<p>I truly believe that Constructivism is a solid belief and decent approach towards students’ opinions and attitudes. ‘"Constructivism" is a philosophical viewpoint on how the mind forms and modifies its understanding of reality.’ (UMPERG, 2006). At the same token I am also a firm believer in the Objectivism approach as I am a math teacher who needs to be the expert teacher rather than the facilitator. This split view might be frowned upon by extremists in each field but, it is my belief that in order to be a successful teacher, in any curriculum, we must embrace the positive qualities of each approach.</p><p>I am not saying a traditional model of knowledge is not as important but I find that when we allow science to become a part of our every day thinking we are able to let students own their own work. I’ve found that the traditional model is flawed in assuming that students can only make the concepts learned part of their epistemology through direct instruction (absolute knowledge). Because I am looking at the role of math in science, I also feel that the role of facilitator is just not enough for math. The epistemology of math is a gray area to me. What I mean by that is that students can experience their world by observations (i.e. senses and daily interaction lead them to their own epistemology). While with math, a student’s interaction with numbers is limited in their daily interactions. Of course finances is a daily occurrence but how many people think about the slope of a street while driving it? What about how a building uses geometric properties to sustain the weight of its infrastructure? Therefore, as a math educator I am building an epistemology, not trying to deconstruct and then rebuild one. Now, this isn’t to say I am deconstructing false math concepts students hold, because it’s sometimes amazing to discover what concepts students try to apply to new material.</p><p>Case in point, I felt very connected with the Physics work from MISEP Summer 1 now that I can use some of that experience to explain my world around me. I don’t think its some abstract intangible thing or beyond my realm. It’s also made me look forward to the Physics 2 course in the program. The “traditional approach” side of me took over when examples of the physics problems were worked out ad nausea by Larry, Bill, and Avi. I found by knowing the formulas (mathematically) I could understand the physics properties of the situations. Therefore, I was able to understand my world and be able to explain them at the same time. I didn’t feel like I would be rehashing material when it would be my time to explain it in a classroom setting, but instead I would be able to share my experiences. I also felt that since I have a connection to the material I will bring my emotional “excitement” to it. Students have the ability to sense this and find motivation in it, especially when I give reason (usually scientific but also real world everyday implications) for the math material they are learning.</p><p>I think we all found that learning in Larry’s class we had to suspend our beliefs and we could not grasp certain concepts as we were not so willing to let it go, this follows the pedagogical reasoning of constructivism:</p><p>Restructuring world views requires much effort. I suppose this is what attracts me to education so much, the allure of a difficult challenge. Telling students that they are wrong is one of the easiest ways to alienate them; not only are you telling them that they are wrong but everything most people have told them are wrong. This usually doesn’t sit well, so herein lies the challenge of restructuring our students’ minds without turning them off to the educational process altogether.</p><p>So what is science? This question has been and always will be debated as long as time has existed. Some people feel science is merely anything humans will explore; others feel that there is structure to this “exploration”. At any rate, I believe science is when humans try to understand their world by being able to observe and quantify physical evidence. Once again, this brings me back to embracing the traditional views of education because it is based on the more quantitative nature of looking at unknowns. It’s almost comical that it takes equations to help me understand the world when I could just look up and see the world in action. It seems that since I hold this self-awareness knowledge I am open to different types of students’ needs and ability to learn differently from each other.</p><p>The “why” is a different question that can be debated as well. Thomas Kuhn believes scientists are constantly attempting to create a paradigm shift; this is usually done for recognition (people can become VERY famous or VERY rich) or sometimes they are trying to disprove prior science they feel is incorrect. Kuhn says paradigms help scientific communities to bind their discipline to help the scientists to create avenues of inquiry, formulate questions, select methods with which to examine questions, define areas of relevance, and possibly establish/create meaning (Kuhn, 2006). However, this always boils down to using one area of knowledge (Mathematics) to explain another area of knowledge (Science). How is it that we knew how far away the earth was from the sun hundreds of years before we were even able to be in space? How is it that we were able to explain the circumference or depth of the earth without ever really measuring those things? And so, mathematics has always been a very crucial part of the learning of science. I myself have never been in a science course that did not use mathematics; I can’t say the same for English or Social Studies.