Blog #7

Fiore, G. (1999). Math-abused students: are we prepared to teach them?. The Mathematics Teacher, 92(5), 403-406.

This article was written for the purpose of trying to show teachers that they need to know and understand the limits that all of their students have in their abilities with math. The author, Fiore, recounts an experience teaching a college math class. He tells about a couple students that struggled in his class. After talking to them, he found out that they had been "math-abused", meaning that their previous math courses have hurt them either emotionally, mentally, or both. In one of the cases, it was the teacher that caused so much abuse to the student. Fiore emphasizes our need to understand the students and to be careful not to "abuse" any of our students.

I agree with the author and think that it is critical for us not to "abuse" our students. Like the students in Fiore's math class, teachers can hurt the students they teach. It can be as simple as calling on students that you know are struggling and don't know the answer, or making fun of a student in class. It is the people that teach from the beginning that cause someone to like or dislike a subject, depending on how that person's experience turns out. A suggestion, given in the article to help understand the students, was to ask each student to write a paper talking about their previous encounters with math. This way the teacher knows where each students is coming from and can carefully plan lessons around the needs in the class. This can help the students learn more efficiently. Lastly, the students deserve an environment that they feel comfortable learning in. Teachers owe it to their students to make their classroom a place of welcome, not hostility. Math is a subject to be understood and enjoyed, not dreaded.

Blog #6

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

In this article,the authors puts a teacher's knowledge into two categories: knowledge of mathematics content and knowledge of pedagogy (teaching). The authors talk about the need for teachers to use their knowledge about the content and pedagogy to sufficiently to teach the students, and they stress the importance of teachers adapting their knowledge to the way their students learn. They also need to be able to interpret their students solutions and adjust the content and teaching style to assist the students. An example of this was given with a teacher that asked a group of 6th graders to complete a problem dealing with ratios. After looking at the student's results, the teacher noticed that most of the students didn't fully understand their task and got the question wrong. The teacher then noted that she needed to change the content for the students that next week and that she also needed to teach this concept differently. By being willing to adapt our knowledge to the current situation, will help us more accurately help the students we will teach.

I agree that teachers need to change, or adapt, their knowledge for their students. When teachers change their knowledge of pedagogy, they are sometimes able to help their students learn better. When students get more of an understanding of math it may improve their confidence in being able to do math. Changing the way the teachers teach is not the only thing. Teachers can change their knowledge of the content. By doing this, they can more change the way the content is explained. This change may help students apply the content outside of class and in the real world. It can also give students a greater appreciation for math because of the way was learned. Teachers need to able to adapt their knowledge at all times for their students. When they do this, they become better teachers.

Blog #5

In Warrington's article, she points out a lot of advantages to constructivism. She talks about how the students are encouraged to think deeply about the math concepts. With this they learn to invent their own methods of solving problems. In the very beginning, they learned to make rules for solving the simple fractions. This helped them solve the harder questions. Another advantage is that the students got involved. They all worked together to solve the problem given to them. This helped build unity in the classroom. The students also learned to find mathematical relationships. When they were solving a question, they noticed that you can double the fraction so the numbers are more simple to work with. They, in essence, learned to construct their own relationships of fractions.

Although there some advantages, there are a couple of flaws with this method of teaching. One of the biggest is that it can take a long time to solve a question. For example, the students tried to solve 4 2/5 divided by 1/3. After a long time of thinking and working on it, they finally found the solution, but this process wastes valuable time. Another disadvantage with constructivism, is that the students have disagreements. Some of these disagreements take a long time to resolve and they can also waste class time. Not all of the material needed to be covered, can be with a constructivism. Although there are a lot of advantages to constructive learning, there are some disadvantages that need to be looked at also.

Blog #4

Constructivism is a term used in von Glasersfeld article. He talks about constructing knowledge instead of obtaining it. He does this to emphasize the importance of us building it ourselves. The more knowledge we build, the better we understand it and the more we will want to learn. To construct knowledge, we need someone to help us build a base, being a teacher. With their help, students can learn to create their own scaffolding of knowledge.

One thing that I would like to do in with constructivism in my math teaching, is giving the students hands on activities. They learn basic concepts, but they are the ones that put it into action. In my jr. high we were taught what geometric shapes were the strongest and then used that knowledge to build a bridge. By doing that, I learned by myself what worked best and I constructed my own knowledge on how to do things. It gives the students a chance to experience things by themselves and have fun. This is a constructive outlook, because you give the students a basis of how to do something and then they decided how they are going to build off of their knowledge.

