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In light of the magnitude of this unit, the 8th grades students certainly went above and beyond the usually "8th-grade level" material. Some of the math concepts are normally taught at the 9th-10th grade level. From a cognitive view point, students became experienced in thinking about data around us at an intuitive young adult level.

In the probability section students were able to view probability not only in situations that involve chance (games, lottery, etc.), but also through a much broader picture (sport predictions, medicine, stock market, consumer ads etc.,). The students displayed some interesting thoughts in understanding the transition between experimental and theoretical probabilities. For example, approximately 70% of the students were able to conclude that although theoretical probabilities stem from a sample space, they are also created by a continuous number of trials. Some students had explained that if someone had been playing darts for quite some time and are aware of how "good" they are. They will tend to use their scores has an average of their ability. This average which is only based on practice, is almost viewed as a theoretical, even though their performance will fluctuate. I thought this to be very insightful in the sense that the whole point of experimental probabilities is that even though they may not be proven or hold much mathematical weight, they indeed often times are thought of as "theoretical". In many circumstances we do not have anything to rely on other than experimental results.

The notion of chance and luck in the first part of this unit was a bit of a challenge. In the introduction the students really tended to related the two as one. Most students had some kind of understanding that probability tells you the "chance" of getting something. However, they never saw chance as a mathematical arena. In other words although you could find out your chances of winning, actually winning, is a matter of luck beyond your control. The backgammon scenario seemed to have brought the concept of probability into light. Approximately 75% of the students began to realize another way to view "the luck of the dice". They began to believe that really analyzing your chances in any game could actually act as a strategy. In other words, once the die is rolled, the values can be used in your favor. Moreover, students also realized that although one can determine the likelihood of a certain event occurring, sometimes taking a chance exactly means what it is intended to mean-go for it!

The law of large numbers brought out some interesting insight as well. Students saw that in board games a player will only get one chance or one shot at something. Thus, even though the probability of an event happening could be something like 1/18, the likelihood of that particular event happening in one shot, really is not worth the worry. Therefore as they began exploring the law of large numbers, approximately 65% of the students had a handle on the fact that it would take a certain number of trials to have an outcome occur. Some students (about 30%) were able to take the experimental probabilities that their classmates got and figure out the average experimental probability-which in fact came fairly close to the theoretical. About half of the students caught on to the fact that certain experimental probabilities would be slightly off due to the fact that every student rolls the die differently. Thus, after performing the experiments, approximately 80% of the students did find faith in the theoretical values. The students not only learned that the more trials one performs the closure they arrive at the theoretical value but more importantly they were able to think about what this means in a given context, such as the game of backgammon. Through the experimentation they were able to conclude that if they could calculate a theoretical probability, it could be used to see what would happen over the long run, without having to rely on experimentation alone.

Approximately 85% of the students were able to deduce that though some events may not have theoretical probabilities, if experiments were carried out consistently, in general they would know what would happen. For example, in the coke can situation, the students knew more often than not it would land on its side, regardless of the fact that there is no theoretical claim behind it. By the thumbtack discussion, they finally realized that to say a thumbtack has a 50/50 chance of landing on its side vs. top actually does not make sense since they do not know how often each would actually happen. However, they knew that due to the mechanics of a thumbtack, it would probably land on one side allot more often than the other. Moreover, they quickly were able to come up with many situations that only hold experimental values. This realization, I do not think happens in the adult population. In other words experimental probabilities are every where.

The survey section of this unit was a remarkable transition from experimental probabilities. Students had the practical experience of understanding exactly why their results were viewed as "experimental". Every survey presenter was more than prepared to discuss how their results would change if geography, population, sample size were varied. They also realized that many of their questions could not represent the general norm because their is so much room for variance. In other words, asking people in New York what their favorite sport is could not hold the same for people in Canada. Students were extremely proud of their work. So proud, that they did not want me to take their work with me! I think what students really felt was that this was something they did on "their own". More over, students understood how their surveys played a role in the bigger picture. For example a question that asks what your favorite kind of music is, could very well target the radio stations, music agents as well as the music industry on what kinds of records are going to sell. They were able to take their survey a step further to discuss some things they would need to do in order to make their results more significant. When I began student teaching, I told the student that you only know something if you can teach it. Well, during the survey presentations the students were able to almost assume the role of the teacher and ask some interesting questions about the data collection and the conclusions that were discussed. Almost every student saw the importance of interpreting the results in the context of who was surveyed and how the survey was performed. This came to light in the discussion of the articles in Appendix L. Interestingly enough, during the articles discussion, one class began talking about the Trident commercial: 4 out of 5 dentist recommend Trident chewing gum to their patients who chew gum. At that age level, students know enough not to believe commercials. However, they really never took an in depth look at how certain commercial can be faulty. Students were able to deduce that 4/5 means that 80% of dentist will recommend Trident. However, they knew to ask, really where 4/5 came from and exactly how many patients discuss with their dentist the kind of gum they chew. Thus, they applied their learning to everyday events. Yet, how many really take the time to understand how such statistics can be faulty.

