In the article "Hot Wheels" from Mathematics Teaching in the Middle School, an activity involving hot wheels cars is described. The students in the classroom are given a hot wheels car and they are told the scale between a hot wheels car and real car is 1:64. They are then told to figure out the dimensions of the real car based on their hot wheels. Students are not told exactly how to do this so they must use problem solving in order to complete the task. After doing this the students can see that the scale is not completely accurate because the dimensions on the real car are nothing close to an actual real car. So, the students must try to figure out why their dimensions do not match up to those of a real car. Meanwhile, the teacher is providing all the questioning for these students to guide them to the solution.
Postive things of this article and activity are clearly seen when you compare it to the process standard problem solving from Principles and Standards. Problem solving should incorporate multiple topics and this activity does that. It covers similiar figures and proportional reasoning. The activity also shows the real life application of mathematics. Most students would never think they could use math with hot wheels cars, but after doing this activity they can see that and start to wonder what else they can apply math to.
In this article the students did not have all of the knowledge necessary to complete the activity. They knew how to measure things but through the activity they also learned about proportions. That is a key concept in problem solving, that the students can continue to build their knowledge through problem solving. The students were able to reflect on this activity and understand how they could use mathematics to find the size of a real car from their hot wheels car. By allowing them to reflect the teacher helped the students to recognize even more about mathematics.
Thursday, January 28, 2010
Process Standard: Problem Solving
After reading the information on Problem Solving in Principles and Standards I can understand what a crucial part the teacher plays. So much of what students get out of problem solving is based on the teachers preparation and chosing the right problems. The more opportunities students have to work on problem solving they better they will be at it and understand it. Learning problem solving techniques is going to help students approach problems in multiple different ways and also help them in all aspects of school and life. Problem solving does not have to just involve math and the more topics that are covered in a problem the better. Students do not have to possess all the knowledge necessary for a problem. If the problem is chosen correctly students can learn math while solving their problem. If teachers can anticipate the value of the problem they are chosing before giving it students will be able to get the most possible out of the problem.
An idea that had never occurred ot me before involves young children's questions they ask. Students are always asking questions about why something happens or how something occurs and teachers can take these questions and turn them into mathematical problem solving. This way teachers would know the students are interested in the problem and would more then likely get the most knowledge from the problem.
With all the different kind of problems that can be used for problem solving it is important for students to know multiple strategies for solving them. The more strategies available to the students the more likely they will be able to solve the problem. I know from experience this is true because I am in a problem solving course right now. With all the different kinds of problems we get assigned not one strategy works for all of those problems. Sometimes it works better to draw a diagram, but other times it is best just to write it all out.
It is also important to make sure that students are reflect on their problem solving skills and strategies. If time is taken to reflect on what worked and what did not work, students will hopefully be able to avoid those strategies that can be misleading and get right to an effective way of solving the problem.
An idea that had never occurred ot me before involves young children's questions they ask. Students are always asking questions about why something happens or how something occurs and teachers can take these questions and turn them into mathematical problem solving. This way teachers would know the students are interested in the problem and would more then likely get the most knowledge from the problem.
With all the different kind of problems that can be used for problem solving it is important for students to know multiple strategies for solving them. The more strategies available to the students the more likely they will be able to solve the problem. I know from experience this is true because I am in a problem solving course right now. With all the different kinds of problems we get assigned not one strategy works for all of those problems. Sometimes it works better to draw a diagram, but other times it is best just to write it all out.
It is also important to make sure that students are reflect on their problem solving skills and strategies. If time is taken to reflect on what worked and what did not work, students will hopefully be able to avoid those strategies that can be misleading and get right to an effective way of solving the problem.
Wednesday, January 27, 2010
Video: Lessons on Variables Grade 4
The activity the students were doing in this video was creating a variable machine. At first they were told to start with A=0 and B=1 and so forth for the alphabet. Then they were to figure the score for their name and for their whole group. Once they had done that the teacher had them recreate their variable machine to try and get the highest possible score as a group. This activity was a great tool for the teacher to show the students how variables can effect outcomes. By changing the variable you can get a whole new result. By doing this activity the students were able to visually see how variables effect solutions.
