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In-service program documentation
Reflecting and sharing on the on-going field experiences (D6.6)
Notes from the "round robin" sharing session that occurred
after the implementations of the first illustrative unit
Note: Because of the generative nature of the sharing discussion we have chosen to keep these notes intact. Thus we have included the segment referring to the implementation of the "Remodeling" unit despite the fact that we no longer use this unit as one of our illustrative units. [These notes, along with the 'list of concerns' and reports of the "round table" discussions were sent to all participants.]
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| Area unit - participants who implemented the area unit as their first unit share their experiences and concerns |
| Remodeling unit - participants who implemented the remodeling unit as their first unit share their experiences and concerns |
| Tessellation unit - participants who implemented the tessellation unit as their first unit share their experiences and concerns |
AREA:
Meghan (& Sue): The students adjusted very well to the different form of learning -- though a few are still struggling. In a recent journal entry about discovering PI, for ex., many students said they were very surprised with all they came up with, on the whole the journal responses showed that the students were able to demonstrate an awareness of and appreciation for the inquiry process used. Meghan and Sue felt the whole experience discovering the relationship between circumference and diameter was great.
Sue: A concern has to do with working with a large blended class; some students are very weak with skills with fractions, for ex., and even if they are practicing those a lot in the context of computing area & in dividing circumference by diameter, etc. some students still they have trouble - we're looking for some ideas for what to do in these cases since we don't want to overdo it for the other students.
Maureen: My students (6th grade self-contained) really reacted very well with the area unit; they really enjoyed math (they said so, and this is the first time her students have said so!); Students have a lot less problems with distinguishing area and perimeter, because they have some images to get back to remind them about how to measure -- a great plus! Students also came up with great definitions and had no difficulty recognizing "book" definitions on the test. They did a lot of interesting writing, too, and it is nice since I teach writing as well and this gave a good beginning for integrating writing in various subject areas. Concern: this was a "peak", how can it be sustained for the rest of the year? Another concern: It's difficult to plan when you do not know what the students will really respond! But I am realizing that it is OK to lead them sometimes.
Diane H (Charlene): Advantage of being two teachers in the class -- one more person to observe, help students, etc.
Diane: (6th grade) Have integrated math and language arts. At the beginning she was really concerned about beginning with area, when the students did not review fractions prior to that. But it really worked well; it helped to have some "warm-up" with some exercises with fractions to refresh some ideas. The students were told at the beginning that they could build their "dream rooms", and now they are doing it, and it is very exciting; the students had to put everything on graph paper, which really helped them to notice what could not work (ex: a swimming pool that took most of the room!). Now feels the unit integrated basic skills beautifully ! Concern: what to do the rest of the year, if one does not want to go back to traditional?
Sheila: (self-contained 7th grade special ed.) Unit went for three week and went very well. Concern: I felt they needed more directed problem solving, prompting and clues than I expected to give. A few students had a very solid handle on area, and that created some confusion for those students when at the beginning of unit they had trouble recognizing how the formulas they knew fit with what they were doing now -- but in the end it worked out. The students came up with their own definitions and variables, and that went very well; but, when it came time to go to the formulas written in the book, and recognize/translate what they did in the standard form, it was difficult. The rocket was one of the last, culminating activity, but seemed to go very well. At the end the students seemed to be pretty happy for them.
George: (high school geometry) Kids were very comfortable with the unit (some of the other teachers weren't!). It seems important that the students can actually "see" real rectangles, figures, etc. with the stated dimensions so that they get a sense of measures. They used centimeter graph paper, because this way they could be precise and use calculators -- even high school students hate fractions! "Discovering geometry" by Michael Serra - Key Curriculum Press is a great textbook/resource to use in this kind of teaching geometry, at almost any level.
