May Journal *from MTMS August 2009

Stephan, M. (2009). What are you worth? Mathematics Teaching in the Middle School (15)1, 16-23.

Teaching seventh grade students the concepts of integers and operations can sometimes be a challenge. Many different approaches have been used to teach integers, but the abstract concept of integers leaves students with an incomplete understanding of the concept. This concept is typically taught using a number line and adding inverses. It has been improved to use plastic tiles and a mat to group positive and negatives. For example, the class in this journal explored temperatures in Pennsylvania. When a temperature increased from a negative degree, students would place that specific number of degrees in tiles in the negative work area and repeat this process with the positive degree. The students would then cancel the opposites. However, this still leaves students with an inadequate understanding of the reasons behind why the steps are used to solve a specific integer operation. The teacher in this article demonstrated a new way that she taught her students this abstract concept of integers...by determining how much people are worth. Using Oprah Winfrey as an example, who had a net worth of $1.5 billion dollars, the concept of net worth was discussed with students. The concept of debt was then brought up during class. The teacher introduced the concept and asked the students if they think that she has to pay money on her house and other things such as her studio. The children then came up with other possible things that Oprah probably owed money for, and the teacher wrote the responses on the board. Once students understand key terms and how integers work in this setting, students will be able to determine how much net worth a person has. Students can explore and solve the problem in whatever way they like, but most students add the assets, add the debts, "and subtract one total from the other total." This leaves students with the understanding that debt takes away from the overall net worth someone has. Once students understand this, they can work into more complex problems. A dating game was used in the article where three bachelors were introduced. Each bachelor had a positive number for something, a negative number for a loan, a positive number for income, etc, and the students would have to use these integers to solve which bachelor had the greatest or least net worth. This is where students are introduced to symbols and can actually put meaning behind the work and operations. The teacher then has the students create their own net worth statement for a specific person. This can be assessed to see if the students understood the concepts.

When I was in school, we learned integers through symbols first, and then worked on a number line to determine subtraction/addition for integers with real numbers. However, this always confused me. In my head, I always thought of negative numbers as money. I think that by using an example like this for students, it helps them realize negative numbers are a part of everyones lives...even the rich and famous. This also can help students realize the thought process behind developing a good savings plan as well as a good budget plan which can benefit them in real life. Developing the concept in a concrete form and moving to the abstract idea of symbols seems to be more beneficial in this case. I think that if i had used this as a lesson in school in seventh grade, I would have understood integers and integer operations much better.

Manipulatives Blog

1. How do you hold every student accountable?

Using manipulatives to support mathematical teaching can be incredibly effective. To manage student accountability, I think that teachers should make sure that the different types of manipulatives are introduced. This way students have the opportunity to learn how to use the manipulatives correctly and apply the different manipulatives to their concept skills. Another way I think that students can be held accountable with manipulatives is through allowing students to freely explore and use the manipulatives during a free period or down time during a learning center. By allowing students free access to different manipulatives, students will feel more comfortable with the material during class time. Another way to manage manipulatives during class time would be to assign a group leader to the table to ‘manage’ the manipulative tools. Dividing roles in groups helps to give each individual child a role during instruction. By giving each child a role, they are held accountable to both their group and themselves.

Manipulatives are sometimes viewed negatively when they are viewed as ‘playing materials.’ In a sense, I completely disagree with playing being a negative concept with manipulatives. Children love to play and explore. In a journal article I read earlier this semester, a sand play table was used in a classroom. Children were able to go ‘play’ at the sand table and communicate with their friends (incorporation of process standards). Once the students became comfortable with the table and the communication at the table, the teacher would introduce new material to the table to go along with concepts he/she was teaching. For example, the teacher would put different sized beakers in the area with the sand. The teacher would allow the students to first explore the new objects through play and communication. Eventually, the teacher noticed that the students would compare the different amounts of sand that each type of beaker could hold. Students began to explore mathematical concepts on their own. As the students played at the table, they began to question certain math concepts. The teacher slowly but surely would go to the play table and pose questions to the students and listen to their reasoning (another process standard). Concepts were explored through a simple station where children were able to play and communicate. I believe manipulatives should be used this way. Students need the ability to play, explore, and communicate. Eventually, students begin to question which leads them to a more exciting role in the math learning process. Teachers can then guide the students with questions or comments regarding the students’ questions. Eventually concepts are constructed and developed over time. Process standards are incorporated naturally into this process.