</p><p>The real way to look at this is to ask how math education allows for students to better themselves. Will it help with critical thinking skills? Will it foster excitement about the sciences? Will it help them understand and evaluate current science news? I believe that if math education can do those things for students’ than it embraces exactly what is positive about math and science and can only be beneficial. I also feel that if we as teachers pick one view (strictly Constructivist versus Objectionist) than we are closing our minds to different students and their abilities. It might make more sense that using Objectivism might be a better tool for teaching critical thinking skills since math is more concrete. On the other hand it doesn’t sound like Objectivism would really rally the troops for current news science.</p><p>I have found that when I thought about using math in science classes that I realized collaboration in my school was very low. I have never once teamed up with a science teacher to possibly co-teach or even align my standards with their standards. I feel like I have a scientific mind because I am decent with numbers. I can recall what brought me to be an educator in the first place: Eleventh grade wasn’t really a time in my life that I felt like I had any real goals. My dad was pressuring me to go into the Air Force Academy because I made a comment when I was nine that I wanted to be a pilot. I even had a House of Representative member’s signature for my admission into Colorado Springs; the problem was, it was a nine year olds dream, not a sixteen year olds. I was eventually going to let my dad down, but it was even worse because I didn’t have any other goals to present. The thing about life is, goals sometimes present themselves. Sitting in my Elementary Functions class wasn’t always easy to do the last period of the day. So, being the typical student I was, I decided having a conversation with a classmate next to me (while sitting in the front row) would go unnoticed and be non-interruptive to class. At this point, Ms. Little took notice and didn’t take this lightly. I was told to come to the board to teach the material “since you know it so well”. I was stubborn and cocky, so I took the challenge. To my own amazement, I knew what I was doing! I’ve never thought math was my strong point. It was just something I did in my studies because its always been there. I actually had other classmates asking me questions and I was answering them. Ms. Little was amazed that I had taught the material and some kids who wouldn’t pay attention in her class had perked up to ask me questions. I felt like light bulbs were going on… I became proud of something I did in high school. At that point, I had something to replace Colorado Springs with.</p><p>I decided it was really easy for me to get my general education master’s degree, but it wouldn’t make me a better teacher for it. I needed something that challenged me and allowed me to bring something back to my students. I was really bored myself in high school and if it wasn’t a challenge it wasn’t worth learning. I also felt that too many students’ in my classes would always ask “Why are we learning this? Why is this important?” and I often didn’t have the answers to it. I felt if I had a science education background I would be able to make the bridge connection that my students were not getting. I also felt that if I could integrate science with math than I was able to justify my master’s degree. I wasn’t going through the motions for sake of the state mandates but for sake of the essence of the educational process.</p><p>My teaching unit is going to be on Theoretical Probability. The process of computing future events based on logic is a skill often used in science. This would be on task with the Core Curriculum in the School District of Philadelphia. The 9th grade Algebra I course allows for an entire Chapter on Probability and this fits in nicely with science connections. I think that the concept is challenging as it tests some of my students weaknesses in fractions and counting principles.</p><p>For the assessment of Theoretical Probability I will have to use benchmark testing scores and Terra Nova prep materials as required by my school. However, I use for my own purposes a quiz that emphasizes the important skills that are useful for the students later. This is a, self made, ten question assessment that not only allows students to show development but allows me as a teacher to see areas of weakness. I also don’t have to worry about Benchmark turnaround times which have been less than expedient. Other than those constraints I felt that I would like to avoid a multiple choice exam as usual and attempt to reach out to the students in a more hands on approach. Students are going to create a “game show” concept in which we will use Theoretical Probability to help us “win” the game show they’ve created. It also moves away from Objectivism and more towards my Constructivist approach in the math classroom. The rubric for the game show will actually be created by the students and what they would think is important in the process of helping us win at these “games of luck”.