Susan (& Lynne):(6th grade -- "regular" classes/ no advanced kids, no blended either) Neither of the teachers was sick/absent during the unit, and that seemed a crucial thing with this kind of teaching! The choice they made as a school (since other teachers would do area again in 7th and 8th grade) was that at the sixth grade level they would focus on developing the basic concepts of Area and Perimeter and it worked very well. Manipulatives (even simple ones like post-its and small ceramic tiles) were a key to the success of the unit. There were kids that were absolutely convinced that the hypotenuse of a right triangle was the same measure of the long leg, as they could picture "straightening it out"; it was crucial to have manipulatives to measure and convince themselves that it was not the case -- and that was a very good thing for them to do. It was interesting that when they first got into perimeter of a bulletin board, many students went on happily multiplying the dimensions -- they did not think! The final assessment (where there were many questions that asked for explanations and not just numerical answers) went very well. A revision they would make: They went too easy for too long, i.e., they did rectangles for too long -- the complex shape should come much earlier, the first or second day ! Concern: One parent called complaining that it was too easy for her kid in my class; a few more complained in Lynne's class; they have to think about what to do to challenge the best kids in the class; it seems more difficult to do that in this kind of teaching, because it is less individualized than traditional. Concern: There were a lot of opportunities to bounce off with -- measurement, classifications of geometric figures -- but it is difficult to choose what to take on because of time constraints.
Claire: (First class -- 6/7/8th grade) I took the best things from the Tessellation Unit and the Area Unit and incorporated them in a new unit Investigating Circles -- because some of the kids in the class had had the other 3 units. I used the structure of the 3 units to apply to this topic -- which also has a lot to do with area. Defining words and developing precise vocabulary was an important goal. Defining circle was more difficult than one may think; they did a lot of measurement -- of angles, distances, etc.; the students gave directions to draw things; I often reworded kids' observations and discoveries by using precise vocabulary and we then discussed why it was important to do this. I did not think they could derive the formula for the area of the circle, even after some good discovery activities about PI - but I expected them to try anyway because I believe that it is critical for kids to develop formulas. They attempted to approximate the area circle and used a square outside and a square inside; it was difficult for the students to understand the concept at first, but they were all thrilled when they discovered that the area of the circle was less than the square outside, i.e., less than 4 times the radius square and then made the connection to their empirical discovery that the areas were more than three times the area. Concern: the kids still do not well on test and I can't figure out why.
Discussion:
Julie: Interested in how to plan to coordinate what to do in Area unit in 8th grade, if the area unit was already done in 7th grade.
Raffaella: In Brighton, we decided to focus on developing basic concepts and only the area formula of rectangle and triangle in 6th grade, and focus on developing area formulas in 8th grade.
Claire: There is a lot you can do to expand -- for example, for area you can go into surface areas. I always have this problem because every year there is a new batch of new 6th graders.
Judi: Other teams have raised the problem of coordinating across grades.
Raffaella: It is really crucial to do so for this and other topics in the curriculum.
Sheila: I would have liked to give a pre-assessment to see where each of the kids is coming from, but could not think of any that would not give the unit away.
Debbie: Doing the fish or the rocket early could act as a pre-assessment since every child can actually do something with it, carefully looking at what they do could tell a lot.
Sheila: I did not do it first because I was worried that it would frustrate the students, and was concerned not to do so with students with a history of failure. It was probably more that I was nervous than being a problem with the students.
Susan: We saw students being able to count many squares and still be happy doing it at the end; so it does not seem that they would have problems doing the complex shape very early. Discussing real life situations was also a key point to help the students really understand differences between area and perimeter -- though students still are confused at first, they now have a way to think through and try to check.
Barbara: As Debbie said, students can show a lot of what they know or not if they are asked to explain what they are doing -- and that could serve as a great form of initial assessment, for ex. verbalizing as they attempt to find the area of the rocket.
Susan: It was crucial to plan and comment on what was happening with Lynne; the accountability with another teacher was a great professional experience. A problem that I would like to solve is how to have good sharing without having everyone show their work -- because it takes so much time! -- but the students really loved to go to the overhead and show what they did!
Sue: They try to monitor the varied approaches students are taking and sometimes they only have one of each shared. Then they have students write in the journals their individual solutions and approaches, then the next day the teacher would summarize or ask for any of the "missed" approached to be shared.