Reflective Task 1: Describe the nature of the evidence of student learning , e.g., answers, explanations, questions, and so on.
I thought as the activity progressed it became more obvious the students were understanding and truly learning what the teacher wanted them to know. The evidence for this was seen in how long their responses were, actually being willing to share their responses, and going further into an explanation without probing from the teacher. In the beginning of the activity the students were obviously hesitant to offer any information because they didn't really understand the purpose of the activity. But as the activity progressed and they began to feel more comfortable with variables they began to really offer their thinking behind their answers. Once the teacher got to the part about explaining their points for the name "bear" she barely even had to ask the students extra questions to get more information out of them.
Personal Reflection 1: What kind of questions predominate in your classroom? Single answer? Short answer? Explanation?
In the classrooms that I am a student in the questions use to be more short answer and single answer. However, now that I am in many more upper level courses in college the questioning is more about the explanation. These types of questions are difficult for me because I have never really been asked to explain my reasoning behind my thoughts. I think this is a more effective approach to questioning because it makes the student draw on all the knowledge they have and recount many things in order to verbally say what they are thinking. By relooking at ideas it helps students understand even better. In the classrooms that I have observed in as a teacher many of the questions are single answer but then the teacher adds on the explanation question. It seems most students get flustered with this because they think they can give a quick simple answer but then are asked for more and aren't prepared.
Lesson Analysis 2: Describe what the teacher does to support learning while students are working in groups.
One way the teacher supports learning while the students are in groups is walking around the room to all the different groups. She interacted with the students and continued to ask questions while they were working in their groups. She was trying to make the students think about the method they were doing while they were doing the method. She made sure to ask the groups why they chose their method and if it benefited everyone in the group. Also, if she notices a group doing something not on the same page as what she was thinking she will call that matter to the attention of everyone in the class. This way the whole class is still learning even though they are working in groups.
I really enjoyed watching the video clips on this topic. At first I was a little confused because the clips didn't go in order of the actual lesson, but once I saw all of the clips the lesson made sense to me. I thought it was a really creative way to teach students about variables. I thought it also showed me great ways to question my students. It seems so many people just worry about getting the right answer and by watching this teacher with her students I realized how important it is for the students to understand how they got the right answer.
Reflective Task 1: Describe the nature of the evidence of student learning , e.g., answers, explanations, questions, and so on.
I thought as the activity progressed it became more obvious the students were understanding and truly learning what the teacher wanted them to know. The evidence for this was seen in how long their responses were, actually being willing to share their responses, and going further into an explanation without probing from the teacher. In the beginning of the activity the students were obviously hesitant to offer any information because they didn't really understand the purpose of the activity. But as the activity progressed and they began to feel more comfortable with variables they began to really offer their thinking behind their answers. Once the teacher got to the part about explaining their points for the name "bear" she barely even had to ask the students extra questions to get more information out of them.
Personal Reflection 1: What kind of questions predominate in your classroom? Single answer? Short answer? Explanation?
In the classrooms that I am a student in the questions use to be more short answer and single answer. However, now that I am in many more upper level courses in college the questioning is more about the explanation. These types of questions are difficult for me because I have never really been asked to explain my reasoning behind my thoughts. I think this is a more effective approach to questioning because it makes the student draw on all the knowledge they have and recount many things in order to verbally say what they are thinking. By relooking at ideas it helps students understand even better. In the classrooms that I have observed in as a teacher many of the questions are single answer but then the teacher adds on the explanation question. It seems most students get flustered with this because they think they can give a quick simple answer but then are asked for more and aren't prepared.
Lesson Analysis 2: Describe what the teacher does to support learning while students are working in groups.
One way the teacher supports learning while the students are in groups is walking around the room to all the different groups. She interacted with the students and continued to ask questions while they were working in their groups. She was trying to make the students think about the method they were doing while they were doing the method. She made sure to ask the groups why they chose their method and if it benefited everyone in the group. Also, if she notices a group doing something not on the same page as what she was thinking she will call that matter to the attention of everyone in the class. This way the whole class is still learning even though they are working in groups.
I really enjoyed watching the video clips on this topic. At first I was a little confused because the clips didn't go in order of the actual lesson, but once I saw all of the clips the lesson made sense to me. I thought it was a really creative way to teach students about variables. I thought it also showed me great ways to question my students. It seems so many people just worry about getting the right answer and by watching this teacher with her students I realized how important it is for the students to understand how they got the right answer.