</p><p>UMPERG (2007). University of Massachusetts Physics Education Research Group Website. Retrieved December 21, 2006. http://umperg.physics.umass.edu/topics/constructivism</p><p>Kuhn, Thomas S.. (2007). In Encyclopædia Britannica. Retrieved December 21, 2006, from Encyclopædia Britannica Online: http://www.britannica.com/eb/article-9002756 Lesson Plans: Theoretical Probability</p><p>Objective(s): </p><p> Students will be able to demonstrate their ability to calculate Theoretical Probability for real situations.  Students will be able to define probability related vocabulary.</p><p>Standard(s):</p><p> S.D.P. - Literacy, Science and Mathematics 2006-2007 Course of Study High School, Page 6.  S.D.P. – Alegebra I - Timeline Curriculum Guide, Page 28.  P.D.E. – Standards 2.1A, 2.7B, 2.7D, 2.7E.</p><p>Background:</p><p>Theoretical probability can sometimes be a tricky concept for students. My experience with students and fractions is that they are truly afraid of them. This makes any type of probability a somewhat daunting task. With this activity, students are going to be engaged in a fun activity so they can “forget” about the fraction woes and focus on the concept of probability.</p><p>The connection of probability to math and science is a strong one. I will be using the book “Empire of Chance” by Gigerenzer , Daston, and Swijtink. This book helps tell the “story” of probability in science over the past 300 years. In this book we find that the quantification of science helped create several paradigm shifts within the science community several times over.</p><p>Classroom Activity:</p><p>Discussion of the number cube (die), coin, spinner, bags, deck of cards, and other items of chance will be the foundation of the topic. Once students are well versed in the description of those items (i.e. a deck of cards has four suites, 12 face cards, etc.) than we can start thinking about our chances of picking it. This leads to the direct instruction portion. 1. Start simple. Talk about using a coin and what are my chances of getting heads and tails. Define probability as the ratio of favorable outcome OVER total outcomes. Show students that the sum of all possible outcomes is equal to 1. Define sample space as the set of all possible outcomes (i.e. {Heads, Tails}).</p><p>2. Continue with more complex situations, like a single die. Figure out a probability table for two die. What does our sample space look like? {1, 2, 3, 4, 5, 6}</p><p>3. What happens with multiple dice and coins? If I have two coins, what is the size of my sample space? And what does it look like? (Here students begin to see that the number of outcomes for one item times the number of outcomes for another is my sample space size). 2 possible out comes for one coin and then 2 more for the second coin is 4 possible outcomes for 2 coins tossed in a row. {HH, HT, TH, TT}. A die and a coin has 12 possible outcomes, 6 sides on a die times 2 sides of a coin equals 12 possible outcomes {1H, 2H, 3H, 4H, 5H, 6H, 1T, 2T, 3T, 4T, 5T, 6T}. Using Tree diagrams to organize our outcomes is most helpful.</p><p>4. Students should practice various problems until they have the concept of sample space, sample size, and how to create a fraction that depicts the possible outcomes of a given situation. This unit should take about two days and has various handout worksheets for classwork and homework.</p><p>5. After the second day, the unit quiz is given for understanding of the concept. On this day, students will be given the unit group project. Students will be shown a clip from the television show, Deal or No Deal. It is important to show the students that people are making a decision based on their odds (which we defined as Theoretical Probability). The project is for them to create a “game show” or to reenact one they have seen before. The major difference is, they must explain to the contestant through every step what their odds are of winning or moving on through their particular game. I will also hand out the rubric that details the point values of their grade. Students are already assigned groups in the beginning of the year based on Cooperative Learning Strategies and student achievement levels.</p><p>Reflection:</p><p>Since I am limited in videotaping in my school, I have included a brief description of several of the groups’ presentations to the class. Overall, we had a great time with these presentations and I will definitely be doing this activity again. Next time, I may give the unit quiz after we do the projects or give a post quiz to see if student understanding has increased. I may also include another component of the project, where the student’s have to write a reflection on how they came up with their idea, issues they faced, and other meta-cognitive questions.