Blog #3

"Benny's Conception of Rules and Answers in IPI Mathematics" is a journal written by Elrwanger. In this article Elrwanger spends time with Benny, a student of this IPI program. He notices that Benny has misconceptions of rules with fractions and decimals. For example, Benny thought 2/10 was equal to 1.2. Through questioning Benny, Elrwanger found that Benny had made up rules. This reasoning Benny had made pointed out that students need a relational understanding and a strong one from the very beginning. If they don't have this understanding, then they can become lost and make up their own rules. Benny must not have fully understood a concept, so he made up a rule by himself. Benny had been doing this program since from 2nd-6th grade and he had a lot of misconceptions to unlearn and had a lot of new conceptions to learn.

Relational understanding is really key from the beginning, even now. It is hard to change a student's ideas about a concept once they have made up their own rules, and especially if they have been using this rule for many years, like Benny. I feel it is necessary to really get with the students and make sure they fully understand something. I remember my teachers teaching an idea/concept and then we had a worksheet to enforce the concept. To see if we had understood the concept, they would go over our worksheets herself. A lot of my teachers tried hard to make teach relationally because it is easier to come up with concepts, then having to memorize them all. If you don't give students a good base from the beginning, it is going to be really hard for them to build off of misunderstood information.

Blog Post #2

Thinking back on my years at school, I remember learning math different ways. Somethings I learned without knowing why it worked and others gave me real life application. These different teaching methods are discussed in Richard R. Skemp's article called, "Relational Understanding and Instrumental Understanding". As he goes through the article, he explains what Relational and Instrumental mean, what their advantages are, and how they correlate to each other. Relational Understanding is being able to understand and identify the real life application. This could be as simple as understanding why we use a specific formula to find the circumference of a circle and where we got that formula from. The advantages of this are that is is easier to remember, it can motivate the student to want to learn more, and students can may try to understand new material relationally. The disadvantages are that is takes to long to understand, the teacher has to cut out material, and somethings don't need to be understood. In contrast, Instrumental Understanding can be described as rules without reason. We do something, even though we aren't sure why we do it. For example, we multiply width and height to get the area of a rectangle. Some students don't understand why they are doing it, other than the reason that their teacher told them to. The advantages of this understanding are that it is easier to teach and understand, the rewards are immediate, and the students can get the right answers quickly. The disadvantages are they can be over-loaded with information, the teacher cannot tell the understanding of the concepts, and the students only remember the information for the next test and then lose it. Although Instrumental and Relational are different, they are also very similar. While learning Relational Understanding, you also learn about the Instrumental part of it, whether or not you actually understand it. But is not entirely the other way around. A teacher can almost easily teach Instrumental without teaching any Relational. I think that the best way to teach, is to include both ways of teaching. A good mix of both can help the students and not give them an over-load of material or keep them wondering how it really applies.

Writing Assignment #1

Mathematics is an interesting subject. It is so set, yet so mysterious, due to limited information. I think we can define mathematics, as the numerical system that makes the earth work. It provides us with the information on how fast things move, or how to build a bridge. It lays the foundation for science.

I have found that I best learn mathematics in a structured environment. The more instruction and direction a teacher gives, the better I do. For me this works, because I get less distracted and more focused on the tasks to complete. Plus, I know what the teacher expects of me. It gets me more into the class and I feel the class is more well behaved also, so there are more opportunities for the teacher to explain concepts and not have to control the class behavior.

I really don't know how my students will learn the best. I think it really depends on what type of class you have. I like to have structure, but students can get intimidated with too much and not feel comfortable asking the teacher for help. I do want to have a lot of structure when I teach, but if that doesn't work for my students, then I will probably change to help them learn the best way they can.

Previously I was an intern at a jr. high and I was able to really observe the teacher. I found that the best ways the students learned the concepts and were more apt to learning them, was if we could tie the concept to a real life situation. If the students understood why they would be needing the information later in life, they liked to learn it. Hands on activities were also big factors in helping them get excited. If they knew they were going outside to measure something or just observing objects, they got excited and wanted to learn more. If I was the student, I would want to know why something worked or why I needed the info and so I understand where they are coming from. Plus field trips and so much fun.

I think that one of the worst practices in school is teaching the students that what they learn right now, will not matter later on in life. In all reality it may not, but a lot of times, the students pick up on those things and decided not to learn other things because it "won't matter". I think it is important for the teacher to emphasize that all the concepts the students learn will matter and it is important to learn them.