Return to list of materials about "Providing classroom support during the implementation of an illustrative unit"
In-service program documentation
Providing classroom support during the implementation of an illustrative
unit (D6.5)
Excerpts from facilitators' personal logs/ journals during the field
experiences
The following excerpt from a facilitator's personal journal reflects many of the issues involved in providing classroom support. In this series of entries the facilitator discusses her concerns about offering too much criticism, especially when she has not been able to visit the classroom on a regular basis; her feelings of "guilt" about leaving the teachers with a lot of work to do as a result of her suggestions; how the facilitator may also become intrigued enough to engage in an inquiry of her own; how having observed a previous set of activities allows her to draw a different conclusion about a very directed set of activities (different from what she might have concluded about the lesson in isolation); and, how important it is for the facilitator to be assessable and keep the communication open.
Excerpt from facilitator's personal journal reflecting on her experiences working with two participants who were implementing a Tessellation unit
(10/4)
[...]
I wanted to record some thoughts about the Tessellation experience as well.
First of all, I have not been able to observe it and give feedback as I
had hoped. This is part of the problem, because not being there on a regular
basis I feel more uncomfortable criticizing things or suggesting radical
departures from their plan. It also sort of happened that the two times
I did that (when they were doing the "triangle lab" as part of
their modeling of testing conjectures with the statement "all triangle
tessellate", and then yesterday as they were moving to the next activity
of testing the conjecture "all quadrilaterals tessellate) my feedback
had the result of making them do something radically different from their
plan (in the first case, to make sure they discussed the "proof"
that triangles tessellate by using the property that the sum of the angles
of a triangle is 180; yesterday, by suggesting that they reconsider doing
a lot with quadrilaterals and polygons before the project and, more specifically,
to rethink what they wanted to do with quadrilaterals); so I feel bad that
it almost seems I do not like what they are doing when it is not the case.
Indeed, I feel that they have put together a very interesting and more coherent
plan for Tessellation than they did last year, and they have mastered the
art of making explicit to the students what they are doing and why, and
are also very good at facilitating class discussions and pointing out connections
-- all very crucial elements that are not easy at all to develop! At the
same time, I still feel that they often "overdo it" in trying
to have everything ready for their students, without letting them explore
and discover and trust that most of the "curriculum" will get
covered. Today I am going to visit their class again to see what happened
with the suggestion I gave them yesterday, to make sure that I can support
them a little bit more this time, rather than "give the suggestion
and leave them".
Feeling "guilty" about leaving the teachers with just a vague suggestion that would give them a lot of work, as well as intrigued by the problem myself!, I also decided to work a little on my own on the problem of whether indeed any quadrilateral (convex or not) would tessellate. It was an interesting exercise, since it required some thinking about manipulatives (I had a great deal of trouble just sketching in this case!) as well as working from a "simpler case" (that of irregular diamonds) and trying to generalize from that. I am going to share my examples and manipulatives with them today after class, but I also hope that I can communicate my belief that it may be important for them (the teacher) to "know" the solution to the problem, but at the same time it is not crucial for the students to actually have them solve the problem -- rather, I wish they will agree to leave the problem open so that the students will really be challenged and maybe some of them may choose this question or a variation of it for their project. We'll see!
(10/5)
As a brief follow-up on yesterday's journal, I would like to record that yesterday, as I visited one of the teacher's classes, she had put together a really good lesson. The lesson opened with a great open-ended warm-up exercise (about thinking whether "all quadrilaterals tessellate" is true or not, and then draw either a counterexample or a confirming example) and its discussion, followed by a more structured activity where (with the help of drawing already there) the students were asked to give justifications for why rectangles and parallelograms tessellate, and then if and why they needed after that to try with squares or rhombi (though rather structured and guided, this activity felt good because of the previous warm-up exercise and discussion, and raised a lot of good and spontaneous discussion about relationships between squares and rectangles, etc., as well as was conducive to observations about angles formed by parallel and transversal lines), and finally by a "note-sheet" summarizing the key families of quadrilaterals and their relationships (which felt like a good synthesis/review at this point, rather than at the beginning of the lesson). As I said, altogether a great lesson that also showed how these teachers are really able to listen to and process my criticisms, and understand the idea of "giving the big picture" etc., though I think they tend to get too caught up with the details in the concern of "covering the curriculum".
However, this experience took its toll, especially as it followed a similar experience the previous Monday -- as I mentioned in my previous journal. These teachers were on the phone at 10:30 the previous night discussing their plan for the next day -- something nobody can feel happy about! One teacher in particular was quite upset about that, and they both asked to "talk to me" a moment after class. We had a good, if brief, conversation where they shared their discomfort and I assured them once again that they should always feel free to acknowledge my suggestions or criticisms but decide for whatever reason (especially logistic ones!) that they are not going to take it on for the next day/activity. We all agreed that we all knew that, but it is part of their "nature" to find it very difficult not to respond to criticisms of this sort -- especially when they realize they agree with it and there would be a better way to do the lesson! And this is true, so even if I think our conversation reassured them of my genuine acceptance of whatever they want to do with my suggestions, I think I should be a little more cautious/sensitive about what and when I make them in the future. At some point one of the teachers pointed out that it never happened to other people in the team that their plan got disrupted the day before -- and she is perfectly right! I did honestly respond to her: "because you are the most open!", and she agreed, but again I think this calls for a bit more sensitivity on my part.