2. Why not “hands-on”, why is it “hands-on, minds-on?”

When students become engaged in the manipulatives they are working with, they are stimulating their minds. Rather than just playing with blocks at a table, students are communicating and manipulating materials. Students naturally begin to question ideas on their own through this process. However, if good instruction is provided before the introduction of manipulatives, students can then use the manipulatives to enhance the instruction. If students have been learning about fractions in the classroom and then pattern blocks are used as an exploration manipulative, students will begin to apply their learning to the manipulatives. Students will start to question, “I wonder how many triangles will fit into a hexagon.” Students then work and communicate with each other and construct their concepts in a more concrete manor. Manipulatives are not going to save mathematics. Teaching still needs to be effective and teachers need to stimulate interest in the subject and show real-life application. When students have strong instruction, they can succeed in the use of manipulatives. Students will build onto the foundation through exploration. This is where students being to use “hands-on, minds-on.” When students use manipulatives in this way, they will only strengthen abstract math concepts into more concrete representations.

3. Process Standards

When students have choice and control in the manipulatives that they use, students can apply reasoning and proof to show how that specific manipulative was used to deepen the understanding of an abstract idea into a concrete representation. Manipulatives and representation work cohesively. Using concrete representations of abstract mathematical concepts and ideas helps students to see where the idea or abstract material becomes a concrete concept that they understand with meaning. Manipulatives shift students’ thinking from basic memorization of rules and facts and more into concept construction. When students get to use manipulatives, they deepen their understanding of specific concepts. The more representation methods that the children can apply using the manipulatives, the more constructed these concepts will be shown through the students. Students also get to use communication and problem solving as they work through using different manipulatives. As teachers, we need to give up some of the control with students and allow them to explore more freely. By doing this, students learn to think on their own and build onto prior knowledge to create a foundation for greater concepts. Instead of the teacher being viewed as the ‘experts,’ the students begin to think they are the experts. When they take responsibility for their own learning and become engaged through the use of manipulatives, educational concepts can be constructed at greater levels. Connections can also be made through the use of different subjects such as art. When students learn concepts in mathematics such as slides, flips, rotations, etc. students can create tessellations. Tessellations can be viewed as both artistic and mathematical. When we use manipulatives for content material such as slides, flips, rotations, turns, etc., students can visually see the concept as well and physically manipulate the object through exploration.

Errors Blog

Going through the math errors document was probably my favorite and most beneficial activity this semester. When working through numerous types of mathematics concepts, it is really interesting to see how some students make mistakes. I had never imagined teachers actually sitting and looking at an error to see where a student made a mistake; however, I found it to be fascinating to see the ways in which some students logically think and apply methods that do not work for specific problems. When teachers take the time to look at how the student made the error, they can begin to see where a specific student needs some additional guidance. Students often are taught the formulas or told ‘this is just how you do it,’ which does not enforce any core concepts. Going through the errors enforced why the process standards are so important. When the content standards are appropriately used in the classroom, students are able to see concepts in more depth. Instead of just working though problems, students understand the meaning behind the problems and are less likely to make errors such as the ones presented. Instead of the student focusing on the ‘rule’ or ‘formula,’ they can recall how to do the problem through reasoning and proof. Students then are less likely to use incorrect rules or short cuts to simply solve the problem.

Learning different methods for teaching subtraction, additions, multiplication, division, and using fractions is something that was specifically beneficial to me. Growing up I learned the formulas and did not learn meaning behind the formulas. When I try to recall when one fraction is bigger or smaller than another, I often get confused. Learning to visualize fractions as portions of a whole is crucial. Students in younger grades often have a hard time determining which fraction is bigger or smaller and how to add, subtract, multiply, and divide them. Learning the meaning behind these concepts and using models to help students visualize these concepts is definitely reinforcing.

