Teaching Children Mathematics. Promoting Equity Through Reasoning. By: Mary Mueller and Carolyn Maher
The beginning of this article talks about the gap between white students and minority students in the area of math. It said that 34% of white fourth graders scored at or above "proficient" while only 5% of black students and 10% of Hispanic students. Some factors for this that the article mentions is low expectations and classroom environment. An after school program was started that took 24 African Americans and Latinos and did math with them. The environment was set up for the students to have open ended questions, work collaboratively, and justify their answers. Students worked with exploring fractions using Cuisenaire rods. Teachers did not judge or evaluate the students responses so they felt much more apt to responding and trying to get their fellow peers to understand their reasoning. The students ended up coming up with proofs for their way of justification by default and were able to reason through their responses and others' responses. A lot of time teachers think the best way to deal with minority-race and inner-city students is to have a very procedural classroom. They do this because they think that these students need structure, which they do to a certain extent. However, we cannot limit their minds, we need to open opportunities up for them to explore different ideas. Some things that will help teachers to create an environment condusive to problem solving are: give choices, differentiate, made ideas public, select the best task and tools, and hold high expectations.
I found this article to be very interesting, because I had no idea that minority students were still lacking so far behind the white students in mathematics. I thought this article was very helpful in showing how as teachers we can still help these students be successful in math. It seems like it would be so great for these students to have open-ended discussions as long as they are taught how to stay on task. One point I was not exactly clear on with this article was if white students were given these open-ended opportunities while the minority students were not. I know that in my classroom I will allow the same opportunities for all my students in the classroom no matter where they come from. If I recognize a student may need some extra help I will try to provide that for them rather then just changing my expectations of the student. I think it would be really good to have this same type of environment the article created in my own classroom as well. While I know everyday cannot be like that, I would like to try and have at least one day a week where the students could explore different responses to questions and reason their way through their explanations. I think this really helps them to grow in mathematics and fully understand what is going on with topics.
Sunday, May 2, 2010
Tuesday, April 27, 2010
Manipulatives Blog
I think one of the main ways to make sure kids are being held accountable when using manipulatives is for the teacher to make sure and circulate the room. Obviously this leaves some lag time that the students could get off task or only let one student do the work, so more would probably need to be done. Maybe each student could be held accountable for documenting their work with the manipulatives. Like we did in class with the manipulatives we had to write down what we came up with using the manipulatives. Something like this could be done with the students in our classrooms as well. Another way to hold the students accountable for the manipulative activities is to do one on one evaluations. The teacher could have each student model the activity they were suppose to do for them again in an assessment type format.
I believe that the reason people say "hands-on, minds-on" now instead of just "hands-on" is because of how interactive the manipulatives are now and how much it can help students understand concepts. If the student is working with manipulatives they are more likely to be engaged both physically and mentally rather then if they were doing a worksheet or listening to a lecture. "Hands-on, minds-on" helps recognize how engaged the students are with their activities and hopefully learning even more.
The process standards fit in very nicely with the use of manipulatives. Students must problem solve when working with manipulatives. They may have never seen a particular manipulative before so they must problem solve how to actually use the manipulative and work through the activity. The students will also be using communication as well if they are working with a partner. They will have to communicate their ideas to each other to be able to properly use the manipulatives. Reasoning and proof can also be addressed with the use of manipulatives. The students will need to reason through how they are working with their manipulatives and then prove what they have done works. Connections are also prevalent with the use of manipulatives. It is very easy to incorporate many different content standards with the use of just one manipulative and students will be able to recognize how they all work together. Finally, representation is used with manipulatives by the students representing different problems with their manipulatives.
I believe that the reason people say "hands-on, minds-on" now instead of just "hands-on" is because of how interactive the manipulatives are now and how much it can help students understand concepts. If the student is working with manipulatives they are more likely to be engaged both physically and mentally rather then if they were doing a worksheet or listening to a lecture. "Hands-on, minds-on" helps recognize how engaged the students are with their activities and hopefully learning even more.
The process standards fit in very nicely with the use of manipulatives. Students must problem solve when working with manipulatives. They may have never seen a particular manipulative before so they must problem solve how to actually use the manipulative and work through the activity. The students will also be using communication as well if they are working with a partner. They will have to communicate their ideas to each other to be able to properly use the manipulatives. Reasoning and proof can also be addressed with the use of manipulatives. The students will need to reason through how they are working with their manipulatives and then prove what they have done works. Connections are also prevalent with the use of manipulatives. It is very easy to incorporate many different content standards with the use of just one manipulative and students will be able to recognize how they all work together. Finally, representation is used with manipulatives by the students representing different problems with their manipulatives.
Monday, April 26, 2010
Errors Blog
I think one of the most informative things we did this semester was looking at the Errors in class. It was so helpful to actually sit and look at what a student did and try to figure out what they were really thinking when completing the problem. So many times in school I felt like teachers would just simply mark when I got an answer wrong and never really looked at what I did wrong in my computation. Doing these errors in class I was able to get practice on analyzing students responses. I think this will really help me in my classroom be able to identify what is really going on with my students.
Another helpful thing from these errors was learning different ways to teach my students mathematics. In 107 and 108 we got a taste of the different ways to try and look at the math problems and solve them but I feel like with this we went more in depth. I was able to see even more ways to do math for my students. When trying to teach students the traditional way to do addition or subtraction, if they do not understand it I now have a way to try and help them understand it. I will be able to present these problems to my students in a way that actually makes sense to them so they do not just think of math as a subject they just don't get and can't do. After completing these errors I will be to adapt to my students and accommodate their needs in mathematics.