I must say, though, that in terms of a learning experience there is nothing like something like this to really bring home the nature of what is still lacking in their otherwise excellent plans! The power of it (compared with putting the suggestion aside for the next year) is that they really can see the power of the alternative once they do try it in class -- as they did this time. Last year, for example, I mentioned the same criticisms about this specific part of their plan (and the previous one about triangles we discussed last Monday) and though they acknowledge them at the time, these comments were mostly forgotten by the time they got to plan these lessons. Yet, I still agree with them that the timing is the crucial issue. So, I have taken this opportunity to ask them if I could join them for some more long-term planning -- since the problem also is that I do not have much other opportunity to know their plans before hand and thus provide timely feed-back; we'll see if we can manage this this time.
The following excerpt from a facilitator's personal journal clearly demonstrates the ownership that is felt by the facilitator when they actively support the planning of participants units. This excerpt also shows how important it is for the facilitator to be present in the classroom to hear students thinking, to spend some time working out and reflecting on the mathematical issues which surfaced in the classroom, and to discuss these ideas with the teacher.
Excerpt from facilitator's personal journal reflecting on her experiences working with two participants who were implementing an Area unit. The thoughts expressed here were subsequently shared with the two participants.
(9/22)
There are so many things going around my mind after almost a full week of observations in three different classes, that I do not even know where to start. Yet I feel the need to put in writing some of these thoughts and reflections before I lose them! I also hope this way not to take over too much of the meeting time with MY observations, since there will be so many teachers who will want to share the great things that are taking place in their classes...
[...]
Starting Monday, I have been observing the area unit that Sarah and Linda are implementing in their 6th grade classes. I have been especially interested in observing this unit with regularity because I have not worked in a 6th grade classroom before (so I often do not know what to expect from students of this age). Also, the area unit Linda and Sarah have planned is very different from any I had worked with before, because it focuses more on developing the basic conceptual foundations to area, rather than on area formulas; and since over the summer Linda and Sarah invited me to their planning meetings, I feel a part of the team. So, I feel very much in the same position as the teachers themselves, in that every day I am going in very eager to see how our ideas will play out and not knowing what to expect and, in turn, I get out with hundreds of ideas about what else could be done with the unit.
Overall, I think we have all been very pleased with how the students have reacted to the activities we planned for them. They have been quite engaged with each of them, without a complaint, and with very few exceptions do not seem to find any problem with the material. In only one class three or four students, when asked by the teacher, said they thought this was "too easy"; otherwise, no complaints has been voiced one way or the other. Things have taken longer than what planned on paper, but that did not present a real problem since it was somewhat "expected", and so neither Linda nor Sarah have so far felt "pressed for time" or bound to the schedule set in the original plan (though in some cases some aspects of the activities have been dropped because of lack of time within a class period -- but I'll come back to this point later). I find it especially nice to see most students being eager to give their answers and go to the overhead to show their work, not seeming to be afraid to make a mistake (I have to ask Linda and Sarah if this is typical of ALL 6th graders at the beginning, because I certainly did not observe that in most of the "traditional" secondary classes I have been visiting), and getting really involved whenever there is a hands-on activity.
(By the way, I am amazed daily at how Linda and Sarah are able to organize the logistics of these hands-on activities -- each with plenty of different manipulatives! -- so that the students very orderly move around, get their things set, etc. with minimal disruptions; I even went through a fire drill when all the students went out of the classroom orderly and in silence, and then returned to their class 5 minutes later and began their lesson as if nothing had happened! I still have not discovered the magic and how it works, but you can bet I am observing very carefully and taking note of how these experienced teachers are managing their classes! I also want to say that, usually, I am very suspicious when classes seem "too disciplined", but so far I have not been able to sense that the discipline is getting in the way of the students being spontaneous or participating in class at all! Rather, it seems to enable them to use class time very efficiently and productively.)
In the midst of all these positive things, there have also been (of course) some others sort of "nagging" at me. I think that I have just come to some understanding of what I feel is missing or I would like to change the next time around, so I will try to articulate it in the remaining part of this journal.
In part, thinking back to what two of their colleagues said and my comment earlier in the journal, I think I can attribute it to the unsettling feeling of doing this new thing for the first time. There is no way we could have got it right the first time, and indeed it is only as I see the activities we "imagined" (as we planned) taking shape in the class, with real kids, that I can fully perceive their potential and value -- and sometime I also become aware of different goals that they could help to achieve, that I had not thought about at the time of planning (and, of course, this means that the activity itself was not set up to achieve that goal and probably would not do so fully this time around).
