Mathematics can be implemented into many different areas of the world, like: stores, prices, outdoors, collecting data, etc. This article discusses the different ways that educators have implemented different mathematical skills and learning examples into the great outdoors. When students are engaged in something that is related to the real world (real world situations) the students have a better understanding and grasp more out of what they are learning. The trails that are talked about in this article present many different tasks that teachers can present to students. For example, trails display different sequences that students could pick up on and need to draw or develop into a mathematical problem.
Also discussed in the article was how the students were creating their own trails that would be walked at a later date in the semester. I found this to be quite interesting because the students would be in charge of every piece of their trail. They would create the directions used and how many steps walked would need to take place. The students were also aware of the different shapes that were around them and had to measure them to in different ways.
English, L. D., Humble, S., & Barnes, V. E. (2010). Trailblazers. Teaching children mathematics 16 (7), 402-410.
Monday, March 22, 2010
Assessing Understanding through Reasoning Books - Article
In today's society, people are taught that there is usually one correct answer and that this answer should be a yes or no response. This is what children of all ages are thinking, so they do not want to think and challenge themselves to what a teacher could be asking for. In this article, it was talked about that teachers are assuming that their students know what he/she is talking about and that they will remember everything they are taught over a short amount of time. But this is far from the truth; many students do not know what the teacher is talking about without many examples and follow up questions. Many students are more focused on getting an answer and do not know how or why that answer was chosen. The students needed an answer and wrote one down without have a justification for the answer they chose.
When teachers are trying to assess what the students have taken away from a lesson, they need to be able to have the students explain what they have done. Students need to be able to create an argument and justify why they worked out a problem the way they did. The teacher needs to help the students with this process and help the students develop the deeper thinking skills needed to communicate the mathematical terms and procedures correctly.
The reasoning books that were used by the teacher were filled with all sorts of mathematical problems; all of the problems needed to be justified and communicated in a way that would be understood by the teacher. Some of the problems presented didn't have a lot of information, but the students had to be able explain without help how they came up with the different answers.
Roberts, S. K. & Tayeh, C. (2010). Assessing understanding through reasoning books. Math teaching in middle school 15 (7), 406-414.
When teachers are trying to assess what the students have taken away from a lesson, they need to be able to have the students explain what they have done. Students need to be able to create an argument and justify why they worked out a problem the way they did. The teacher needs to help the students with this process and help the students develop the deeper thinking skills needed to communicate the mathematical terms and procedures correctly.
The reasoning books that were used by the teacher were filled with all sorts of mathematical problems; all of the problems needed to be justified and communicated in a way that would be understood by the teacher. Some of the problems presented didn't have a lot of information, but the students had to be able explain without help how they came up with the different answers.
Roberts, S. K. & Tayeh, C. (2010). Assessing understanding through reasoning books. Math teaching in middle school 15 (7), 406-414.
Wednesday, March 3, 2010
Video - Lesson on Graphs
1. What are the mathematical ideas in this lesson and how significant are they?
While watching this particular video, I noticed that the teacher was incorporating many different ideas into her lesson. She began the lesson by showing her students the problem on the overhead projector and reading it aloud to them. Then she posed the question about why the problem had an 'x' and what that 'x' meant. By the teacher asking her students these questions the students are able to think about what it meant here and why it would be in the problem. Many of the students had answers about what they thought it was and what they believed was meant by the 'x'. This is significant because the teacher is explaining to the students why it is important and how the students can develop an understanding for what they are trying to find. The 'x' represents a number, any number, that can be determined by completing the math problem. Graphing the problem is also an idea that the teacher was trying to represent. By having the students graph the problem, they are able to visually see what number or numbers can be represented by the 'x'. By graphing, the students can eliminate numbers that the 'x' cannot be, which will help them formulate a number that can be represented.
2. What evidence is there that students have learned the mathematics being taught?
The evidence that is present in the videos, is that the students are able to explain in great detail how they came up with their answers and how they created the graphs. The one student who was explaining the graph and how the ski company's profits changed over the course of five plus years, was able to show the rest of the class how she came up with her numbers and how her graph represented the numbers that she had come up with. The students were able to create their own problems, which shows that they understood and learned what was important and needed to be present in the problem.
3. How effectively are student mistakes addressed and misconceptions dealt with?
From the clips of video that I watched, the teacher seemed to address the students mistakes and misconceptions every time. This way the teacher was able to see where the students were getting the mistakes or misconceptions. If the teacher had let the mistakes or misconceptions continue not only would it hurt the students because they would be learning the wrong information, but it would also hurt the teacher because she might have to reteach something over again. This would hurt the teacher because she could have stopped the students from continuing these mistakes in the beginning. Also be addressing the mistakes and misconceptions the teacher is showing the students that others in the class are making the same mistakes and that it is alright as long as they are addressed in an orderly way.