Another helpful thing from these errors was learning different ways to teach my students mathematics. In 107 and 108 we got a taste of the different ways to try and look at the math problems and solve them but I feel like with this we went more in depth. I was able to see even more ways to do math for my students. When trying to teach students the traditional way to do addition or subtraction, if they do not understand it I now have a way to try and help them understand it. I will be able to present these problems to my students in a way that actually makes sense to them so they do not just think of math as a subject they just don't get and can't do. After completing these errors I will be to adapt to my students and accommodate their needs in mathematics.
Technology Blog
I found the week we spent focusing on technology really interesting. I have worked with Geometer's Sketchpad before but it had been awhile so it was nice to have a refresh of the different kinds of things you can do with it. I think the best part of the Sketchpad is being able to show students the properties of geometric figures. They would be able to actually see how angles stay the same, or lengths stay the same. I feel students will actually understand these concepts better because they will actually be able to see it rather than just telling them this is what it is. I will definitely try to incorporate Sketchpad into my teaching even if it is only using the free preview version. I think all students should be able to utilize this technology.
The calculators we used in class were also really interesting. I had never seen a calculator that was able to do all the different things that calculator was able to do. I think it would be really important to make sure students really understand how to work with fractions. The calculator will be very nice for when students have mastered their understanding of it, but if introduced at the wrong time I think it may hinder there understanding. Obviously though I think that these calculators are very helpful and a good look into what the future will be like for our students.
Another piece of technology we have really utilized in this class is the smart board. I had heard about smart boards before coming to this class but had never seen one actually used before. I think it is an incredible piece of technology because it can make things so interactive for students. I think by having the students actually work on the original document from the computer makes it more exciting for them and like they are really contributing. I'm sure sometime in the future there will even be personal smart boards for every student on their desk.
The calculators we used in class were also really interesting. I had never seen a calculator that was able to do all the different things that calculator was able to do. I think it would be really important to make sure students really understand how to work with fractions. The calculator will be very nice for when students have mastered their understanding of it, but if introduced at the wrong time I think it may hinder there understanding. Obviously though I think that these calculators are very helpful and a good look into what the future will be like for our students.
Another piece of technology we have really utilized in this class is the smart board. I had heard about smart boards before coming to this class but had never seen one actually used before. I think it is an incredible piece of technology because it can make things so interactive for students. I think by having the students actually work on the original document from the computer makes it more exciting for them and like they are really contributing. I'm sure sometime in the future there will even be personal smart boards for every student on their desk.
Wednesday, April 7, 2010
April Journal Summary #2
Mathematics Teaching in the Middle School. Map Scale, Proportion, and Google Earth. By: Martin Roberge and Linda Cooper
This article talks about how children are able to at a young age realize the relationship between things on a map and their actual size. However, only 23% of middle school students could correctly find the distance between two cities on a map. They believe there is this gap because there is so much focus on just setting up the proportions and not realizing what they are representing. Teachers should focus students' attention on the context of the problem so they can develop an understanding of the relationship between values in a proportion. They had students take aerial maps and find ratios and scales for them. The students looked at baseball fields and football fields. They then extended their map scales to creating proportions. They then wanted students to apply map scale within a proportion to explore new situations. The teacher then placed numerous aerial pictures of famous landforms and asked different questions about the pictures. This activity really helps students to understand the relationship between values as they construct proportions.
I found this article really interesting because I have worked with Google Earth before. Since the teacher got their aerial shots from Google Earth I was interested in how they used them. I also think making the students work through proportions and map scale this way would be very beneficial. I think in math we tend to just give students equations and things to plug different numbers into that they do not understand why they are doing it. By having the students figure out the map scale on their own first, they are able to then understand why we make proportions and where they come from. If students understand where the proportions are coming from hopefully they would be less likely to make mistakes in their scaling. Hopefully we could increase that 23% who do it correctly to about 50%. An adaptation I would maybe make to this activity is to let the students use Google Earth themselves and pick a picture to scale on their own. This way they could know another use of technology and explore a little more on computers.
This article talks about how children are able to at a young age realize the relationship between things on a map and their actual size. However, only 23% of middle school students could correctly find the distance between two cities on a map. They believe there is this gap because there is so much focus on just setting up the proportions and not realizing what they are representing. Teachers should focus students' attention on the context of the problem so they can develop an understanding of the relationship between values in a proportion. They had students take aerial maps and find ratios and scales for them. The students looked at baseball fields and football fields. They then extended their map scales to creating proportions. They then wanted students to apply map scale within a proportion to explore new situations. The teacher then placed numerous aerial pictures of famous landforms and asked different questions about the pictures. This activity really helps students to understand the relationship between values as they construct proportions.
I found this article really interesting because I have worked with Google Earth before. Since the teacher got their aerial shots from Google Earth I was interested in how they used them. I also think making the students work through proportions and map scale this way would be very beneficial. I think in math we tend to just give students equations and things to plug different numbers into that they do not understand why they are doing it. By having the students figure out the map scale on their own first, they are able to then understand why we make proportions and where they come from. If students understand where the proportions are coming from hopefully they would be less likely to make mistakes in their scaling. Hopefully we could increase that 23% who do it correctly to about 50%. An adaptation I would maybe make to this activity is to let the students use Google Earth themselves and pick a picture to scale on their own. This way they could know another use of technology and explore a little more on computers.