</p><p>Group I: This group did a game show called “Is It Worth A Spin?”, the idea behind the show was for a large wheel to be placed in front of the room. The wheel has 26 slots on it. Each slot has a color placed on it (various colors red, green, blue, yellow, orange, black, white and brown) and when we first start the game, each slot has either a blue, red, yellow, or brown placed in it. Once a contestant is picked, they are told they are playing for the grand prize of a 100 grand! The rules were simple; students were told that they need to figure out the theoretical probability of spinning a certain color. If they could figure it out, then they were allowed a spin. If the spin landed on a primary color, than they were still in the game. If they landed on anything else, it wasn’t worth a spin. Students made it to certain levels and they were awarded minor prizes (bag of M&M’s, Now-and-Laters, etc). 5 spins was all it took to win a 100 grand, and the panel of judges were there to confirm with a whiteboard tablet that the probability was correct or incorrect. After each spin more non-primary colors were added to the wheel. This group handled about four students in one period. Their math calculations were spot on and they handled the flow of the game with precision. They lost some points for not giving the probability of their contestants’ chances for each step until the last spin. Only one other student made it to the final spin and did win… a 100 grand candy bar. This was very well planned and was a lot of fun for the entire class.</p><p>Group IV: This group did a game show based on 1 vs. 100, where there were trivia questions and the whole class played. Everyone was given a marker board and told that they were part of the show. The students were told that they need to answer questions on the board with either A, B, or C. The contestant was told he would write his answer on the board. It seemed very unprepared or unrehearsed at times. Group members didn’t seem to know their roles or were being coached by other members. The contestant was told that he had 0/30 probability kids beat. This was incorrect, because that means he had zero chance of winning the game. When a contestant got the correct answer and students in the class had a wrong answer his “odds” of winning changed. For instance, if the contestant got the question correct and 8 students in the class got it wrong, than the “probability” changed to 8/30 (4/15). This is incorrect; instead the students were giving the progress of students knocked out. I let the game go a few times until some class members asked how this was probability. The group was given great feedback from the class and was also given pointers on how they could fix the concept, including a completely new format. The students were given a chance to represent in 3 days.</p><p>Group VI: The final group requested to go last which is odd because I told the class I usually mark the later ones with a “finer comb”. The group did almost a direct translation of Deal or No Deal called Bagged Lunch. The group set up 30 brown lunch bags and placed dollar amounts on pieces of paper on the inside. The outside had a number from 1-30 and they were placed on a series of 5 shelves. From the beginning this looked really good and I was very happy with the way kids were prepared. A student was brought up and asked to pick a lunch bag. They were told the amounts ranged from .01 cent to 2.00 dollars (enough to get a school lunch). Students were then told step to step what their odds of getting a cheese pretzel was (they cost 1.00 at school), for instance if their lunch bag held a dollar or over, they would have a 3/5 chance of getting a cheese pretzel. This was really fun and kids had prices of the food amounts and kept comparing their probability to those amounts. The children brought in a boom box to play show sound effects. They set up a cell phone system that when the offer was being made to “trade” their bagged lunch they would have the lunch lady call (another member in the group). There were a few mathematical errors; mainly they kept forgetting to reduce to lowest terms. Since the kids only brought in three dollars to play with, I decided to supply the rest of the money for a few more contestants. Overall, this was by far the most excellent presentation; it wasn’t original but didn’t have to be.</p><p>Theo. Probability Project Rubric</p><p>In your learning groups, you must perform a game show (original or a variation of one on TV) that uses probability. You must give the contestant his chances of winning in the game step by step. This was they can make an informed decision on whether to continue or not. The following rubric is a guideline of what I am looking for. There will be peer reflections at the end of you presentation and will be used by me to give you an overall grade. This is worth 45 points (2 QUIZZES!). If you have any questions while designing this presentation you should come to me right away.</p><p>10 points for organization: Does your presentation flow? Make sure you practice this before you do it. Does it make sense to the audience? Does everyone understand their role in the group? Does the audience understand what everyone in the group is doing?</p><p>15 points for presentation: How elaborate did you make it? Did you make it visually appealing? Did it feel like a game show? Did it spark excitement from classmates? Were people involved and interested?</p><p>20 points for correct math: Did you reduce all of the numbers properly? Did it take too long to calculate? Did the probability make sense? Was it relevant to the contestant winning?</p><p>Make sure you ask yourself all of these questions while practicing this project. Let mom, dad, or anyone else watch it. Let them give you feed back before you commit to this.</p>