Looking for patterns in student errors and putting meaning behind the errors can help teachers focus on areas in which students are having difficulties. When we realize the mistakes the students are making, we can help address the problem and put more emphasis to the meaning of the mathematics concept. Teachers can help students put an end to misunderstandings when they realize the pattern behind the error.

Technology Blog

Technology is becoming better and better is what seems to be exponential time. As a future teacher, it is important to look at how technology can positively and negatively affect a classroom.

Our generation today has so much experience with technology. Growing up with a computer, and learning how to play around on programs has been beneficial. When I look at the older generation, I can generally say that many adults feel less comfortable with technology than my generation. Most adults are less likely to spend time experimenting with learning new programs; although this is not always the case. This proves to be beneficial for the younger generation teachers who are entering school districts and know how to learn programs on their own. During my Novice teaching experience, my teacher had me spend an entire day teaching her how to use different technology programs. Knowing how to learn, adapt, and change with the times is important as a teacher. With technology expanding so quickly, it is important that we learn how to figure things out on our own or collaboratively.

With technology expanding so rapidly, it is hard to keep up with all of the new and fancy programs. Students in today's society are used to spending time on computers and using different programs. They are learning how to work the technologies we are using just as quickly. However, I believe that technology can prove to be a negative factor in education if used in the wrong way. Although technology is seen as the new 'fast pace way to life,' technology can actually waste a lot of valuable time for learning as well. When technologies fail, or computer programs do not sync, technology can prove to be a hinderance. Programs are changing and becoming better every day, which means formatting for each program is now different. From computer to computer, technology can prove to be a challenge. Also, when the internet fails, the computer freezes, or any other technical difficulty is faced, technology can waste a lot of time. It is important if we, as teachers, use technology in our classrooms, that we do so in ways that value the learning opportunities of our students. Two examples of valuable educational technologies would be the smart board and the geometer's sketchpad program.

The smart board is an incredible presenting tool that students typically get excited to use. There are many opportunities that the smart board provides that can enhance certain educational topics. My very favorite math technology is Geometer's Sketchpad. For inquiry learning, I believe this is the most valuable math technology program available. Students can explore different math concepts from beginning learning experiences to very advanced learning experiences. I have personally worked on basic geometry concepts in this program as well as advanced calculus ideas. I had a teacher at Bradley who provided guided learning worksheets that went step-by-step through directions on how to create a specific geometry idea or concept. Once we had used the direction to complete the activity, we had to experiment with the shape/design/measurements to figure out what the diagram represented and meant. I learned how to complete many geometry proofs using this program. This is very beneficial, because sometimes students focus more on the formulas rather than the concept. When students get to figure out the concept on their own while visually seeing representations of something they created, they take more value in their learning process. Overall, this is my favorite program. I think this provides many valuable learning opportunities for students and demonstrates one great way that a teacher can apply technology in the classroom.

Calculators are another technology that can be used in the classroom. They can also be great for students to use and check their work. Calculators now can visually show representations of ideas. Students can even explore concepts on calculators to prove how a certain formula or concept works.

I do not believe that technology can 'create the learning experience,' nor do I believe it can solve all the problems for education. I think that when used in the right opportunities, technology can prove to be very beneficial. Math applets provided by Illuminations and NCTM can help students work through inquiry learning to deepen their understanding of concepts. When teachers use technologies that are appropriate, educational experiences can be enhanced. Teachers really need to learn when and when not is appropriate to use technology, find a balance for how much technology should be incorporated, and relate to both the positive and negative aspects of technology in the classroom.

April Journal Article MTMS

Obara, S. (2010). Constructing spatial understanding. Mathematics Teaching in the Middle School (15)8, 472-478.