April Journal Summary #1
Teaching Children Mathematics. Asmorgasbord of assessment options. By: Kathy Bacon
This article is focused on student-centered classroom assessment. Assessment should support the learning of useful mathematics and provide useful information to both teachers and students. Teachers should look at assessments to see where next to go with their students and should really use assessment as a guide for the next phases of learning. For assessment to really be effective teachers need to use a variety of techniques, understand their goals, and think about what is really being developed. Math should be a progression of learning. Van de Well provided an idea of the different levels of learning and how children should move through the process. Level 0: Visualization, Level 1: Analysis, Level 2: Informal Deduction, Level 3: Deduction, Level 4: Rigor. These levels help to assess a student's level of geometric thinking. An example was then given of three students at different levels of their geometric thinking. Through each child's reasoning the teacher was able to help the student build and go further into their geometric thinking. This article really stresses how assessments should allow creation of products that show their thought process. The assessment option should be thoughtfully matched to the achievement target.
I thought this article gave a lot of insight into assessment. I had never really thought of assessment as a progression. I can see now why it is so important to have insightful assessment for math. One thing about the article I didn't really like was all the examples given. I did not see how they were related or how they showed progression. I wish they would have given more examples of what the students were taught, what the assessment was, and then where the teacher went from there. I would have also liked to see how the teacher analyzed the assessment results a little bit more. Overall, I though the article was helpful in making teachers think more about assessment but I do not think it was presented in the best manner.
This article is focused on student-centered classroom assessment. Assessment should support the learning of useful mathematics and provide useful information to both teachers and students. Teachers should look at assessments to see where next to go with their students and should really use assessment as a guide for the next phases of learning. For assessment to really be effective teachers need to use a variety of techniques, understand their goals, and think about what is really being developed. Math should be a progression of learning. Van de Well provided an idea of the different levels of learning and how children should move through the process. Level 0: Visualization, Level 1: Analysis, Level 2: Informal Deduction, Level 3: Deduction, Level 4: Rigor. These levels help to assess a student's level of geometric thinking. An example was then given of three students at different levels of their geometric thinking. Through each child's reasoning the teacher was able to help the student build and go further into their geometric thinking. This article really stresses how assessments should allow creation of products that show their thought process. The assessment option should be thoughtfully matched to the achievement target.
I thought this article gave a lot of insight into assessment. I had never really thought of assessment as a progression. I can see now why it is so important to have insightful assessment for math. One thing about the article I didn't really like was all the examples given. I did not see how they were related or how they showed progression. I wish they would have given more examples of what the students were taught, what the assessment was, and then where the teacher went from there. I would have also liked to see how the teacher analyzed the assessment results a little bit more. Overall, I though the article was helpful in making teachers think more about assessment but I do not think it was presented in the best manner.
Thursday, March 25, 2010
Assessment Article
Article: Assessment design- helping preservice teachers focus on student thinking
For this article it talked about how to make tests and quizzes the best way to find out the most information on a student. Tests and quizzes can be just a way to check memorization and facts but it can also really look at how a student is thinking. The right questions must be asked though to really pull the information from the student. If a teacher wants to just look at the students steps in a problem they can have short answer questions where students write out their whole process of the problem. This way teachers can see where a student might go wrong or if they just made a minor mistake. Using tests and quizzes to have students use higher level thinking can be difficult. Teachers must make sure they are asking the right types of questions. This takes a lot more time rather than copying and pasting some simple math problems for them to do. They recommend teachers to use lots of different types of assessments. All the kinds of assessments that are out there are beneficial in some way. While tests and quizzes have been the main focus of most math classrooms for awhile that doesn't mean you should no longer use them at all.
For this article it talked about how to make tests and quizzes the best way to find out the most information on a student. Tests and quizzes can be just a way to check memorization and facts but it can also really look at how a student is thinking. The right questions must be asked though to really pull the information from the student. If a teacher wants to just look at the students steps in a problem they can have short answer questions where students write out their whole process of the problem. This way teachers can see where a student might go wrong or if they just made a minor mistake. Using tests and quizzes to have students use higher level thinking can be difficult. Teachers must make sure they are asking the right types of questions. This takes a lot more time rather than copying and pasting some simple math problems for them to do. They recommend teachers to use lots of different types of assessments. All the kinds of assessments that are out there are beneficial in some way. While tests and quizzes have been the main focus of most math classrooms for awhile that doesn't mean you should no longer use them at all.
Monday, March 22, 2010
March Journal Summary #2
Mathematics Teaching in the Middle School. The Value of Guess and Check by: Shannon Guerrero
This article is about using guess and check to help students truly understand word problems and be able to better represent and understand quantitative relationships. Most students are very intimidated by word problems because they do not always understand what the problem is asking and then do not know how to set up the equation to go along with the problem. By guessing and checking their answers they will hopefully be able to reason their way to the correct answer. Guess and check allows students to use logical reason and problem solving to come up with the correct answer. The article also suggests using a guess and check table so students can see where their guesses take them and can decide whether to make them larger or smaller. Guess and check encourages reflection on the original problem, development of algebraic statement, and determination of the reasonableness of a solution. This can also be used at any grade level.
I found this article pretty interesting because I have never really thought of guess and check being an actual strategy. It's nice to see how guess and check could actual help a student to understand a math concept. When I was a student I always hated when my teacher's told me to just guess and check an answer. In my classroom I feel I will want to explain to my students how guess and check can help them. If I had this in my classroom I may not have been so turned off from it. It's good that this article was written to show teachers a way to help make word problems not as intimidating to their students.
This article is about using guess and check to help students truly understand word problems and be able to better represent and understand quantitative relationships. Most students are very intimidated by word problems because they do not always understand what the problem is asking and then do not know how to set up the equation to go along with the problem. By guessing and checking their answers they will hopefully be able to reason their way to the correct answer. Guess and check allows students to use logical reason and problem solving to come up with the correct answer. The article also suggests using a guess and check table so students can see where their guesses take them and can decide whether to make them larger or smaller. Guess and check encourages reflection on the original problem, development of algebraic statement, and determination of the reasonableness of a solution. This can also be used at any grade level.