Susan: One thing that worked was also to give tape and have every student post their work on the board once and then look across all of it for similarities and differences.
Susan: Lynne and I felt that we could not use the book, all had to be teacher created; but there were good things occasionally in the book, it could even give some credibility to what we were doing -- now we know it is not really necessary to ignore the book completely.
Meghan: (Back to the issue of how and when to use a complex shape) They did the rocket the first time (and several times thereafter) to develop the basic concepts, etc.; then used it later to apply the formulas. They also used different units -- for ex., paper clips vs. rulers, etc. -- and recomputed the rocket using these different units, discussing issues of accuracy, etc. We felt it was very important and valuable for the students to define their own measuring devices.
George: How concerned have you been with accuracy?
Meghan & Sue: In our class the kids decided how concerned to be and when. Kids themselves defined "half-clip", and were happy with it.
Susan: We did talk of real-life situations when you do or do not need to have accuracy (ex: students reported of parents not really computing when they have to paint, etc.); also did some approximation of irregular shapes. Lynne and I also plan to continue to refer to remodeling of the school that is going on, to continue to review area concepts within other units - rather than do everything within this unit.
Sharon: Turns the topic to tests and asks Claire about whether her tests are still traditional or what.
Claire: I tried all different kinds and the kids still seem to have problems with tests in general.
Sheila: I had the same problem when I taught in the City. Could it be something to do with not studying at home?
Claire: I do give a lot of homework -- not everyone does it, but it's better than before. Many students seem to even forget their is a test, even after continuous reminders. Engagement is a problem (they may be engaged in class activities but not at the test time) -- though it does get better as the year progress, after the students have adapted to such a new and different school. Yet, these same students CAN engage in other school (usually extra-curricular) activities -- so it is not just because they don't have home support, etc.
Deb: Why don't you do some collaborative tests, because it may be difficult for students to work individually in the test when they have worked in pairs, etc. in the whole unit.
Claire: You always tell me this, and I hear you... I still do most tests individually, but I think about that. Also, the test is only a small part of the portfolio and overall assessment...
REMODELING:
Sharon: (Hilton high school; Course IA) We started with Remodeling, as a way to do the first unit of review of arithmetic operation. Overall, I am not very happy with it. The kids liked it, because it was different than usual. Everybody worked on the same project with numbers I gave (so that they had mixed numbers) -- but most students just wanted to round off, as one would approximate in real life anyway. They used real-life information, etc. -- but they insisted on rounding and approximating because that was real-life -- and they were right! But we did not cover what they needed to cover in arithmetic operations. There were also problems of continuity, because of freshman events canceling classes, individuals skipping classes, etc. Problems with giving appropriate homework; the students are doing a lot better with their homework now that they are back to traditional homework than during the Remodeling unit. It is difficult to have them taking notes, having notebooks, etc., since they did not start doing that from the beginning.
Charlene (& Laurie): (Hilton high school; General Math) It was difficult to start the year, because the class was too large and had to be split, etc. So they had to wait till this was settled, and just started Remodeling now. They are doing a lot of measurement -- measuring things with their body, pencils, etc..
Laurie: The students seem very comfortable. It is very nice to do this blending with Charlene. The students are feeling comfortable with math -- which is something that these students have never felt. I am really happy with what is going on.
Charlene: The students are really collaborating with each other and learning from each other. When they are done with measuring with different measuring tools, we will raise questions about what are appropriate measurement units to help them gain an appreciation of standard units. A main concern: the time this is taking; but you seem to need it to do things well; will the kids get bored?
Diane: I had the same concern, but now I feel I am covering so many things within this unit that I am not so concerned about time any more.
Charlene: When I called a student to question her absence she said that this is her favorite class and she would not want to miss it -- and that is not usually the case. It is a great thing that this course does not have a Regents exam and I can create the final test as I want so it can reflect what I've done - (no other math teacher is teaching the same course either).