Friday, January 22, 2010
Issues with Current Mathematics Curricula
Reading the article Middle-Grades Mathematics Standards: Issues and Implications, I came across several important points regarding the math curriculum throughout the country. The first point the article brings up is that there is a disparity among states pertaining to when certain topics are introduced and how often they are subsequently referred to. The article uses the example of similarity in angles/shapes, and shows how states introduce this concept and very different times. Introducing topics at different times leads to students in some states mastering concept at a later age which will inevitably impact later education.
The article also states how important it is for the curriculum from one grade to be compatible with both previous and future grades. With material from previous years, there must be a balance between re-teaching and building on a concept. It is a waste of time in the classroom to completely re-teach a topic that has already been extensively covered, but at the same time students may need a refresher before moving on to a more advanced implementation of a topic. It is also integral that educators build upon previously learned concepts at an appropriate rate and difficulty since synthesis of previous learnings leads to learning of more complicated material.
Another difference between many states' math curriculum is the emphasis on performing a task as opposed to justifying which method to use before performing a problem. While some states require students to simply create and use a histogram to solve a story problem, other states require the students to pick which graph to use, justify their choice, and then proceed to solve the problem. Obviously the justification process is an important critical thinking exercise and helps students truly understand the material, but many curricula do not use this concept when stating guidelines.
The final issue with state curricula that the article talks about is the lack of clarity in many state guidelines. Oftentimes they use several verbs to speak of the same action, or use unclear or ambiguous language to describe a topic. Since these standards are the basis for all the material covered in a students time in the classroom, it wold be extremely beneficial for these to be made more easily readable.
This article shows how important the curriculum is to the study of mathematics, specifically in the middle grades. Topics must be integrated between grades, there must be an emphasis on evaluating a problem rather than simply doing it, and there is a need to simplify and clarify the current guidelines to ensure students are taught in a uniform manner.
The article also states how important it is for the curriculum from one grade to be compatible with both previous and future grades. With material from previous years, there must be a balance between re-teaching and building on a concept. It is a waste of time in the classroom to completely re-teach a topic that has already been extensively covered, but at the same time students may need a refresher before moving on to a more advanced implementation of a topic. It is also integral that educators build upon previously learned concepts at an appropriate rate and difficulty since synthesis of previous learnings leads to learning of more complicated material.
Another difference between many states' math curriculum is the emphasis on performing a task as opposed to justifying which method to use before performing a problem. While some states require students to simply create and use a histogram to solve a story problem, other states require the students to pick which graph to use, justify their choice, and then proceed to solve the problem. Obviously the justification process is an important critical thinking exercise and helps students truly understand the material, but many curricula do not use this concept when stating guidelines.
The final issue with state curricula that the article talks about is the lack of clarity in many state guidelines. Oftentimes they use several verbs to speak of the same action, or use unclear or ambiguous language to describe a topic. Since these standards are the basis for all the material covered in a students time in the classroom, it wold be extremely beneficial for these to be made more easily readable.
This article shows how important the curriculum is to the study of mathematics, specifically in the middle grades. Topics must be integrated between grades, there must be an emphasis on evaluating a problem rather than simply doing it, and there is a need to simplify and clarify the current guidelines to ensure students are taught in a uniform manner.
The Curriculum Principle
I had never really thought of curriculum being a principle of mathematics. After reading what was stated in Principles & Standards for School Mathematics I feel I have a better understanding of why curriculum is so important for mathematics. It is very important for the mathematics curriculum to be taught in a way that shows how all things math are interconnected. By showing students how everything is related it will hopefully help them to have a better understanding of math concepts. Teachers should also be teaching the important mathematics. Teachers should be teaching the concepts that will lead to other more involved concepts, help them appreciate math more, and help them in math and outside of math. It is also important for teachers to realize that these important concepts are always changing depending on what is going on in the world. I had also never realized how important it was to communicate across the grade lines for math. Teachers should be talking to each other so they can know what level their students are at so they do not re-teach things they already know or move into a topic that is not connected to what they learned the previous year.
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