While watching this particular video, I noticed that the teacher was incorporating many different ideas into her lesson. She began the lesson by showing her students the problem on the overhead projector and reading it aloud to them. Then she posed the question about why the problem had an 'x' and what that 'x' meant. By the teacher asking her students these questions the students are able to think about what it meant here and why it would be in the problem. Many of the students had answers about what they thought it was and what they believed was meant by the 'x'. This is significant because the teacher is explaining to the students why it is important and how the students can develop an understanding for what they are trying to find. The 'x' represents a number, any number, that can be determined by completing the math problem. Graphing the problem is also an idea that the teacher was trying to represent. By having the students graph the problem, they are able to visually see what number or numbers can be represented by the 'x'. By graphing, the students can eliminate numbers that the 'x' cannot be, which will help them formulate a number that can be represented.
2. What evidence is there that students have learned the mathematics being taught?
The evidence that is present in the videos, is that the students are able to explain in great detail how they came up with their answers and how they created the graphs. The one student who was explaining the graph and how the ski company's profits changed over the course of five plus years, was able to show the rest of the class how she came up with her numbers and how her graph represented the numbers that she had come up with. The students were able to create their own problems, which shows that they understood and learned what was important and needed to be present in the problem.
3. How effectively are student mistakes addressed and misconceptions dealt with?
From the clips of video that I watched, the teacher seemed to address the students mistakes and misconceptions every time. This way the teacher was able to see where the students were getting the mistakes or misconceptions. If the teacher had let the mistakes or misconceptions continue not only would it hurt the students because they would be learning the wrong information, but it would also hurt the teacher because she might have to reteach something over again. This would hurt the teacher because she could have stopped the students from continuing these mistakes in the beginning. Also be addressing the mistakes and misconceptions the teacher is showing the students that others in the class are making the same mistakes and that it is alright as long as they are addressed in an orderly way.
Monday, February 15, 2010
Math Applet - Grades 3-5
Math Applet grades 3-5:
http://standards.nctm.org/document/eexamples/chap5/5.5/index.htm
Summary:
Collecting and understanding data is the main concept in this activity. This particular example used weather as a main source, but any other type of information can be used. Students also learn how to label and create the graphs or spreadsheets. These graphs and spreadsheets can be compared with one another to see what every person interpreted and understood. There are different ways to intemperate mathematical equations, and students can have different answer but these answers could all be correct.
Critique:
The ideas given in this example were really good, and I liked how they could be manipulated and changed depending on what is being taught. Different tasks that will be learned are: how to collect data, organize the data and create graphs of the information collected. All these tasks are important in the development of today's society. These applet also allows students to create different types of data and interpret them differently. After the students have collected the data they can create different mathematical problems and equations that have to do with data.
http://standards.nctm.org/document/eexamples/chap5/5.5/index.htm
Summary:
Collecting and understanding data is the main concept in this activity. This particular example used weather as a main source, but any other type of information can be used. Students also learn how to label and create the graphs or spreadsheets. These graphs and spreadsheets can be compared with one another to see what every person interpreted and understood. There are different ways to intemperate mathematical equations, and students can have different answer but these answers could all be correct.
Critique:
The ideas given in this example were really good, and I liked how they could be manipulated and changed depending on what is being taught. Different tasks that will be learned are: how to collect data, organize the data and create graphs of the information collected. All these tasks are important in the development of today's society. These applet also allows students to create different types of data and interpret them differently. After the students have collected the data they can create different mathematical problems and equations that have to do with data.
Math Applet - Grades K-2
Math Applet - Thinking Blocks K-2
http://www.thinkingblocks.com/
Summary:
Thinking Blocks was developed by teachers as an interactive activity that students could use to work on many different mathematical ideas. Students are able to work on and with addition, subtraction, multiplication, division, and ratios. Different types of mathematical problems can be practiced, such as part whole and word problems. You are able to compare and contrast the ideas in the practice activity, with one another. When working on a problem, if someone keeps getting stuck or confused, you can watch a video with examples and complete instructions on how to complete the problem. The videos also highlight important information that will help students see exactly what is important in the mathematical problem.
Critique:
Students are able to pick a what exactly they want to work on, a teacher can also provide the student with what they need to work on. The think blocks also allows students to practice at different levels the same activity. I liked how the videos about the problems were shown next to the actual problems, that way a student can still be looking a the problem and watching an example of how it is done at the same exact time. There are also many different mathematical procedures that take place, which makes this applet worth while for teachers and students. There are many different types of mathematical terms and procedures that are taking place through the thinking blocks. As a teacher, you can also create and print out examples that you think up. This can allow teachers to create problems that are created for the intent of actual students, who might need help in a particular mathematical area.
http://www.thinkingblocks.com/
Summary:
Thinking Blocks was developed by teachers as an interactive activity that students could use to work on many different mathematical ideas. Students are able to work on and with addition, subtraction, multiplication, division, and ratios. Different types of mathematical problems can be practiced, such as part whole and word problems. You are able to compare and contrast the ideas in the practice activity, with one another. When working on a problem, if someone keeps getting stuck or confused, you can watch a video with examples and complete instructions on how to complete the problem. The videos also highlight important information that will help students see exactly what is important in the mathematical problem.
