In this document we have collected selected artifacts from the planning of the Olympics unit -- including excerpts from lesson plans at various stages of development, facilitator's and teachers' journals, examples of readings and other instructional materials that were used and/or created by the teachers, etc.. These documents are intended to complement the presentation featured in the video entitled "Planning a new inquiry unit: A case-study," by providing illustrations of what actually took place at key stages of the process of planning this unit.
These materials have been organized according to the key stage of the planning process identified by our case-study of this experience -- as summarized in the next page. For each of these stages, we have first of all provided a "reminder" of how this stage played out (on grey background). The documents identified in this list are reproduced immediately following it.
SUMMARY OF KEY STAGES OF THE PLANNING PROCESS: Stage 1. CHOOSING A THEME:
Stage 2. FIRST ELABORATION -- Exploring the math potential of the theme:
Stage 3. FIRST DECISIONS:
Stage 4. FURTHER ELABORATION:
Stage 5. "PRELIMINARY" PLANNING OF THE UNIT -- before unit starts:
Stage 6. "DAILY" PLANNING -- as the unit is implemented:
Stage 7. PLANNING FOR FUTURE IMPLEMENTATIONS:
|
Stage 1. CHOOSING A THEME
HOW THE THEME WAS CHOSEN:
| Stage 1. CHOOSING A THEME Available resources |
Document A
Source: Document create a posteriori
"Olympic Medal Counts: A Glimpse into Humanistic Aspects of Mathematics" by Raffaella Borasi in Arithmetic Teacher, November 1989.
Starting with the puzzling fact that U.S. and Italian media report Olympic medal counts in different ways -- and, thus, would rank nations' accomplishments at the Olympics differently! -- this article examines a number of different, but equally legitimate, ways to "order" countries in terms of medals won, and consequently raises questions about the complexity of order relationships and the degree of subjectivity that is often associated with even "mathematical" rankings.
"The Olympics: An Integrated Unit of Study" by Mary Anne Rossbach (unpublished manuscript), April 1992.
Here the teacher of this unit reports on her daily lessons, including some sample student/teacher dialogue, worksheets created for the unit, samples of students' work and her own reflections on the experience.
"Including students in the planning process" by Mary Anne Rossbach (unpublished manuscript), April 1992.
Here the author discusses her experiences with involving her students in helping to plan units for the year and advocates for student voice and student choice. In particular, the author reflects on her experiences involving her third grade students in planning a unit on weather and another on the Winter Olympics. She includes a very helpful and inciteful list of what she learned about planning with her students.
"Current Events: A Gold Mine of Opportunities for Multiple Connections" by Raffaella Borasi, Lisa Grasso Sandridge and Edith Kort (unpublished manuscript), February 1993.
This article presents an example of a themetic unit based on the 1990 Census, taught in an 8th grade math class, and also discusses key elements of this unit that could be generalized and used as guidelines in the creation of other math units based on current events.
Stage 2. FIRST ELABORATION -- Exploring the math potential of the theme:
Also, see Document F to read about a teacher's thoughts at this beginning stage of the planning of the unit.
| Stage 2. FIRST ELABORATION Brainstorming and exploring possible questions for inquiry |
Document B
Source: Brainstorming done in writing by the facilitator and distributed to the other team members as a stimulus for discussion
WHY THINGS ARE MEASURED:
WHAT IS MEASURED:
NOT SURE IF "WHAT" OR "HOW":
HOW IS MEASURED -- i.e., what UNITS are used:
HOW IS MEASURED -- i.e., what TOOLS and APPROACHES are used:
WHAT IS COMMON IN PREVIOUS EXAMPLES:
WHAT IS DIFFERENT IN PREVIOUS EXAMPLES:
OTHER QUESTIONS/ ISSUES THAT COULD BE RAISED:
| Stage 2. FIRST ELABORATION Brainstorming possible student activities |
Document C
Source: Brainstorming done in writing by the facilitator and distributed to the other team members as a stimulus for discussion
| Stage 2. FIRST ELABORATION Brainstorming potential curriculum topics/goals for the unit |
Document D
Source: Brainstorming done in writing by the facilitator and distributed to the other team members as a stimulus for discussion
(TO BE COMPLETED)
| Stage 2. FIRST ELABORATION Brainstorming potential curriculum topics/goals for the unit |
Document E
Source: List generated by the planning team at a meeting, as recorded in the facilitator's journals on the experience
| Stage 2. FIRST ELABORATION |
Document F
Source: Excerpt from Cindy Callard's shared journal
I am really looking forward to our measurement unit on the Olympics. Raffaella has some great ideas and I'm really looking forward to sitting down again and making some further plans (although I'm very apprehensive about all of the work there is going to be!). I think one of the things we need to do next is to really think through what our goals are for the unit and what kind of parameters, in terms of time and activities, we are going to set up. Some thoughts:
Stage 3. FIRST DECISIONS:
Stage 4. FURTHER ELABORATION:
| Stage 4. FURTHER ELABORATION Collecting and examining potential resources |
Document G
Source: Excerpt from the facilitator's journal recording the highlights of this planning experience
Denise (one of the teachers) had found a number of sources about measurement and the Olympics that she now shared with the rest of the group:
Cindy (another teacher) had also started to collect newspaper articles related to the Olympics, as she liked the idea of having students collect these and use them somehow in class -- in analogy to what done in the Census unit (see my article on this experience).