I found this article pretty interesting because I have never really thought of guess and check being an actual strategy. It's nice to see how guess and check could actual help a student to understand a math concept. When I was a student I always hated when my teacher's told me to just guess and check an answer. In my classroom I feel I will want to explain to my students how guess and check can help them. If I had this in my classroom I may not have been so turned off from it. It's good that this article was written to show teachers a way to help make word problems not as intimidating to their students.
March Journal Summary
Teaching Children Mathematics. Identifying Logical Necessity by: David Yopp
This article is about teachers being able to identify where students are wrong in their logical reasoning. By teachers understanding the logical necessity of arguments they can better find the exact spot students are faulty at in their arguments. An example the article gives is with factoring. The student understood that numbers could be factored down to the same numbers but did not realize they had to be completely factored for their argument to make sense. By working back through the statements the student made the teacher was able to see that they understood part of the concept, but just messed up at one point in it. The author feels it is very important that preservice teachers know strategies for logical necessity and then gives a few different strategies they can try.
I think this article could be helpful in the classroom. It seems to me that logical necessity would help teachers to really understand what their students are getting rather than just giving counter examples to show their arguments are incorrect. However, I felt this article was a little confusing in it's explanation. It was hard to fully understand what I needed to do to use logical necessity in my classroom. I think it is good though that this article was printed though so that teachers are made aware of this strategy. It would be nice to know more information on how helpful it really is for students. In my own classroom I will make sure to have my students walk through a step by step reasoning of any conclusions they draw. This way I will hopefully be able to see where they are getting off track.
This article is about teachers being able to identify where students are wrong in their logical reasoning. By teachers understanding the logical necessity of arguments they can better find the exact spot students are faulty at in their arguments. An example the article gives is with factoring. The student understood that numbers could be factored down to the same numbers but did not realize they had to be completely factored for their argument to make sense. By working back through the statements the student made the teacher was able to see that they understood part of the concept, but just messed up at one point in it. The author feels it is very important that preservice teachers know strategies for logical necessity and then gives a few different strategies they can try.
I think this article could be helpful in the classroom. It seems to me that logical necessity would help teachers to really understand what their students are getting rather than just giving counter examples to show their arguments are incorrect. However, I felt this article was a little confusing in it's explanation. It was hard to fully understand what I needed to do to use logical necessity in my classroom. I think it is good though that this article was printed though so that teachers are made aware of this strategy. It would be nice to know more information on how helpful it really is for students. In my own classroom I will make sure to have my students walk through a step by step reasoning of any conclusions they draw. This way I will hopefully be able to see where they are getting off track.
Tuesday, March 2, 2010
Video Blog #2: Lesson on Graphs Grade 7
The activity students were doing in this video was creating a story based off of a graph given to them by the teacher. The students were to see how the two variables interacted with each other. The teacher then also had students look at sequences. They were suppose to be able to see how functions are made.
Lesson Analysis 1: Identify several strategies the teacher used to orchestrate and promote student discourse in this lesson.
During the lesson the teacher would try to rephrase questions that students did not understand the first time around. She also would try to make the problems more relatable to them. She would turn the situation around to put them in it so they could think through the problem more easily. The teacher also did a good job of breaking the problem down even more. Rather then focusing on the whole overall solution she broke down where the students were getting confused so that then they could understand the whole problem.
Reflective Task 1: What specific actions could the teacher have taken to improve the effectiveness of learning when students are working in groups?
The teacher stated in her interview after the lesson that she allowed students to pick their own groups, so they were working with their friends. To improve the effectiveness of learning in the groups she could have assigned them their groups in a different manner. She stated that she does this usually but took a risk in letting them work with their friends. Another way to improve it would be to have some of the higher level thinkers with the lower level thinkers so they could maybe help teach each other. It would also help for her to walk around the room while they are working in their groups to make sure they are staying on task and getting the most they can out of the activity.
Reflective Task 2: Describe how the teacher's questioning, and the manner in which student responses are handled, contribute or do not contribute to a positive classroom learning environment?
I thought the questioning the teacher did was a good way of getting students to follow the process they should. She asked building questions so that their knowledge continued to grow. I thought the way student responses were handled did not contribut much to a positive learning environment. Students would just shout out answers and no acknowledgement was made on the correct answer or who made it. If a student did not know the answer to the question I do not think they would have been able to pick it up. One way that there was a positive impact is that she did not negatively respond to any of the responses she did hear. This would help students to feel more confident in trying to answer more questions.
I thought this video was very interesting to watch. It seemed that the teacher had a good understanding of how to build on knowledge students already had. I found it very interesting to see how in depth her lesson plan was and all the different aspects of it. She did not keep the students doing the same task the entire class period which helped to keep the students attention. I think I would be able to use the information from this video in my own classroom someday.
Lesson Analysis 1: Identify several strategies the teacher used to orchestrate and promote student discourse in this lesson.
During the lesson the teacher would try to rephrase questions that students did not understand the first time around. She also would try to make the problems more relatable to them. She would turn the situation around to put them in it so they could think through the problem more easily. The teacher also did a good job of breaking the problem down even more. Rather then focusing on the whole overall solution she broke down where the students were getting confused so that then they could understand the whole problem.
Reflective Task 1: What specific actions could the teacher have taken to improve the effectiveness of learning when students are working in groups?
