The textbook readings for this module begins the overview of probability. I believe the easiest way to introduce students to the idea of probability would be to use the coin example given in the text (pg. 87). If you flip a coin, there are two choices that the coin can land on (heads or tails). However, the coin can only land on one of the choices at a time. Therefore the coin can either land on heads or tails one time out of two possible outcomes (1/2).
For problem A5 part C, I believe that it would be possible to increase your skills at the game push penny that was used for an example. I understand that you will never be able to exactly know where your penny will land; however, in my opinion you would be able to determine how much force to use when pushing the penny which would increase your chances. Hope, do you agree or disagree that you could acquire a greater game-playing skill for this push penny?
I thought problem B2 was very interesting and brought up a misconception many people have. “Suppose you toss a fair coin three times, and the coin comes up as heads all three times. What is the probability that the fourth toss will be tails?” I believe our heads want to say that there is a small chance that the coin will come up tails because the coin has landed on heads all three times, when in reality, the coin still has a ½ chance of coming up tails.
I also thought that the tree diagram in part C was an interesting way to map your data. I don’t think that I’ve ever seen this done before. I thought it was a great way to see your outcome of tossing coins.
A Whale of a Tale Article:
Below is my example of a probability line chart.
- Ms. Kincaid wanted the students to make predictions about their experiment on the basis of mathematical probability. Discuss preconceptions that students exhibited about tossing dice even after discussing the mathematical probability. Discuss the instructional implications of dealing with these preconceptions.
-Ms. Kincaid started the video by asking if students remembered what mathematical probability was. After she made sure the students understood and had no misconceptions, she asked students what were some possible outcomes they could have if they rolled 2 dice. Ms. Kincaid let students raise their hands and tell her numbers as she wrote the numbers down on the blackboard. Once the students had all of the possible outcomes written down, they discussed different ways they could roll the dice. The students write down all the possible outcomes that could lead to the numbers they had already written down. This helps the students become very aware and understanding of the experiment.
- Were these students too young to discuss mathematical probability? What evidence did you observe that leads you to believe that students did or did not grasp the difference between mathematical probability and experimental probability? At what age should probability be discussed?
-The students were not too young to discuss mathematical probability. They were very smart and completely understood what was happening to their experiment. I loved the comment one student made after observing the data collected that 7 would be the most likely outcome “because it has the most ways you can make it.” I believe that it is important to start discussing data with every age, even in kindergarten. Obviously you will not be able to go very far with the topic, but all children can really catch on the things that are more likely than others.
- The teacher asked the students, “What can you say about the data we collected as a group?” and “What can you say mathematically?” How did the phrasing of these two questions affect the students’ reasoning?
-The phrasings changed the student’s answers by a great deal. The first student that was asked to say something about the data as a group described how the graph reminded him of a rocket. When the students answered what he could say about the data mathematically, his answer changed to 7 would be more likely based off of the chart.
- Why did Ms. Kincaid require each group of students to roll the dice thirty-six times? What are the advantages and disadvantages of rolling this number of times?
-I am honestly not sure why the number 36 was picked for the number of rolls. (Hope did you catch why?) I think that maybe the number corresponded with the groups and added to a nice even number that would be easy to view.
- Comment on the collaboration among the students as they conducted the experiment. Give evidence that students either worked together as a group or worked as individual.
-The students worked together in groups based on their seating arrangements. One person was in charge of the dice, one person was in charge of recording, etc.
- Why do you think Ms. Kincaid assigned roles to each group member? What effect did this practice have on the students? How does assigning roles facilitate collaboration among the group members?
-I believe Ms. Kincaid assigned roles to each group member to help keep the experiment organized and help each student pay attention to detail on their specific job. By students having a certain job, they will be required to keep track of their job, but also to explain their information to their group members which helps with communication skills and working together.
- Describe the types of questions that Ms. Kincaid asked the students in the individual groups. How did this questioning further student understanding and learning?
-I liked that Ms. Kincaid asked students questions to figure out the possible outcomes and which number they believed would be more likely before they completed the experiment. This really helped the students to understand what would be happening during the experiment and let them test whether or not their predictions were correct.
- Why did Ms. Kincaid let each group decide how to record the data rather than giving groups a recording sheet that was already organized? When would it be appropriate to give students an organized recording sheet? Discuss the advantages and disadvantages of allowing students to create their own recording plans.
-I believe that Ms. Kincaid let each group decided how to record their own data so she could see the different ways students knew how to record data already. I believe it would be more appropriate to give students an organized recording sheet when they are older and collecting data on their own and not in groups. I think it was an advantage to have students make their own because you could really see how they were thinking and quickly observe the students that might need more help on recording data. One of the disadvantages would be that the student’s data could become very unorganized and hard to read.
For further consideration….
Knowing what you now know about probability concepts in the elementary school, how will you ensure that your students have the background to be successful with these concepts in middle school?
-I will begin introducing probability with fun activities, such as the dice roll video, and making sure students have a good understanding of things that are more or less likely to happen.