Private Universe Project in Mathematics: Thinking Like a Mathematician video 4

This video introduced a mathematician who demonstrated some of the real world uses for her skills. I also liked that she did not really enjoy mathematics until she entered the 9Th grade (Algebra) and the abstract concepts pulled her in.
The video then went on to classroom instruction and the students were introduced to the Towers of Hanoi. They began problem solving and were attempting to figure out how many moves it would take for 100 discs. They began small and worked up from a few discs to more and more discs. They began to identify patterns and this was the key for them to solve the problem because they were able to use the patterns to solve the problem. After figuring out how many moves it would take for 100 discs they were then tasked with determining how many seconds and how many years it would take to accomplish that task. Their ability to come up with and identify the patterns was pretty impressive and the students figured out it would take about 2 billion years to move 100 discs.
Next the video cut to a high school teacher and her methods of instructing students. She "encouraged her students to think." She did not force feed students answers but allowed them to investigate and problem solve. She also demonstrated to her students that it was acceptable to have different strategies to solve problems. She developed a community of students who accepted each others methods and strategies.
Finally, the video went back to the original core of students and demonstrated how they went back to previous problems and expanded upon them and how their strategies used to solve these problems evolved.

Color Tiles Lesson Plan

The lesson I chose for this week is Squares Within Squares. This was taken from the manipulative disc. The students are to build a 10 by 10 square and identify as many squares within the larger square as they can. I chose this because my students recently worked on a similar problem but on a smaller scale. They had to find how many square were within a series of three squares that were intersecting.
This lesson requires students to use patterns, problem solve and think beyond the obvious. Some square will be very easy to find while others will be much more difficult. It will also require the students to develop a strategy to record the squares they have found so they do not duplicate squares. I also believe this lesson could engage all the students in my class. It will entice the higher achieving students into solving the problem and also should not frustrate the lower achieving students.

Virtual Manipulatives Triominoes

When I first saw this manipulative it appeared to be an interesting puzzle or activity to try out. After beginning I noticed this did not present any problems or activities to work on. It was simply allowing me to connect triominoes and connect pieces for creating designs.
Triominoes are three sided peices with three different colors at each vertex. You can not connect pieces without the colors matching at the sides of the triominoes. The manipulative does allow you to flip, rotate and move the triomino. I was unsure of how this could be used with students but after working on it for a little while I thought that maybe students could be presented with a design and they would have to recreate it using the triominoes making sure they mathced sides so the pieces would connect (problem solving). Overall I really did not like this specific manipulative because I would have preferred it to present a problem or activity to be worked on.

Lesson Plan Choice

The lesson I will be using this week is the lesson named "Don't break the Wagon." This is a strategy game where the students will use a game board and pattern blocks. Their goal is to force the other player to place the last piece on the board and break the wagon. Students will learn to use strategic thinking and spatial reasoning.

Priveat Universe Project In Mathematics Video 3

Problems and Possibilities: Inventing Notations

This video demonstrated how giving students a chance to make their own mathematical notations without providing them with standard mathematical notations can be very surprising. They naturally develop their own system of documenting their possible outcomes. The teacher should be willing to discover the notions the students invent. Look to see how they represent their ideas and the teacher should also learning from the student's. We should take the time to listen to and view what they do. Each group within the video had a different way of reporting their findings. It was interesting to see a teacher express her concern when her students did not initially solve the problem using a list but actually used drawings and other representations She was initially intimidated because they did not solve the problem in the way she visioned they would. However, they still came up with valid answers using the drawings. Teachers must be in tune with what students are interested in and what they would like to do and how they would like to solve a problem. Our way is not the only way.
It was cool to see how some students used blocks to represent pizzas and others used lists. Some students used symbols inside circles and others just used abbreviations. I also like the fact that the researchers intentionally used two topping with the letter P to force students to think beyond just using the first letter of the topping. It really forced them to plan out their problem solving technique. The student who used the 1's and O's to make combinations of pizzas in a chart format was really amazing. He was also able to connect the pizza combinations with the 4 tower combinations.
These videos show that students do not need to be force fed mathematical concepts but given the freedom they can learn concepts on their own. It appears a much more natural way for students to learn and they can develop and discuss their own ideas. "We must give student the opportunity to show us" what they can do.

Virtual Manipulative Pythagorean Theorum

I chose the virtual manipulative geometry 6-8 proving the pythagorean theorum. In this I was provided 4 triangles and a square and had to fill in two different shapes. This word in turn prove the pyahtgorean theorum to be accurate. The first shape was a square (c) and the second shape was two squares of different sizes placed together (a and b). The four triangles and square that I was provided were the exact same size and would in turn fill in both of the shapes. The second shape (a and b) was much easier for me to figure out than the other shape (c). Shape c required manipulation and rotation of the square in order to fill it in. After some time I was able to figure it out and solve the problem. It was a really cool manipulative and I was recently working on this with my students and I really see this as a useful way to ensure the students understanding of the theorum.

Here are some of the patters my kids created.

This was nothing formal I had planned. I pulled out the pattern blocks from my bag while I was doing homework for another class. I looked up from my book and my kids had began making patterns. I grabbed the camera and took some pictures. The next day we sat down and made some more. My six year old insisted on fixing some of my designs because he felt that they were broken and my nine year old (Tyler) was much more patient and consumed with making designs.

Virtual Manipulatives Six Peg Puzzle

The six peg puzzle was another fun puzzle. Starting with four pegs was pretty basic and advancing up to six and eight required a little more problem solving. The main goal was to keep the pegs alternating until they reached the end of the board. Once that was done the puzzle became pretty basic. This is another great problem solving tool that is fun and intriguing. I think these would be even more beneficial if they were able to have actual manipulative's and not just the virtual one however, it is much easier to provide a classroom the website than numerous puzzles.

Virtual Manipulatives Towers of Hanoi

After explaining to my wife the directions and goal of Towers of Hanoi she began moving pieces. During the process she expressed frustration and felt at times it was impossible to solve. I believe she felt I was setting her up on a game that could not be solved. Once she felt comfortable that it was possible and a real game she immediately began working to solve the puzzle. From that point on she solved it very quickly with little frustration. She ended up solving the puzzle on her second turn in 15 moves. This game may initially be intimidating but once people get a grasp for it then becomes rather fun. It is a game I can definitely see my students enjoying.

Private Universe Project Video 2

In watching video 2 it was interesting to see how teachers approached the problem of the towers and the various strategies they used to solve the problem and justify their answers. It was thought provoking when the presenter stated that many teachers are very tense, nervous or even become angry with they way he presents his lessons. They are not prepared to justify their answers or verbalize their methods. He stated they are expecting to be told if they are right or wrong. He also noted the importance of having children express themselves and verbalize their thinking in class and how many kindergarten teachers are far more comfortable in his workshops than other teachers. Kindergarten classes tend to focus on collaboration and communication. As students advance in grades it becomes more of a listen to and respond to the teacher individually.
The students in the video were able to grasp the tower concept and appeared to be more open to justifying or proving their answers then the adults. However, the students had a harder time accepting the way in which their peers solved the problem. What was really amazing was how the students were able to advance their strategies beyond 4 and 5 towers and up to ten towers.