| Stage 4. FURTHER ELABORATION Deciding on key activities/components of the unit |
Document H
Source: Excerpt from the facilitator's journal recording the highlights of this planning experience
Though not in the more orderly and logical order in which I am going to report them here (for sake of clarity), we identified and elaborated upon the following activities:
| Stage 4. FIRST ELABORATION Matching list of goals and key activities |
Document I
Source: Document created by the facilitator based on discussions that took place in the planningteam
Stage 5. DETAILED "PRELIMINARY" PLANNING OF
THE UNIT
(i.e., before the implementation of the unit started):
Also, see Document N to read about a teacher's thoughts
at this stage of the planning of the unit.
| Stage 5. "PRELIMINARY" PLANNING Detailed plan of the first day of the unit |
Document J1
Source: Excerpt from the facilitator's journal recording the highlights of this planning experience
We are now at the stage of elaborating our overall plans and creating actual daily lesson plans. I started feeling not very sure about whether at this stage group work would really be very productive -- my tendency would rather be to have individual attempts at working out a lesson plan, followed by feedback and modifications from the rest of the group... Yet, I found it very interesting to follow the dynamics of our conversations as we generated more detailed ideas of what we could do in class, and argued over how to word certain questions or assignments, and also occasionally changed more radically the plans as a result of working out the implications of certain activities. Indeed, I get more and more convinced that a teacher needs to work out the details of how activities will play out in class and can benefit by discussing those ideas with others, not only because if you do not pay attention to those details even a potentially good lesson can fail, but also because sometimes one could distort the original ideas through their misunderstood implementation (I am not explaining myself well here, I'll try to do better in person).
One major modification in the overall plan that we made at the beginning of this meeting was to rethink the "teaching about how to draw statistics graphs" (i.e., bar graphs, pie graphs, etc.). Initially Denise and Cindy had thought of devoting a couple of lessons teaching this topic "explicitly" prior to the Olympics unit, so that the students could use this information in their class activities and projects, and also cover one more topic "officially" in the curriculum. Raffaella proposed to keep the goal of providing the students with some experience and explicit instruction on how to create certain types of graphs, but instead of doing this "out of context", to connect these lessons to "making sense" of the students response to the survey on students' interests in the Olympics (see activity 1 in previous journal). Everybody agreed on this idea, and decided to keep it into consideration as we designed the questions in the survey (so that we could create data that would be amenable to the kind of graphs we wanted the students to learn to create).
Then we moved to planning in detail the first day of the Olympics unit -- a day that would be actually scheduled about a week before the official beginning of the unit, and would be used to set the stage for the unit itself and begin some preliminary activities (such as collecting information on the students' interests, beginning to collect newspapers articles on the Olympics, and most importantly "prepare" the students to the unit). Cindy and Denise had already thought about what they wanted that day to accomplish and how it should look like:
(a) an "introduction" to the Olympics, to be achieved through a brief tape or article
(b) brainstorming on different winter sports -- to remind the students of what sports and events will be part of the Olympics
(c) a discussion on how math is involved in the Olympics -- to make the students appreciate why we would be doing a unit on the Olympics in a math class in the first place, and alert them to "see" the math potential/ connections when viewing/ reading about the Olympics
(d) an introduction to the activity of collecting newspaper articles (to communicate its rationale, expectations, and logistics)
(e) filling out of a survey (designed to collect information about students' interests in the Olympics that may help our planning and the future formulation of projects and groups)
Even with such a clear plan for one lesson, it took the group a long time (and quite a bit of discussion!) to actually coming to a point where Cindy and Denise felt prepared to implement the lesson in their classes.
First of all, we spent quite some time discussing what we wanted to say to the students about the collection of newspaper articles, since we soon realized that in order to do so we actually needed to decide what we wanted to do with these newspaper articles in the first place, and then how we could actually orchestrate their collection and use. The idea of using such articles had come from reading about the success of this activity in a unit on the Census (see report of this experience among our background material). We all liked the fact that collecting articles would help the students connect what we were doing in class with real life events, could be a vehicle of involving parents in the unit, and also could help raise some of the questions about the Olympics we would like students to pursue. Yet, when we began to think about what this would mean, we realized the risk of becoming soon inundated by articles on the Olympics (since it is likely that there would be a lot more than it was the case for the 1990 Census). We brainstormed several ideas about how to deal with this: create a bulletin board in each class where the articles should be posted, create some folders organized by topic or sport, have each student keep a folder... As we seem to get even more entangled on the logistics of this activity, we went back to what we really wanted to do with these articles, and concluded that: (a) we wanted to be able to choose a few to read in class -- for "warm-up" activities in the first few minutes of class, to introduce certain issues/ raise questions, or to respond to specific questions; (b) we wanted the students to be able to use these articles as resources when they were doing their projects. We then concluded that it would be probably a good idea to keep the articles organized by sport, either on bulletin boards or in notebooks, perhaps also assigning some students the responsibility to keep these notebooks or bulletin boards in order. The teachers also decided that they would begin collecting articles on their own and start thinking of potential uses (maybe even already creating some warm-up activities).
We then spent quite some time trying to choose and carefully articulate the questions for the survey. Once again, this required us first of all to make more explicit our goals (i.e., what we really wanted to know from the students and why, as well as keeping in mind the use we wanted to make of these data as a context/ vehicle to engage the students in the creation of some statistical graphs) -- since we wanted the survey to be short, clear and to the point. It took us several rounds and editing before we agreed on the four questions to the included in the questionnaire (see attached). Note that question 1 (what sports they are most interested in) is stated in such a way that would easily allow us to create frequency tables, bar graphs, even circle graphs -- and in fact we may need some of these representations to have a sense of what events the class is mostly interested in and we may want to focus on as a class. A summary of the responses to question 2 and 3, instead, is probably likely to take the form of a list of ideas (where in some cases it may be important to know WHO gave a certain answer, even more than a numerical count of how many people gave the same answer). The results to question 4 (who they would like to work with) should not even be made public. (NOTE: we thought it would be important to have questions for which doing the graphs may not make much sense at all).