Critique:
Students are able to pick a what exactly they want to work on, a teacher can also provide the student with what they need to work on. The think blocks also allows students to practice at different levels the same activity. I liked how the videos about the problems were shown next to the actual problems, that way a student can still be looking a the problem and watching an example of how it is done at the same exact time. There are also many different mathematical procedures that take place, which makes this applet worth while for teachers and students. There are many different types of mathematical terms and procedures that are taking place through the thinking blocks. As a teacher, you can also create and print out examples that you think up. This can allow teachers to create problems that are created for the intent of actual students, who might need help in a particular mathematical area.
Wednesday, February 10, 2010
Classroom Characters Coach Students to Success Article
The article I read was, Classroom Characters Coach Students to Success, in the Teaching Children Mathematics magazine. This article discussed different ways to incorporate characters into the learning environment and have these characters be beneficial to the students. The article also discussed a 1-2-3 Time that seems to work with many different teachers. The 1-2-3 Time allows students to go to different stations that incorporate mathematical skills in 3 different ways. These different ways could include, but are not limited to: computer activities, reading and writing experiences, or instructional material.
The coaching aspect of the article, was a computer related character that was part of a mathematical program. This character would help encourage the student to continue to work and try different computations on a particular math problem. The math problems on the computer were relevant to what the students are learning and are at the appropriate level for the individual.
Math learning games are another way that allow students to have a hands-on experience with math and to get a better understanding for how something is meant to be. Math learning games allow the students to work in groups to create a solution that everyone can agree on. Math learning games also provide students with the ability to think outside of the box, they are able to formulate many different answers for one particular problem, but many answers could be correct.
Edwards, S., Maloy, R., and Anderson, G. (2010). Classroom characters coach students to success. Teaching children mathematics 16 (8), 342-348.
The coaching aspect of the article, was a computer related character that was part of a mathematical program. This character would help encourage the student to continue to work and try different computations on a particular math problem. The math problems on the computer were relevant to what the students are learning and are at the appropriate level for the individual.
Math learning games are another way that allow students to have a hands-on experience with math and to get a better understanding for how something is meant to be. Math learning games allow the students to work in groups to create a solution that everyone can agree on. Math learning games also provide students with the ability to think outside of the box, they are able to formulate many different answers for one particular problem, but many answers could be correct.
Edwards, S., Maloy, R., and Anderson, G. (2010). Classroom characters coach students to success. Teaching children mathematics 16 (8), 342-348.
Dividing Fractions and Problem Solving Article
I read the article entitled, Dividing Fractions and Problem Solving, from the February issue of Mathematics Teaching In The Middle School. This article talked about many different ways to use fractions in the classroom and how to enforce the ways that fractions are used. There are many different ways to model fractions; one way is to use picture and another way is to use words. When using words to address fractions, the teacher needs to make sure that they are explaining what they are trying to get across to the students. Fractions can be stated in more than one way, and a teacher needs to make sure that they are explaining to their students the different ways these fractions can be stated and use them in the correct way. The teacher also reinforced the fractions by having the students create fraction circles that were of different values, but also different colors.
The article also talked about two different models that are used when talking about fractions. One model is the Measurement Model; in this model students are given an exact number that is going to be used in a problem, and they need to figure out the exact answer. Where as in the second model, the Partitive Model, the students are trying to determine how many groups of something can go into each group. Both of these models work with whole numbers, fractions, partial numbers, and remainders.
When using fractions and learning about them, I find it easier to work with something that is hands-on or visual. Many of the examples in the article refereed to different figures that had student work and examples of how they got to different answers. Fractions are sometimes a difficult concept to grasp, so teachers need to stick with it and help every student. Talking about the part-whole model allows students to see what fractions are doing and how they are used in the real world.
Cramer, K., Monson, D., Whitney, S., Leavitt, S., and Wyberg, T.(2009). Dividing fractions and problem solving. Mathematics teaching in the middle school 15(6), 338-346.
The article also talked about two different models that are used when talking about fractions. One model is the Measurement Model; in this model students are given an exact number that is going to be used in a problem, and they need to figure out the exact answer. Where as in the second model, the Partitive Model, the students are trying to determine how many groups of something can go into each group. Both of these models work with whole numbers, fractions, partial numbers, and remainders.
When using fractions and learning about them, I find it easier to work with something that is hands-on or visual. Many of the examples in the article refereed to different figures that had student work and examples of how they got to different answers. Fractions are sometimes a difficult concept to grasp, so teachers need to stick with it and help every student. Talking about the part-whole model allows students to see what fractions are doing and how they are used in the real world.
Cramer, K., Monson, D., Whitney, S., Leavitt, S., and Wyberg, T.(2009). Dividing fractions and problem solving. Mathematics teaching in the middle school 15(6), 338-346.
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