Module 7

Hey Jess!
Hope you're having a good week! What did you think of this week's module? I thought it was a good review and a pretty fun way to remember probability! I got some great ideas for teaching and discussing probability in the future, especially the video about rolling dice.

Textbook Reading
While students can figure out theoretical probability before experimenting, it can be neat to have them experiment first! They can then compare their results, the experimental probability, to the theoretical probability. This way, they can make their own predictions beforehand and they will have no prior expectation as to what will happen; the comparison afterward can serve as more of a reflection. Do you think it is beneficial to still find the theoretical probability after completing an experiment if they did not find it before?

On the Illuminations website, the experiment I explored was "Pandemics: How Are Viruses Spread?" Essentially, the students will manipulate different variables to determine the spread of a virus. As they manipulate these 4 variables, graphs will appear to show the growth of the virus with their data. It begins with one "person" being infected, and based on the variables entered, the disease will either spread or die. There is a great link to explore vocabulary and related topics to viruses, which connects this activity to both science and language arts and helps the students learn more about the activity they have completed. Virtual experiments are fun because the students are able to experiment with things that they never normally would be able to. Obviously, the students would never really be able to manipulate the spread of a virus, so this is something that they can see on the computer and watch how it would hypothetically effect the world around them if it were to happen. It is a fun way for them to learn and apply their knowledge to activities they would never be able to see at hand in the classroom. The only downside I can really see to virtual experimenting is that technology is not perfect, and some things may be a bit confusing and take a little longer to navigate at first, or glitches may occur. Are there more negative sides to virtual experiments that you see?

A Whale of a Tale Article

First off, I loved this article! It was a great way to integrate math and literature by using a book to base this chart, and it seemed like a really fun and inclusive activity for the students. Below is an image of my probability line. I chose to base mine on me and my dogs, which you know is pretty typical of the both of us. I know this is a little silly, but it does demonstrate a probability line!

Dice Toss

• 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.
• The students discussed possible sums they can come up with rolling dice. When one student said that they could roll a one, they discussed why that would not be a possible sum. They then discussed the many different ways that each sum could be rolled and came up with all possible combinations. This led them to their preconceptions. They looked at which sum had the most possibilities and decided they those numbers had a higher probability of being rolled. Instructionally, this shows the teacher that the students understand the combinations created when rolling dice and it allows them to predict what will happen so they can reflect on their predictions afterwards. 
• 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?
• No, I do not believe these students were too young. They showed understanding in explaining their answers, like the one girl who said that 7 was most likely to be rolled because it had the most combinations. I do not think there is a set age to begin discussing probability, but I think that easing students into it early on will help make this an easier concept for them to learn. As a class, they went about this in a drawn out way writing each and every possible sum. Older students may not need to do this, but now these students get a visual representation of what is probable, and the actual dice rolling allows them to see that mathematical probability is not always what occurs. 
• 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 student's original answer was speaking strictly visually about the shape that was made, but as soon as she asked about the data mathematically, he was able to use vocabulary he knows about probability and tell her which item is more likely and why. This shows that they can not only make a connection to the image of the data, saying it looks like a rocket, but they can also interpret what it means for the probability of the dice and this shows their understanding of the project as a whole. 
• 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?
• There are exactly 36 combinations that can be rolled when rolling dice. Rolling it 36 times will allow them to compare to all of these possible combinations, since there is an equal chance of getting each combination.
• 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 individuals.
• These students definitely worked together as a group, as each of them had a specific job. Their teacher sat with them and observed the different ways their group was collecting data, but everyone in each group was participating in either rolling the dice or collecting the data. 
• 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?
• Assigning roles allows students to each feel that they are doing an essential part. No one will  over-power the others and no one will be left out. Without any one of them, the experiment could not be completed as needed and they know this, so they focus on doing well on their task at hand.
• Describe the types of questions that Ms. Kincaid asked the students in the individual groups. How did this questioning further student understanding and learning? 
• She mentioned to one girl that her recording was not very organized. This commentary encouraged the student to record her data in a better way. She asked them simple questions like "what did you find?" to allow them to interpret their results out loud, and "did you expect this to happen?" to see if the students could compare their results with the original mathematical probability. She also allows them to reflect on their past experiences with probability when asking if they are surprised.
• 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.
• In not assigning specific recording methods, she allowed them to work together and decide as a group which was the best method. While this may have led to messier results, it let the students reflect on what could have been a better way to record this data. If he had wanted all of the students to have the same recording method or if she wanted this to go a little faster, she could have given the a pre-organized sheet. While this took a little longer and was messier for some, it encouraged higher order thinking and decision making rather than teacher-led instruction. When they made the graph as a class, they were able to see that that specific method of data collection leads to a much neater and appealing visual that can be easily interpreted.
• Would you use something like this in your classroom? Do you think this activity could be modified to be an introduction to probability for younger students, or a harder probability task for older students?

For Further Consideration...

To ensure that my students gain the background knowledge of probability that will be needed when they enter middle school, I will plan for activities that can physically show them how probability works, engaging them and using manipulatives. Having them make predictions mathematically and then comparing their results will show me if they understand the basic concepts of probability. Asking open-ended questions, like "what patterns do you see?" and "what can you tell me about this data?" will give the students the chance to show me their knowledge on probability concepts, as it did in Ms. Kincaid's class in the video.