The correlation unit was a little more mathematically demanding for the students. I knew with such young students, the only way they could get a handle on this section would be to focus on the meaning of the material. Though the students may not have mastered the material as expected at a 9th or 10th grade level, at least 80% of them have at least developed some important concepts about the best fit line, the equation of the line, and graph prediction. First an foremost at this age level, students do not have a large degree of experience with equations. Thus, when the investigation of predicting from the equation of a line began, I asked students to think about such an equation as a way of writing a picture. Students viewed it as a blue print for what their line looks like. Having begun this exploration with this view, the students knew that breaking y = m(x) + b piece by piece would allow them to understand what each variable really meant.

Defining slope came from what students saw it as: "steepness". As I began drawing the triangles to show what slope meant, about 40% interpreted it as how fast something was moving up or down a line. Now, even though this is not the mathematical description of slope per say, the students had the right idea of what slope means to a line. It is in fact a rate of change. I also required them to define slope according to the data. Though they were unable to come up with this on their own (since it is new material), they were very interested at what this value meant when looking at a graph. Most students will learn slope as another formula to use when graphing a line. However, these students learned it not only as a process but as a defined purpose. As student gained more experience with finding slope approximately 60% of them realized that one could choose any two points on the line to find the slope. In fact, I thought I would literally have to teach them that concept but they were quicker than me.

The y-intercept was a very interesting experience for the students as well. At first finding the place where the line crossed the y-axis seemed easy. However, what happens when the line does not reach the y-axis? Many students (about 75%) ended up automatically extended the line. When I asked them why they did this they explained that it seemed like the right thing to do. Upon further discussion, I realized that students saw a graph as a continuous grid. In other words they explained to me that although we only took reach heights and jump heights of our class, we can still assume that taking heights of another younger or older group can also be included in the graph. In other words, they saw that the data can be taken of anyone and plotted on the very graph they were looking at-regardless of where the best fit line was drawn. Upon discovering this they also understood that it is a good idea to extend the line whenever possible to predict further data. Thus, sometimes you do not need to use the equation of the line in order to predict.

When I introduced the best fit line, I asked students to think of it as a way of capturing your data. Throughout the next couple of lessons, I would often quiz the students and ask them to tell me the purpose of the best fit line. What is so interesting is that the students did not use my words per say. Rather, approximately 75% of the students saw the best fit line as the average of the points. Now, I allowed students to take on this view. However, I and my cooperating teacher sat down and thought about what the students were gathering about the best fit line. We had realized that the best fit line is actually capturing where the average amount of points are on the scatter plot. Moreover, at a higher math level, the best fit line actually shows the average distance points are away from a line! This was an amazing analysis on the students behalf.

From a graphical interpretation, approximately 80% of the students were able to understand their own point on the scatter plot as it relates to the best fit line. They knew that if their own point was above or below the line, it was because they were taller or shorter than the average classmate. They also saw how some short people had a high jump height, and where their points were located from the majority of the points (outliers). This was an important insight for the students because they finally understood how "the exception to the rule" fits in with correlations. In other words, the students are able to explain that although you have some short people who can jump high, there is a general trend that seems to say the taller you are the higher you can jump.

The more practice they were given piecing everything together the more they could understand how to check to see if their answers made sense. This is a very valuable and important point to mention. Thus far, and especially in their future, students will use the idea of checking their answers by performing substitution. However, in correlations, checking the answer means asking oneself does it make sense? Approximately 65% of the students were able to arrive at an answer and than reflect on their graph to see if indeed the answer they got made sense with their original data.

Due to the mathematical content, many students were having problems with the basic kinds of skills that are necessary in mastering this material (i.e. solving for an equation, understanding that a/b means that (a) will go in the box, while (b) is your divisor, and remembering which variable go where in the equation). These kind of skills seemed to come together with practice. Aside from this, these 8th graders gained some amazing insight into material that they will get into once again next year. Their caliber of understanding is exactly what will make it easier for them next year.

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