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in-service program
Selected artifacts from the in-service program (B2)
Examples of participants' written reflections on their field experiences
The two written reflections included in this section were written by the participants in this implementation of the in-service program at the end of their field experiences. Since they are offered as complementary examples of what teachers chose to record in these reflections, we invite you to read both papers, which were written by a special education teacher and a middle school mathematics teacher, respectively.
Final reflection by a special education teacher (2 pages)
It is easier, for me, to start with my thoughts about the teaching process. The greatest changes I have made have been in the areas of planning and assessment.
As a special education teacher, I have always been concerned that my students were not learning the same information as the mainstream students. Through using the process oriented approach, I have become less concerned with that because my students have had more enriching experiences than the mainstream student (with the exception of Earl's).
My planning has evolved greatly. I have gone from the "what's the next chapter" approach to "word problems are important, what math can I cover in that unit." I have found that I am now starting with a general topic, and figuring out what different types of math may fit into that topic. That has worked well because it forces me to plan a unit, not a chapter. I feel much more organized having an outline, with objectives and subtopics to refer to. I'm not making concrete plans. I will take my outline, start with an introduction of the overall unit, and see what the students come up with. This leads me back to my outline, where I can plug the students' ideas into the subtopics I have, and maybe add or eliminate subtopics that did not come up. I am also able to lead the students in a certain direction in a way that they feel that they discovered something wonderful, which they did. I feel much better about the way the lessons run when I can guide them, rather than force them to work. It is also beneficial to use the outline because a student may come up with a subtopic, or idea that I would not relate to that topic.
Team planning is another aspect of planning that is different for me. I have found that simply verbalizing an idea or problem to someone, makes planning much clearer, although more time consuming. Thoughts become more clear, because someone else has to understand them. I have enjoyed and learned from team planning so much.
The other aspect of teaching that has changed dramatically for me is assessment. I always had this fear of not having enough "grades" in the book at the end of the semester. I have found that different things can be graded. For example, I now have my class using a behavior rubric to monitor themselves. Every few weeks I will have them reflect on their behavior in a journal and that becomes part of their grade. I also have become much more comfortable with the fact that a final project is as valid (if not more so) than a final test. For example, at the end of our tessellation unit, we made a classroom quilt. It would have been much easier for me to give a geometry test along with a journal entry. The making of the quilt allowed the students and I to really show what was learned. There was a rubric to guide them, so they knew exactly what was expected. Although I have used rubrics before, I had not used them in math class. It was easy to evaluate the projects because the rubric made it almost a yes/no situation. The students also performed very well, partly because they knew exactly what was expected and what needed to be done by when to get a good grade. I have also found that having the students grade themselves is helpful. It lets me see how the student feels about their own work and efforts.
Even though projects are a lot more work, for teacher and students, I feel they are a very valuable experience. I expect to use them in the future, along with journals and quizzes, as a major portion of my assessment. I have found that I am more willing to try new ways of assessing, and am not relying on test grades to validate what happens in my classroom.
My biggest problem now is that educational changes are happening too slowly for me. I keep hearing "portfolio" and "real life experience," but our report cards don't allow for a true representation of those forms of assessment. I feel like what we're doing is wonderful, but we're not getting all aspects that need to change, changed! It's the same thing that always happens: the district wants change, but not too fast and only if there is a committee formed to work it out before you try anything.
I'm not the only one who has had to make some adjustments. My students have also made a lot of growth and progress. At first they asked if we could just do what everyone else is doing, book work. It took a few weeks, but we're all fairly comfortable with the way things run. I have seen some very different effects on learning in my math classes this semester. One class, blended 8th grade, was very open and willing to try new things. They ask the most wonderful questions! During an area lab, one student asked me why we didn't just give them the formula and have them practice it. I told him to wait until the lab was over and ask again. At the end of the lab, he said he could answer his own question: because he understands where the formula came from, and even found other formulas. He thought he would be able to remember it better too, having found it himself.
To me that summarizes the changes in the learning process that have happened. The students have come to realize that math is not an absolute. There is more than one way to go about things and more than one answer is possible. I have watched students go from very passive, concrete learners to active, explorers of mathematics. This type of classroom has also made the students less aware of learning disabilities because every thought is valid, and every answer that can be explain and makes sense is right. Our learning disabled students have benefited a lot from this program. I feel that is because, in part, they are no longer worried about "getting it wrong." They can explain themselves, verbally or in writing. Math is no longer memorizing and computation only. So much self confidence has been built in these students that, I feel, it carries over into other classes and daily lives. When I hear a learning disabled student who has failed math for years say, "this is fun, I understand it," I know that a process centered approach is what is needed to have success.
The changes that have been made in my teaching and in my students' learning are far beyond what I had expected. The growth and understanding are greater than what some of my students were expected to be able to do when they graduate! Making the "how" more important than the final answer has made the students more involved and in a way, a more important part of the learning process.
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Final reflection by a middle school mathematics teacher (3 pages)
Teaching with the inquiry approach has been very exciting. The reason why I am so committed to inquiry instruction is because it fulfills the needs of individuals more than the traditional approach that taught to objectives. I am confident saying that it fulfills the needs of individuals because some of the most important lessons that I taught were during the first week of school before we discussed anything about math. I don't think that this approach could work without the use of cooperative learning groups so I spent a great deal of time helping the children to determine what they found important and how they wanted to be treated. I don't think that many of the children have experienced what it is like to be in a community of learners and what it is like to work effectively in a group. In retrospect, I can honestly say that the children had ownership in the classroom and they were more focused on learning than on the "bureaucracy of schools."
