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2 |
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5 |
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7 |
a) Introduction to the Olympics unit (3-4 days) |
8 |
b) Who's fastest? (4-5 days) |
10 |
c) Who wins? (3-4 days) |
12 |
d) Medal counts and ranking issues (2-3 days) |
14 |
e) Issues about measurement and its history (3-7 days) |
15 |
f) Project development |
18 |
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:
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 decided 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.
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.
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: 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.
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.
(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.)
Note: goals have been identified using normal font, while activities addressing them have been identified in italics.
In what follows we have tried to give a more concrete idea of the design of our Olympics unit by reporting, for each of the segments described in the overview, more detailed plans for each key activity and their rationale, followed by supporting materials such as the hand-outs prepared by the teachers to support class/homework activities and/or the reading materials distributed to the students. In a few occasions, in order to give a better idea of what the lessons entailed, we have also reported excerpts from the report of what actually happened in one of the classes where the unit was implemented, samples or summaries of students' work, key materials used for planning, and/or excerpts from the teachers' and/or observers' journals (NOTE: all of these artifacts have been collected in the "Selected Documentation" at the end of the packet).
Because of the nature of the unit, as well as our goal of inspiring other people to create their own unit taking advantage of our work (rather than repeat our experience or evaluate its implementation), what follows will not always be in the exact order in which it happened in class and may occasionally deviate from what we were actually able to implement.
NOTE: Several of the handouts included in the detailed plans of the unit are labeled as "warm-up". A typical routine in the classes in which the unit was implemented was to start most lessons with a short seat-work activity that would enable the students to quiet down and get focused on the topic of the lesson and/or do some additional practice on the topics addressed the previous day. Throughout the unit, the teachers chose to continue with this routine, but at the same time they tried to design the warm-ups along the theme of the Olympics as well as the mathematical topics discussed in the lesson. Similarly, it is important to note that it was the practice of the teachers who designed and implemented this unit to rely heavily on hand-outs as a way to structure their plans and students' activities, and provide materials for the students to go back to as needed; at the same time, these handouts were not used as seatwork that the students were expected to do in isolation, but rather as a springboard and concrete support for group work, class discussions and/or activities, etc.. The Olympics unit could be easily implemented without making use of most of the handouts included; however, we believe that the handouts included in these detailed plans could also be useful to teachers assuming a different organzation for their classrooms, as they clearly articulate components and requirements of specific activities that need to be communicated to students and/or taken into consideration when developing one's lesson plans.
DAY #1: Introduce the nature and scope of the unit.
Rationale: Present students with a preview of the upcomping Olympics 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.
Plan:
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 Olympic sports (write each class' ideas on the board and transfer to one master newsprint copy at end of day -- see Document 1 for a copy of this student-generated list).
Activity #3: Hand out article on a brief history of the Olympiad and discuss. Pass out list of the Winter Olympic Sports (see Document 2 for a copy of this reading) to see if any were missed 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 (see Document 3 for a copy of this student-generated list). 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 fo this unit that will investigate a question that they have about the Winter Olympics. Their homework is to complete a survey that asks them about their interests in the Olympics (see Document 4 for the text of this survey, and Document 5 for two examples of students' responses). They will also have an opportunity to earn extra credit by collecting articles about the Olympics (1 point an 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.
DAY #2, #3, #4: Introduce/review various statistical graphs in the context of analyzing data from the Olympics survey
Rationale: Since we wanted students to eventually be able to use a variety of graphs and other ways to represent and analyze data in their projects, we planned this activity in order to provide an opportunity to review and/or introduce these tools in a meaningful context--i.e., the analysis of the data collected in the Olympics survey. A complementary goal for this activity was to learn about the students' own interests and prior knowledge related to the Olympics (so that the students could use that information in selecting their partner, and so that we could take the students' interests into account as we planned/implemented the rest of the unit).