Students need spatial sense and understanding when using two-dimentional shapes and three-dimentional diagrams. Sometimes, students experience difficulties with spatial understanding becaues it is not intuitive. Students need practice visualizing two-dimentional nets and their transformations to three-dimentional figures. In order to help students understand how shapes are creating through folding, curling, etc. to a net, teachers often use geometry technology software. One teacher used a triangular pyramid and a cube and asked the students to create the nets to these figures. Using Geometer's Sketchpad, the teacher showed the students how to construct a net and modeled construction as well as use of the Sketchpad. Students were then able to explore and create their own net to try to figure out the net for the solids she presented earlier. Another approach the teacher took to developing spatial understanding was to pass out congruent three-dimentional pieces of a shape and had her students try to develop the net for a specific figure. Students often started solving this problem by using a square out of the pieces given to them in order to start building their net. Students were able to see that by creating two identical pieces from equilateral triangles into the form of a trapezoid, that they could construct a tetrahedron from these two 3-D figures.

Working with students to help them develop spatial understanding between 2-D and 3-D shapes will benefit them in their understanding of geometry. By having this type of understanding as teachers, we will better be able to use hands-on activities to teach our own students how to better comprehend spatial sense and how it applies to different forms of geometry. When students use inquiry based learning such as the type given in this lesson, students get to explore different nets and models as well as incorporate technology into the curriculum. This overall will help students to develop spatial sense.

April Journal Article TCM

Treahy, D.L. and Gurganus, S.P. (2010). Models for special needs students. Teaching Children Mathematics (16)8, 484-490.

Co-teaching is a specific type of teaching where a special education teacher and a classroom teacher work collaboratively in order to develop a more cohesive progam for planning, teaching, and assessing/evaluating. By using collaborative efforts such as co-teaching, the classroom becomes unified and represented as a whole unit. In one specific classroom, a classroom teacher and special education teacher taught mathematics together. The classroom teacher would teach her mathematic concepts or lesson while the special education teacher would ask her to clarify specific points or to repeat important concepts that may need to be heard again. By doing this, the two teachers incorporate a strategy similar to the reading strategy, think-alouds. These two teachers also take turns teaching important concepts and share the role of instructing in the classroom such as a team-teaching role would present. By doing this, students are gaining a deeper understanding of difficult concepts. By repeating difficult material and clarifying it in different ways, students get to see different aspects of the problem. The article lists a number of different models for teaching a classroom of integrated learners; these include:

Team: shared role between the two teachers

Alternative:

• one teacher works with small groups for re-teaching, enrichment, pre-teaching, etc. while the other teacher works with the rest of the class
• teachers use think-alouds to clarify and to help train special needs students to work through the problems rather than applying a random formula or idea that they have learned; (lack of metacognitive skills)
• great for hands-on learning
• great for working one-on-on or in small groups to clarify concepts for struggling learners

One teach, one assist:

• allows one teacher to teach while the other teacher walks around and observes and helps students who need it
• can beneift students with behavior issues
• helps to focus more on special needs students; gifted learners, special needs, ELL
• allows individual assistance for students
• great way to monitor progress

Stations:

• content is divided into different stations and each teacher teaches a specific station
• great way to incorporate the NCTM standards
• teachers can focus on an area of their strength to benefit deeper understanding for students
• stations often incorporate real-life integration
• allows for monitoring and reinforcement
• can help prevent math anxieties early on

Parallel:

• the teachers plan cohesively but then each teacher teaches a half of the entire class
• teachers must plan for all areas of learners
• teachers must think about how all students learn and how to relate to all students

As a teacher, co-teaching would be something that would be benefical in the classroom. It provides rich learning experiences for all areas of learners. During these types of instruction mentioned in the article, teachers use models and clarification so that all students have the ability to understand concepts. Co-teaching helps create a common instructional language that all areas of learners can understand. This helps to unify the classroom and help students become more engaged in the materia. C0-teaching provides a rich learning opportunity for a wide-variety of students. The talents and energies of two teachers can be effective in a class of different types and styled learners.

Observations with Checklists

Cole, K.A. (1999). Walking around: getting more from informal assessment. Mathematics Teaching in the Middle School (4)4, 224-227.