Laurie: I am doing this unit in my other special ed. class, too, and the students there seem to be surprised that they can do what they do for ex. understanding and using formulas.
Claire: (To Sharon) What math do you think that your Course IA students would need to know to help them in Course I?
Discussion:
Sharon: Some of the students did review their skills... I wish now that I had introduced right away the calculators, before the unit (I just did it), because I begin to think that if the students have not got these skills so far, I have little hope to make it happen now, and instead I could focus on other things once the skills problem is out of the way -- and now even the Regents exam allows the use of scientific calculators.
Debbie: In the last issue of Math Teacher there was an interesting article from a Course I teacher that involved a building project that I am now thinking of using for my Remodeling unit.
Judi: If one wanted to use the Remodeling as a review at the beginning of the year, it could be used quite effectively as a way to get at issues about estimation, deciding when you need to be accurate or not, then if the activities are chosen carefully, you could find situations where accuracy is necessary and so it would motivate the more precise arithmetic skills as well as activities which require estimation skills.
TESSELLATION:
Debbie (& Brenda): First year Debbie is working in a blended classroom -- it was difficult at the beginning for Brenda to find her role, but now it is great to have her in the class. It is interesting that the students thought that all the things they did at the beginning on "what is math?" and "definitions" was part of Tessellation, and that it was going on forever. Debbie's response was to bring the unit to the point where they identified a need for some math tools, and now she is teaching those tools a little more directed, and then will return to the tessellations questions which prompted the need and have the students use the tools to continue the exploration of tessellations. Concern: starting the year with this kind of unit is a bit unsettling, it would be easier to start more traditional. New focus this year: work on defending things -- so it seems like spending less time on Tessellations specifically and more on things that would help them over their whole math experience.
Brenda: Since this year they are in a new school, numbers have brought blending to be a necessity. I was worried about how the special ed. students would do in this new situations -- but it is easy in Debbie's class, because of all the techniques she is using. My role is more behind the scenes; I can give extra support to some of the students in their reading classes as well, reinforcing what was done in math class in this context. Concerns: LD students who are not good at writing and verbalizing have more difficulty with journals, definitions, etc.; especially if they are better in math than in writing and reading, and they find it very difficult and less comfortable; I am not sure that the students have yet seen connections between Tessellations and real math -- i.e., Tessellations were fun, but what had it to do with math?
Claire: Why not ask the kids that very question: what is the connection between what we are doing with tessellations and real math?
Julie: I did that, and forced the students to be very specific in this assignment (not just say "it has to do with shape...") -- and the students did some really good thinking. In fact this was precisely the assignment which told me "they are getting it !"
Sheila: That was my concern, and why I did not to the unit myself -- what if the kids do not see this as math, and I myself am not sure about that...
Claire: As the teacher becomes more comfortable with this question [ and to do this it is critical to ask to yourself the question: what does this have to do with math? AND work to answer it] you can really reach a depth and math richness you did not have before.
Denise: ( have a math class of 6 kids)-- the small number is a problem to get good discussions. There was a very good discussion on the definition of diamond, ultimately referring to the dictionary to help them with clarity. The found it described as a lozenge shape and initially accepted this as the term to describe the shape they were considering but then a student brought back counterexamples of lozenges that were not diamonds. This was a great experience. Concern: students are journaled out, as they do it in reading, science, social studies and now math -- there is a real problem of coordination. Examples of journals: what do you think is the most important shape we talked about today/draw...
Claire: Maybe you should just call these learning logs rather than journals, and not give up on them, because this articulating their thoughts is so crucial!
Julie: That's what I am planning to do.
Barbara: It is interesting that the students would not complain about doing homework or thinking in more than one subject; but they are not used to writing. But it may not be so different...
Susan: Lynne and I began the year with a "what is math" activity, where they wrote autobiographies, and it was very successful... Maybe we should talk about these kind of experiences, too, since many of us did it.
Meghan: Maybe with such a small group, the journal could be more an oral event, and it could be either taped, or recorded as just one entry by the teacher after that.
