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Entry #7

Gilbert, M. J, & Coomes, J. (2010). What Mathematics do high school teachers need to know? Mathematics Teacher, 103(6), 418-423.

As the title implies, the article discusses what teachers really need to be able to do and know when teaching mathematics. According to Gilbert and Coomes, having a greater understanding of how and why students are doing specific tasks in a certain way is superior in importance to knowing how to do advanced level of mathematics. Responses from different students are provided in the article after being posed with the following proportion problem, “Lincoln Elementary School pairs second- and sixth-grade students as ‘study buddies.’ What is the ratio of second- to sixth-grade students if 2/3 of the second graders are paired with 3/4 of the sixth graders?” Each of the responses published were different, and not all of them were correct. The authors used this to display that it is far more important for an instructor to be able to see things through multiple different view points and be able to determine what is correct or incorrect and why that is so. According to the authors, being able to teach mathematics extends beyond being able to solve the specific problems. Teaching mathematics requires knowledge of how to solve problems but, it also, and most importantly, requires the knowledge and ability to recognize new ways of things or find errors and correct them.

I also believe that it is of greater importance to understand in depth the material to be taught rather than being able to do difficult upper level math. In the article, a teacher that they worked with was quoted essentially saying that she would have preferred to spend her time covering in depth what it is she would be teaching rather than taking upper level math classes. I too have had this issue tug at me. My high school did not offer anything above calculus 1 and statistics. I feel that since I will not being teaching linear algebra in high school and definitely not in middle school, that it is verging on waste time. It may be interesting but since I won’t be teaching in it, I am not sure that it is worth studying. I have found that throughout my entire life I have just done things in math without a sound understanding of what is being done. I agree that it is important so that I will be able to know why things are the way they are in mathematics. I would not consider myself a valuable educator if I could not answer questions of why we use a specific rule or procedure. Lastly, it is important to have the ability to understand approaches by different students so that they can be appropriately addressed. An approach that is successful could be useful in teaching in the future, and an approach that was unsuccessful needs to be corrected so that lasting misunderstandings are not created. It is more important to me to be able to teach and understand adequately what I will be teaching than it is to be able to do highly sophisticated mathematics.

Entry #6

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

The article written by Terry Goodman had a few main points that were closely intertwined. His main objective was to demonstrate that students often take the initiative and gain a desire to learn more, when a topic of interest is introduced. He was having students work on probability and decided to use basketball free throwing percentages to have his students calculate what probability certain theoretical shots would have. Goodman started the problem out very simply by asking his students if a given player had a free throw shooting percentage of 60% and had a one-and-one free-throw opportunity, what the probability of the player scoring 0, 1, or 2 points would be. After the students calculated those probabilities he continued to challenge them by asking them to create different probabilities in different situations. It came to the point that his students were suggesting to him what different things they could calculate and discover. Through telling the story that he experienced in his classroom Goodman wanted to demonstrate that students do have a desire to learn more. Goodman wanted his readers to realize that if you take something from the “real” world and draw the mathematic concepts out of it, students can be very interested and hard working.

In my opinion the article was not very controversial and the point he made is something that we are all fairly comfortable with. It is important to give students opportunities to apply mathematics to things that they find important. Unfortunately most people spend their school life wondering why they have to learn the math that they do. Using life experiences gives them an example of one of the many situations that they may want to use math. By implementing this kind of instruction on a regular basis, in which life situations are analyzed, students may begin to find meaning in math. Goodman also allowed them to explore his scenario beyond exactly what he had planned out. This allowed the students to discover on their own what they thought might be important. When students begin to explore, I believe that they enjoy their learning more since it is what they are curious about rather than what the teacher feels they should be learning. When students enjoy their learning I believe they succeed much more frequently. One last thing that can be learned from Goodman’s model is to remember not to underestimate what students of any age are capable of discovering and learning.

Entry #5

There can be many advantages to a classroom with a constructivist mindset like the one put for by Warrington. We see that all of Warrington's students were confident in their learning. Each of them created ideas and took responsibility for their own understanding, we learn that the students never gave the typical response of “I don't know how to do this,” indicating that instead of giving up they were willing to keep moving forward. We also see that in addition to creating ideas, these students have come to the desire to have a sound understanding of what everyone else believes. The class discussion comes to a point where only one student does not agree with what the class believes, Warrington tells us that she would not accept the class’s theory until it made sense to her. Something that Warrington does not directly say but eludes to when explaining that one of the students had assigned meaning to a basic arithmetic problem, is that the students gain an understanding of what the number they are using really mean and how they relate to “the real world.” A constructivist classroom yields nicely to a solid, relational understanding.