This article describes how informal assessments such as observations with checklists can be used to gain, organize, and document information about students' performances and academic growth in order to produce coherent stories of student progress. Teachers walk around the room to gain types of informal assessments on student learning and progress. By doing this, it helps teachers to receive immediate feedback of how students are thinking and working on problems. For example, a teacher described in the article walked around the room in order to view students' thoughts about a math problem they had been working on. When she walked around to each group and talked with them about their thoughts/ideas, she round that come children had confused meters and feet in their problem they had been working on. By observing students informally, teachers can gain a lot of information about how students are working and certain misunderstandings students might have. Teachers can also introduce new ideas and pose more in-depth/higher-order thinking questions to students. The teacher in this article continued to walk around the room during different days throughout the weeks and found that her questions she had posed to certain groups had been effective toward their learning. Students were able to use their journals to further explain their mathematical reasonings. Using checklists can help us to organize our observations as we walk around the room. Checklists can focus on mathematical tasks, students' communication, using math language, and their ability to work well with others in the classroom. Checklists help to observe students and find out how well students are using the mathematical concepts and standards in the classroom.

March Journal Article TCM

Wallace, A. H., White, M.J., and Stone, R. (2010). Sand and water table play. Teaching Children Mathematics (16)7, 394-399.

Sand and water tables provide many educational opportunities for younger children. Social atmospheres and play time help students with their natural emotional and cognitive development. When teachers provide opportunities such as a sand and water table for play, students get to explore using different types of manipulatives. Teachers can help engage students in more valuable learning opportunities using different types of manipulatives at the play table in addition to creating different learning environments. For example, teachers can designate specific tasks at the table, but can also allow free exploration for students. By providing students with different beakers and 3-D shaped objects, students can explore different concepts such as equal values, volumes, shapes, etc. This article explains that it is important to allow free exploration first so that the children become socially comfortable as well as get used to some of the manipulatives. Once free exploration has been done, the teacher can introduce different concepts. This might mean that the teacher would point out the numbers on the side of a beaker and help guide the students to figure out what the numbers mean. After the concept introduction, the teacher can use application of the concept. Students then get to choose their own manipulatives/materials and explore concepts through concepts that have been introduced. The final stage is the evaluation stage. The teacher informally can evaluate students the entire process. Some teachers use checklists based on whether or not children are observed using different mathematical concepts. Other children are evaluated on specific concepts and whether or not they can complete specific objectives.

One of the activities described in the article was a dinosaur dig. Students got to play the role of archaeologists to dig for dinosaur bones and artifacts. The students then were asked to sort their artifacts based on what they believed 'fit together' by their characteristics. The students then created a graph based on their real artifacts and compared the data that they had dug from the sand/water table. We use this activity in the pre-school that I sub for and the students LOVE it. Students like getting to play the role of an dinosaur expert and love to dig up artifacts and dinosaur bones. This activity provides great mathematical concepts for children such as qualitative and quantitative data as well as sorting things based on their characteristics. Also, students get to use graphs to show the results of their data. Overall, I think that using the sand and water play table is a great idea because it opens up many opportunities to integrate the NCTM standards as well as a lot of inquiry-based teaching.

March Journal Article MTMS

Roberts, S.K. and Tayeh, C. (2010). Assessing understanding through reasoning books. Mathematics Teaching in the Middle School (15)7, 406-413.

Reasoning Books are very beneficial to students learning and understanding mathematics. Often times in math, students are most focused on finding the correct answers or quick solution to problems or questions rather than gaining understanding. As teachers, we can help guide students in the right direction by starting reasoning books with our students. Reasoning books should be started at the beginning of the year and teachers should model how to create reasoning books by going over good samples, adequate samples, and poor samples. Through modeling a good reasoning book, students will be able to better develop their reasoning book skills. These reasoning books are great for students' communication skills and helps students focus on the bigger pictures in mathematics. In these books, students will write responses and reasonings to specific mathematical problems. In these books, students will use data, drawings, reasonings, justifications, etc. to explain a mathematical argument. These reasonings should be able to be read by anyone and should explain exactly how the answer or solution was found. Reasoning books help students to answer the more in depth questions about math such as "how do we know?" and "will this always be true for every problem?" etc. The ideas in the book should focus on the important concepts in mathematics that students have difficulty understanding, concepts that provide a bigger picture, and ideas that relate to real-life context. By creating reasoning books, students will value mathematical learning through reasoning early on in the semester. This book will also help students focus on the conceptual side of mathematics rather than just formulas and correct answers for the entire semester/class.