Though Denise and Cindy knew pretty well what they would like a video or article to do as an "introductory activity" (see (a) above), we left the meeting not sure about what this article or video could be -- since the material that they had able to collect up to that point did not really seem very satisfactory. (And indeed, sometimes it is finding these materials that could take a lot of time -- if one finds something satisfactory at all!)
Brainstorming about the various winter sports the student knew about, and beginning to generate a list on newsprint of these sports (see (b) above), seemed a pretty straightforward activity -- and an important one, since we wanted students to have that information as a reference before they wrote their answers to the survey.
How to orchestrate a productive discussion on "the math in the Olympics" was instead the object of some brainstorming and discussion, since the teachers were concerned that the students would not come up with anything very interesting at this point. Obviously, the content and quality of the video/ article shown in (a) would be crucial for this (and we still did not know what this would be!). We generated several ideas, without really deciding on any of them:
We also realized, though, that we really did not need to get much in depth with this at this point in time, though we wanted at least to raise the awareness that there is indeed a lot of math embedded in the Olympics, and sensitize students to looking for it.
At this point, Denise and Cindy felt they could go ahead with putting down the plan on paper (see attached) and hamnmering out the last details. The rest of the meeting was spent discussing what to do for the explicit instruction on graphing (it did not take us a long time to decide to devote about 2 days to this -- 1 day for frequency and bar graphs, and 1 day to circle graphs -- doing essentially the same things that the teachers would have done when teaching this topic in statistics the other years, using data mostly from question 1, and pointing out why data from the other questions would not really be appropriate for these kinds of representations) and how to organize the first few days of the activity "who is fastest".
| Stage 5. "PRELIMINARY" PLANNING Detailed plan of the first day of the unit |
Document J2
Source: Document create by Denise Anthony and Cindy Callard
GOALS: Present students with a preview of the upcoming Olympic Unit. Encourage students to begin thinking about how math is used in the Olympics and questions they may have about the Olympics that they would like to investigate.
Activity #1: Show 3-minute video clip of an introduction to a show on the Olympics to spark students' interest.
Activity #2: Class brainstorm about Winter Olympics Sports (write each class's ideas on the board and transfer to one master newsprint copy at the end of the day).
Activity #3: Hand out article on a brief history of the Olympiads and discuss. Pass out list of Winter Olympic Sports to see if any were missing in brainstorming. Lead to...
Activity #4: Pose question: How is math used in the Olympics? Students generate ideas with their partner and then share with group. Tell students we will be continually considering this question throughout the unit so they should try to keep it in mind!
Activity #5: (Closure) Tell students that they will be
working with a partner on a project for this unit that will investigate
a question they have about the Winter Olympics. Their homework is to complete
a survey that asks them about their interests in the Olympics (see below).
They will also have an opportunity to earn extra credits by collecting articles
about the Olympics (1 point per article, up to 5 points). In order to earn
the extra credit they will have to highlight the place in the article where
they found math is used. These articles will be put up in the classroom
as a resource for others.
Olympics Unit Survey: Answer each of the questions below. Please take the time to answer every question carefully as we will be using your responses to help us develop the unit to include your interests! 1. What Winter Olympic sports do you enjoy watching:
2. Tell us about a Winter Olympic sport that you know alot about (the athletes, the sport itself, etc.) 3. If you were a sport reporter assigned to the Winter Olympics, what questions would you be interested in investigating and reporting about?
4. For a final project for this unit, you will be working with a partner to investigate a question that you have about the Winter Olympics, List TWO people below that you could work with on this project. (NOTE: This project will require you and your partner to spend time together out-of-class to work on it.) |
| Stage 5. "PRELIMINARY" PLANNING "Daily sketches" for each main segment of the unit |
Document K
Source: Teachers' first written preliminary plan for the unit
Introduction -- DAY 2
Using the data from the kids survey about the Olympics and what they might like to learn or research, the lesson on day 2 will center around creating graphs and tracking frequency of responses about their favorite sports.
Introduction -- DAY 3
Again using student data, we will continue to develop different ways to dispay the data. Today we will focus on the circle graph.
Introduction -- DAY 4
The use of a line graph to represent data over time. Not applicable to sports data but better perhaps to show the Olympics medal count over time or to show the times of a skier at different gates throughout the race.
The use of distortion in graphs to misrepresent data and the idea of how to lie with data. Deceiving graphs and how they are constructed.
Segment #1. Who's fastest?
Resource -- Article about Human Achievement.
Using data we have found regarding different athletes and the speeds they average, students will be asked to do four activities:
DAY #1 -- Estimate the speed of several athletes in different types of events from roller skating to ice skating to luge to skiing. Rank those athletes from fastest to slowest. Estimate their speed in mph. Report this data in a table.
DAY #2 -- Given a list of track events and the world record speeds in table form in meter per second units, the students will practice converting these units to mph to compare the speed of the previous day's athletes to the speed of runners. Discussion of why these speeds were reported in meters per second and not miles per hour.
DAY #3 -- compare the data of day #1 and day #2 in graphs. Create bar graphs to present this info. Discussions of instantaneous vs. average velocity.