The teacher stated in her interview after the lesson that she allowed students to pick their own groups, so they were working with their friends. To improve the effectiveness of learning in the groups she could have assigned them their groups in a different manner. She stated that she does this usually but took a risk in letting them work with their friends. Another way to improve it would be to have some of the higher level thinkers with the lower level thinkers so they could maybe help teach each other. It would also help for her to walk around the room while they are working in their groups to make sure they are staying on task and getting the most they can out of the activity.
Reflective Task 2: Describe how the teacher's questioning, and the manner in which student responses are handled, contribute or do not contribute to a positive classroom learning environment?
I thought the questioning the teacher did was a good way of getting students to follow the process they should. She asked building questions so that their knowledge continued to grow. I thought the way student responses were handled did not contribut much to a positive learning environment. Students would just shout out answers and no acknowledgement was made on the correct answer or who made it. If a student did not know the answer to the question I do not think they would have been able to pick it up. One way that there was a positive impact is that she did not negatively respond to any of the responses she did hear. This would help students to feel more confident in trying to answer more questions.
I thought this video was very interesting to watch. It seemed that the teacher had a good understanding of how to build on knowledge students already had. I found it very interesting to see how in depth her lesson plan was and all the different aspects of it. She did not keep the students doing the same task the entire class period which helped to keep the students attention. I think I would be able to use the information from this video in my own classroom someday.
Monday, February 15, 2010
Math Applet Review #2
"Understanding Distance, Speed, and Time Relationships Using Simulation Software" from NCTM, e-Examples for 3-5 grade students.
http://standards.nctm.org/document/eexamples/chap5/5.2/index.htm
This math applet helps students to understand functions and representing change over time. Students can place two runners at different points on a running line. Then students push the play button and see how fast the runners run and when they cross paths. There is a graph that is formed from where the students placed the runners and the students can pause to see where each runner is at certain points. The applet is very appealing to the eye because each runner has a different color that represents them so the student can clearly see which runner is where at all times. The students can easily slide the runners to different positions and change the size of their strides. There are also follow up questions for students to consider and help make sure they are understanding functional relationships.
This applet seems to be very helpful for students to understand functional relationships. By being able to manipulate the runners in multiple ways students can really visually see these relationships. If students do not see something the first time they can continually change the placement of the runners for any sort of question they may have. This seems like it would be a good supplemental tool to teaching functional relationships. Teachers can have students answer the questions listed after the activity to see if the students really got what the activity was trying to do. This applet seems to have the potential for the student to grow in this math concept. The student is able to play with different ways they can affect the relationship.
http://standards.nctm.org/document/eexamples/chap5/5.2/index.htm
This math applet helps students to understand functions and representing change over time. Students can place two runners at different points on a running line. Then students push the play button and see how fast the runners run and when they cross paths. There is a graph that is formed from where the students placed the runners and the students can pause to see where each runner is at certain points. The applet is very appealing to the eye because each runner has a different color that represents them so the student can clearly see which runner is where at all times. The students can easily slide the runners to different positions and change the size of their strides. There are also follow up questions for students to consider and help make sure they are understanding functional relationships.
This applet seems to be very helpful for students to understand functional relationships. By being able to manipulate the runners in multiple ways students can really visually see these relationships. If students do not see something the first time they can continually change the placement of the runners for any sort of question they may have. This seems like it would be a good supplemental tool to teaching functional relationships. Teachers can have students answer the questions listed after the activity to see if the students really got what the activity was trying to do. This applet seems to have the potential for the student to grow in this math concept. The student is able to play with different ways they can affect the relationship.
Math Applet Review #1
"Add like Fractions with Circles" from Visual Fractions
http://www.visualfractions.com/AddEasyCircle.html
The objective of this math applet is to help students understand how to add fractions together. By using pictures of circles with parts shaded in the interactive site helps students to see what the fractions are they are adding and what their sum is. Students are given a picture of circles divided into a different number of sections. They have to enter what the number is that the circles are representing. Then students are given a second set of circles and have to enter the number those are representing. Each set of circles is also shaded in with a different color. With both pictures still present the student then is told to find the sum of the two sets of circles. The answer is shown in the circles as well once the student has submitted their answer and it combines the two colors used to still show the two different numbers in the answer. The student can then submit a report of all the one they have gotten wrong or right to the teacher so they can check the work. There is also a button for explanation. If a student does not understand why something is the answer they can click the explain button and it will explain how the program got the answer it did.
This applet seems pretty good at helping students to see what fractions really look like and helping them understand how to count out and figure out the fraction. However, if a student enters a fraction wrong for just one of the addends it doesn't really explain why it is wrong it just tells them to look at it again. It also does not register this as a wrong response. So it could take a student 5 tries to get the addends right but they get the sum right away and it looks like they got the whole problem correct. This applet doesn't seem to promote much in the growth of math content. It basically only shows students if they understand how to make fractions and add them. If a student doesn't understand this concept they are not going to be able to learn how to from this applet. The feedback the program gives for an incorrect sum is simply if the response is too large or too small from the actual answer. This applet would be good for a teacher to use to see how well their students are doing with fractions. From the report of the activity the teacher will be able to see how much more time they need to spend on fractions with their students. This applet cannot be used to try and teach those struggling with fractions.
http://www.visualfractions.com/AddEasyCircle.html
The objective of this math applet is to help students understand how to add fractions together. By using pictures of circles with parts shaded in the interactive site helps students to see what the fractions are they are adding and what their sum is. Students are given a picture of circles divided into a different number of sections. They have to enter what the number is that the circles are representing. Then students are given a second set of circles and have to enter the number those are representing. Each set of circles is also shaded in with a different color. With both pictures still present the student then is told to find the sum of the two sets of circles. The answer is shown in the circles as well once the student has submitted their answer and it combines the two colors used to still show the two different numbers in the answer. The student can then submit a report of all the one they have gotten wrong or right to the teacher so they can check the work. There is also a button for explanation. If a student does not understand why something is the answer they can click the explain button and it will explain how the program got the answer it did.
