Polygon Properties: What is Possible? - Article

This article had a lot of information in it about all different types of polygons and geometric shapes. There are three levels of geometric thinking (visual, descriptive, and informal deduction) that are developed and strengthened over the grade school years. Level 0, visual level, the students are stating what a shape is and use nonverbal usage. Students make their discussion and thoughts heard aloud based on what they see and have seen in the past. Level 1, descriptive level, the students will expand on what a shape is and how it is categorized. The example in the article states, "a triangle is no longer a triangle because it looks like one, but because it is a closed figure that has three sides and three angles." This level shows how the students knowledge on the shape and knowledge from the class has helped them. Level 2, informal deduction, students are able to come up with a logical property for why a shape is in the category it is placed in. This was interesting to me, because I never knew that these levels existed. But I like how I they build on each other and promote higher thinking.

One activity that was presented in the article was, Shape Sorting. This activity gave each group of students a bag full of different shapes both regular and irregular. Then the students had to sort them, using any sorting technique they wanted. It gave the students the control in the sorting process. Once they were sorted in the first area, the students would sort them again from the groups they were originally in. I liked how the students each had a different view on what they thought the next sorting area should be. This activity shows so many opportunities to talk about and describe shapes of all kinds.

I liked how with the Shape Sorting activity the teacher made connections to the students about what was happening and what they needed to get done. The students would create their own shape, of any kind, and have to sort it into one of the groups that were created in the activity. They also had to explain why the shape was in this category. (Many process standards were being met in this activity!)

Another activity that was discussed was Riddle Activities which was something new to me, but I found that it was a good way to learn about shapes and shape types. The students were given riddles about a shape and had to determine which shape it could be. The students used Geoboards to show what the shape could be and to show "proof" of an answer. Riddles started out simple and sometimes obvious, but become generally harder and more difficult. The students would use trial and error to determine the shape, and in some cases an answer was not found. Students gradually found out that not every riddle could be solved. I like how this activity has the students working and thinking, even if the riddle is not solved.

Robichaux, R.R. & Rodrigue, P.R. (2010). Polygon properties: What is possible? Teaching children mathematics 16 (9), 524-531.

Manipulative Activities

This was a completely different day, over all the other days we have had in the classroom this semester. Today was the first day where each and every person in class was able to participate in the different manipulative activities that were around the room. I seemed to learn, like each and everyday in class, more about how to properly use and incorporate these different activities to best suit the students.

The one major thing I noticed about doing these different activities is that the teacher would constantly be walking around the room to make sure each student was helping work with the different manipulative. I think it could be hard to determine how to hold every student accountable, so having a check list or just observing the students might be enough. I also think that by having each group member at a manipulative table switch off who was writing down the information was another great idea. This way when the papers are collected, the teacher can see exactly who was writing and see the different handwriting analysis on the paper.

All types of games and activities have to have some kind of catch phrase to attract the buyers to them. But the phrase, "Hands-On" has been eliminated and has been brought back to life with, "Hands-On, Mind-On." I think that the, "Hands-On, Minds-On" is a much better way to promote a product. This way it is an activity that is done with your hands, but at the same time you are thinking of all the possibilities that can come with these activities. For example, the unifix cubes or the snap cubes, those were both hands-on and mind-on. You obviously had to use your hands to figure out different ways to use the cubes, but you also had to use your mind to see how and why certain situations took place. There is more thinking involved when something is mind-on because students not just playing with the manipulative.

The process standards are used in many ways with all the different manipulative activities. Students and teachers will be able to determine different types of reasoning and proofs to try and prove a point about how a manipulative could work. Communication, this will take place all the time, when students are talking to one another about their ideas for how a manipulative could work. There are always many different ways for them to be used. Connections could be made between subjects or lessons being taught, and they could be made about the different types of manipulative activities being used. For example, I was making connections between the unifix cubes and the snap cubes on how they were similar, yet different from one another. Students will be able to problem solve and become more aware of the problem solving when working with hands-on minds-on activities. They allow students to think in a more abstract and out-of-the-box way that can help them develop. Representation, how was this not used in the manipulative activities? Every thing that was taking place with the manipulative activities was being represented in a different way and was able to mean something else at some point.

