Renee Tysinger, 7/18/16, Blog Entry #4

Before Mandelbrot, mathematicians believed that mathematics should be represented by straight lines. Mandelbrot on the other hand thought outside of the box and received much criticism for doing so. Rather than assume that the straight line attribute must be true, he asked if mathematics could be represented using rough curves instead. This "What If Not" strategy lead to great discoveries beyond that of the mathematics field, such as movie animation and even the fashion industry (NOVA, 2008).

Surprisingly Mandelbrot did not always enjoy mathematics. This all changed when one day in class he discovered that he was able to geometrically visualize algebraic problems with very little difficulty. This allowed him to see things that others could not. Mandelbrot eventually went to work for IBM. At the time IBM was having difficulty with signal noise across the telephone lines being used to transmit data. Mandelbrot graphed the data and noticed that there was a consistent pattern despite how far he zoomed in or out. This pattern reminded him of the monster curves that he learned about as a child (NOVA, 2008). The computer technology at IBM allowed him to run the pattern millions of times. This was something that no one had been able to do before.

I enjoyed learning about fractals and how they can be found in nature and other fields besides mathematics. For myself personally, the movie animation was by far the most interesting part of the entire video. I never realized how much math went into creating the graphics. Shortly after watching this video my husband and I watched a movie. I preceded to tell him about the fractals that had been used to create the mountains in the background. I don't think he was as amused as I was despite all of the math that he uses in the Navy haha!

Renee Tysinger, 7/17/16, Weekly Problem #4

What three consecutive even integers have a sum of 150? Can you find three consecutive odd integers that add up to 150?

Strategy #1

1st integer: x --------------------------- 48

2nd integer: x + 2 --------------------- 50

3rd integer: x + 4 ---------------------- 52

x +  (x + 2) +  (x + 4) = 150

3x + 6 = 150

      - 6      -6

----------------

3x = 144

---     ----

3         3

x = 48

Strategy #2

150/3 = 50 ----- middle number

3 integers --> 48, 50, 52

The first strategy that I implemented to find the three consecutive even integers involved an algebraic equation. I used an expression to represent each unknown even integer. I let x represent the first unknown integer. Since the second integer must be both even and consecutive, I let x + 2 represent it. the reason that I added 2 is because all even integers are 2 values apart. For example, 2 and 4 are even consecutive integers. For the third integer I used x + 4. This is because the third integer must also be even and consecutive in relation to the first 2 values. For instance, take the consecutive even integers 2, 4, 6. 4 is 2 away from the starting value and 6 is 4 away from the starting value. Next I added the three expressions together and set them equal to 150. After combining like terms and using inverse operations to isolate the variable, I discovered that the first integer would be 48. This means that the other 2 consecutive integers would be 50 and 52.

For the second strategy I used simple division to find the average, or the middle, of the three values. I divided 150 by 3 since we must divide the sum of the values by the number of values in order to determine the average. This resulted in an average of 50 for the middle value. The other two consecutive integers thus consists of the consecutive even integers to the left and right of 50. This strategy leads us to the same 3 numbers as the first strategy, 48, 50, and 52.

I was unable to find 3 consecutive odd integers with a sum of 150. The sum of 3 odd integers will be odd and 150 is an even number. To test this I chose 2 sets of 3 consecutive odd integers near the 3 consecutive even integers listed above. The first set is too small and the second set is too large.

47 + 49 + 51 = 147                           49 + 51 + 53 = 153

Renee Tysinger, 7/11/16, Blog Entry #3

     I have found that many of my students have trouble being able to start a problem. Once they are given a starting point they can usually work from there. I also have students that tend to get stuck in the middle of a problem and will give up. They lack the confidence and persistence to keep trying at that point. I think the four corners and a diamond organizer could help in both of these situations. I am not available, nor or other students, during an assessment or the EOG. The graphic organizer would help them to organize their thoughts, make connections, and identify missing information (Zollman, 2012). My favorite part about the graphic organizer is that it is not intended to be a linear activity. Our students do not see the different approaches that we may try when we first encounter a problem. They only see well thought out steps that appear to be easy for us because we are the teachers or maybe because they think we are smarter than them. If they are unable to solve problems with the same ease and linear fluidity they can become frustrated and even give up. The graphic organizer can help them move away from this mentality.
     I plan to use the graphic organizer for my unit challenge problems at the end of each unit. There are 3 levels of problems each worth an assigned number of points based on the difficulty. It does not matter which problem that they choose as long as they have earned at least 5 points by the end of the quarter. I will be requiring them to submit the four corners and a diamond graphic organizer with each problem. This will help them to be more successful at solving the problems (especially the more difficult ones) and it will allow me to see their thought process.