Plan: Using data from the Olympics survey, create frequency tables, bar graphs and pictographs to represent responses about favorite sports, and (MAYBE) circle graphs (NOTE: depending on the students' prior knowledge of these graphs, these lessons may contain some direct instruction, though in a meaningful context). Discuss elements of good graphs in this context. Some additional practice on using these graphs would be done in homework assignments. Also introduce the use of a line graph to represent data over time (applying this to some data connected with the Olympics, since it would not be appropriate for any of the data collected in the survey). Show deceiving graphs and discuss the idea of how to lie with data.
DAY #1, #2 and #3: Exploring the question of "Who's fastest" in various sports under the teacher's direction
Rationale: This activity is intended mainly as a model--i.e., it should enable students to experience the variety of avenues for exploration that could be opened around issues connected with the Olympics, and thus develop an image and some clear expectations of what the students themselves could do in their project around a question of interest to them. At the same time, this specific question was chosen because it provides the need and opportunity to review/introduce a number of concepts and mathematical tools--alternative measurement systems and units of measure, appropriate choice and use of unit of measure, conversions among units--that could be useful as the students do their own projects, as well as explicitly address some of the content the teachers are expected to cover in their 8th grade course.
Plan: Introduce this segment by reminding the students that later in the unit they will be doing a project where they will have to develop a question of interest to them about the Olympics and answer it. In preparation for this activity, in the next few days the class as a whole will be modeling this process on the question, "Who's fastest?", selected by the teacher. As a catalyst for thought and discussion, give a list of various sports and ask students to rank them from fastest to slowest by guessing/ estimating the top speed they think can be reached in each sport (using mph, since this is the unit of measure the students are probably most familiar with and have an intuitive sense of) -- see handout reproduced in Document 6); then provide a reading providing information about actual speeds reached by various athletes in the sports considered (see excerpt of this reading in Document 7) and ask students to compare their guesses about top speeds and ranking with the actual data reported in the reading (see homework sheet reproduced in Document 8). A class discussion following this activity (where the teacher should try to build on the students' own questions and contributions as stimulated by the previous activity) could lead to some first clarifications about what speed is and how it can be measured (this could include also distinguishing between average and instantaneous speed, discussing different units of measure and their relationship, etc.). In consideration of the fact that the metric system will be used in the Olympics (because of the international nature of this event), the teacher will also bring up the need to understand the metric system and "get a feel" for measures expressed in this system as well as learn how to make conversions. Further practice on conversion to and from the metric and U.S. systems will be done by the students in some meaningful contexts (for example, trying to make sense of how fast certain athletes go in various sports, and what unit of measures may be most appropriate in each case), using a conversion chart and calculators for their computations (see homework sheet reproduced in Document 9 and Document 10 as examples). Questioning how fast not just the "best" athlete, but athletes "on average", are in a given sport may also invite the application of the notions of mean, median, mode and range (see worksheet reproduced in Document 11 as an example). In order to show how one could go beyond the data considered so far, and raise and address different questions, the teacher could raise the question of, "Why would athletes achieve such different speeds in different sports?" and address it after assigning the reading of an article on "The Physics of the Olympics" (Kilgore, 1988).
(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. As these ideas are generated, they could be recorded on newsprint.)
DAY #4 and #5: Synthesize and reflect upon what was done while analyzing the "Who's fastest" question, focusing on both process and product
Rationale: The goal of these final days in the segment is to pull together what has been done up to this point to address the question of "Who's fastest", especially pointing out the need sometimes to go deeper than just collecting and representing data (i.e., looking "behind the scenes") in order to really address one's question. Articulating and looking explicitly at the process followed could also provide a concrete way to introduce the requirements and expectations for the students' own projects, using the "Who's fastest" question as an example.