I really enjoyed doing these activities, I feel like I know more about them and how to use them in the correct way and how to give proper instruction (when it is given) to the students.

Technology Reflection

When didn't we use technology this semester? We have used so much technology everyday this semester, some of which I have never used before. The smart board was the coolest new piece of technology that I used this semester. Smart boards were a lot easier to use than I had anticipated. It was interesting to see all the different things you could do with the smart board. I was happy that I stayed in the classroom a few weeks ago when the other education classroom came in, because I was able to see all the different things that could be done on the smart board, that is not directly related to mathematics. It was cool to see that a smart board could be used with all different students at all different grade levels. There was so much that could be done on the smart board. I thought it was interesting to see that the smart board could use sound effects to say different letters.

Computers, another piece of technology that we used each and everyday. Computers have so many possibilities for teaching and for showing many examples for mathematics. I liked how much work we had to do on the computers. I felt that the way of communication of Google Docs was a little hard at first, but as the semester went on I began to understand it a little more.

Just recently we used calculators, which I have personally used since I was in grade school. But the type of calculator that was used was a little different from what I had back in those days. So these calculators where interesting to see and use. I was hoping that we would see a calculator that would hook up to a computer and see how it is used in that form.

Problem Errors Reflection

I really liked the problem errors that we worked on. I felt that it was good to see how students can come up with different answers to general math questions. I also found it interesting that if a teacher gives too much information about a particular math problem, the students could just be using the wrong rules on the problem. I think that doing this will help me to space out information about a given topic, so students are not overwhelmed with too much information.

I liked how the entire packet of error problems, gave a large perspective of what could happen with a real student. Also the variety of problems that were given also gave me an understanding of what was happening and how to teach the material in a better way. I believe that the packet hit on a lot of math topics, that seem easy and understandable to everyone, but that do have a way of getting confusing. I felt that seeing all these different ways was helpful and gave insight about what to look for as a future teacher.

I think that one element that I would have liked to see throughout the packet, would be how two students interpreted the problems. I know it would make the packet longer, but I believe it would help us in the long term. This way we can see two different ways that a problem was interpreted by the students.

Map Scale, Proportion, and Google Earth

This article dealt with a lot of different elements to the world outside of math, and the world inside of math. This article took the concept of distance and approximation to a whole new level. Students needed to use Google earth and the different formulas and elements learned in math to measure real life images of the world. Students learned many different terms that would relate to the math terms, but also related to the maps that they were looking at. For example, scale was used to show the different ways to weigh objects, but then it is also used to determine the different distances on a map (Google earth images). Students also found it interesting to find their own homes and complete measurements on a familiar location.

I really liked how the students were using Google earth to find the different mathematical elements being studied. I found this to be a great real life situation concept that students could use later on in their lives as well as at the present time. It was interesting to see how the students learned the different forms of the scaling. For example, the students noticed the differences of the inside and outside of a football field. This helped show that measuring and knowing mathematical formulas is beneficial to students understanding of the world.

Roberge, M. & Cooper, L. (2010). Map scale, proportion, and Google earth. Mathematics teaching in the middle school 15(8), 448-457.

A Foxy Loxy and a Lallapalagram - Article

This was a very interesting articles pertaining to how there is a different language in the mathematical world, to the world of other subjects. Many of the students discussed in this article did not have English as a first language, many of them were learning it as a second or third language. So when the teachers would ask for specific information about a particular shape some students would answer in their native language. Students were able to say what a shape was, but were not able to speak about what the shape was mathematically called. For example, students knew square, but when the teacher turned it and held it by a corner, the students called it a diamond. Students were also not able to comprehend what was being explained to them when the teacher was explaining details to them.