DAY #4 -- Discussion: What affects speed? (drag, lift, inertia) Talk about a few sports of the winter games where athletes might be concerned about these aspects to create more speed.
DAY #5 -- What new questions do we have about speed? What math have we uncovered in speed? Writing assignment.
Segment #2. How do we decide who wins an event?
Issues to be dealt with in this segment include:
DAY #1 and DAY #2 -- Students in the judge's chair
Students will see the U.S. Nationals skating and play the role of the judge. We will keep a table of the data to determine the average score. We will compute averages of mean, meadian and mode, as well as averaging by dropping the highest and lowest score.
DAY #3. Measurement of body sizes (height, wrist, waist, etc.).
Graphing and computing the average size of our class data.
DAY #4. What affects who wins? What new questions could we explore? What about how ties are handled?
Writing assignment.
Segment #3. Project developement.
2 days spent developing criteria for project through modeling a research question and the process of developing a project.
Criteria established for the project so far include:
Modeling ideas -- ages of participants or number of participants per country
Segment #4. A History of Measurement
Historical terms like cubit/ yard/ span/ etc. How precision developed over time? How terminology helps with precision? When and how is precision applied to different sports in the Olympics? Speed skiing, Luge, Skating, Biathlon.
Segment #5. Olympic Conclusions
The Medal Count reporting or misreporting
Overall awarding of medals (ties)
Should the medals be reported by type or by total?
Lying with statistics
Segment #6. Project Workdays and Project Presentations
| Stage 5. "PRELIMINARY" PLANNING Project expectations, structure and support |
Document L
Source: Document created by the facilitator based on the team's discussion of the first ideas articulated in the preliminary plan for the unit
Expectations and criteria:
Ideas for possible projects:
| Stage 5. "PRELIMINARY" PLANNING Revision of "Daily sketches" for each main segment of the unit |
Document M
Source: Document created by the facilitator based on the discussion of the first written preliminary plan for the unit that occurred in a team meeting
Introduction see Day 1, 2, and 4 from Denise and Cindy's "old overview"
1. Who's fastest? (5 days)
NOTE: Since one of the main goals of this segment is to model how questions can be generated and pursued, throughout the lessons the teachers will try to highlight whenever possible ideas for individual projects could be generated, and record those on newsprint.
DAY #1 (T 2/1):
Give a list of various athletes (from Denise's book) and ask students to guess who is fastest, and also rank them from fastest to slowest. Guess speed of each athletes, and report those guesses in a table form. Compare these guesses/ rankings with the actual data reported in Denise's book, presented by the teachers in a similar table form. Compute range of these data.
DAY #2 (W 2/2):
From Denise's book, give a list of track events and the world record speeds (table form, reporting meter per second units). Students will practice converting these units to mph to compare the speeds of athletes discussed the previous days with the speed of these runners. Discussion of why these speeds were reported in min/sec and not mph, to introduce awareness of metric vs. American measuring systems (one may connect this discussion with the international nature of the Olympics; maybe raise some historical questions, though they would not be addressed here)
HW: practice on converting
DAY #3 (R 2/3):
Compare the data of day 1 and 2 using graphs. Create bar graphs to represent this information. Bring up again issues on measurement systems and their history as appropriate. Discuss average and instantaneous velocity (maybe comparing speed in various skiing events, vs. speed skiing).
HW: Reading on history of measurement (probably ch.2 from Precision book)
DAY #4 (F 2/4):
Discuss what affects speed (so that students go beyond collecting data, to really trying to answer some questions and get to the WHY of things), by asking students to guess why athletes would achieve different speeds in various events. Maybe connect this with the SLOPE involved in various winter events (to connect with previous unit on graphing equations). More generally, talk about a few sports of the winter Olympic games where athletes might be concerned about what to do to create more speed.
HW: Reading article on the physics of the Olympics
DAY #5 (M 2/7):
Reflection on the process and the math uncovered. Identify and add to the questions generated on speed, and summarize what learned about them. Discuss what math was uncovered in the process. Share articles on the Olympics that students have contributed to that point and identify the math in them. Review ideas for possible projects already collectd on newsprint, and try to generate and record other possible ideas for projects.
HW: reflective writing assignment
2. How do you decide who wins an event? (4 days)
NOTE: Issues to be dealt in this segment include: 1) subjective vs. objective scoring (ex: figure skating vs. speed skating); 2) multiple runs and elimination runs (ex: luge; hockey); 3) raking by judges in comparuison to performances.
DAY #1 and #2 (T 2/8 + W 2/9):
Students will see a segment of the U.S. Nationals figure skating and play the role of the judge. We will keep a table of the scores given by each student and discuss how to determine the final score. We will compute range of individual scores as well as "average score" by using alternative methods: mean, median, mode, average after dropping lowest and highest score.
DAY #3 (R 2/10):
Generate a list of how other events are decided (ex: skiing, speed skating, ski jumping, luge, bobsled, biathlon, hockey). Compare these various situations to raise and discuss the questions: How do you decide who wins an event? How are ties handled in each case? What affects who wins? What new questions could be generated and explored? Once again, throughout this discussion, new ideas for projects will be recorded on newsprint (and perhaps also observations about math in the Olympics)
DAY #4 (F 2/11) (SUB DAY!):
Introduce circle graphs by using the scoring data generated in day 1 and 2.
HW: Bring in tentative questions each student is interested in exploring for their individual projects
NOTE: Olympics start Sat. 2/11
3. Project development (2 days)
DAY #1 and #2 (M 2/13 + T 2/14):
Introduce expectations and criteria for the individual projects; look at newspaper articles to identify what kind of data students may collect for their projects and how they may be reported; give time to pairs to share, elaborate and finalize their questions.