Plan: As a class, identify what we have learned about speed and mathematics, respectively, as a result of the previous exploration of the question "who's fastest". Students' contributions will be recorded on the board or on newsprint (perhaps using some conceptual map diagrams to help the students see the relationships between the items generated), with the teacher asking for clarification and elaboration as appropriate (see Document 12 for a brief account of this experience and the conceptual map created by the class). The consideration of some elements that the students may not have been able to articulate at this point, or further elaboration of elements they have already mentioned, could be elicited if necessary by the presentation on overhead of excerpts of articles that address some specific issues in this segment, trying to help students go back to what done and summarize what learned about speed and/or math. At this point, the students will hopefully have gained a good image of what an inquiry on a question related to the Olympics could look like, so that it will be meaningful to present to them some specific directions/ requirements for the project they themselves are expected to do. This framework should also be used to identify and reflect on the key elements of the process followed in the inquiry on "Who's fastest" just concluded as a class (see Document 13 for a record of how the teacher used the proposed framework to summarize the result of the inquiry around the question "Who's fastest?"). (NOTE: It would be important at this point to give some time to the students to discuss with their partner in class what question they might like to explore for their project and how they could go about it, as this would enable the students to make explicit connections between what done in this segment and the project they are expected to develop--see Project Development segment.) (MAYBE) Give a reflective writing assignment at the end of this section.
NOTE: This segment could occur prior to the beginning of the Olympics, or while the Games are in progress and the students have already begun working on their own projects.
DAY #1 and #2: As a class, explore the question of how figure skating is scored
Rationale: This activity plays the role of providing another "model" of how to develop and pursue a question about the Olympics, dealing with quite different issues than those involved in the "Who's fastest" question. At the same time, it could also provide an opportunity to review/ introduce the notions of mean, median, mode and range, as well as to introduce the consideration of how the use of math (through complex scoring systems) still cannot eliminate issues of subjectivity and fairness.
Plan: In order to appreciate issues related to subjective vs. objective scoring, students will first read an article in class in pairs that explains some key elements of figure skating and how they are taken into consideration by judges in determining athelete's scores (see Document 14 for a copy of this reading); next they will see a segment of the U.S. Nationals figure skating (previously taped by the teachers) and play the role of the judges. After the scores given by each student have been collected and tabulated, alternative ways to determine the final score will be generated and discussed. The range of individual scores as well as the "average score" can then be computed by using alternative methods: mean, median, mode, average after dropping lowest and highest score (see Document 15 for a handout created to help students organize the tabulation of these data). (This situation could also be used to introduce circle graphs if they have not already been introduced.) This experience could then be used to develop a discussion on the subjective vs. objective nature of scoring and the "fairness" of the system employed to decide who wins a figure skating event at the Olympics.
DAY #3 and #4: Synthesis and reflection on what was done while exploring the question of how different events are scored
Rationale: To really provide a model for the students' own project, the previous activity needs to be followed up with an elaboration of the question itself (for example, comparing scoring in figure skating with how the winner is determined in other sporting events and raising issues about objectivity, fairness, etc. in the use of numbers in these situations) and, even more importantly, with a reflection on what was learned from these activities both in terms of content and process.
Plan: Drawing on the students' own knowledge of various sports, as well as what they have read up to this point, identify and discuss how the winners in other winter sports events--besides figure skating--are decided (for example: skiing and speed skating are based on time, but athletes can be disqualified if they fall or miss a gate; short track skating is based on speed, but judges sometimes make very subjective decisions about whether or not an athlete will be disqualified that may override who finished the race first; ski jumping scores are not only based on the length of the jump but also on the style of the jump, and more than one jump is considered; luge and bobsled events are based on time--but rather than having only one race or taking the best time over a number of races, these events are decided by averaging the times obtained over a number of races; biathlon scores are based on an "objective" combination of time spent skiing and number of targets missed--but how these two measures are combined required some decisions in the first place, and it would be interesting to explore how the "weight" given to speed in skiing versus accuracy in shooting may favor some athletes over others; hockey's scores are based on goals only--which may seem simple enough--but there are complex rules about how to decide ties and how teams compete with each other in order to determine the final winner). (Note: Some of the above situations could also provide the context for further meaningful practice of mean, median, mode, range, etc. -- see for example Document 16) Scoring procedures in various Olympics events could then be compared keeping the following questions in mind: 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 and observations about how math is used in the Olympics could be recorded on newsprint, and connections could be made with the various components of the project the students are expected to complete.