This applet seems pretty good at helping students to see what fractions really look like and helping them understand how to count out and figure out the fraction. However, if a student enters a fraction wrong for just one of the addends it doesn't really explain why it is wrong it just tells them to look at it again. It also does not register this as a wrong response. So it could take a student 5 tries to get the addends right but they get the sum right away and it looks like they got the whole problem correct. This applet doesn't seem to promote much in the growth of math content. It basically only shows students if they understand how to make fractions and add them. If a student doesn't understand this concept they are not going to be able to learn how to from this applet. The feedback the program gives for an incorrect sum is simply if the response is too large or too small from the actual answer. This applet would be good for a teacher to use to see how well their students are doing with fractions. From the report of the activity the teacher will be able to see how much more time they need to spend on fractions with their students. This applet cannot be used to try and teach those struggling with fractions.
Monday, February 8, 2010
Journal Summary: February (2)
From the journal Mathematics Teaching in the Middle School I read the article "Transitions from Middle School to High School: Crossing the Bridge". The article discusses the different challenges students may face when transitioning from middle school to high school. Students have difficulty with the alignment of instruction and curriculum. If the students start to fall behind after transition from elementary to middle school they will end up falling even more behind going from middle school to high school. The students could also have difficulty with the content from middle school to high school. In high school everything is more focused on a specific area like algebra and geometry, where middle school covers multiple topics. The third challenge is the physciological and social factors of moving to high school. Students can be unmotivated and want to just fit in with their friends. Ways to help ease the transition process for students is to have collaboration between middle school and high school teachers, try to have the content be structured so students don't get lost, and make the classroom culture be easy to address social factors. Finally the article talks about the importance of creating a learning community to find the best ways to help with the transition for students.
Since I also read the article on transitioning from elementary school to middle school I was already a little familiar with some of the topics covered in this article. Similar ideas from this article and my other one is working with other teachers from the other grades to know what students already know and need to know. The idea I took most from this article is creating a professional learning community. I think this would be important to incorporate into your professional career. By creating this community it will help teachers to be accountable for helping students with their transitioning. It is also beneficial because it causes teachers to evaluate how they are doing at helping the students. Most times it seems teachers come up with plans to help students but then never come back to it to see how it really worked. If the learning community continues to meet and evaluate their ideas and plans the better we will become at helping students be successful.
Since I also read the article on transitioning from elementary school to middle school I was already a little familiar with some of the topics covered in this article. Similar ideas from this article and my other one is working with other teachers from the other grades to know what students already know and need to know. The idea I took most from this article is creating a professional learning community. I think this would be important to incorporate into your professional career. By creating this community it will help teachers to be accountable for helping students with their transitioning. It is also beneficial because it causes teachers to evaluate how they are doing at helping the students. Most times it seems teachers come up with plans to help students but then never come back to it to see how it really worked. If the learning community continues to meet and evaluate their ideas and plans the better we will become at helping students be successful.
Journal Summary: February
From the journal Teaching Children Mathematics I read the article "Transitions from elementary to middle school math". This article addressed some of the main things transitioning students struggle with in math and how teachers can help make this struggle less for them. The main place to start in helping the students is for the teacher to be aware of what the students are going through. Students are seeing a difference in instructional method, work expectations, and general difficulty of material. Teachers need to be sensitive to what the students are going through, stress the importance of the teacher-parent relationship, and recognition that the transitioning process is ongoing and not a single step. Teachers should also be communicating across grade levels. It would be beneficial for 6th grade teachers to observe the 5th grade classroom their students will be coming from and vice-versa. This way each could try and incorporate something the others classroom does, such as independent bell work to begin the class, into their own classroom for a short time so it is not as much of a shock when the students move on. A lot of times schedules conflict for this though and maybe teachers could simply videotape their classroom for a day and discuss what they are doing with the other teacher. Sometimes this does not even happen and what a teacher could try and do is watch a general videotape on that grade level to try and get a sense of what may be going on in their classroom. And finally another way to communicate between grade levels is to show examples of student work to show the kind of quality that was expected and material.
I thought this article was very interesting and put a lot of stress on the teacher to help make the transition easier on the students. The article was sure to point out that just because these students are transitioning schools does not mean they have to fail in any area. I also really liked that the article focused on teachers communicating between each other. In many of my other classes they stress communication between teachers and this article just helped to emphasis that point. A teacher could definitely take this into their everyday classroom. They could try to find a cooperating teacher in the elementary school that feeds into the middle school or vice-versa and try to find out more about each other's classrooms. Even if a teacher doesn't have time to try and contact someone else they can find resources online themselves. Whenever they have a free moment they can try to find corresponding videos to help them be better educated on what their students are going to know or need to know. I also think the teacher can be able to identify how different their textbooks are. Especially in the middle school, if the textbook has too much information and small print, maybe try and print it larger for the students a few times, or provide supplemental representations as well. I think this article is very important for teachers to read and be knowledgeable on because it seems too many times teachers place the blame on students for not succeeding during the transition period.
Schielack, Janie and Cathy L. Seeley. (2010). Transitions from elementary to middle school math. Teaching children mathematics. 358-362.