     Brown and Walter (2005) suggested a problem solving strategy that consisted of 4 levels; list the attributes of the given, give some alternative attributes using the "What If Not" strategy, pose questions, and analyze the results. Dr. Wiles used a version of this strategy while working on Fermat's Last Theorem. Since he knew that he could not possibly determine the number of elliptical curves, Dr. Wiles went a different route. He used level 2, the "What If Not" strategy, to try representing them as Galois representations instead.

     I am the most impressed by the difficulty of the math involved in proving Fermat's Last Theorem. I could not even begin to comprehend the math that they were implementing. I am also impressed that Dr. Wiles worked so tirelessly on his own on the same problem before reaching out for help. I feel like I would have gone searching for a second a pair of eyes much sooner. I cannot imagine being that persistent.

Renee Tysinger, 7/11/16, Weekly Problem #3

For the first two strategies I used proportions to find the number of missed questions. Since the student answered 85% of the questions correctly, they answered 15% of the questions incorrectly. Using proportions I was able to determine that 15% of 60 is 9 missed questions. For the second strategy I found the number of correct questions first. Once again using proportions, I determined that 85% of 60 is 51. The student therefore answered 51 out of the 60 questions correctly. I used subtraction to find the number of remaining questions that were missed.

For the third strategy I used an equation. I determined the number of correctly answered questions first. Once again this left the student with 9 missed questions remaining. The equation could have also been used to find the number of missed questions using 15% instead of 85%.

For the last strategy I determined that 10% of the 60 questions was 6 questions. I did this by moving the decimal point one place to the left. I need 8 sets of 10 to reach 80%. I multiplied 6 times 8 to get 48 questions. I still needed 5 more percent to reach 85% and I knew that 5% would be half of 10%. By dividing 6 by 2 this gave me 3 more correctly answered questions. By adding 48 and 3 the student gets a total of 51 correctly answered questions. Once again this leaves us with 9 incorrect questions out of the 60 questions on the test.

Renee Tysinger, 7/5/16, Weekly Problem #2

     The first strategy that I used was division. As a middle school math teacher, I teach basic operations with word problems. Division is a very fast and simple approach to this problem. The second strategy uses manipulatives to make groups of 8. This represents 1 pencil per student. Each student would get 3 pencils since there are 3 groups of 8. I feel that this strategy could be useful in elementary school to introduce basic operations and for the visual learners. The third strategy uses a double number line. I personally do not like this strategy but it is in the 7th grade curriculum. I have also seen it used in elementary school. The double number line is intended to introduce equivalent fractions and unit rates. The double number line begins with your starting information which in this case is 24 pencils and 8 students. It does not matter which unit goes on top. From there I continued to divide the number of pencils and the number of students by 2 until I reached the solution. All 3 strategies successfully led to the same solution.

Renee Tysinger, 7/5/16, Blog Entry #2

Questions asked in the video:
How do you accurately find longitude at sea? Can the stars, planets, and moons be used to accurately find the longitude while at sea? How can you make a clock that will be accurate at sea? How do you create a pendulum that is not effected by the motion of the ship or environmental factors? Could a pocket watch keep accurate time? (PBS - NOVA, 1998)

Discussion:
In the "The Art of Problem Posing" it mentions in Chapter 3 that some mathematical questions may go unanswered simply because our views are restricted (Brown and Walter, 2005). Narrow views can limit our thinking and make certain questions and approaches seem absurd and impossible. This proved to be the case in the video longitude video. At the time, clocks were not reliable enough to accurately tell time at sea and thus the astrologists felt that it was absurd to even consider them as an option. It seemed impossible that a clock could withstand the motion of the ship as well as the other environmental factors that occur throughout a voyage. John Harrison on the other hand disagreed with this view and did not let the current limitations of the clock affect his vision. He spent years creating and testing clocks to overcome the limitations and to create the first reliable clock at sea. Brown and Walter (2005) also mention that modifying a question will bring it into line. I try to remind my students to use this technique when they get stuck on a problem. Unlike John Harrison, many of them will give up if they do not find the answer immediately. John Harrison was having trouble creating a clock that would accurately tell time at sea. He tried several different models but something was missing. Rather than quit, or continue to create models that weren't just right, he decided to modify his question. He was no longer concerned with finding a pendulum to withstand being out at sea. This led him to the pocket watch and thus the answer to the longitude problem.

