by math. I simply liked it because I felt that I did it well. Once I began teaching I realized my experience was lacking. I can remember instances when a student would solve a "text book" problem in a way entirely different from the "text book" solution. the students' methods were often easier and did not require the fancy algebra we had been practicing. What was upsetting to me was that I had never used my own creatively to solve problems in that way. When the class would respond, "That's so much easier. Why can't we just do it that way?" I said they could; but I had no idea about how to teach them to approach problems in this way. I was so used to attacking problems without thinking. I would just identify the type of problem and use the procedure I learned to solve that type of problem. This was my first indication I needed to change. [...]
Now with hope that it is possible to make mathematics meaningful, I need to focus on how to do this. NCTM Professional Teaching Standards (1991) clearly defined for me how I should approach planning and implementing my lessons. Having this document to read and re-read helps me to stay focused on the objectives of teaching mathematics in a meaningful way. The first standard of this document states that "The teacher of mathematics should pose tasks that are based on sound and significant mathematics." It is to the NCTM Curriculum and Evaluation Standards (1989) that I look to for the articulation of sound and significant mathematics. In an interview for a math teaching position I stated that the NCTM standards are the bible of teaching. Though I knew of these standards before taking this course, I did not fully understand their meaning and application. The area inquiry unit, cost of a pet activity, and the tessellation video helped me to gain insight to what implementing the standards should look and feel like both as a learner and teacher. The principles of an inquiry approach embraces the standards.
Having taught an inquiry unit I realize that it requires more time than a traditional unit. Experiencing for myself the effects of inquiry on "real students" was what sold me on the concept. Often I have read stories about amazing things happening in classrooms around the country. Truthfully, it is sometimes hard to believe. I am tempted to attribute the reported success to unusual circumstances such as, it was in a selective private school. This was something I had to experience myself before I would believe it could work. Teaching the area inquiry unit helping me to begin synthesizing my newly acquired knowledge. Throughout my unit I refined the structure I provided my students with. Seventh graders need help staying focused on their task. However, I did not want to lead them to my idea of the answer. Creating an appropriate structure takes more skill than I first realized. I think I took for granted what you provided as structure for our experiences in class. Writing the final report and presenting my experience to the class helped me to reflect on the unit. By doing this reflection, I began to see possibilities for improvement and enrichment of the unit. I discovered that I could connect evaluating algebraic expressions, the order of operations and the study of fractions to my area unit. Writing my reflections on the unit also gave me time to forget my frustrations and recognize the rich mathematical thinking my students were engaged in. When I looked back on the notes I wrote about the way my students approached finding the area of various shapes without formulas, I was impressed with the level of thinking that went on. It was also helpful to remember which students really depended on the cutting and pasting for their conceptual understanding. These notes helped me to realize how concrete some of their thinking was and that I should watch for growth toward more abstract thinking. Listening to my students think has helped me meet their needs. It is now a priority for me to take time and listen to my students think through a problem.
I have a new appreciation for a variety of instructional strategies. I have read in the past that students should be writing in mathematics and keeping journals. Units this semester I had no idea of how I should use them. Now it seems like such an obvious way to help students articulate what they are thinking as they solve problems. [...]
I also gained an appreciation for cooperative learning and class discussions. In all of the readings about classroom episodes and the activities we engaged in, except the card trick, students worked together. From personally experiencing this kind of cooperative learning I believe I learned more by being able to collaborate with others than I would have if I worked alone. I enjoyed learning in this way. I also felt that I benefited from being asked to reflect individually in writing after learning experiences. This helped me to synthesize my own meaning from the ideas shared in class. This combination of group work and personal reflection allows students to benefit from one another's insights and at the same time provides the teacher with an assessment of what each student learned from the experience. I would like to build this into my own teaching. [...]
In conclusion, this semester has helped me to re-conceive my view of mathematics. I see mathematics as a much more interesting and stimulating discipline. I see the potential for creatively in mathematics classes. I feel much more in tune to the affective issues that contribute to or take away from meaningful mathematical experiences. With time I will be able to develop strategies to help foster in students more positive and productive views of mathematics. Through my experiences with inquiry and changing teacher and students roles in the classroom, I intend to step back in the classroom and become a facilitator and guide while my students take the fore front and explore cooperatively open-ended problems. This is an exciting and challenging time to be a math educator. I took forward to realizing the ideals presented in this course, even though I am fully aware of the resistance I may encounter.
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