## Conference MaterialThe Game of Pig
6-12 Mathematics
"The Game of Pig"
By F. Joseph Merlino The Greater Philadelphia Secondary Mathematics Project December 18, 2001
1. Please provide a brief description of the instructional unit, including approximate number of hours of classroom instruction involved. The instructional unit is called These concepts are contextualized in the unit problem, which involves determining the best strategy to win the most points in the The concepts implied in solving The
2. What did you expect would be the challenges for teachers in teaching this unit? For teachers new to the
A companion content problem for new teachers is to see how a particular concept is developed over time and related to other concepts within the unit. Teachers do not as yet have a "cognitive map" of either the flow of the unit or its content. As a result, teachers tend to teach each day’s lesson in isolation without providing students opportunities for insight into the interconnections of activities and topics. It is like pounding each note when one is first learning to play the piano rather than making music, or like self-consciously taking awkward steps rather than dancing fluidly.
New IMP teachers have difficulty linking the individual parts of a day’s lesson into a coherent whole, and then linking the many daily lessons into a larger unit. Moreover, they don’t see the relationship of the Organizing a Lesson for Conceptual Understanding: Perhaps the most difficult challenge for new IMP teachers is to teach a unit like 1. Posing a problem or question to the student, 2. Eliciting individual intuitions and prior student knowledge, preferably in writing, 3. Having them experience a physical simulation or model of the problems, 4. Encouraging students to individually reflect upon and write about their experience, 5. Directing students to exchange their reflections and ideas, and/or doing another activity in small groups, 6. Sharing results as a whole class and discussing similar and conflicting analyses 7. Repeating steps 1-6 with a more complex problem or further extension of it.
Most mathematics teachers will say their goal is to produce "independent thinkers." At the same time, teachers find it a personal challenge to let go and allow students to think and be independent, even within a unit specifically designed with such reflection and independence.
One of the best questioning methods is for teachers to literally sit down at a group table and do a Q & A session with the students. The teacher, who should probe further based on student responses, does the questioning. For example, in
A related challenge for teachers is to pose questions that are not immediately answerable but are left "pregnant." The whole
Indeed, if genuine student understanding of
Successfully teaching a unit like The goal of our professional development is to help teachers achieve "flow" with their students. Unless students have had a reform curriculum in at least middle school, most 9
3. Therefore, what were the challenges in providing professional development for this specific instructional unit? The PD challenges are both general and particular. They are general as regards the whole IMP program, and particular as regards the specific instructional unit e.g., the
4. How was the professional development structured to meet these challenges? The professional development is actually a package of training components rather than just classroom type of in-service. These components consist of the following: - Each IMP teacher receives
__twelve hours of in-service in each IMP unit__, roughly two full days per unit, taught by a fellow teacher who is an experienced IMP teacher. - Part of the Pig unit training is also to be
__trained in all the other IMP units__to deepen teachers’ content knowledge of probability as well as to understand how the IMP curriculum is structured. - Each teacher can enroll in a five-day, IMP-based,
__graphics calculator course__. - Each teacher is encouraged to
__actually teach the unit__they have been being trained in, at least for one semester. - Each teacher is provided
__a mentor who visits their classroom__and conducts before and after conferences with the teacher via e-mail and in person. Teachers are mentored 20 hours the first year. - Each teacher and mathematics department is encouraged to meet each week to discuss common concerns, insight and experiences in the program. On-going collegial dialog involving focused reflects on practice is an essential PD component.
- Each mathematics department is encouraged to arrange peer teacher visitation with the school.
- Each mathematics department is encouraged to allocate resources and time for teachers to attend IMP-related professional conferences, such as NCTM meetings, regional IMP users conferences, and National IMP Teacher Leadership conferences.
- What did a typical teacher experience in professional development prior to
teaching this unit for the first time? **How was a typical teacher supported as s/he taught the instructional unit for the**? first time- What did a typical teacher experience in professional development about this unit
after teaching this unit for the first time? - How many hours of total professional development did the activities in a. — c represent?
Teachers will have gone through at least 2 days of IMP in-service in the previous IMP unit, Teachers are supported usually through in-classroom mentoring by an experienced IMP teacher from our LSC project and special school departmental meetings, either during or after school.