Dana: This may be a really nice alternative to still get at articulating ideas, but taking away the burden of writing.
Deb: A similar problem is there with cooperative groups -- if they do it in all subjects they may get really tired with it. But one may also just "repackage it" -- ex: when my kids were working in pairs they didn't count that as groups. I think it helped that they did not have to move their seats to do it since the rows were always pushed together in twos..
Barbara: There may also be difficulty for some students to appreciate that math homework can take different forms; this also means that writing should not be extra work but in substitution, as a different vehicle to do the homework.
Sheila: Importance to talk to parents about that -- because they may not understand our assignments; some parents do not believe that they have to do writing for homework!
Susan: That happened in one class, when the students were asked to measure something at home.
Frank: We started with "what is math" to help the students to appreciate that math is much more than what they thought, and to appreciate Tessellations. But it is also true that the students feel that they have done enough with it for the moment -- though they will come back to it when doing some transformation geometry. It may also be important to expose the students to different learning/teaching approaches throughout the year. Concern: not always happy with what the students write.
Barbara: This may have something to do with the way the writing assignment prompt is given -- and it may take a lot of thought (and experience!) to prepare good prompts/assignments that would really make the students think and write in the directions we hope.
Frank: Some students seem to have real problems in writing, no matter what support you try to give them. But different students may react better to different techniques.
Aimee: (observing Denise's T. unit) Very impressed by seeing the T. unit unfolding, and seeing how some "curriculum" math is integrated with the unit -- at first I was also wondering about what Tessellations had to do with math. When the students were introduced to the project they were supposed to do for Tessellations, I was worried about whether they could do it! But the students came up with at least 10 good questions to explore! Concern: as the students work in groups, I do not know how to answer requests for help, what to say and what not; and this may have something to do with knowing what your goals as a teacher are for the unit and particular activity. A surprise was that the math book they use had something about Tessellations -- which helped to validate the use of it!
Earl: Giving the textbooks out to the kids seemed to make them more comfortable (even if they do not use it, and are not expected to even bring it to class). Amazed at the quantity of geometry that was covered. At first I was very worried about covering the curriculum, but not now after seeing what came out. I'm very happy with what is coming up.
Heather: (observing Cindy's T.) Cindy alternates a lot between large group discussions and small group work. Cindy is very good at orchestrating the large group discussions so that a lot of math is covered (ex: in figuring out the angles in a regular polygons, they covered a lot of the geometry of traditional 8th grade and Course II, and in a very more meaningful way). It was very neat to see students convincing each other and disagreeing; some kids brought play-dough to create manipulatives to explore Tessellations in 3-d! Concern: in a blended situation, when the special ed. teacher is new/temporary and does not know what the goals of the unit are, what can you do? if the special ed. teacher and/or resource room teacher is anxious about the whole thing, even in the resource room situation s/he may not be able to help the students.
Judi: This seems to be a common problem when the partner (whether math or special ed.) has not been part of the project. Nancy, who participated in the Summer Institute but does not have a class of her own, she only sees math kids in resource room, has pointed out how important it is for her to have come to the Summer Institute because she can really help the students that are in Earl, and Sue's math classrooms. It is also a problem even for people who went through the Summer Institute but may not have time to plan together during the year.
Claire: We may have the same problem with parents trying to "help" the students at home. I write notes to parents saying "this is a problem solving unit, please do not solve the problem for them!", and it seems to work. And one could do the same for resource room personnel.
Kay: My goal is to engage my students (half from last year, half new), make them feel better about math, not so discouraged. In Tessellations, everyone had something to offer. We branched off to many things I did not anticipate, which made them feel that knowledge was not all with the teacher. They seem to think a lot more now, when asked to do something in math. In the test, when asked what are five things they learned, they all wrote that they liked math better. Manipulatives are crucial. Very pleased with the fact that the students have become more confident. No concern now, but there were many times when she felt like giving up, but the team meetings really helped!