HW (for whole duration of the Olympics): Watch relevant games and collect data for individual project
4. History of measurement (3 days)
NOTE: While activities about measurement systems and their history are going on, we may need to devote some class time to discuss students' questions about the project and/or give some class time for partners to get together, since most students would at this point be collecting data for their projects.
DAY #1, #2 and #3 (W 2/16 + R 2/17 + F 2/18):
While issues about history of measurement and different measurement systems would have already been raised, we can now go back to them in more detail with some readings and activities. One major activity will be to have students identify various standard American unit of measure that are based on body parts (ex: inch, span, foot, etc.), and then measure such body parts, tabulate data collected, compute mean, median, mode, range and discuss these results. (Students may also be asked to line up in order of height, to show visually what the median would be). The issue of precision in measurement could also be raised and discuss (see readings) in connection to the question of how speeds, distances, etc. are actually measured in various Olympics events.
NOTE: Winter break, while the Olympics game are still going on
5. Follow-up activities and project preparation (5 days)
DAY #1 (M 2/28):
Teachers model how projects should be put together on one question of their choice (still to be determined)
DAY #2, 3, 4 and 5 (T 3/1 -- F 3/4):
1/2 class devoted to pairs working on their projects; 1/2 class developing activities connected with Medal Count (see Raffaella's article) and discussing issues of raking and "lying with math".
6. Project presentations (5 half days)
DAY #1, 2, 3, 4 and 5 (M 3/7 -- F 3/11)
Half class devoted to project presentations (5 minute per pair); half class begin probability activities.
| Stage 5. "PRELIMINARY" PLANNING |
Document N
Source: Excerpt from Cindy Callard's shared journal
I felt a lot better after last week's meeting and deciding a little bit more on our goals for the unit. I always feel better knowing where we're headed a little bit! I was also glad to outline some of the lessons as well. The unit seems much more tangible to me now that we have more to work with.
I still have a little concern about the students generating their own
questions--ones that are broad enough to do some research and gather some
information on, yet not too huge. I also wonder how easy/difficult it will
be for them to generate questions at all. I've been trying to think why
this is unsettling to me. I've decided that I think that I am uncomfortable
with raising questions myself, and I kind of transfer this feeling
on the students. When Raffaella mentioned us generating our own list of
questions that we have about the Olympics, I was a little stumped! I enjoy
watching the Olympics, but have never really questionned them. And
actually, I think this is why the style of teaching promoted throughout
the project is sometimes difficult for me. I always liked math because I
liked the structure, organization, linearity of the problems I solved in
highschool and college even (although definately not in graduate school!).
And now I am trying to teach and plan in a style that is not as linear.
I see my own thinking process not so much as branching outwards, but as
more logical and step-by-step. And I see the teaching style that we are
trying to do as much more branching outwards. It is not as comfortable for
me as the logical math lessons I have presented in the past. And not that
I am comparing to Audrey and Denise (and actually Raffaella and Edie) in
a negative way for myself (we all have strengths and weaknesses!), but I
see them as being much more outwards (rather than inwards) thinkers and
so this teaching style is not as "far away" from their own
thinking style. Don't get me wrong, I think this style is great, and the
benefits for students far outweigh my "uncomfortableness," I just
have a hard time adjusting personally! (Change is not easy for me either-or
for others!).
Stage 6. "DAILY" PLANNING (i.e., as the unit was implemented)
Also, see Document Q to read about a teacher's thoughts at this beginning stage of the unit.
| Stage 6. "DAILY" PLANNING An example of detailed "daily" planning |
Document O1
Source: Excerpt from the facilitator's journal recording the highlights of this planning experience
Denise, who had been browsing among the various sport magazines covering the Olympics looking for ideas for the "warm-up" activity for both the next day, found something interesting that she wanted to share with the group. She had found an article on the Biathlon (a sport mostly unfamiliar to the students, and presenting interest for the way it is scored -- an issue that would be addressed in the following segment) reporting, among other information about the sport, the facts that the athletes are expected to hit a target of a diameter of 4.5 inches with a 22/100 inch bullet, standing 50 meters apart. She was surprised and intrigued by this information, as she had not realized how difficult in would be to hit the target in those conditions (even without considering that one was doing so after having been cross-country skiing as fast as they could!). This gave her the idea that it would be worthwhile for the students to "get a sense" of what these measures were.
To "operationalize" this plan, we still had to discuss what materials to give the students (so that the measurements would be reasonable, yet would present some problem solving tasks) and how to word the directions for the task assigned to each group. Both elements were discussed by the whole group. As we reviewed what would be necessary (and/or useful) to measure distances such as 50 meters, 22/100 inch, 4.5 inches, and to draw models of the bullet and the target, we suggested to provide the students with a number of tools in the front of the class (compasses, circle templates, rulers, tape measures, graph paper, yard and meters sticks, calculators) but let them choose what they thought was most appropriate for their task. We also suggested a possible wording for the written directions to be handed in to each group, that Denise recorded (see enclosed copy of the warm-up with all directions). We all felt at this point that the activity was now ready to be implemented in class.