As a teacher, this provides a valuable context for student learning and teacher assessment. By viewing the reasoning books, teachers can clearly see exactly how the child was thinking and where they need help at in their thinking. As teachers, we often overlook ideas that we assume our students know. When reasoning books are introduced, students have the opportunity to explain their understanding of the concepts and how they got to their final conclusion/answer. We can guide students by modeling examples of good reasoning entries, showing them adequate examples of reasonings, and also reasonings that need extra work. When students understand how to write clear reasoning entries, it will greatly benefit their conceptual knowledge for math. This provides a great context for teachers at the beginning of the year and will provide students with higher expectations for learning rather than just 'finding the correct solution.'

Video Analysis #2

http://my.nctm.org/eresources/reflections/index.htm




1) tell the purpose of the activities in the video

The purpose of these videos is to explore and strengthen the relationship between graphs and tables with seventh grade students. By exploring meaning behind graphs and tables, it will help students strengthen relationships among ordered pairs, linear patterns, and the rules that accompany these patterns. The teacher starts out by having them explore relationships between graphs and tables. Students make up stories to go along with their graphs. Then the students explore making tables from graphs and creating tables based on linear functions. Once students have an understanding of the linear functions, they try to create their own.



2) answer at least 3 of the questions posed

• Describe how appropriate you think the primary task in this lesson is for developing an understanding of the mathematics being taught.

I think that the primary task in the lesson is very appropriate for developing an understanding of the math that is being taught (functions). Obviously the teacher wants to have her students investigate graphs and the different values that graphs can represent. Rather than giving her students a table and having them graph the values, she starts backwards and has her class create a story based on the graph given to them. From there, the students present the material to the class and explain their stories. The teacher's goal is to have her students understand the relationship behind the graph so that they can form a table using the values and reasoning behind the graph. This is very appropriate because it leads her in to teaching students how to look at x and y values and find unknown values as well. She goes right into teaching students about sequencing and infinite values through increasing and decreasing numbers in a sequence. The method she uses in teaching is investigative and her goal is to have her students creating sequences by the end of the class to stump either her or the other students. Students will gain a valuable concept understanding when they use communication, representation, reasoning and proof, problem solving, and connections. The teacher is using all of the standards through teaching her lesson.

• What specific actions could the teacher have taken to improve the effectiveness of learning when students are working in groups?

The teacher could have provided a guide sheet so that the students stayed more on task. These could have listed things such as the 'subject,' what method they were using, what dollar amount they were starting with, etc. Instead, when she discusses these things with the groups, it seems as if it is a little chaotic and they do not have a complete understanding of the 'subject' of their problem and their methods for application. If the students had written them down when she came to discuss their problems with them, she would not have had to lead or guide them so much in the questions that she was asking. Also, rather than have them work with their friends, ability grouping sometimes aids in difficult concepts such as sequencing. Students could also have had the opportunity to communicate with other groups to find out the methods they were using to solve their problems/create their problems.

• Describe how you use the evidence you collect about what students have learned to modify your teaching

If students do not seem to be grasping the ideas and concepts for a specific subject matter, I would have to modify my teaching to adapt to their needs. If most of the children in the classroom do not understand a subject, it may be beneficial to start again where they are having difficulty and review areas where they are having trouble, or provide a more valuable learning opportunity through an activity, exploration of ideas, communication, etc. Students may need hands-on materials, visualizations, etc. to aid them in the develpment of difficult concepts. I really liked how the teacher put the error up on the board and exploring why that error is wrong. I think that by doing this, students get a more in depth understanding of where they went wrong, how they can fix it, and the method for correctly solving problems.

3) explain your thoughts on the overall use of the video
The overall use of the video was important because it showed a good example of teaching graphs and linear functions to seventh grade students. Rather than throwing equations and formulas out at the students, they explored the meaning behind graphs and tables and the functions that accompanied them. Through group work, communication, and problem solving, students were able to create problems of their own at the end of the lesson. This shows how I can better structure my teaching as I enter into the teacher education world. Using strategies that focus on the standards really benefits students in their learning process and provides a strong foundation for meaning behind concepts.