I found it extremely interesting in the way that the students explanation of information to the teacher. They would state the different principles that a rectangle had, and the teacher would try and have them elaborate on what they stated. This way the students understood that they needed to use mathematical terms that were descriptive of what was being learned. For example students needed to explain what 'straight' meant and how they could expand on what they are trying to say.

Wilson, J. (2010). A foxy loxy and a lallapalagram. Teaching children mathematics 16(8), 492-499.

Assessment - Tests and Quizzes Article

The article that I read had to do a lot with creating assessments and how they work with the students and the teacher. Quizzes should be given throughout a lesson so a teacher can see how students are doing with the material that is being taught. If the teacher just gives a test, then the students do not have any practice, besides work from the class, about what is being taught. Also, quizzes should be planned and unplanned throughout the lesson. Students should know about quizzes and be prepared for them, but they should also not know about them, so the students should always be keeping up with the material. Tests, on the other hand, should always be planned and the students should have plenty of time for questions and practice of the material. Both tests and quizzes should be made up of all different types of problems: open-ended questions, word problems, and basic problems. The article I read focused on the open-ended questions and how those types of questions allow students to think critically. For example, if you went into a store and bought $17.57 worth of groceries, and you gave the cashier a $20.00 bill. How many different ways can the cashier give you your change? This is considered an open-ended question because there is not just one correct answer. Students will be able to come up with many different answers for the statement to come true.

Leatham, K.R, Lawrence, K., Mewborn, D. (2005). Getting started with open-ended assessment. Teaching children mathematics. 11 (8), 413-419.

Trailblazers - Articles

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.

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.

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.

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.

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.

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.

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.

Problem Based Learning - Review of a Website

According to the website, listed below, problem based learning can come in many forms of communication to those learning it. Problem based learning in many respects can be based on real world experiences and real world issues that are talking place. This way students are more willing to learn what is being taught to them. Real world situations also make the students feel as though it is worth their wild to learn about it because it could be useful one day. In many respects, students are learning on their own with problem based learning, and only use the teachers help when they need it most. Some teachers are afraid to use problem based learning in their classrooms because the students would not be learning the tasks and skills needed for the future. With students working on their own, they are becoming more discipline and are more willing to try new things. Students are able to think outside of the box and implement ideas on how to solve a problem.

The article/website that I choose to use, didn't have many activities listed in it. But the activities that is does provide talk about how they used real world problems and examples with the students. This way the activities had meaning for the students and for the teacher. Everyone was able to participate in the activities and they developed new ways to create problems, by using the same objects and activities from before.

http://www.pbli.org/pbl/pbl.htm

Problem Based Learning - Sakai Link

There are many different types of learning styles, and these styles work differently for the students that are in classrooms. When it comes to problem based learning, the students are more in tune with what is being presented to them. Instead of looking to memorize facts about a subject, problem based learning provides students with the skills to make meaningful answers and discussions about a subject. Students also learn more social and developmental skills because they are thinking in a newer way that provides them more opportunities to talk and discuss a situation. Generally speaking, teachers divide the students up into groups of four or five and have them work on an assignment together. By working with other students, the students are able to collaborate and develop new ideas about how to work with a problem that is presented to them. When a teacher is teaching the class with problem based learning, the teacher should have the students help him/her create the problems that will be on the exam/test. This way the students know what they need to work on and what they need to work harder on. Problem based learning also provides students the opportunity to think outside of the box and come up with many different ways to answer any given question. This type of learning never leaves a student, but teachers don't use it in their classrooms at all times.