Earl: (7th grade) Decided to jump into it right away. It is a different approach, and the kids may be a lot more comfortable with it than we may be! The kids are really very adaptable. When asked to write about what they were doing in math they were quite positive, they all felt that they were doing math. During the Institute he had concerns about the manipulatives (where/how to get enough), but then settled it by having the students construct some of the figures and then duplicate, etc. The kids' involvement is incredible, much more than I have ever seen; they are attentive even if they are crowding, speaking loud, etc. I see the Tessellations unit winding down now and getting naturally to the area unit. There are so many different teaching and learning approaches imbedded in an Inquiry Unit as the result of the many different things that need to be done that I am not concerned about alternating inquiry units with more traditional approaches. Concern: what do you do if you need a sub? what are you telling these people to do?
Julie: (8th grade) Concern: starting the year with the Tessellations unit, how do you establish your classroom routines (taking notes, notebooks, etc.), expectations, etc.? This year, I decided that all these study skill expectations had a place in the Tessellation Unit too, so I asked students to take notes about Tessellations etc., and it worked. So now, after all, I think I will keep doing the Tessellations Unit at the beginning of the year -- because I could see from students' responses in the "what is math" activity that it is important for them to start the year by seeing math differently.
Deb: (7th grade) I carried the students' responses to "what is math" a little further, I put them up and the whole class discussed them. Then they chose one as their first note to put it in their notebook. Then used the experience discussion the definition of math as a point of departure for the Tessellations unit -- by saying that they would act as mathematicians in this unit. -- and that worked very well, they immediately got in the right spirit. Also added the request to come up with real-life example of the nonsense definitions for ex. find a real-life 'nestle' - this worked well as a model for doing the first tessellation assignment. This year I also taught Tessellations for the first time in all my classes (instead of only the Modified class) -- an important step to break out of the district "mode". This got us to ask other teachers in the school to get into the Tessellations unit. Some parents in accelerated classes have complained, saying their child is more used traditional teaching -- probably the problem was more with the parents than the students; but this is an issue that needs to be addressed.
Julie: Interestingly, it seems like to extreme ends of the spectrum seem to raise concerns about the new approach, the lowest functioning kids and the accelerated kids.
Claire: Yet, they are the ones that benefit the most in my class.
Linda: It is really hard to get the conversation going if there aren't enough kids in the room, enough "good thinking" that can get the conversation going. So the special ed. students blended really benefit from this. Comparing two self-contained classes, one of 15 students and a small one, the small one has problems, the other is working well; a great thing that happened in both classes, though, is that the kids really got involved when doing some Tessellations -- once the kids did not want to leave, and one said "this is the only thing I might ever like in math - let me stay !"
Carol: (7th grade) I already made notes about what things I would do differently next year. Nothing went the way I expected it to go. I started very well, but then sort of lost my focus. But it's okay, since I learned a lot about my kids personality (who was comfortable with this kind of thinking, those who were not, those who felt bold enough to speak, etc.), though I am not sure I would do it as the very first thing in the year. Some kids still seem not to have a clue of what happened; I am curious to see what will happen with them in a more traditional unit. There were also some very positive instances, even from students who fought the unit -- better than I have ever seen in 4 years of teaching; so even just that made it worth it. And I see that now they are working much easier with angles, etc.
Raffaella: reports on Cindy and Denise's use of modeling with the questions "all triangles Tessellation" and "quadrilateral Tessellation" to introduce some math tools that would make the students' own research questions more sophisticated and interesting.
Judi: But there is also a lot of value not "funneling" possible questions too much, as kids in Claire and Steven's classes the first year came up with interesting questions that had to do how changing the definition would effect what counts as a Tessellation than studying typical geometric figures, for ex. What if the definition said 2 shapes ? or What if color did matter ?
Barbara: Value of reflecting with other professionals about what happened in class and how it could be changed -- this is something that often we do not do because we work alone, and it is easy not to have time, but it is crucial for professional growth. Everyone is encouraged to think back on what they have done so far, especially the planning process.
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