| Stage 6. "DAILY" PLANNING An example of detailed "daily" planning -- Instructional materials created/gathered |
Document O2
Source: Teacher-generated hand-outs
| Stage 6. "DAILY" PLANNING An example of detailed "daily" planning -- Instructional materials created/gathered |
Document O3
Source: Teacher-generated hand-outs
| Stage 6. "DAILY" PLANNING On-going revisions of overall plan |
Document P
Source: Document created by the facilitator based on the further team discussions of the preliminary plan for the unit
Initially, we had intended this meeting to be a very brief review of the revised written plan for the Olympic unit prepared by Raffaella on the basis of the previous meeting, and then give time to the teachers to actually go on preparing their detailed daily plans for at least the first few days of the unit. This plan, however, had to be changed because due to several "snow days" plus assemblies, etc., the students had missed so many classes that we would not be able any more to keep with the original plan and be ready by the time the Olympics would actually begin. (This is a point to keep in mind when planning units based on "current events" -- while it is always the case that one may need to adjust one's original schedule of the lessons in a unit because of external events like assemblies, special programs, etc., this may not cause too much problems if the unit can just be extended for the missing days; however, when you are trying to fit a unit around a current event, this cannot be done, especially if some of the activities need to be timed in connection to actual components of the event)
We found the task of "refitting" our plan so as to take into account the lost school days very frustrating, as well as difficult, because we really liked best our original plan and felt that now we needed to make a compromise by cutting components that we really thought should be there, or by giving less time to activitie, with the result of giving the students less time to really get involved in such activities or assimilate what we wanted. However, we felt we had no choice at this point, and tried to do the best we could -- the attached revised plan was the result of about an hour of discussion on how to best make the compromise. While we went back and forth about possible alternatives, in our final revision we decided to operate according to the following strategies: (a) as much as possible, move ahead what is not absolutely necessary for the kids to know or have done before they have to formulate and start working on their projects -- i.e., before the Olympics start; (b) try not to give less time to activities, but rather if necessary cut some activities altogether and do the rest well; (c) since we could not extend the unit for ever, and the students had already had the experience of doing presentations in the Tessellation unit, we thought that we could save time in the end by substituting the originally planned presentations with the creation of a newspaper contaning as articles all the reports from the students' projects.
According to these guidelines, and considering the fact that given an additional assembly day and a whole day meeting on the grant project the teachers would have a total of 5 days (+ one day with a sub) before the beginning of the Olympics, here are the major changes we ended up making:
| Stage 6. "DAILY" PLANNING |
Document Q
Source: Excerpt from Cindy Callard's shared journal
Well, the Olympic Unit has officially begun--and with it has come all
of the joys, frustrations, and work overload of a new unit. Denise, Audrey,
and I have already spent afternoons commiserating about all of the work
and effort that goes into this type of thing! I think what strikes me too,
is that there are many different levels of planning. And while we can do
one type of planning with Raffaella and Edie, when they leave there is another
piece left to do--the actual daily lesson (warm-ups, activities, and homework)
to keep a class of eighth graders interested and on task! A practical impossibility!
Denise, Audrey, and I are now in that "day-to-day" planning mode.
We have a general outline of the unit, know where it's heading, what math
we want to incorporate (basically all of our statistics chapter with some
measurement pieces as well), and how long we would like it to last
(note the underlining!). So away we go! The giant snowball has started rolling
behind us, we just need to keep moving so we're one step ahead of it!
Stage 7. PLANNING FOR FUTURE IMPLEMENTATIONS
| Stage 7. PLANNING FOR FUTURE IMPLEMENTATIONS Reconstructing key components of the unit/writing an overview |
Document R
Source: Excerpt from the Olympics Unit Report created by the facilitator in collaboration with the teachers after the end of the unit
This unit was conceived as a "thematic" unit designed around a current event. As such, our main goals were to build on the students' interest in the Olympics so as to uncover the use of mathematics in that real-life situation, to introduce/discuss some relevant mathematical tools, and to provide the students with the opportunity to inquire about a question of their own, making use of math as appropriate. More specifically, given the nature of this situation, we decided to focus on issues of measurement and statistics.
In the spirit of an inquiry approach, the structure of our unit included:
(a) a beginning section where the students would explore some specific aspects of the Olympics in activities designed and led by the teacher, so as to reveal some interesting mathematical aspects of this situation, generate ideas about possible questions the students themselves would like to pursue, and model the process they could follow to pursue such questions;
(b) the introduction of mathematical concepts and tools that could help the students address their questions, especially during the beginning section and then as appropriate throughout the unit;
(c) an individual project where the students would pursue questions of interest to them related to the Olympic Games, collecting appropriate data as the Games progressed and using this data as well as other sources in order to address their questions;
(d) a culminating activity where the students would synthesize and share with others the results of their projects;
(e) reflections on the math uncovered and the significance of what they were doing, possibly at the end of each segment.
The planning of our unit also had to take into consideration the fact that the students could collect data for their projects ONLY during the Olympics Games (a 2 week event). This had considerable implications for the scheduling of specific activities and segments.
Taking all these elements into consideration, our Olympics unit was comprised of the following elements/segments:
Students' project development:
(NOTE: Though this component developed throughout the unit, parallel to the other segments described below, we chose to describe it first in this overview, since the nature of the students' project informed several activities in the unit, even some that developed prior to the students actually working on their project.)
Though we wanted the students' projects to develop around their own questions and interests about the Olympics, from the very beginning we were also concerned with the articulation of some clear expectations about the project itself. Though we considered a number of alternative options--such as individual vs. pair or small group projects; presentations of the results to the rest of the class vs. some final written product; etc.--we dedcided to ask the students to work in pairs, addressing a question of their choice with the ultimate goal of producing a newspaper article summarizing their findings that would be "published" in a class Olympics magazine to be shared with the rest of the school; we expected each article to include an articulation of the question/issue they had pursued, as well as their findings supported by appropriate data and represented using appropriate graphs. Class time and some specific activities were designed at different points in the unit to help students understand what the project required them to do, generate and refine the question they wanted to investigate, develop a plan to address their question, organize and analyze the data collected, and write their final report. We also thought of a possible culminating activity where each student would choose, read and critique a number of articles from the collection produced by all the classes who did the unit--to better appreciate and make use of their peers' efforts in the projects--though we did not end up using this idea in our implementation. (Note: The newspaper never materialized due to time and energy limits of the teachers.)