It is necessary to discuss the "bureaucracy of schools" that the kids experience to better understand why the inquiry approach is so valuable. We all know that many educators feel that the bureaucracy is taking away from quality instruction. The dreaded "D" word is heard every day: DOCUMENTATION. I think the students experience the same thing. Everything that they do is paper work. This process of filling out papers is most detrimental to the study of mathematics because the students do not get to see the actual modeling of mathematics. Figuring out the area of a rectangle has little meaning, while figuring out the area of the classroom to purchase a carpet is valuable; figuring out which numbers to substitute in a formula has little meaning, while creating a formula to communicate how you solved a problem is valuable; and manipulating fractions in isolation has little meaning, while working with linear measurements in the classroom that are not whole units is valuable. I think many students have given up on math because all they see are symbols that don't have meaning beyond themselves. Since students have had few opportunities to see mathematics in context they have become disgruntled with completing meaningless paperwork. How do we know that the students may find it useless paperwork? I think they express their opinions about this all the time with two simple questions: Do we have to show our work? And what are we ever going to use this for? The students do not feel that mathematics is applicable to their lives and they do not understand the process of mathematical thought and justification. I think the children should be praised for these questions. We all want to know the reasons why we should embark on a new task and we all want to know how it is valuable to us. It is time for educators to stop telling children why they need it and to begin showing them why they need it.
This is why the inquiry approach is so valuable. The students discover for themselves why these things are so important. They see mathematics in context so they know what they are going to use it for in their lives and they show the work that is require because they want to share their results. I think many of the students are beginning to see that there is not just one right answer and they are beginning to understand the answers that they are producing. I am pleased to see some of the students understanding that there is a reason why the answers are correct other than because the teacher, the book, or the calculator said it was correct.
Since the beginning of the year I have completed several projects with the students that have made mathematical thought more clear to them. We have completed units on tessellations, creating area formulas, drawing floor plans, building a shelf, and measuring area and perimeter. In each unit there were many different issues that appeared and some issues that reoccurred.
The most frustrating of the reoccurring issues was the students' inability to accept any ambiguity. I feel that their frustration is the product of their previous six years of public education and our culture. The students are used to being given information and having to demonstrate an ability to retrieve that information. If you retrieve the information the answer is correct and if you can't retrieve the information the answer is incorrect. The students find it very difficult to take a risk and offer their own opinion. They believe that if you don't know what the answer is that there is no way to go about finding it. I think that if the students are repeatedly asked to do things for themselves that this will change and that they will become more independent in their learning and their life skills.
Before starting our first unit of the year, tessellations, the students worked in groups to discuss their values and what kind of rules they thought would be necessary to make our community a safe place. We proceeded with similar team building and group skills for a few days before beginning the tessellations unit. I chose to do the tessellations unit first in an attempt to show the kids that they can use a logical thought process to come to some truth that they are comfortable with. Since tessellations was a topic that they had never seen before they did not have any predetermined conceptions about their ability to handle the material.
We started the tessellations unit by determining a definition of a dog. As the students created definitions I pulled out stuffed animals and pictures of other animals that fit their definition. It was a good way to show them that everything they had accepted as being true could be challenged and it showed them how difficult it was to articulate a statement with our limited language. At the end of this lesson we discussed how they felt about the process. I found this to be very helpful for the students and myself. It gave the children a chance to express their frustrations and feelings about how difficult it was to convey their ideas to somebody else and how difficult it was to justify their opinion with details. It was helpful to me because it revealed many of the learning styles of different students in the class. The intuitive, creative students were easily identified and the structured thinkers were as easily identified but less vocal. This lesson helped me to learn how to deal with the feelings of different students and it created a safe atmosphere. Since we were defining a dog they approached the lesson as if it was some kind of a game. It made many of the students feel uneasy when they were not told that they had a correct answer but the discussion about how they felt during the process was more valuable to them than being told that they were correct. For many of the students, I think that it was the first time they were able to share their feelings and feel validated doing so.
As we progressed through the unit we often stopped to discuss the process and how we felt about what was happening. In an effort to help the students reflect I often used newsprint to recall ideas. We would record concerns that we needed to discuss and we would generate lists of vocabulary words that were being used during conversations. These lists were used several times in our discussions to show the students how there is a need for basic knowledge and vocabulary words and they were used to show that our knowledge was not exhaustive in nature. I also found it helpful to record their thoughts because they were convinced that anything that was written down was useful and they would elaborate on their thoughts and justify their answers. The students enjoyed this unit because they were given notes and records of our conversations instead of writing during the class. They enjoyed the time they were given to discuss their thoughts.
The second unit we did was on creating area formulas. I think the students enjoyed the unit because we didn't focus on correct units. I try to choose one of the NCTM standards to stress in each unit and this unit's emphasis was on notation. My goals was for the students to understand that the letters and symbols that were used as variable could be anything but they were usually chosen for a reason. The student recognized that formulas were shortcuts and that we had to determine when they were helpful formulas -- this was based on when the formula worked for all the cases. To stress the concept of a variable representing something else I let the students choose any variable they liked. I modeled the process by using a star and a smile face instead of "b" and "h" to find the area of a rectangle. It was exciting to watch the kids creating formulas with football times basketball or hairbrush times flower. Thus children felt empowered that they could do as they chose and come up with a reason why they chose certain variables. After the fun wore off they started to recognize that they could make their formulas different to adequately communicate the process they used. They recognized that symbols like star and smile face need to be explained if they were going to be effective.
I think the students have found this approach enjoyable and they will continue to grow in mathematics and as individuals. Although using this method of instruction is more time consuming and more difficult to assess, I find it efficient in developing thinking skills. I will continue to instruct my classes in the spirit of inquiry and I will later take on the challenges of how to assess students' work with this method of instruction.
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