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.