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Contents:
I. MOTIVATION & RATIONALE	3
A. Teaching philosophy	3
B. Why Statistics and Probability?	4
C. Summary of the unit	5

I MOTIVATION & RATIONALE

A. Teaching Philosophy

The National Council of Teachers of Mathematics has been a driving force in the mission to improve the quality of school mathematics. The instruction of mathematics today encompasses an environment where students adapt an active role in their own learning. Thus, the learning of mathematics transcends from a transmissive model towards a reconstructionists framework. A transmissive model is one that views knowledge as objective, where the teacher is an authority who uses directive pedagogy (E. Swartz, 1992). A reconstructionists model views knowledge as socially constructed through interaction. The teacher presents knowledge as a problem-posing pedagogy, not a banking approach (E. Swartz, 1992). In this model, mathematics is presented as a subject that is not necessarily pre-determined. Rather, it is a subject where one can assume the role of an investigator or explore, forming hypothesis and testing outcomes.

My teaching philosophy is one that exemplifies the teaching of mathematics through constructionist view point. This involves viewing students as active learners who are capable of developing their own sense of understanding of mathematical content. Students are encouraged to utilize what they know or perceive to further investigate, explore and modify their understanding of mathematical phenomenon. According to Magdalene Lampert, "Knowing mathematics in school therefore comes to mean having a set of unexamined beliefs, whereas Lakatos and Polya suggest that the knower of mathematics needs to be able to stand back from his or her own knowledge, evaluate its antecedent assumptions, argue about he foundations of its legitimacy, and be willing to have others do the same" (Lampert, 1992). The teachers role is to organize collaborative and critical activities that allow students to communicate and express their thoughts. The discourse in a mathematics classroom should be directed by both the teacher and the students. Moreover, the students should be given opportunity to provide examination and cross examination of one anothers thoughts and solutions. Value is placed on EVERY students' questions, arguments, beliefs and suggestions.

Incorporated in this unit is a pedagogical style that is reflective of the constructionist model of teaching mathematics. Students are active participants and explorers in their learning. This exploration involves mathematical discovery, "hands-on" activities, and real world application. Moreover, social interaction and effective communication plays a dominating role in how the students construct their thinking and understanding of the mathematical content. The following practices were utilized in this unit in order to promote this teaching philosophy:

• . Interesting and relevant topics. Student-centered communication can be facilitated through engaging activities of interest to the students. Moreover, this particular unit is directly reflective of real world phenomena.
• Open-ended questions. These types of questions allow students to construct their own responses and encourages divergent views as well as creative thinking. As teachers we need to make a conscious effort to limit the short answer questions or "fill in the blank" kinds of discussion. The discourse in the Documentation and Implementation section of this unit focuses on tapping into students' thinking, using both open ended questions, and questions that allow students to be reflective of their thoughts and insights. Questions of this kind will begin with phrases such as:

Explain to me why you think....?

How is it that..........?

What lead you to your conclusion?

You said....meaning what?

How do you see this pertaining to my question?

Show me what you mean by...?

Do you think that....? Why?

• Writing in mathematics. Writing in mathematics opens an avenue for students thought processes as well as problem solving strategies. Moreover, this method allowed students the opportunity to bring their metacognitive understanding to writing. Appendices D, G-J, and S are reflective of this task. The questions consists of asking students to write out the kinds of meanings they have acquired thus far in the unit.
• Cooperative and collaborative groups. Small groups allow for opportunities to explore ideas. Whole group discussions can be used to compare and contrast ideas from the smaller groups. All students will have the opportunity to play the role of the "presenter". Students became active participants in the sense that they were taught the importance of listening to one another and assist one another in processing their thoughts and arguments.
• Physical materials: The use of materials in mathematical tasks promote communication among students by serving as a natural stimulus for talking. In this particular unit, students used a board game to explore probability constructs.

B. Why Statistics and Probability?

This unit has been designed to bridge a connection between statistics and probability. There is perhaps no other branch in the mathematical sciences that is as important for ALL students , college bound or not, as the study of statistics and probability. A knowledge of both these areas are necessary in order for students to become intelligent consumers who can make critical and informed decisions. Even a cursory glance at newspapers shows the extent to which the language of statistics and probability have become a part of everyday life. This particular unit examines the question of why statistics and probability were created. Both subject areas can be viewed as tools that are necessary to predict a multitude of events (from game theory, to weather forecasts, stock market, medicine etc.). Moreover, statistics and probability offer students an opporuntunity to examine how it is that mathematics can portray a curriculum that is open to alternative solutions, personal opinions and arguments as well as defined to have a purpose for examination. Students should be given the opportunity to explore how statistics and probability can be used and misused and how misconceptions can lead to erroneous decision making. Examining a multitude of probabilities and statistical scenarios and drawing conclusions is a concept that perhaps a large number of adults have not mastered, nor have been granted the opportunity to explore. According to Huff: "So it is with much that you read and hear. Averages and relationships and trends and graphs are not always what they seem. There maybe more in them than meets the eye, and there may be a good deal less" (p. 8, 1982). Because of society's expanding use of data for prediction and decision making, it is important that students develop an understanding of the concepts and processes used in analyzing data (NCTM 1989, p. 105).

The study of statistics and probability arises from a need to make informed judgments about uncertain events and data around us (Moore, 1989). For example, to establish an appropriate inventory, the manager of a shoe store must decide which sizes and brands to restock. The weather bureau must issue reports that indicate the likelihood of rain, sunshine, hurricanes, tornadoes, and other conditions. Consumers are bombarded with often conflicting claims about product quality from which they use to make informed purchasing decisions. The medical arena conducts thousands upon thousands of experiments in order to assess the effectiveness of a drug or the likelihood of contracting certain diseases. According to Moore, "interest in teaching statistics is certainly due in part to recognition of the place that working with data plays in everyday life and in many occupations. Its is increasingly common to teach mathematical topics that are of direct use, rather than to select topics simply because they lead to later topics in mathematics. Statistics is such a topic (1989).