NOTE: We planned to use this segment after the Olympics Games were concluded and the students had collected the data for their projects and were devoting some class time to working with their partners on their projects. Thus, we envisioned what follows to be half-day activities that would occur parallelly with the preparation of their projects. However, it would also be possible to slightly modify this activity--by using partial medal counts after a few events rather than just the final ones, or final medal counts from past Olympics--and use it as an additional and/or alternative model before the students began working on their own projects.
HALF DAY #1, #2 (and possibly #3): Comparing different ranking of nations based on the Olympic medals won
Rationale: Realizing that there is not just one "right" way to rank nations on the basis of the Olympic medals won is likely to surprise students and make them question whether mathematics is as "objective" and "cut-and-dried" as they thought. It will also help them begin to realize that you can "lie with mathematics" even when you may be able to provide a perfectly acceptable and mechanical explanation of how you arrived at your results. This situation can also provide an ideal opportunity to have students develop and justify their own algorithms, both in terms of their correct execution and appropriateness. (See Borasi, 1989, for an in-depth discussion of the math and pedagogical potential of this situation).
Plan: Present students with the final ranking of national achievements at the Winter Olympics based on medals won as reported in the U.S. media (where the criteria used are: total number of medals (PRIMARY), number of gold (SECONDARY), number of silver (THIRD), number of bronze (FOURTH)) and Italian newspapers (where the criteria used are instead: number of gold medals (PRIMARY), number of silver (SECONDARY), number of bronze (THIRD)). Ask them to compare and explain the two rankings, and then discuss which of them is "fairer" or what alternative ranking could be even more fair (see handout reproduced in Document 17). The generation and "defense" of alternative ranking systems can provide a real "need" for students to clearly explain and justify their algorithms to their classmates, as well as introduce some interesting mathematical problems, such as those involving weighted averages (an issue that could be connected back to the way some events are scored, as discussed in the "Who wins" segment). (NOTE: We found a newspaper article reporting the prize money offered to athletes as incentives for winning gold, silver or bronze medals by some countries or companies, and used this reading as a further stimulus to discuss the relative "worth" that could be assigned to different kinds of medals.) A good discussion could also be generated about the "objectivity" of mathematics per-se and the ethics involved with using mathematics appropriately. (See the report of a 10 minute discussion on alternative medal rankings, reproduced in the Document 18, to get a sense of the kind of ideas and issues students could raise in this kind of discussion).
NOTE: This segment (or some of its components) could take place at almost any point in the unit after the first modeling experiences and the viewing of some Olympic events have provided the students with various examples of situations involving measurement. Depending on the time available, the teachers' curriculum constraints and priorities, the students' own interests and the questions they may have spontaneously raised, some or all of the following activities could be used. (These activities have been listed in one possible sequence. In most cases, this order could easily be changed.)
1. "Getting a sense" of common measures (1-2 days)
Rationale: It is a common complaint that most students do not have a real sense of even the most common measures--something that could be very detrimental to their ability to estimate appropriately and to reason through conversions and other kinds of problems involving measurement. By using the concrete and (for many students) more familiar context of sports events, students could come to a better intuitive understanding of what some basic distances, times, weights, areas, etc. look like (perhaps even using both metric and U.S. systems).
Plan: While discussing specific Olympic events, or in the context of exploring questions such as "Who's fastest?" and "Who wins?", we wanted to be alert for and capitalize upon opportunities to involve the students in "making sense" of the measurements that were involved. What follows is a collection of examples that we developed in the context of the specific questions we chose to model with the class and/or in response to specific interests raised by some students:
2. What, why and how we measure (1-2 days)
Rationale: By comparing various situations involving measurement (and ranking) in the Olympic Games, the students could become aware of the variety of things that can be measured (lengths, time, artistic performance, team performances, etc.), as well as of the fact that what and how we measure depends essentially on the ultimate purpose for taking that measurement. An explicit comparison of what is common in all these different situations could also help students come to appreciate the more general and fundamental principles of measurement.