I thought this article was very interesting and put a lot of stress on the teacher to help make the transition easier on the students. The article was sure to point out that just because these students are transitioning schools does not mean they have to fail in any area. I also really liked that the article focused on teachers communicating between each other. In many of my other classes they stress communication between teachers and this article just helped to emphasis that point. A teacher could definitely take this into their everyday classroom. They could try to find a cooperating teacher in the elementary school that feeds into the middle school or vice-versa and try to find out more about each other's classrooms. Even if a teacher doesn't have time to try and contact someone else they can find resources online themselves. Whenever they have a free moment they can try to find corresponding videos to help them be better educated on what their students are going to know or need to know. I also think the teacher can be able to identify how different their textbooks are. Especially in the middle school, if the textbook has too much information and small print, maybe try and print it larger for the students a few times, or provide supplemental representations as well. I think this article is very important for teachers to read and be knowledgeable on because it seems too many times teachers place the blame on students for not succeeding during the transition period.
Schielack, Janie and Cathy L. Seeley. (2010). Transitions from elementary to middle school math. Teaching children mathematics. 358-362.
Wednesday, February 3, 2010
PBL Review: Step 3
The first PBL I looked at was titled "Lounging Around". This PBL was set for grades 7 and 8 in the middle school. Students were told the school would be creating a new student lounge and they would be allowed to design and decorate it. They were given a set budget to work with in order to decorate the entire lounge. The PBL incorporated standards from geometry, measurement, algebra, data analysis and probability, and number operations. They also incorporated other subject areas such as English, science, social science, and fine arts. There were mini lessons also incorporated and working with excel.
I thought this PBL followed what I understand to be a PBL pretty well. The question was an interesting topic for middle school students. It also was very open ended and allowed the students to direct their learning. It seemed like there were a lot of standards and objectives covered in one project. I know it is important to have multiple standards and objectives but it seems like almost too much. If all those standards are being covered and the students are really understanding them it seems like the project would probably take even longer.
The second PBL I read was titled "Operation: Redo the Zoo". This PBL was set for 5th and 6th grade students. This PBL lets students redo the zoo. They are given a set budget and timeline of creating the new zoo. Many different subject areas are covered with this PBL and standards. There will be a large use of technology including TV, VCR, Excel, and Word. There are mini lessons as well to supplement the project.
This PBL was a little more involved then I originally thought PBL's should be. Detail is very important but I thought their problem had too many guidelines for the students to follow. I also did not like the question they gave students because some students may not have ever been to the zoo and animals are not necessarily a common interest in students. I think a better topic could have been picked for the age group selected. Or there should be a field trip scheduled to the zoo before the PBL is assigned.
Both PBL's focused on the students following a budget. While this a good topic to cover I think they could have come up with some more creative ideas. Both also used mini lessons to supplement their projects which I think is a good idea. Both were connected to the real world in some way however I think "Lounging Around" was better applied to something they may have to do themselves one day.
From "Lounging Around" I would try to take away some of the objectives and standards. If they are not directly assessed in the rubric then I don't see there being a need for them. From "Operation: Redo Zoo" I would give the students a different problem involving the zoo. Maybe they could create the most eco-friendly zoo possible with no need to follow the budget. Students do not always have to be under a specific budget for every PBL.
It appears that math is the main focus of both PBL's. I think the math expected is fair for the ages of the projects. The math from "Operation: Redo Zoo" may be a little higher then the grade level selected but it could be adapted for younger ages.
I thought the assessments in "Lounging Around" were very good at focusing on what needed to be evaluated. However in their overall rubric I thought they had too much in it. This is probably because they had so many standards and objectives covered but I feel there is too much detail within the rubric. For "Operation: Redo Zoo" I was very impressed with their assessments. I liked they were checking their math skills along the way with the PBL and the final rubric seemed to cover all the areas necessary to tell how well a student understood the project.
I thought this PBL followed what I understand to be a PBL pretty well. The question was an interesting topic for middle school students. It also was very open ended and allowed the students to direct their learning. It seemed like there were a lot of standards and objectives covered in one project. I know it is important to have multiple standards and objectives but it seems like almost too much. If all those standards are being covered and the students are really understanding them it seems like the project would probably take even longer.
The second PBL I read was titled "Operation: Redo the Zoo". This PBL was set for 5th and 6th grade students. This PBL lets students redo the zoo. They are given a set budget and timeline of creating the new zoo. Many different subject areas are covered with this PBL and standards. There will be a large use of technology including TV, VCR, Excel, and Word. There are mini lessons as well to supplement the project.
This PBL was a little more involved then I originally thought PBL's should be. Detail is very important but I thought their problem had too many guidelines for the students to follow. I also did not like the question they gave students because some students may not have ever been to the zoo and animals are not necessarily a common interest in students. I think a better topic could have been picked for the age group selected. Or there should be a field trip scheduled to the zoo before the PBL is assigned.
Both PBL's focused on the students following a budget. While this a good topic to cover I think they could have come up with some more creative ideas. Both also used mini lessons to supplement their projects which I think is a good idea. Both were connected to the real world in some way however I think "Lounging Around" was better applied to something they may have to do themselves one day.
From "Lounging Around" I would try to take away some of the objectives and standards. If they are not directly assessed in the rubric then I don't see there being a need for them. From "Operation: Redo Zoo" I would give the students a different problem involving the zoo. Maybe they could create the most eco-friendly zoo possible with no need to follow the budget. Students do not always have to be under a specific budget for every PBL.
It appears that math is the main focus of both PBL's. I think the math expected is fair for the ages of the projects. The math from "Operation: Redo Zoo" may be a little higher then the grade level selected but it could be adapted for younger ages.
