Private Universe Project-Workshop 4:Workshop 4. Thinking Like a Mathematician

This video was really interesting to me for many reasons. It started off with a teacher introducing the Towers of Hanoi. It proved that you can teach children to learn to problem solve on their own by just giving them the opportunity to explore. I really enjoyed watching them figure out the patterns and how many years it would take to move a hundred disks. This allowed the students to gradually work up to the skill the teacher wants them to succeed at. Part II of the video was a high school math teacher. This teacher’s goal was to teach the students how to think independently. It reminded me of when I was in elementary school and my math teacher would just sit and do problems over and over. After reviewing the concept she would make us go up to the board and each student was required to do a problem on. If we get this problem wrong she would yell and embarrass us in front of our peers. Students would be afraid to be called up to the board. Instead of encouraging us to think on our own her focus was to teach to the books. If this student gave us the opportunity to think on our own like the teacher in the video we would have felt confident to go to the board. I really appreciated the fact that this teacher had an environment conducive to learning from one another.

Virtual Manipulatives-Color Chips gr, k-2

I used this manipulative game on one of the second grade students that I am with for an after school activity. She seemed to understand it and really enjoyed playing it. She was able to complete the computer questions and come up with her own. She really enjoyed coming up with her own because she can add as many big numbers as she can. I really liked this manipulative because students can visually see the pieces and can add them off the screen. Students will really benefit from this activity.

Private Universe Project-Workshop 3:Inventing Notations

After watching video 3, I began to reflect on myself as a teacher. I thought about how many times I expected a student to solve a problem using what I consider the “traditional” way. This video represented what students can do and different learning styles. It was focused on how children themselves can teach other classmates. They were asked to solve the 4 Topping Pizza problem. Students have so much to offer when they are given the freedom and opportunity to solve problems. The most important part was that the students had to identify and vocalize their individual techniques. I also liked that students were asked to prove how they knew they got the right answer. Lastly, I believe that when teachers teach to meet the interest of their students they are more likely to be effective.

Virtual Manipulatives-Complete the Pattern (gr.k-5)

This week I decided to use the virtual manipulative, complete the pattern. I really liked this one because of the color choices it used and because it varied the patterns. In addition, if the student got the pattern wrong it would tell them Oops! This isn't the pattern we had in mind. Change some your colors and try again. I like that it uses positive words so that the student does not get discouraged. Also, if you solve it correctly is says Good Job! The point of this manipulative is that so teachers can see whether or not students know what patterns are and whether or not a student can determine if something is a pattern or not.

Math lesson-Pattern Blocks

The math lesson I chose was the Pattern Blocks, Hexiamonds. This lesson is designed to teach students about what Hexiamonds means, how many different Hexiamonds they can make, and reinforce congruence using flips and rotations. The students will be using green triangle pattern blocks and a pattern block triangle paper.

Private Universe Project-Workshop 2: Are you Convinced?

I am a perfect example of someone not being able to justify how they solved a problem. Many times I can solve the problem but I cannot put into words the mathematical reasoning behind it. This video made me realize the importance of giving an explanation for the answer and knowing if I am sure that I have the final correct answer. As a teacher I often expect my students to do so but never realizing that it is something I struggle with myself. I really enjoyed the way each teacher went about solving the towers. I was able to compare it to how we did it in class on Monday and how each of us had a different way of trying to solve it. It is also interesting to see that students were able to figure out the patterns and were more likely to respond to how they got their answers. They were even able to continue using more and disks for their towers. I learned that it is important to challenge children and ask them after solving anything if they are sure they are correct? I think that will make them double check their work and will allow them to think with reasoning.

Virtual Manipulatives- Peg Puzzle (gr. 6-8) & Towers of Hanoi (gr.3-5)

Algebra gr. 6-8 Peg Puzzle
I used this virtual manipulative on my husband. I first started my giving him the directions and the goal of this assignment. The goal is to switch the pegs on the left with the pegs on the right by moving one at a time. I explained that you can not move backwards and you can not jump over more than one. He first started with 4 pegs. He completed this puzzle with no problem. He than tried the 6 peg puzzle. After the third move he got stuck. The best thing about the activity is that unlike the hands on one from class the virtual notifies you when you are stuck. It states," You're Stuck-none of the pegs are allowed to move." I feel that this is beneficial because instead of wasting time trying to figure out if you can move you can start over right away. After a few tries he was able to solve the 6 peg puzzle. So I challenged him to the 8 peg puzzle. He than realized that there was a pattern to the 4 and 6 peg puzzle he tried several more times to figure out the pattern so that when he begins the 8 peg he can move through it easily. He began to write down each move in order to see if it can be replicated if more pegs are added. When he started the 8 peg puzzle you can see that he was a little frustrated, he solved it once and than couldn't figure out how he did it. He tried over and over until he was comfortable to solve it and explain what the pattern was he used to solve it.

Algebra gr. 3-5 Towers of Hanoi
For the towers of Hanoi I decided to test it on a third grade student from my building. I chose this student because he was in my class during lunch. In the virtual manipulatives game its default is 3 disks. However, for practice I told the student to start off with 2 disks. I explained to the student the directions and the goal. The goal of the towers of Hanoi is to move a stack of disks from one peg to another in as few a moves as possible. I also explained the rules which are you can only move one disk at a time and that you can not stack the larger disk on the smaller one. The student than began to solve the 2 disks. He was able to do it on the first try. He was then notified that Congratulations! You moved the towers in as few moves as possible-3. The student was very happy to see that. He than tried 3 disks. He completed it but unfortunately he did not complete it using the least amount of moves. So he tried a few more times until he did. I was very proud of him. I than asked him if he thought there was a pattern? He said that he wanted to try a few more times so that he can see if there was one. After trying several more times he stated that the smallest peg moves the most out of all of them,and that it moved from the first spot to the last and than from the last spot to the middle and then from the middle to the first again and finally back to the last spot, also that the largest peg only moved once which was from the original spot to the last spot. Unfortunately, we did not have enough time to try 4 pegs.

Private Universe Project-Workshop 1: Following Children's Ideas in Mathematics

This video was very interesting. I liked the fact that as a grad student I was given the opportunity to do the same project in class and then watch a video of younger students doing the same thing. I was able to reflect and see how they thought or what approaches they took compared to mine. I also enjoyed the level of effort the students put into these projects. As an adult I often can have the easy way out, which is giving up but these students even tried when school was over. I can also relate to the students who forgot certain math lessons in certain grades and how the effective teachers and lessons they presented always stand with me. It also really made you see how kids are capable of thinking in complex ways and how they have to be given the opportunity to do so. When you give your students the chance to show you what they can do you will be shocked as to what they come up with. Whenever I plan a lesson, I will always try to give my students the opportunity to strengthen their mathematics problem solving skills. I believe that students learn best from hands on activities which will allow them to begin to think mathematically.