Introduction to the unit: (4 days)
We planned to begin the unit with a discussion about the Olympics (in order to spark the students' interest) followed by a survey of students' interests in this event, designed to try and identify students' interests and questions about the Olympics (and possibly to take those into consideration in the planning of later segments of the unit). This activity would be followed by an analysis of the survey data in class, which would provide the opportunity to introduce/ review in a meaningful context a number of statistical ideas and graphs, while at the same time let students know some of their peers' interests. At this point we would also ask students to start collecting articles about the Olympics and bringing them to class for "extra-points" -- to be used as resources for their projects later on, and, occasionally, as springboards for class activities.
"Who's fastest?": (4-5 days)
This segment was intended to provide an opportunity to "model" a question about the Olympics with the class as well as introduce issues such as appropriate units of measure, conversions, different "averages", alternative ways of comparing results/ measurements, etc. Activities in this segment would include: comparing speeds achieved by the best athletes in different sports, in the same sport but in different events (ex: speed skating on different distances), or in the same event by different athletes; using a variety of mathematical tools to represent and analyze these data. Explanations about WHY such different speeds can be achieved could also be pursued by reading relevant articles. Issues about different units and systems of measurement could also be raised and partially addressed in this context. In order to serve as a true model for the students' own projects, this segment would also include an explicit reflection on its process and content. This reflection would then be used to introduce the project requirements and expectations. (Note: There was a great awareness gained in our implementation as to the complexity of measuring speed and how very different it is for different sports.)
Who wins?: (3-4 days)
Activities around the issue of "scoring" sporting events--from more "objective" ones such as speed skating to more "subjective" ones like figure skating--could provide another model of how one could identify and address a different question related to the Olympics. Exploring the issue of scoring could invite further discussion on the appropriate use of different "averages" and illustrate how the use of numbers may not by itself guarantee objectivity and fairness. (Note: Excellent activities on biathlon simulation and simulated scoring by class of figure skating competition. Also great awareness in general of Olympic procedures for all of us.)
(NOTE: Neither the order nor the content of the previous two segments is crucial to the development of the unit; in fact, any of these could be substituted by the class pursuit of a different question related to the Olympics (possibly even one generated by the students themselves)--as long as it is developed so as to provide a "model" to the students for their project activity; the choice of the specific questions of "Who's fastest?" and "Who wins?", in our specific implementation, was determined by the teachers' own interests as well as the mathematical topics and tools they wanted their classes to "cover" in this unit).
Medal counts and ranking issues: (2-3 days)
After the conclusion of the Olympics (and parallel to devoting some class time to working on their projects) we wanted to develop an activity around "medal counts", asking students to rank countries on the basis of medals won, and surprising them by showing how different countries may produce different rankings based on different ordering criteria (for example, in the U.S. media the countries are always ranked on the basis of the total number of medals won as primary criteria, while in the Italian media the primary criterion used is the number of gold medals...). This activity could lead to a discussion of the mathematical topic of order relations (ranking) and open issues about the "truth and objectivity" of mathematics. It could also provide an opportunity for the class to continue doing some activities related to the Olympics as the students finish working on their projects. (Note: Kids were surprised that we really had an Italian paper and were very interested in this activity.)
Issues about measurement and its history: (3-7 days)
We expected that the experience of watching various Olympic events, and even more explicitly the exploration of questions such as "Who's fastest?" and "Who wins?", would raise several questions about measurement that could be revisited in more depth in this segment. More specifically, we developed a number of activities that could be used to address one or more of the following issues: helping the students get "a sense" of some common measures; what, why and how things can be measured, and what is common to all these different "measurement situations"; how and why measurement systems developed; standard versus non-standard units of measurement--their origin, development, advantages and drawbacks; the metric system versus the U.S. measurement system; precision in measurement--what it means, how it can be decided, and how it can be obtained. (Only some of these activities were actually used in our implementation because of time constraints.) (Note: We had a few discussions that were very meaningful about metric measurement possibilities coming to our country soon--from current local newspaper articles. We did not have time to go into as much detail about measurement history, development, as planned.)
(NOTE: This segment, as a whole or in part, could be used at almost any time in the unit after the first modeling activities and the beginning of the Olympic Games.)
| Stage 7. PLANNING FOR FUTURE IMPLEMENTATIONS Reconstructing key components of the unit/writing an overview |
Document S
Source: Excerpt from the Olympics Unit Report created by the facilitator in collaboration with the teachers after the end of the unit
NOTE: Under the title "Project development" we have collected a number of activities that we planned to help the students develop their project at different stages--i.e., understanding what the project required them to do, choosing the question they wanted to investigate, refining this question and developing a plan to address it, organizing and analyzing the data collected, writing their final report/article. Given the nature of the unit and, especially, the constraint of having only two specific weeks in which the students could collect their data (one of which was vacation!) required careful scheduling of these activities throughout the unit. Thus, though we have chosen to report these activities all together in this section, in what follows we will also identify when each activity took place during the unit.