Applets Reviews

http://math.rice.edu/~lanius/fractions/index.html
Who Wants Pizza? A fun way to learn about fractions.
Grades 3-5

This applet is laid out in a very organized manor. Students can easily explore an introduction to fractions, different methods for using fractions, and practice fractions (addition, equal, subtraction, multiplication). The ease of use of this applet is fairly simple. The organization of the applet is laid out nicely and the practice problems show both visualizations of the parts in comparison to the whole. Children can easily practice their knowledge after reading about the type of fraction use they are interested in. The description for using the fractions for different methods is also very descriptive and shows visualizations to clarify word meaning. At the end of each online section there is a place where students can test their knowledge and submit their answers to see if they were correct. There is also a bonus question at the bottom that helps children think more critically. Students are able to view the correct answer after they have thought about the problem. Overall, this applet can be as challenging as the student or teacher wants to make it. Students can explore different avenues of fractions (addition, subtraction, multiplication, etc.) which gives them different options for practice and challenges.

I believe that students would find valuable learning in this math applet for numerous reasons. As I went through the math applet, it was refreshing to have the information laid out in both words and pictures. Through description using visual fraction bars with individual boxes colored in to represent part(s) of a whole unit, students can easily see how different parts relate to a whole unit. This idea can be challenging for students to understand, and I think that this applet gives a great understanding to the part-whole relationship. This applet also gives a valuable assessment for the students because it provides them with practice questions and an opportunity to submit their answers at the end of their practice. Students get to use the mouse to shade in different fractions that the applet asks for. Through use of the computer and visual aids, students get to investigate fractions and work through their learning on their own. I think that this specific applet provides students with a lot of benefits for learning. Students are able to explore the concepts of different avenues of fractions on their own. They can pace themselves and can work on different areas that they find interesting or a struggle. Students also get to practice their knowledge of fractions through the computer assessment provided. A weakness that this applet has is that students might get carried away with the different amount of options provided. If students have not yet worked with multiplication with fractions, they might want to click on it in advance and it may confuse them. However, this could also provide students with the opportunity to explore different avenues of fractions. The information provided would give them a good basis for beginning exploration on different fraction uses. I liked that the students could relate the fraction use to real-life food situations; which students would love. This applet also provides the teachers with a teacher's page which shows how students can use the strategies in the applets to build upon fraction concepts. The grids provided in the applet can be printed out and used in congruence with the activity as well as be used in part for an assessment. This teacher's page also describes the use of how this applet coincides with the NCTM standards for 3-5th grade students. Overall I found this applet very user-friendly and educational in the use of fractions.

http://standards.nctm.org/document/eexamples/chap6/6.5/index.htm
Pythagorean Theorem Math Applet
grades 6-8

This applet is very user friendly and is to the point and is clear and concise. Basically, students get to explore the Pythagorean Theorem through the use of an interactive figure with two little squares, a triangle, and a bigger square. Each of the two little squares are shaded a color. These little squares can be moved through the click of the mouse to fit into the bigger square. This shows how the different legs of the triangle end up creating the actual symbolic representation of the Pythagorean theorem a^2+b^2=c^2. Students get to use the visualization in order to examine and describe how the relationship in the diagram represents the actual Pythagorean Theorem. Overall this is a very basic applet provided by the NCTM which aligns with student objectives and shows a basic, clear description of the Pythagorean Theorem.

Student learning in this applet is a little less guided than the previous applet described above. This applet is for grades 6-8 and is a little more based on inquiry learning. Students have more of an opportunity in this applet for exploration and learning through their own exploration with minimal guidance. Guided questions are posed so that students can answer them as they use the actual applet. This will help guide them to their overall answer at the end of the exploration process. Students are presented with more inquiry based questions asking them to explore if the relationship found in this applet could possibly exist among any other shapes or figures. This applet builds on prior knowledge and then adds to prior knowledge through challenges presented involving higher-level thinking. The strengths of this applet is that it is clear, concise, and straight to the point. Some students may understand the visualization really quickly and can use the posed questions to explore further investigations, whereas other students can explore the Pythagorean Theorem through proof form rather than symbolic representation. This builds on students' ability to analyze different relationships presented through the subject of geometry and also helps build on their understanding of the actual proof for the Pythagorean relationship. The weaknesses of this applet are that it does not provide a whole lot of guidance. There is no real way for the students to assess themselves based on this applet to see if they are doing the activity right. Depending on what type of activity they are working on, open or close-ended, this could pose as a strength or a weakness. I think that another strength to this activity is that there are extensions at the end that the students can build upon through the use of this particular applet. I believe that this applet could provide more of a student type assessment through the website in order for students to explore their thoughts and then get feedback on their work. However, this can also be a strength based on how the applet is being used. The actual activity in this inquiry based applet could pose as a good assessment method for the teacher rather than self-assessment for the student.