They received training It can vary between 24 hours and 54 —60 hours.
5. Based on PI/project staff observations of a minimum of 3 teachers, in what ways does the instruction in this unit appear to be consistent with the project's vision and in what ways does it is it not? Three ninth grade teachers were observed teaching lessons from the Game of Pig Unit; two from suburban schools and the third in an urban setting. All three teachers were new to the IMP program this year. The first teacher, from one of the suburban schools, began the lesson by working on a homework problem on expected value. In the problem, the students imagine that they have two pockets and that each pocket contains a penny, a nickel, and a dime. They reach in and remove one coin from each pocket and determine the possible amounts they could get from the total of the two coins. The first half of the lesson was devoted to having the students present their solutions to the homework problem. The second half of the lesson involved the teacher instructing the students on how to use the graphing calculator and how to do a frequency bar graph on the graphing calculator. While there is a mathematical segue between the homework problem and frequency bar graphs on the graphing calculator, the teacher made no attempt to tie the two together. The teacher went over questions on the homework, but did not probe students further with extensions to determine if they understood the concept nor did she tie the problem to previous activities. The teacher was directing the students toward the correct answer to the homework questions in a lockstep manner. For the students, there was no apparent purpose to why she was teaching them how to use the graphing calculator to make frequency bar graphs. The second lesson observed was also a teacher in a suburban school. The students, who were lower in ability level than the first class observed, sat in rows facing the teacher. The lesson was also from the Game of Pig but was about a different homework problem on expected value. The teacher had written the questions on the board and students came up to the board and filled in the answers. In going over the questions, the teacher played the major role in this lesson, talking most of the time while trying to get students to answer his questions using the "fill in the blank" format that he had set up on the board. The questioning was task-oriented in arriving at a product or a result, but lacked purpose aimed at developing students' conceptual understanding. The final lesson observed was in an urban school. In this lesson students were working on expected value by playing the spinner game. In this game, "Al" and "Betty" were playing a game with a spinner. Al has 25% of the board and wins $4 from Betty when the spinner stops on his portion of the board. Betty has 75% of the board and wins $1 from Al when the spinner stops on her portion of the board. The students are trying to determine who is more likely to be the winner of this game, and if Al and Betty played 100 times who would be the expected winner. The students worked on the problem with the spinners for the entire period and the teacher did not provide any type of sense-making during the lesson. As in the previous lessons, the aim was to get the answer to the problem of Al and Betty, not to get them to understand that probabilities can have weighted values to them that affect your decisions. In addition, there was no tie between this activity and related activities they had previously experienced. In hindsight this gap was foreshadowed by the instructional purpose written on the board at the beginning of the day: learn how to play the spinners game. In all three lessons observed, teachers did not demonstrate that they understood the content or how the concepts in the lessons they were teaching fit into the concepts in the unit. They tended to zero in on the minutiae of a particular lesson and apparently did not recognize how the lessons fit into the bigger picture of the unit.
6. Assuming that you devoted the same number of professional development hours to this unit again, how would you structure the professional development and why? The issues demonstrated in the observations are difficult to address within the format of the professional development program. Teachers need to change their perceptions of themselves as teachers, their own ideas of learning, and their understanding of how students learn mathematics. During the first year, a new IMP teacher’s classroom practice is usually not very consistent with the project’s vision. Teachers typically dominate the classroom and talk too much. They don’t often allow students to present. They don’t manage groups well. They don’t understand well the mathematical purpose of the unit and don’t have a sense of students’ prior knowledge. They are unskilled and unartful about leading students’ discovery. They grade too many papers and spend too much time on each one. They are very inpatient with themselves and with the students. Some don’t distribute or use the graphing calculators or use the overhead projector. Most teachers do try hard and make a good faith effort to implement the curriculum as intended. Over time, by the end of the first year, many teachers make great strides in their content knowledge and pedagogy. But it can take five years or more for teachers to become truly comfortable and confident with the whole IMP program and skilled at managing a student centered classroom.
A further advance in our professional development package would be to infuse multi-media technology into the training, mentoring and after-school meetings whereby teachers could conveniently access visual vignettes illustrating different aspects of a student-centered lesson that are specific to a particular curricular unit, such as |