http://www.ntlf.com/html/pi/9812/pbl_1.htm

Communication according to NCTM

Speaking and communicating in a clear and convincing way is essential in life. In mathematics speaking and communicating helps the students share examples and ideas as well as making clear something they have learned. Through communication you are able to accomplish four major tasks: 1) organize the ideas you have, 2) use mathematical thinking, 3) evaluate and analyze what you have shared as well as the ideas of those around you, and 4) you use mathematical language. Many students find that mathematics is pointless and not going to help them in the real world. Teachers need to make the math more relatable to students, and make it important for them to learn not only for a test, but also for the real world.
Organizing mathematical communication is important and needs to be clear for students to understand it. Students have been taught mathematics skills from a very early age and the mathematics that they will learn will become more and more complex, so they need to understand how it needs to be organized. Students need to be taught how to formulate questions and then understand how to support their answers to these difficult questions. Students also need to be given different types of examples that are not just worksheets and one type of problem. A teacher should be able to show the students diagrams and symbols that are used in mathematics. This will allow the students to reflect on what is happening and allow them to use some type of higher order thinking for what is happening in the math problem.
Students should constantly be working on how to analyze and understand data that is presented to them. Not all students are going to be on the same page mathematically when they are learning new skills and forms of working, so students need to work with one another to learn and teach a skill that will be around for a long time. When students are working together they are able to bounce ideas off of one another and see what ideas work and which ones don’t. They are able to discuss what is happening in a situation and are able to communicate it in a mathematical way. Talking with other students about a mathematical idea or problem will allow all the students who are having difficulty understanding, more help to develop the area that is difficult.

Communication Article

When people look at an object they can see completely different things. As a teacher, you need to be able to draw or write something on the board that will mean the exact same thing to all your students. If something means something different from each student, the students will not fully understand what is being expected of them or what they are learning. Communicating mathematically is difficult to begin with, so teachers need to be able to communicate these features in a way that the students will understand. Once students reach the sixth grade, the mathematical skills they will be using are more complex than those used in the lower grades. Yet the mathematical skills they are learning in the sixth grade are some of the skills that the students will still use until college and beyond. So teachers need to be able to communicate these skills and terms in a collective way that is not changed in a large way. When a math teacher shows his/her students a picture of a geometric shape or object, the students might see two different objects. One person might see a square where as the other person sees a rhombus. These objects and shapes need to be taught to the students in a way that will make them understand what is being shown to them. If students are asked to draw an object and then explain how many other objects might fit into the original object, they could have difficulty if the object is not correctly drawn. Teachers and their students need to have an open line of communication in every subject, but especially mathematics because many students find it to be a difficult subject. The students an d the teacher need to be able to reflect and discuss in detail about what is being taught and learned by the students. This will help see how deep the students are understanding the material and how well they are comprehending it.

Article on Assessment

When a teacher is conducting an assessment the teacher needs to be sure that every studnet in the classroom understands what is being taught and what everything taht will be on the assessment means. For example if a student is having trouble with their numbers in word form from the numerical form. This could make it difficult for the student to complete an assessment of any kind. This is what the issue was in the article I read, the student had English as a second language and was unable to differenciate from the two different ways the numbers were presented. The article also talked about how a teacher needs to be able to show students the different skills that they will be working on for the test and the different ways that the questions will be presented on the test. Students have trouble with math problems that they have not seen the teacher use in his/her classroom before. This makes it difficult for the student to comprehend what is on the assessment. Teachers should use mock questions that are similar to the ones that will be on the assessment so that the students are prepared for the assessment that will be given to them at the end of the lesson.

Assessment

Assessments help a teacher understand and see where students are in their particular learning area. These assessments can be done in many different subject areas and can be created by the teacher or help from other sources. According to the Assessment Principle reading an assessment should focus on what the students are leraning at a particular time. This will show a teacher what students are understanding and what they are comprehending at a specific time. The assessments should not cover material taht was not taught by the teacher during his/her lesson.

Before conducting an assessment the students need to learn the course content that is projected to be on the assessment. Students need to work individually, in groups, and with the teacher to comprehend and ask questions they might have. Assessments should be ongoing throughout the year and cover many different ideas and skills. The assessments should focus on what the students are learning and what they truly need to know and understand. The way the assessment is presented should be done in multiple fashions. There should be open ended questions, with other questions are performance based. This will help the teacher see what a student is able to understnad and in what form they understand it best. Every student is different and will learn in a different way.