Blog Entry #7

Peters, S.A. (2010). Engaging with the art and science of statistics. Mathematics Teacher, 103(7), 496-503.

In the article, "Engaging With the Art and Science of Statistics," Susan A. Peter's main idea was to define what field of study the practice of statistics fit into. In her discussion, and as inferred by the title, she claimed that statistics does not merely use mathematics and science, but that it is an art as well. She mentioned that statistics involves mathematical techniques and theories that help to collect and analyze data. This would be evidence that it is a mathematical science. However, she also talked about how creating graphs and displays for the data that is collected requires an artul touch in order to represent what was found accurately.

Statistics definitely involves both mathematical and artistic techniques, however I do not believe that this singles out statistics from any other math field. All math concepts involve some sort of art procedure whether it be in graphs, pictures, story problems, etc. Therefore, while I agree with her argument that statistics is an artisic, mathematical science, I do not think that makes it unique from the other math fields.

Blog Entry #6

Goodman, Terry. (2010). Shooting free throws, probability, and the golden ratio. Mathematics Teacher,103(7), 482-487.

In Terry Goodman's article, "Shooting Free Throws, Probability, and the Golden Ratio," his main idea is to show the difference it makes in a child's learning when topics of interest are used as object lessons. In this particular situation he chose to teach probability and the golden ratio by explaining through free throws in a basketball game. He started out with a word problem where a certain player had playing statistics that showed she made free throws 60% of the time. If this player was in a situation to shoot a one-and-one free-throw (where if she makes it she gets to shoot again, but if she misses she stops) the instructor asked the students what the most likely outcome would be--scoring 0 points, 1 point, or 2 points. After predicting this through various procedures, the instructor continued to challenge them by switching up the given situation several times. Throughout the article he displayed the different procedures that children were using to solve these problems and expressed that they were soon coming up with their own ways of solving them rather than coming to him for ideas. By displaying the thinking process of the children in his class he showed how children really do have a desire to learn and it can be easily encouraged when topics of interest are intertwined in the concept material.

I feel that Goodman's topic is brought up often when discussing mathematical teaching procedures. Although it is a popular idea, I am not sure that it is practiced as often as it should be. I feel that teaching children mathematics with real world objects and activities that they enjoy and participate in often can really have a deep effect on how they learn the material. A topic of interest is a way for them to break the ice with the new concept and tackle problems in a way where they can use knowledge that they already have acquired. It is also a way for them to visualize the problems, and this is helpful because most children at that age are very visual learners. Although this article didn't bring up any huge issues in the math world, I believe that the author did make a good point in illustrating that "explorations such as this one have the potential to help students further develop their mahtematical understanding, skills, and problem solving."

Blog Entry #5

Teaching mathematics to students without first telling them the procedures or even the right answer can be very helpful to a child's education according to Mary Ann Warrington. This method of teaching gives students the opportunity to construct their own knowledge with what they have been taught previously as well as through heated mathematical debates. Warrington spoke about how this kind of teaching environment gives the students more confidence and willingness to take risks by solving problems with their own devices. She talked about how constructivism brings forth intellectual autonomy. When children are encouraged to think for themselves they will develop much more successfully.

Although teaching in a constructivist manner has many advantages, there are some disadvantages that should be looked at carefully. If a teacher sits passively in the classroom observing the students as they construct knowledge, they could start to form false methods or procedures. Warrington stated that the teacher must understand each child's thought process and be able to carefully determine when and how to direct them to deeper and higher levels of thinking. During her experiences in the classroom, for example, when the class was working through the problem 4 2/5 divided by 1/3, the majority of the class was agreeing with a false procedure. If the one student who had done it correctly hadn't spoken up or figured the correct process out, then the entire class would have been in agreement with an incorrect method. Although constructing knowledge is extremely effective if done correctly, there is no way to force the children to come up with the right methods or procedures on their own. If they cannot figure it out on their own than the teaching method is useless.

