ESCOT Summer Workshop 2000
ESCOT Summer Workshop 20000 || Math Forum Workshops

Day 1 Summary

Getting Acquainted
The participants met in Ashton House to have breakfast, reacquaint or be introduced to one another. After breakfast the group walked to the Educational Materials Center. Jody demonstrated a game which required a participant to toss an orange (The only sphere that we could find!) to another participant and ask them a question. Everyone participated and a wide range of questions were asked.
The opening activities were followed by an overview of ESCOT presented by Jeremy. The ESCOT project is almost two years old. It was designed to bring people together who could integrate software with a strong curricular focus. It consists of an ESCOT PoW that involves integration teams working together to develop problems with applet components. Students who worked the problems have typically come from teachers involved in the project and project staff have served as mentors. It is an exciting project because it brings together a geographically diverse and group with varied talents. Feedback on the project was very positive. National Science Foundation (NSF) support has been important for the project.
Looking at Last Year's Problem Sets
Steve Weimar and Jody facilitated small group work focused on reviewing last year's problems and links to the NCTM Standards. Groups were self-selected such that there was an even distribution of roles represented in each group.

This was followed by sharing with respect to each of the problems:
  1. Pirates and Diamonds
          Part I || Part II || Part III || Part IV
  2. Scale n' Bowl
          Part I || Part II || Part III || Part IV
  3. Llama
          Part I || Part II || Part III
  4. Rock, Paper, Scissors
          Part I || Part II || Part III || Part IV
  5. Earthquake
          Part I || Part II || Part III || Part IV
  6. Shoelaces
          Part I || Part II || Part III
  7. Pi Machine
          Part I || Part II || Part III || Part IV
  8. Search and Rescue
          Part I
Key points of consideration include:
  1. When technology enhances teaching
  2. Identifying the mathematics and possible revisions for strengthening the mathematics.
  3. Support necessary for teachers
  4. Interestingness of problem context
  5. How much support should technology supply to the student
A discussion followed focused on the problem issues
  1. Which math concepts are important?
  2. Clarity of problem
  3. Problem posing strategies
  4. Interface issues
  5. Benefit of technology
  6. Connection to NCTM Standards
  7. For whom are the problems effective?
  8. Context in which teachers would use ESCOT PoW
These were some thoughts that emerged:
  1. Are there ways to develop technology so that the teachers can easily use it as a supplement to their curriculum?
  2. What kinds of supports need to be in place to enable teachers to work effectively with enhanced problems?
  3. Do the problems have too much text for the teacher? for the student?
  4. There are different models of usage and these are also informed by reading level.
  5. Where does open-ended exploration fit in?
  6. What is the role of the story in the problem and does it lead to engagement or is it a distraction?
Overview of Evaluation of PoWs

Ann Renninger, Claire Feldman-Riordan, Sarah Kate Selling and Wes Shumar overviewed findings of the ESCOT Project, Year One. Ann linked an evaluation of the ESCOT Project to the larger Math Forum evaluation. In particular for that evaluation she and her students have been evaluating the impact of Forum services including the Problems of the Week on students' mathematical thinking building on Schoenfeld's 1992 discussion. They have developed a method to code mathematical thinking and applied this to a study of the ESCOT PoWs. Briefly mathematical thinking refers to students' abilities to connect to problems, the strategies they use in problem solution and their autonomy or need for support.

Findings from studies of the ElemPoW, MidPoW, and GeoPoW indicate that over a ten month period students make more effective connections to problems, are more strategic in problem solution and more autonomous in their work with the problems.

Student's mathematical thinking was found to change in their work with the ESCOT PoW as a function of time, however, there is no difference as a function of instructional style.

Qualitative data from focus groups with students indicates that students really liked the ESCOT PoW although they were frustrated with the technological complications. Students who had done both the MidPoWs and ESCOT PoWs thought they were easier because they were interactive. They thought that the ESCOT PoW was harder and more mathematical than the ElemPoW. They had to explain the answer and "...that's what makes it harder. Sometimes it helps but it is also confusing because it brings writing into math."

Three reasons for technological difficulties:
  1. It didn't go live at the right time.
  2. The students had trouble submitting.
  3. The students' submission were too late for mentoring.
It should be noted that the attempt to match problem sets of the ESCOT PoW and MidPow is difficult. The type of problem that needs an interactive component is hard to visualize. The elimination of the visual component in the MidPoW means that the MidPoW by definition is not parallel to the ESCOT PoW.

In general students were more likely to make effective connections to ESCOT PoW problems than to the parallel MidPoW problems. With the exception of one of the MidPoW problems on which students were more strategic than they had been on the parallel ESCOT PoW there were no differences with respect to strategies.

Students on the MidPoW were more autonomous than students on the parallel ESCOT PoW with the exception of the Earthquake problem which students in the ESCOT PoW more autonomous than students on the MidPoW.

Suggestions, based on patterns that emerge from coding:

Problem construction/Mentoring
  1. Students are likely to get at least a portion of the problem correct since ESCOT PoWs ask several smaller questions. As a result connectedness scores have been high.
  2. The difficulty level of the problems is low, usually Level 1 or 2.
  3. Some problems don't ask for a rationale.
  4. There are a lot of strong students who do not resubmit; on the other hand, mentors are more likely to challenge students to improve even when the problem is correct than we've seen with other PoWs coded.
  5. Of students who do send in multiple solutions, many are sending them within a short time period (i.e., all 4 parts of an ESCOT problem in one day; and then answered 2 parts of the following month's problems in one day.
  6. Compared to PoWs, the ESCOT PoWs appear to be over mentored (see first mentor response, example 5 Llama I) and/or the problems have a lot of information in them; as a result, there is so much scaffolding that students are not challenged to figure out what the problem is asking them to do.
  7. In writing the problems, need to specify what you want the students to do. If you want them to use a graph, specify this in the problem. It is also important to allow for different kinds of "correct" answers. (So, if a graph is the goal, say: There are multiple ways to solve this problem, here we would like you to use a graph.)
  8. There is a need to strike a balance with scaffolds in the problem. In the Rock, Paper, Scissors problem, students needed to be asked to tweak their answers to the conceptual question posed, so that they were set up to understand later concepts. Given that the move is to two-part problems, the Search & Rescue problem becomes a good model because of the scaffolds that are built into it.
  9. If students only write a little, they may actually only be able to read a little. If students get back a long answer, they are likely to think that they have missed something.
  10. When information is inserted into the email, there is no resubmission. It may be the case that for the students, inserting information into the email following what they did is hard for them to follow. Kids do not tend to get very many emails.
Which questions could be reused?
  1. Students thought that the Llama problem was the best because the simulation was real. When they could connect to the problem they did and they had insightful things to say, i.e., "there's only one biggest."
  2. Pi problem. Hardly anyone did this; it is a great problem.
  3. Earthquake problem, although students need to be told that they need to use the graph. Could change the numbers so students from previous year wouldn't simply be able to use work done before. One point was left out: students needed to explain the meaning of several lines. There was a red line segment that needed to be interpreted.

        Note: kids' interpretations aren't always very good. Writing can trip them up. It may be possible to build a model into the problem (an applet) that models interpretation for them, similar to that in the Shoelace problem where they learned about the y-intercept.

  4. Any problem that is reused: be sure to ask student to move from doing the activity to generalize the point to principles. Also, be careful about making the problem repetitive.

    For example, Pirates and Diamonds as a 4 part problem was repetitive. As a two part problem, it might be good. Similar point, re. Scale and Bowl. Could have had them stand to the right and figure out the angles in order to generalize. (Could write isomorphic problems re. rebel soldiers and cannon shot; pool- golden rectangles)

    In Search and Rescue, the 4 parts train you, there never was an expectation that you'd generalize about what was learned.
Wes Shumar presented: What makes an effective integration team?
  • Clear goals shared by all members of the team
  • Clear roles balancing the jobs of teachers, developers and facilitators
  • Group processes which facilitate a sense of community or belonging are the result of clear goals and roles.
Worldview - Issues that must be negotiated between teachers and developers:
  • What does effective learning look like?
  • What constitutes a good problem?
  • What technologies do teachers and students have at their disposal?
  • How do they relate to those technologies?
  • Shared meanings - the group works together and builds trust so that there are ideas that are held in common
  • Virtual and physical interactions - what's the balance?
  • How are communication problems negotiated by the integration team?
Math Topics Discussion
A list of middle school math topics was suggested but after discussion we came to the conclusion that we would like to organize by NCTM Content Standards. Time was spent considering the different original problems discussing the matches that we saw to the Content Standards. Next we considered which problems from last year were worth considering for this year. We followed a process of having the problem nominated and seconded in order to be added to a list. Sometimes only one week of a four week problem was nominated and sometimes more. We considered all possible combinations before listing it.

Jody asked that the conversation turn to either what problems people brought with them. Several participants shared math problems:

Hollylynne's Marble Problem
Holly offered to share a math problem that she brought. She said that her interests lie in the teaching and learning of probability so that is what she brought along.

Holly drew four pictures of bags filled with marbles (two colors). The ideas of the problem included getting at different experiments with replacements, making predictions on probability of picking the marbles, comparing the bags of which you have a better chance of picking out a black marble and bringing in another bag to discuss ratio and proportions. "How many more white marbles do you need to get the same probability as in bag b?"

Vicki asked how technology played a role.

Holly said that you can experiment with the actual putting of the marbles in the bags. Remember it is just an idea. Possbilities include leaving it totally open-ended or making it have the same chance as another which would get us into equivalent bags.
Ken's Russian Peasant Multiplication Recollection
Ken went next with a new idea. In a recent Math Forum newsletter there was a feature on Russian peasant multiplication. So take 12 and 8. But I don't want to multiply 12 and 8, so I will multiply 24 and 4, but I don't know my 24 tables, so I'll multiply 48 and 2, which is 96. Ken went on to explain the method including the idea that it involves binary multiplication.

Jody suggested it might be good for a regular PoW and not even need to have any technology.

Chris said that the role the technology could play would be algorithm visualization.
Steve A.'s Idea
Steve A. suggested Australian house modularization. Start with a model: What's the volume and surface area? Now double the number of cubes. Now what's the volume and surface area? And so on. The technology would help us once we get into large models.

Another idea that Hollylynne said that Steve A. had told her about was the idea of 3D geometrical visualization. He explained a sprinkler problem.
Chris' and Seth's Tetris Idea
Chris said that he and Seth had an idea getting back to the idea of Tetris inspired by Leslie, who complains about students' difficulty with fractions. This idea specifically involves adding fractions. Thinking you could have a container, have a pallette with a 1/4 bar, a 1/3 bar, a 1/2 bar, and a full bar. The students must complete a row. (A row is a unit). One idea would have the computer start off, meaning the students are given a 1/3, and then the student would finish off the row. Perhaps have an optional clock?

Bill brought up that could take the game farther to include percents, decimals, and fractions. Can have a bar factory as well? Ken suggested that you are presented with 9/16. Now you are presented with a bunch of bars in your tool kits. You have to give the sum as you go toward your goal.
Janet Bower's Idea
Jody projected the suggestions Janet Bower brought up. The first one has to do with playing pool and figuring out where the ball should hit. Math used included incoming and outgoing angles. The student must make a prediction of where the ball will end up after it hits a side.
Monday, August 14, 2000
Written by Lynne Steuerle

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