Of course, like everything, there are bound to be disadvantageous also. Some disadvantageous I came across while reading come mostly from trying to find a balance for this constructivist teaching style. Towards the end of the paper we see that the children asked Warrington “which answer is right?” Since Warrington does not include whether or not that question is answered, it concerns me that the students may never come to a sound understanding if they do not happen to reach the correct answer in their discussions in class. I would think that Warrington would correct the theories of the students if they arrive at an incorrect conclusion, but since this doesn’t happen in the paper, I am left to wonder how far in the wrong direction the class discussions would be allowed to go. If there is no guidance by Warrington in the discussion, because the students are to be their own mathematic authority, the students could very easily construct false ideas about any subject, like we saw with Erlwanger’s study of Benny. If discussions going in the wrong direction are not corrected, great confusion would result. This confusion would not only affect them with their immediate learning but it is highly probable, like in the case with Benny, that students will continue to have a difficult time with future learning. Although constructivism leads to a relational understanding, a classroom with such extremist views may not lead to correct knowledge.

Entry #4

Von Glaserfeld’s theory of knowledge is based on experiences we have and understandings that we created based on those experiences. Von Glaserfeld talks about “constructing” knowledge, he doesn’t talk about “acquiring” or “gaining” knowledge. He does this because both terms, acquire and gain, have an implication that there is some standard of truth out in the world and we have to get that from someone or something else, but according to von Glaserfeld, our knowledge is constructed based on what we have experienced and how we perceive things. Our past experiences will continually affect the understanding that we embrace as knowledge. Von Glaserfeld talks about knowledge as being a theory because there is no way for any individual to really know that what they “know” is correct. We believe it to be true but it is not necessarily proven facts. We believe that the knowledge we have constructed is true until we encounter things that challenge our understanding and thus our perceived truth is no longer truth, and we construct a new knowledge for ourselves.

Constructivism is something that can play a huge role in classrooms. Since each student has had different experiences and has gained different knowledge, based on constructivism, it becomes very difficult to teach a large group of children. For this reason it becomes increasingly important to have a reliable way of evaluating a students understanding of a given concept. One way this evaluation could occur would be to have some sort of peer instruction. A simple way of doing this would be to have students explain the answer, and the process of getting the correct answer, on homework assignments to the class each day. Now, obviously every student would not be able to present every single day but a rotating schedule could be implemented. In my opinion this has many benefits, it gives students the opportunity to explain what they did, gives you as the instructor an opportunity to correct any misunderstandings they may have, it gives the rest of their peers the opportunity to hear another perspective, and finally when someone explains their work I believe it helps solidify the concept in their own minds.

Entry #3

Erlwanger's disaster study of Individually Prescribed Instruction, titled "Benny's Conception of Rules and Answers in IPI Mathematics," is a paper examining the IPI system. Erlwanger's paper is in support of having a strong foundation in mathematics, something we would call a relational understanding. Erlwanger explains that Benny would have done much better if he had been able to know why certain answers were correct and which processes truly worked. Erlwanger says that although Benny had a "master of content and skill," he did not have an understanding of concepts. A couple of the flaws in the system contributed to this lack of understanding. The first is that the key has only one correct answer so Benny was led to believe that his answers were correct just not in the correct form. Another flaw is that because the teacher is removed from instruction, evaluation of Benny's processes is missing. Instead of someone being there to see that he is doing something incorrectly and then helping him to fix mistakes, Benny is led to the false assumption that he is doing perfectly fine. The above issues go back to not having an instructor to help Benny, or any student, gain a relational understanding, an understanding where the rules make sense and weren't just made up by some guy who spent his entire life writing math rules.

Erlwanger's study supports the idea that there needs to be sufficient, personal relationships between instructor and student. In Benny's case he was working on his own and did not receive any guidance to correct his errors. I have tutored and helped in many classrooms. Most frequently the teacher asks me to be available to answer questions and just walk around the room to see how everyone is doing. I found that very rarely did students ask for help when they most needed it. As an aid in these classes, I had to learn to recognize the behaviors of the different students and see when they were struggling. This is something that a computer can not do in today's classroom. Only a human can see a look on someone's face and be able to recognize that as being a look of someone in need. Without getting to know each of the students it is difficult to help them progress. Since most students will not seek out help it is necessary for there to be a way for instructors to evaluate work done by students. After this evaluation takes place, it is crucial that the errors are discusses and corrected, otherwise like Benny, a student will believe that everything they have done is correct, because no one has told them otherwise.