Journal Article Summary

A fifth grade teacher started her math class by posing a problem solving question to her classroom. In order to solve this word problem, students had to think about different amounts of items they needed in order to decorate a town square that was a certain size. After the students read the problem and felt a bit confused, the teacher allowed the students to get into groups to work together. By letting the students work together in groups it eased a bit of their anxiety. In this group, the children were able to reason different ways to solve the problem through the communication of their ideas. When students express their ideas through communication, both written and spoken, children gain a deeper understanding of the subject and concepts that they are learning. Literature can be used in the classroom and can benefit application to real-world expreinces for students. Communication in the classroom can build realtionships among students which will allow them to feel safe to express their ideas and thoughts. When students express their thoughts it opens up more learning opportunities for everyone else.

I really liked the probelm solving method that the teacher posed in her classroom. By allowing students to get into groups and work together through communication, they were able to find different methods for solving the problem. After the students discussed methods, they were able to write the best method down in their writing journal. Through communication the students were able to speak and listen about different ways that the problem could be done---this shows how students benefit from communicating with others about a specific concept. The teacher then took every groups 'best' way to solve the problem and wrote them on the board. The students were then able to see other methods and how they could revise their own method. By revising and editing previous methods, students are learning how to enhance their thinking and communication skills. I also really liked the 'phone a friend' concept because it really teachest students both the skills of listening and speaking. The student asking the question to a friend gets the opportunity to speak and listen to another person's thoughts. The person who explains his/her thoughts is getting to verbalize their method which in turn will benefit their communication skill as well. Overall, I think that it is important that we create safe environments in our classroom where students feel comfortable to discuss and express their ideas and thinking processes. By doing this, students will gain a deeper understanding for mathematic thinking and learning.

Fello, S. E. and Paquette, K. R. (2009). Talking and writing in the classroom. Mathematics
Teaching in the Middle School 14(7), 410-414.

Communication-Standards for School Mathematics

Communication is very important in mathematics. Students not only learn through communication themselves, but also learn through the communication of their peers. Through communication, students should be able to learn and develop skills such as:

• Organizing thoughts in order to explain them to others
• Communicating clear and developed thoughts to others
• Reflecting on other people's methods for solving problems
• Expressing concepts correctly

When students begin to use conversation with their mathematical language, it deepens their understanding of the subject. When a student can successfully communicate a concept with explanation to another person, it shows that they can really understand a specific concept. In the classroom, communication can be used in a number of different ways. Students can participate in discussion groups on open ended mathematical questions and even use technology such as computers or calculators to guide their discussion. Other types of communication in the classroom can be done through puzzles, building objects, diagrams, written explanations, etc. Children at a young age should be taught to reason and explain their methods for solving math problems. As children get older, they will become better at explaining their reasoning through the developed organization of their thoughts. In order for children/students to be able to communicate effectively, they will need to feel safe in their environment. This means that as teachers we must make our classrooms a safe environment for our students to present ideas and questions. At young ages, children typically see things only from their own view. Mathematics can be a difficult subject because they must look at the ideas and views of other people. As teachers, we need to guide and support our students in learning how to see things from other people's perspective. By doing so, we are then able to teach our students to think more deeply about specific concepts about mathematics and also guide them to be more successful communicators.

I think that making our classroom community feel like a safe place for students to share ideas is the first step toward successful communication. I can remember many math classrooms where I felt too stupid to raise my hand and ask a questions for the fear of what my teacher would say to me. In order for our students to communicate their ideas and questions effectively, we must create a safe place where they can learn. This may mean that we need to strengthen the community within our classrooms before we share ideas. Students need to learn and value everyone else's ideas and exploration for the subject of mathematics. Because math can be such and abstract study, students have the opportunity to explore new ways of thinking and solving problems. When we teach students to value and listen to other opinions and ideas, we can deepen our sense of understanding for the subject of mathematics. Our students are then able to relate to new ideas and methods for solving problems in which they might not have thought about on their own. I really liked the idea of students getting involved in discussion groups where they are able to explain how they were able to come up with an answer to a math problem. Other students in the group benefit from this type of group discussion because they can explain the way that they came up with the solution as well as disagree with how another person completed a problem. If students have to work out disagreements together, it can help them find new ways to solve problems as well as learning correct ways for completing a problem with their peers. When students communicate to each other in group work, they are better able to learn from each other through new methods and disagreements. Leaving our classroom open for communication and exploration will give our students better opportunities to understand the overall concepts and ideas of mathematics.