I thought the assessments in "Lounging Around" were very good at focusing on what needed to be evaluated. However in their overall rubric I thought they had too much in it. This is probably because they had so many standards and objectives covered but I feel there is too much detail within the rubric. For "Operation: Redo Zoo" I was very impressed with their assessments. I liked they were checking their math skills along the way with the PBL and the final rubric seemed to cover all the areas necessary to tell how well a student understood the project.
PBL Review: Step 2 (article: How to Buy a Car 101)
"How to Buy a Car 101" from Mathematics Teaching in the Middle School was a very interesting article. In the article the PBL assignment of buying a car is assigned. Students were told Mr. Jones needed to buy a car but he could only spend a certain amount of money a month. He also needed a dependable car because he drove a long way to work each day and his wife wants a car that looks nice. The students had to take this information and research to find Mr. Jones a car. The students were given 2 weeks to complete the project and then presented their choice to the class in a powerpoint. Local car dealerships also came to the school and showed students possible cars they may want to choose. This helped to make the PBL come even more to life for the students and peek their interest. The teacher was also able to cover 4 state standards from this one assignment. The article also felt the most beneficial aspect of the PBL was being able to draw students into the problem who normally get bored and detached from problems in the textbook. The article also gave tips for teachers who are doing a PBL for their classroom for the first time. It suggests that is a good idea to make an example of the final product so that you as a teacher understand the process. It also says to involve students in the making of the units for the future. If you ask what they are struggling with they will be able to help come up with ideas to help themselves. And finally the article says to not do the project for the students. PBL's make the teachers take a new role in the classroom and you have to try the best you can to let the students do the learning without taking it over from them.
I really liked this article and activity. I thought the activity was very interesting, especially for middle school students. They are beginning to get to the age of almost being able to drive so their interest in cars is increasing and this is a possible situation they could one day be in. Some ways I would maybe improve the activity is to try and have the students try and contribut more to the making of the problem, and also not to show them an example. I think it is a good idea for the teacher to do the problem themselves but they should not show it to the students. By doing this it takes away from the students own originality and almost makes it easier for them to complete the PBL. It was good that the teacher gave the students a set deadline for the PBL and it seems the teacher did a good job of sitting back some and letting the student direct the classroom more for the project. The teacher was able to point the students in the right direction for resources and this helped the students to be successful. I think this was a very strong activity for problem-based learning because it used a real-life situation, made the students figure out what they knew and needed to know, truly figure out the problem, find numerous cars that could be a solution, and then decide on the best one for the problem. There is no set right or wrong answer but the students can still be successful for the project.
Flores, C.A. (2006). How to buy a car 101. Mathematics teaching in the middle school 12(3), 161-164.
I really liked this article and activity. I thought the activity was very interesting, especially for middle school students. They are beginning to get to the age of almost being able to drive so their interest in cars is increasing and this is a possible situation they could one day be in. Some ways I would maybe improve the activity is to try and have the students try and contribut more to the making of the problem, and also not to show them an example. I think it is a good idea for the teacher to do the problem themselves but they should not show it to the students. By doing this it takes away from the students own originality and almost makes it easier for them to complete the PBL. It was good that the teacher gave the students a set deadline for the PBL and it seems the teacher did a good job of sitting back some and letting the student direct the classroom more for the project. The teacher was able to point the students in the right direction for resources and this helped the students to be successful. I think this was a very strong activity for problem-based learning because it used a real-life situation, made the students figure out what they knew and needed to know, truly figure out the problem, find numerous cars that could be a solution, and then decide on the best one for the problem. There is no set right or wrong answer but the students can still be successful for the project.
Flores, C.A. (2006). How to buy a car 101. Mathematics teaching in the middle school 12(3), 161-164.
PBL Review- what is it and where do I use it?
Problem-based learning deals with a poorly structured problem that is related to current events or student interest. Students are to figure out what they know about the problem and what they need to find out about it. From this point the students try to really define what the problem really is. This is a very important thing for the students to complete because if they don't truly understand the problem they won't be able to complete it or come up with sensible solutions for it. Students should then conduct research and begin to come up with many different solutions to their problem. Then the group should reason what solution is the best for their problem.
Problem-based learning can be used in any classroom. It helps students acquire factual knowledge, become good at general concepts in the classroom, and learn ways to solve a problem that can be applied throughout their life. Completing problem-based learning problems helps students to become better problem solvers, and be active learners. Rather than having a teacher lecture and tell them what to know and find out the students do that on their own. By having the problem apply to real-life situations helps the students be more invested in the problem and really want to find solutions. It is important for the teacher to act as a facilitator for the students. The teacher should not be giving the students answers but pointing them in the right direction of where to find the answers. PBL's also help students learn how to work in groups. Since the problem is so poorly presented all students in the group are going to have to do research to try and find out what the question is really wanting to know.
Problem-based learning can be used in any classroom. It helps students acquire factual knowledge, become good at general concepts in the classroom, and learn ways to solve a problem that can be applied throughout their life. Completing problem-based learning problems helps students to become better problem solvers, and be active learners. Rather than having a teacher lecture and tell them what to know and find out the students do that on their own. By having the problem apply to real-life situations helps the students be more invested in the problem and really want to find solutions. It is important for the teacher to act as a facilitator for the students. The teacher should not be giving the students answers but pointing them in the right direction of where to find the answers. PBL's also help students learn how to work in groups. Since the problem is so poorly presented all students in the group are going to have to do research to try and find out what the question is really wanting to know.
Thursday, January 28, 2010
Problem Solving Article: Hot Wheels
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.
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.
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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