In order to understand the various components of this crucial element of the unit, it may also help to articulate how we characterized and organized these projects for ourselves:
Projects expectations and criteria:
Ideas for possible projects (to eventually suggest to students if necessary):
(a) Project requirements and expectations
DAY #1 and #2: Describe the nature and requirements of the project using the first "model" as an example
Scheduling: At the end of the first "model" (in the case of this implementation, the "Who's fastest" segment), and prior to the beginning of the Olympics Games (i.e., before the students need to collect their data).
Rationale: The students need to have clear expectations about what they are expected todo in their project, yet they may not fully understand the requirements until they can make sense of them in light of an example (provided by the first "model" designed by the teacher and conducted with the students' participation in class--e.g.: Who's fastest); a clear project structure can help students organize their work and feel confident even while engaging in a very open-ended activity.
Plan: Introduce expectations and criteria for the individual projects, possibly with the support of some handouts that the students could continue to refer to as they prepare their project (see Figure 1 at the end of the document). Illustrate the project's main components and requirements by referring to the model provided (see Figure 2 at the end of the document).
DAY #1: Soliciting and collecting first questions of interest to students
Scheduling: At the very beginning of the unit, in the context of the introduction to the Olympics and the survey
Rationale: To make students start thinking about the Olympics and to gather some ideas about their beginning interests, both to support the detailed planning of the unit and to have a starting point for students to further elaborate for their project.
Plan: A survey following a first class discussion (see Introduction, Day 1)
DAY #2 (and maybe #3): Providing students with an opportunity to generate worthwhile questions to be pursued in their project and/or further elaborate on such questions
Scheduling: Following the first "model" and the detailed description of the project's requirements and expectations
Rationale: Once they better understand what is expected of them in the project, students need materials and structure to help them identify, refine and select worthwhile questions to pursue in their projects.
Plan: Distribute some reading materials that could help the students generate further ideas about their question and ways to address it (such as a special newspaper insert on the Olympics provided by the local newspaper to schools upon request, and/or other articles from magazines, etc.). In order to enlarge the pool of ideas the students can choose from or take inspiration from, possibly have a follow-up class discussion where possible questions for projects are brainstormed and discussed. Have students identify their question (eventually recording their decisions in writing--see forms reproduced in Figure 3 and Figure 4 at the end of this document) and possibly discuss it with their partner (this could also provide the teacher with an opportunity to go around and help students refine and/or redefine their question to make it more mathematically interesting).
DAY #4: Providing students with an opportunity to develop and get feedback on specific plans to explore their questions for the project
Scheduling: This activity should occur after the students have begun collecting data on their projects but also before the students leave for vacation, so as to check on their progress, further elaborate and finalize their plans and make sure they will be able to complete their data collection when on their own.
Rationale: Students need some structure and feedback to help them develop a good plan to pursue their questions.
Plan: Follow-up on students' progress on their project could be achieved by devoting some class time to having them discuss their project with their partner and/or doing some review/practice seatwork, on related math content, while the teacher goes around and provides suggestions. Clarifications on the project's requirements could be made upon request, and new reading material distributed and/or other possible resources highlighted.
Scheduling: This component should occur after the Olympics Games are concluded and the students have collected the data for their projects; since it could be expected that the students will not be productive working on their own in pairs for a whole class period, most of the following "days" will actually be half-days (while the other half of the day is devoted to the Medal Count segment and/or covering measurement issues).
DAY #1: Teachers model how projects should be put together using a question of their choice
Rationale: Modeling this component of the project could really help students better understand how the various components of the project could be put together, building on each other; providing students with a written model for their final project could also be very helpful.
Plan: In class, the teacher goes over what was done with one of the previous "quests" undertaken as a class (I.e., "who's fastest?", "who wins?" or "medal counts") and shows how the results from the various activities the class engaged in could be used to respond to the original question and how these conclusions could be supported with appropriate data; possibly, the teacher could follow-up this demonstration with her own writing of a "final project" on that topic, following the guidelines given to the students, to be shared and discussed with the students during the following class.
DAY #2, 3, 4 and 5 (half-days): Students work in pairs on putting together their projects
Rationale: Students may need this time together in class, since the partners may have difficulty getting together outside of class; this can also provide an opportunity for the teacher to better monitor what the students are doing in their projects and to provide feedback and support.
Plan: [Note: it would be worthwhile to have the students share their "quests" with the rest of the class when they come back from collecting their data, as a way to "get back into" the unit after vacation.] While students work in pairs, the teacher could go around monitoring the work, asking and/or answering questions, inviting students to go deeper in their investigation, etc.
DAY #1 and #2: Producing and distributing the "Olympic journal" along with some follow-up activities to invite students to learn from other people's projects
Scheduling: At the very end of the unit.
Rationale: It would be nice if the students could benefit from the information and insights gathered by their classmates in their different projects; realizing that they can learn from what some of their peers have researched could also be important for students to come to fully value this kind of unit and the learning that it can produce.
Plan: The journal/magazine finally produced by the students would be distributed to students across the school, so that other students could appreciate and benefit from what they have done. An additional culminating activity involving the students' reading (some of) their peers' projects and learning from other's research could be organized around this event. For example, each student could be asked to look through the journal at articles addressing a sport/ question different from his/her own and then write a brief critique of what they learned from this exercise, what they liked in the articles, and what they would suggest for the author to change (NOTE: This activity may be even more meaningful if it PRECEDES the publication of the journal and allows students in one class to provide feedback on each other's final work so as to improve the product that is going to be made public.)
(NOTE: This activity did not take place in our implementation of the unit due to teacher and student exhaustion at this point.)