Video Analysis-Lessons on Variables (4th Grade)

The purpose of the activities in the lesson was to get the students to figure out on their own what variables meant. Instead of the teacher just giving the students the information and diving into a worksheet, she wanted to build the students' understanding of the meaning of the word variables from the ground up. By having the students build the variable machine, they were able to use communication to develop different methods for figuring out how the variables worked. Some students realized that when they changed part of the process that it was not working. In order to figure out how to try something new, they had to figure out where they went wrong. This was the teacher's goal in the activity. By building on knowledge that they have to explore, the students gain a deeper understanding of the concept. From there, the students can develop their concept with a basis of understanding. Her purpose was to get children to understand that variables could represent a variety of numbers through exploration of methods. The students used the variable machine to gain understanding of how changing variables can alter results. By the end of her lesson, the children had communicated their methods for success and had gained an understanding of what the word and concept of variables meant. The teacher did an incredible job achieving her goal with her students in her classroom.

The three questions that I focused on while watching these videos were:

1. What criteria do you use to determine whether or not to use a particular task with your class?

• The criteria that I would base whether or not to teach a particular task would be as follows. I believe that if I am going to teach a task, it needs to be interesting and keep the students' attention. If the lesson is not applicable to the students, then they do not really care whether or not they learn the material in the task. I also believe that a task needs to be aligned with standards and goals. If the task is authentic and can be applied to other situations in life, I believe it should be taught. Assessment is also important to look at when determining whether or not to use a task. If the task can be assessed in an informal or formal method then it is applicable to teach. The task needs to be educational and coherent. I also believe that if a task uses the standards of best practice then it should be taught in the classroom.

2. How do you tend to respond to students' answers to the questions you have posed, and how do you encourage students to ask questions themselves?

• I think that no matter what, students need to feel comfortable and safe in the classroom environment. From day one, students need to know that they are encouraged to ask questions and pose ideas. It is important to make students feel important by being enthusiastic when they ask a question during class. By doing this, students are then encouraged to ask questions themselves. I can remember sitting in class and being afraid to ask a question because my teacher would think it was 'stupid.' No classroom should give this impression to students. By posing questions, sometimes other students in the classroom are wondering the same thing! Feeling comfortable to ask questions by having a teacher that is receptive to questions is critical. Responding to students can happen in a number of different ways. I think that we as teachers need to listen well to what is being asked and then re-phrase the question being asked so that the rest of the class understands the question also. Encourage students to discuss with one-another when it is appropriate and be direct with questions that need to have direct responses. I think it is always important to empower our students by being positive to the answers that students give us to the questions we pose. By doing this, students will feel comfortable in the class. If a student has a completely wrong answer, maybe direct the student to seek help from another student to help explain how they got the answer. If there are shy students in the classroom, build their self-esteem! Ask them questions that you as the teacher know that they will know the answer to. By doing this, it makes them feel more comfortable and safe to answering questions and communicating in the classroom. Presenting a warm, safe environment where the teacher is encouraging and open to ideas and exploration presents many opportunities for students.

3. When during a lesson do you check to see what studetns have learned?

• At the beginning of the lesson there is an anticipatory set. The teacher then provides the students with information on how to do the task. We then model for the students how to do the task. I think at this point, we need to check to see if the students have questions before moving on to the next part of the lesson or task. If students do not have questions, pose higher level thinking questions to these students to get their minds thinking on higher levels. During guided practice is where I think it is important to see what students have learned. We can do this by walking around the classroom to see what and how they are doing. We can visually see what the students are learning and what they are having trouble with. In this particualr lesson, the teacher walked around and could see from converstations whether or not the students were gaining understanding or exploring to gain understanding. If students are not understanding at this point, we still need to provide remediation. The students will eventually learn---some just learn at a faster rate or on a higher level. I really liked how the teacher gave a table that already knew the answers a different higher level question to work on. She encouraged another table of struggling learners to continue to seek the answer. By allowing them more time on a subject that she felt was important, she allowed these students more opportunity for success. We can also assess what the students have learned during the assessment. Assessments need to be varied in order to be authentic. Testing students the same way every time does not show what the student has learned. Students show what they have learned better during different methods of assessment. As teachers, we need to provide them with the opportunities to show us what they have learned through these varied types of assessments as well.

I think that there are many overall uses for this video. Growing up I would have been thrown into variables not even having the slightest idea what the word even meant. I completed worksheets and more worksheets on variables without even knowing the basis of the concept. This teacher had an incredible method for getting the students to build their knowledge from the ground up! These students had an interactive, hands-on lesson that taught them to explore the idea of variables. Through communication and exploration the children worked with eachother to find out how these 'variables' worked. When the students did not understand, they had to move on by finding a reason for why it would not work the first time. From looking at the student work, it was interesting to see how the students crossed out things that worked and circled other parts that did work. One particular child's work starts out really sloppy and divergent, and then as you work down the paper it becomes more convergent and less sloppy. The work done by the child shows how his thinking starts out and how he/she builds on the wrong ideas that he/she has tested. This method teaches children what does not work has a reason attached to it as well. That reason is the basis for how to figure out what to try next. When the children started reasoning, they were able to find methods for how to make their variable machine generate larger numbers. The teacher's method of instruction showed me how math teaching needs to be more explorative for students. We need to help our students build on ideas and concepts instead of giving them the information and then throwing worksheets at them for 'understanding.' These videos gave me new insight to what I have to look forward to through teaching mathematics to my future students.