Plan: Identify with the students as many situations as possible in the Olympics where measurement of some sort is involved. For each of these situations, discuss what, why and how measurement was achieved, pointing out in each case how the "why" determines to a great extent the "what and how"; it would be important not to reduce this discussion to the more obvious measurements (time, length, etc. used to determine the winner) but to also bring in less objective measures (such as those used to score figure skating) and measurements that have little to do with scoring (ex: measurements made to prepare the stage of the event itself, such as skating rink, position of poles in slalom, etc.). It is likely that this discussion will raise some more general questions about measurement. In particular, it may be worth raising the question of what is common to these different situations that makes us call them all "measurement" in order to help students identify key elements of any measurement system such as: selecting and using a common unit of measure, comparing the object to be measured with this unit and/or parts of it, using standard vs. non-standard units, etc. (For more specific ideas, see the list of "ideas about measurement within the Olympics" reproduced in Document 22)
3. How and why measurement systems developed (1-4 days)
Rationale: Questioning how people ever began to measure things could help students better identify and understand the fundamental principles of measurement and also come to appreciate that this (as well as any other branch of mathematics) is the product of human activity.
Plan: Students can be asked to imagine how people in the past may have measured things and, possibly, to write a story describing hypothetical situations when men first began measuring. These thoughts can be shared with the rest of the students (in a "brainstorming" session and/or by reading and elaborating on each story) so as to stimulate a class discussion on the history of measurement. The topic could be further pursued through readings about the history of measurement (there are several children's books on this topic) and/or activities that could help students get a better sense of how some first rudimentary measurement could have been achieved and later improved.
4. Standard vs. non-standard units of measurement (1-2 days)
Rationale: It seems important to us that students reflect on this crucial difference between current measurement systems and more primitive ones and appreciate the value of using standard units. The activity described below could also spark students' curiosity and provide them with a meaningful context to do some concrete measurements using common measuring tools (thus learning to use those tools and getting a better sense of the "meaning" of the most commonly used standard units).
Plan: Once students have encountered and discussed the issue of standard vs. non-standard units of measures in the context of readings and/or brainstorming discussions on the history of measurement, an interesting follow-up activity could consist of having students identify various standard American units of measure that are based on body parts (ex: inch, span, foot, etc.; these could also be identified with the help of some of the readings on the history of measurement previously done by the students) and then measure on themselves these body parts, tabulating the data collected. This data could then be analyzed and "averaged" using various measures (mean, median, mode, range, as well as others) and then compared with the "standard" version of that unit of measure; interesting speculations about the disparities found could then be raised and discussed.
5. Metric vs. U.S. measurement system (1-2 days)
Rationale: As students may occasionally have to intepret information provided in the metric system in their everyday life (either in the context of the Olympics, since this is an international event where the metric system is used, or in other contexts in the future), it may be worthwhile for them to get a sense of the key units of measure in that system, to understand their relation with the U.S. units they are more familiar with, and to be able to make precise conversions withthe help of appropriate technology.
Plan: See activities interspersed in the "Who's fastest" section; some readings from the media about the possibility of adopting the metric system in the U.S. could also be done to spark students' interest in this topic.
6. Precision in measurement (1-2 days)
Rationale: Precision of measurement (i.e., how precise a measurement could and should be, how this depends on the purpose for the measurement and the precision of the available instruments, how to deal with inevitable errors of measurement, etc.) is a crucial element for understanding both the theoretical and empirical aspects of measurement.
Plan: As appropriate throughout the unit, these issues could be brought up and discussed with the students (The different kinds of events in the Olympics, where in some cases the winner is decided on the basis of 1/100 of a second and in others in minutes, are especially appropriate to raise such issues). Further information about how precise measurements of very small things could be obtained could be gained from the reading and discussion of relevant excerpts from books and articles.
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):
1. 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 to do 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 Documents 23, 24, and 25 for examples of such handouts provided to students at different points in the unit). Illustrate the project's main components and requirements by referring to the model provided.
2. Supporting students' in developing questions for their projects
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 Document 4)
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 form reproduced in Document 26) 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). (See Document 27 for a list of the questions students generated at this stage).
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.
3. Organizing and analyzing data; preparing final report
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.
4. Sharing information and reflections
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.)