C. Summary of the unit

The following NCTM standards (p.105, p. 171, 1989) will be reflected in the statistical exploration of this unit. Students will explore statistics and probability in real world situations so that they can:

• collect, organize, and describe data;
• construct, read, and interpret tables, charts, and graphs;
• make inferences and convincing arguments that are based on data analysis;
• evaluate arguments that are based on data analysis;
• develop an appreciation for statistical methods as powerful means for decision making.
• appreciate the power of using a probability model by comparing experimental results with mathematical expectations;
• use experimental or theoretical probability, as appropriate, to represent and solve problems involving uncertainty;
• model situations by constructing a sample space to determine probabilities;
• make predictions that are based on experimental or theoretical probabilities.

This unit is comprised of 3 mathematical sections designed to predict the likelihood of events from both a probabilistic and statistical point of view. The following is a summary and motivation for each section in the unit.

SECTION I

Lesson 1, Introduction to Probability & Backgammon: The introduction to this unit involves allowing students to share their experience with luck, chance and probability. Moreover, students will be able to surface their misconceptions in the field of probability. They will be asked a set of open ended questions that focuses on students perception of how probability plays a role in variety of different scenarios (i.e. carnival games, the lottery, board games, sports, etc.). As students are engaged in sharing their experiences, the introduction of backgammon can surface. Students should be aware that the game will be used to explore if and how chance may play a role in winning the game. The game of Backgammon will serve as a real world context for developing a true understanding of probability. By the time students approach the 8th grade, they will have had experience playing with spinners and dice. The use of Backgammon will serve as a newer manipulative that students will not be so familiar with.

Lesson 2, Backgammon Scenario:

Students will be asked to take a closer look into a specific backgammon scenario. This investigation involves having students examine "what the best move is" at that particular point in the game. They are encouraged to provide the reasoning behind the solutions that they are providing. This lesson allows students to provide a hypothesis on what they think would be the safer move. The set up for this lesson involves students working first in small groups and than moving to presenting their hypothesis.

Lesson 3, Theoretical Probability of the Backgammon Scenario:

This lesson allows students to explore how they would check to see what the safest move really is, thus leading into theoretical probabilities. The development of the sample space and the formula for theoretical values will be arrived through the students' own exploration and former knowledge. Students will be expected to find the theoretical probabilities for each move in the scenario.

Lessons 4-5, Experimental Probability:

The next step is to examine how much truth is involved with the theoretical probabilities that have been determined. For example, what does 18/36 really mean and what would it take to believe in the truth of the value? Students will begin to test the theoretical probabilities and examine the law of large numbers. In addition, they will be able to contrive the formula for experimental probabilities as well as understand the role that such probabilities play in terms of testing the validity of theoretical values. Moreover, they will explore experimental probabilities in other contexts such as research, weather forecasts, and other unequally likely events. The goal is to have students understand that experimental probability can test as well as create theoretical probabilities. Students will also be able to identify where and to what extent are experimental probabilities apparent in our society.

SECTION II

Lesson 6, Conducting Mini Surveys:

Section II of this unit consists of allowing students to conduct their own survey of choice. Their results will be seen as "experimental" probabilities. Thus, probability will transcend from games of chance to its role in statistics and research. Through their own surveys, students will develop an understanding of causation, biases, and sampling techniques and how each can effect the outcome. They will realize, just as they did using the law of large numbers in backgammon, that the closer they arrive to a representative sample as well as a population count, the stronger their experimental results will become. The middle school years are probably the best age group to investigate surveys. The students are very interested in themselves and become even more interested when they are empowered to conduct a survey of their own choice. At an early age, students will begin to learn how to interpret results and critique or analyze conclusion drawn from not only their own surveys, but from research articles, opinion polls, consumer reports and commercials. Students will have developed a sense of how experimental results can indeed act as a tool to predict future results or expectations.

Lesson 7, Article Research and Commercial Analysis:

Students will be given real life articles to reflect and apply what they have learned thus far. The articles present interesting topics to analyze and critique.

SECTION III:

Lessons 8-10. Correlation:

This last section, will take students a step further in examining how statistics can act as a tool for prediction through the exploration of correlations. Students will be given a situation where a detective deduces that the suspect in a crime is tall due to a large footprint that was left in the mud. How close do you think this detectives' assumption is? Students are asked to back up their reasoning when trying to decide the degree of the relationship ( i.e. very strong, moderate, very weak). Students will collect and draw scatter plots of their classmates body measurements. They will learn to interpret scatter plots and to look for linear patterns.

Lessons 11-14, Predicting from the Best Fit Line & the Equation of a Line:

Students will use the classes' reach height/jump height data to examine the purpose of the best fit line in order to predict, describe, check, and compare relationships between two variables. Students will develop a procedural as well as a semantic meaning of the equation of the line (to include, slope and y-intercept). This last section will allow students the opportunity to learn these algebraic concepts through the data they have collected. Thus, meaning is given to each variable in the equation of the line in the context of their own graphs.

This particular unit was implemented to approximately 100, eighth grade students at a suburban school in western New York. The average class size consists of 25 students. The majority of the students will be tracked at the NYS regents level curriculum. However, just as any 8th grade classroom, the students exhibit a variety of mathematical capabilities.

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