Journal Article-Equity

This article relates to the principle of equity in a number of different ways. Equity relates to individualizing instruction so that all students have the ability to learn math, regardless of how they learn the concepts. Involving different mediums of learning to help struggling learners is just one way that equity can be applied to classrooms. The equity principle also shows how we need to relate math to real-life experiences so that children understand the importance of mathematics. This journal article shows how teachers have a hard time relating everyday math concepts to real-life experiences on a consistent basis. Because of this issue, six elementary teachers became interested in developing a study to find different methods for teaching mathematics so that their students could better understand math concepts. These teachers were also interesting in finding new ways to aid the struggling students in their classrooms. In their research, they found that most teachers fall into the category of white and middle class. Teachers that are white and middle class have a harder time relating to the diverse students in their classrooms. By gaining sensitivity to other cultures in our classrooms we can become more understanding of how people learn. During the study, the six teachers conducted research on specific targeted students. They focused on the thinking process of specific students and their math learning as well as their life outside of school. These two factors helped them gain understanding to how these specific children were learning mathematics. For example, one of the teachers focused on a student who had great counting skills compared to other children in the class. When looking at his outside experiences, the teacher found that the child could count to high numbers because he had experience at home counting jewels on his mother's belt. Because of this, the teacher realized that the jewels must have excited the child to learn about counting skills. The teacher could then look at that experience and figure out how to apply similar experiences in the classroom to aid both that student and other students in the counting process. Teachers then took their knowledge and developed a program in their school where they could aid struggling students in their math learning. The teachers developed a system where they chose one struggling student from each class that could then attend a math workshop during the morning. Each teacher was able to take a turn teaching this math workshop while the other teachers taught students from other classrooms as well. This way teachers could get to know the struggling students on a deeper, more individual level. Teachers found that in these workshops, not all students come to school with the same background knowledge or experiences. By working individually with struggling students and finding new methods for teaching them, slowly the teachers began to see improvements in their mathematical understanding. This shows how equity needs to be shown in our classrooms so that everyone has the opportunity to learn through different mediums. I think that it would have been a good idea to incorporate a technology aspect into the struggling learner's workshop to see how it could have affected the teachers' study. I believe students who have not had the same outside experiences with math need to have different mediums to aid their understanding of concepts. Math is already a difficult subject for students to understand, and we as teachers need to realize that our instruction needs to be adapted and individualized for specific students.

Math Principles; Equity

Equity is crucial to mathematics. When I think about the word equity, I think of the word as referring to treating all people with equal importance. In a mathematical sense, I would assume this to mean that all instruction to students should be equal and therefore the same. However, after reading the principle of equity, I realize that equity can mean many different ideas through a mathematical sense. Equity does not just refer to treating all students equally, but it refers to giving each child an equal opportunity to both learn and study the subject of mathematics. This could mean that each child could be learning through different mediums in order to understand math ideas and concepts. Accommodations for students can be made in order to benefit their learning of the subject. With the growing population of diversity in the United States, teachers could find this principle of equity to be difficult to apply to their classroom. This is why it is crucial that we have well-trained educators facilitating the learning of every student. Each individual child, regardless of their background, should be able to learn mathematics and be successful. This means that as teachers we need to find different methods to teach mathematics as well as raise our expectations for our student achievement. By having high expectations for our students as well as being able to facilitate to the needs of individual students, each child will prosper. Technology is a recourse that could benefit a number of different types of learners. Other resources such as hands-on material could aid in the process of teaching students to understand mathematics. It is very important that we as teachers do not just give the answers to our students, but that we aid them in their research and creativity to find out how the process of math works. This way the students will be able to individualize their learning experience and change their previous ways of thinking into a learning process while gaining quality mathematical knowledge. I also believe that it is important as math teachers that we make the students' math learning experience applicable to real-life situations. By doing this, we are then able to show students at a young age that math is important to their life and also their future. Students may enter the classroom with different levels of mathematical understanding, which could be due to numerous factors. Students who do not speak English as their first language, students who have learning disabilities, students who have grown up in poverty, etc., may need special accommodations or additional resources in order to aid their understanding of mathematics. From the reading on the principle of equity, it was proven that all students, when given the proper instruction and aid, can learn mathematics. It is crucial, as teachers that we use the principle of equity in our classrooms so that our students are faced with high expectations and different mediums for learning. By doing this, all students will be able to learn mathematics.