Blog Entry #4

In Learning as a Constructive Activity, Ernst von Glasersfeld speaks of constructing knowledge rather than acquiring or gaining knowledge. In his article he mentions that the knowledge we have is continually built upon by the experiences we have throughout our life. Each experience that confirms our past belief adds onto our "knowledge" and we continue with this sense of truth that we call reality. Our view of this truth and knowledge only changes once we hit a wall and have experiences that contradict what we previously believed. Since our knowledge can always be changing based on what we see as reality, von Glaserfeld considers knowledge a mere theory. Everyone has different experiences, therefore there may not be any knowledge that is entirely correct, but rather viable and capable enough to get us through life successfully.

To use constructivism to teach my students math principles successfully, I would need to find ways for them to experience math on their own. If I gave them examples of activities outside the classroom that they were invloved in (such as sports or dealing with money) and incorporated mathematical principles (such as geometry or algebra), then my students would be able to construct their knowledge based on what they previously knew about these same activities. I also think it would be helpful for them to come up with their own story problems based on their own experiences, that way they can find their own "truth" to the mathematical scenarios.

Blog Entry #3

One of the main points Erlwanger was trying to get across in his paper, "Benny's Conception of Rules and Answers in IPI Mathematics," was the necessity of teacher-student relationships. He wrote about the experience he had interviewing Benny, a sixth grader, who had serious issues in understanding the concepts of decimals and fractions. For several years his misconceptions went unnoticed because of the faulty IPI system which forced kids to learn independently with minimal teacher interaction. By learning everything on these particular computer programs, he was taught to focus on the answers rather than the mathematical processes and concepts. With no teacher to tell him what the meaning and reasons were behind the rules and patterns, he was forced to come up with his own.

I feel that his argument is valid today because teacher-student interaction is crucial to students' education. Although programs and individual work can be very helpful and sometimes even necessary, there must be a balance. If teachers actually work with the students they will be able to notice when students are having difficulties grasping certain concepts. Without this relationship, more and more students will end up like Benny, having false rules and meanings of mathematics coded in their heads.

Blog Entry #2

Richard Skemp's article, "Relational Understanding and Instrumental understanding," discusses the two types of understanding that are widely used throughout mathematics. Relational understanding is knowing the reasons behind the arithmetic. When one is given a problem and understands it relationally, then they know both what to do and why they are applying that specific process. One cannot have relational understanding unless they understand the problem instrumentally as well. Instrumental understanding is knowing the rules, arithmetic and equations used for specific math problems. Although students can get answers right by simply knowing the rules and understanding the problem instrumentally, they will not know the reasons in which those rules are applied. Learning instrumentally can be a much faster way to learn how to solve a problem, but in the long run it is much more difficult to maintain because all it is is memorizing rules. In contrast, relational understanding may take longer for students to grasp and extend the time spent on certain subjects. However, if relational understanding is exercised, students will carry that knowledge with them for much longer because they will have a conceptual foundation that they can build off of.

Blog Entry #1

1. Math. Everyone uses it, but how do you define it? I suppose it would be the attempt of man to form patterns from the numbers, structures and space that we use everyday. There are several forms of mathematics and they can be used for a variety of different purposes. Some may study pure mathematics while others choose to apply it as a science with physics and engineering. Whether we are counting money, building a house, or calculating the force that a spring gives off--we all use math.

2-3. I learn math best when I am given examples of specific problems that I will be dealing with. Once an explanation of the concepts are given, I need to work on problems on my own otherwise I won't understand them fully. I believe that my students would benefit from this method as well. Although working in groups can be helpful, I think it should only be done after each student has time to work out the problem in their own way. Every person works through things differently and I believe it is important to give them that opportunity before providing assistance. Once they have done this, then they can either discuss the problems with classmates or ask me questions that they have. I realize that every student learns in their own way and so questions can be answered specifically for different students' mindsets.

4. It is very helpful when math teachers show different approaches to solve problems and give the students helpful methods to remember the material. The math classes that I learned the most in were taught by teachers who were very interactive with the students and were always willing to work through any obstacle we came across.

5. I think the worst thing for math teachers to do is to give a broad description of the material and concepts and then leave the class alone. Although students should have time to work on the problems themselves, it is useless if the material provided wasn't helpful. My younger sister is in high school right now and constantly needs to call me for help because her teacher never teaches them properly. If an instructor doesn't get to know the students and how they learn things easiest, it can be very detrimental to their education.