Just to give you an example, initially we had planned that in the very first two activities of measuring the student desk with two different units (square 3"x3" post-it's and 1" ceramic tiles) the students would be asked to represent what they did on a grid; if I remember correctly, we had planned to do so so that the students would start to become familiar with using a grid as a representation (since they would find similar diagrams in textbooks and exams) but mainly we thought that this would keep everyone engaged even if their group finished to measure earlier. As the students engaged in this activity, however, it became clear that covering the desk with squares, and even using the shortcut of multiplying the number of squares along the two dimensions to find the total number of squares, did not present a problem, while instead several students asked the teacher for directions about how to use the grid to represent what they were doing -- thus showing that the idea of representing an object with a scale drawing was not immediate to them at all. (Just as an aside, I can't help thinking that this is exactly what we ask students to do in traditional area units right away, at almost any grade level: compute the area of objects represented by labeled drawings on their textbook!) We also realized that once the students had created the representations of their desk using 1 square = 1"x1" tile and 1 square = 3"x3" tile, one of those drawing would look much bigger than the other; this could raise some interesting question about how could the same object be represented with such different drawings (i.e., using different scales). However, we were not able to capitalize on this possibility in any of the classes, because we thought of it too late!
Another thing I have been thinking about has to do with the whole set of activities done in this first week, which all focused somewhat on the area of rectangles (see Linda and Sarah's plan of days 1-4). A doubt that both Sarah and Linda had voiced in the beginning, and I guess kept nagging in the back of our minds, was that perhaps this was too easy, we were too taking too much time over it, since the students would "get" the area formula for rectangles much quicker than this! And indeed, by the end of the first or maybe second activity, I would say that the great majority of the students had got the idea that an efficient way to count the tiles covering a rectangle-shaped object was to count the tiles on two sides and multiply that number (and knew what that meant in terms of adding columns or rows with the same number of squares). At the same time, just as we were doing the various activities (involving measuring real objects with squares and then rulers, representing those objects and measures on paper, as well as computing area of rectangles drawn on paper, with and without a grid) I began to see more clearly that what we were doing was valuable not so much to discover or reinforce the area formula of rectangle (though it certainly was serving that purpose, too) but rather to help the students understand a number of more conceptually difficult issues that are somehow embedded in finding areas (though they also play a role in other parts of math as well), such as:
I am certainly misrepresenting what we did when planning the unit if I say that we did not think of these goals as we planned the specific activities in this first part of the unit, because we did. But I also feel that it was not so clear to us at that time that these goals were to be the focus of this part of the unit, and not just by-products while the focus was understanding the formula for area of rectangles. I think I can best express this as a shift of background versus foreground, if you know what I mean; but in practice, this may be an important difference, as it may influence what we may "cut" or let go when we do not have enough time for all that we had planned for a lesson, and also it may reflect in the way we plan our "synthesis" lessons and/or evaluation instruments. And I really feel now that these issues may be the most challenging for the students to get, and what may really create problems for them in the future if they don't get them. And for this very reason, it may pay to spend time explicitly on them, and to do so when working with a situation that is easily understandable and familiar -- such as measuring the area of rectangles.
This leads me to another point. I keep wondering what the students really think is the point of what we are doing, if they realize its conceptual complexity and thus fully appreciate the value of what they are doing and learning. If you take what we have done so far one activity at a time (as most students are used to do in school math), it is easy to think that the point is to find the solution for that particular problem, and once that is obtained (usually without apparent difficulty, because of the way the problem is set in a sequence, the use of manipulatives, etc.) that's it, too easy! What is not easy, however, is to pull together all the separate activities to try and answer the general question: what is area and how do we measure it efficiently? How can we help students see this bigger picture and make the connections? Spontaneously, both Linda and Sarah at different points felt the need to stop and reflect on what they had been doing (though we had not made it explicit in the plan) and I think that was crucial. I am also thinking how we can make such reflections/ syntheses an integral part of our future plans (making explicit for ourselves what are the key points and connections we are trying to make) and, furthermore, make the students appreciate that such reflection times are more important than getting a specific activity done.
Some of the things I have just talked about also made me think if it was a mistake, after all, to start the unit simply with the activity of tiling a desk, rather than by posing some larger questions about area that the students would be exploring and trying to answer throughout the unit (thus helping them from the beginning to look beyond specific activities and "see the big picture"). I must admit first of all that I was the one not wanting to bring in the word "area" explicitly at the beginning, because I was afraid that this would trigger in the students' mind the idea of having to remember formulas such as "lxw" rather than try to think with their own head on a problem they could solve even not knowing anything from previous classes. I am beginning to change my mind and to think that indeed the way another team started their unit -- asking the students to think of situations where they would want/need to compute area, even before they ever approached the task of how such an area could be measured -- may have a lot to recommend itself.