I think as teachers we can sometimes get stuck in one frame of mind as well. We might teach a particular method because we found it easy, that's how we were taught, or that's how it is presented to us in our curriculum guide. We then expect the students to use these same approaches on their assignments and assessments. Teachers may expect their students to see everything as they do and that is not always the case. I require that my students show all of their work on their assessments not only so I can see that they didn't just guess but so that I can follow their thought process. I have seen some very interesting approaches over the years that I wouldn't have ever considered. I do not take off points if they approach the problem differently. I try to make a point to be open minded but I know that not all teachers are like that.

Exploring Quadrilaterals:
After reading through the article by Richardson, Reynolds, and Schwartz, I considered how I would have approached this problem. I would start the problem using dot paper. I would use dot paper because it is more helpful to have a visual representation in front of me. I would chose the dot paper over the geoboards because I could use different colors to represent the different quadrilaterals. Eventually I would get tired of guessing and testing my way through concrete examples (I truly despise geometry) and I would attempt to find a general rule to identify all of the quadrilaterals. From there I would have moved on to finding the area and perimeter of the quadrilaterals. (I love formulas!) Much like the work in Figure 6 (Richardson, Reynolds, and Schwartz, 2012), I would have to use the Pythagorean Theorem to find some of the exact side lengths.

References:
Brown, S. & Walter, M. (2005). The art of problem posing (3rd ed). New Jersey: Lawrence Erlbaum Associates, Inc.

PBS - NOVA. (1998, October 6). Lost at Sea - The Search for Longitude. Retrieved from http://www.pbs.org/we have/nova/transcripts/2511longitude.html

Richardson, K., Reynolds, A., & Schwartz, C. (2012). Exploring quadrilaterals through flexible approaches. Mathematics Teacher, 106(4), 288-294.

Renee Tysinger, 6/27/2016, Blog Entry #1

As I read through the introduction I found myself agreeing with the authors on a few different occasions. The first is that problem posing enables a deeper understanding (Brown and Walter, 2005). I constantly encourage my students to ask questions after reading through a problem, while solving a problem, and after  solving a problem. I feel that it is important for them to understand how a procedure works and  whether or not their solution makes sense. I also feel that it is important for them to self monitor their progress. All of these things are done through problem posing. I have a list in my classroom of possible questions to ask that we add to throughout the year as a class. I remind the students that this list is only to be used as a reference for there is no such thing as only one right question. Which leads me to the next point that I align with, that students fear they will not be able to come up with the right  answer (Brown and Walter, 2005). I think this is especially true in math. Unfortunately there is not enough time to teach all of the different approaches to solving a particular math problem. I teach my students a method or two but I also allow them to share their own methods. If their method is consistent and brings them to the correct solution then I will give them full credit. I believe this helps to discourage the "right way" syndrome that the authors mention in the introduction.

Reference:
Brown, S. & Walter, M. (2005). The art of problem posing (3rd ed). New Jersey: Lawrence Erlbaum Associates, Inc.

Renee Tysinger, 6/27/2016, Weekly Problem #1

Proportions are a large part of the 7th grade Math curriculum and thus why it was my first strategy to solve the above problem. The size of the car does not change throughout the problem which allows the use of equivalent ratios to find the missing quantity. This is of course not the only way to solve this problem. I have listed 3 other strategies that are common among my students. The second strategy also uses equivalent ratios by multiplying the denominator and numerator by the same number. Since 16 must be multiplied by 4 to get 64 then we must multiply 6 by 4 as well. This strategy results in the same solution as my first strategy.

The third and fourth strategies use unit rates to find the missing quantity. The third strategy determines that .375 of the toy car takes up 1 inch of the row. We can determine how many cars take up 64 inches of a row using multiplication. The fourth strategy determines that 1 car takes up 2 and 2/3 inches. We can find how many times 2 and 2/3 fits into 64 using division. Just like the previous 2 strategies, the solution is 24 cars in a row that is 64 inches long.