Feb. Journal Article TCM

Kilic, Hulya, Cross, D.I., Ersoz, F.A., Mewborn, D.S., Swanagan, D., and Kim, J. (2010). Techniques for small-group discourse. Teaching Children Mathematics (16)6, 350-256.

The way that a teacher facilitates the students in the classroom is very important. Student learning is partially dependent on the way that the teacher teaches and the way that the classroom expectations are set up. When students learn to think critically, reason through difficult problems, and work together through communication, there are more opportunities to understand the basis for mathematic concepts. Through reflection, teachers can try different methods for facilitation in the classroom in order to meet individual classroom needs. Expectations in the classroom should be set up in a way where students really understand that investigating and exploring mathematics is beneficial to their learning process. When teachers use meaningful context and facilitate students in their learning, students gain a deeper understanding. Tips to teach critical thinking were given in the article. These tips included focusing on the students' thinking, listening to their responses while encouraging them to listen to each other, and providing appropriate feedback throughout the learning experience. Many other tips were given, but each teacher can create different methods of their own for meeting the needs of their particular classroom.

As teachers, it is our responsibility to get our students to think at higher levels, provide real-life context for learning, and facilitate meaningful communication. All of these relate to helping our students get the most out of their education. In order for us to maintain productive discussions in our classrooms, the article stated that we needed to model through our own organization and communication. We can use think aloud methods so that the children see how we as teachers think through mathematical problems as well. Also, by showing how listening to others' strategies is important, students can learn to find meaningful methods through others' thinking. By walking around the room and monitoring classroom discussions, we can ask questions that relate to individual needs. I liked the example in the article where the teacher asked why the student chose that walking around the circumference of the circle would be longer than walking the diameter of the circle twice. Instead of telling the student she was wrong, the teacher simply asked how she reached her answer and listened to her thought process. She guided further discussion through other group members by involving them in discussing their methods for solving the problem. Through basic facilitation and questions about reasoning, the student was able to realize her mistake through listening and reasoning. The teacher facilitated the discussion but did not intervene and give the student the answer. It is important to ask the right type of questions that do not lead students into a specific direction of thinking. Creating a classroom where student conversation is important and meaningful is key. To do this, we can make sure that each group is following the expectations that their conversations are explanatory as well as clear in their reasoning. We need to allow students time to think, investigate, and work! Provide students with feedback on their thinking so that they can learn the best ways to learn.

Feb. Journal Article MTMS

Cramer, K., Monson, D., Whitney S., Leavitt, S., and Wyberg, T. (2010). Dividing fractions and problem

solving. Mathematics Teaching in the Middle School 15(5), 339-346.

Many students learn symbolic meaning for concepts without actually learning what the concept actually means. This article show how fraction division learning can be accomplished through the use of problem solving and use of pictures and reasoning. Students learned the basics of fractions at the sixth grade level before moving onto complex fraction problem solving. The students in this article completed an activity with the use of different colored paper circles--- all were the same size. Each of the different colored unit circles were folded into different amounts of pieces to show representation to students that different fractions are still in relation to the one complete unit. Students were able to see visual representations of the fraction parts to the whole unit which made the concept easier to apply to problem solving questions. Once students were able to investigate problem solving questions through pictorial representations, they were able to work on describing their process through language. As the students learned how to describe their picture representations through words and eventually symbols, they were able to understand the use of rational numbers. Students built upon this knowledge and eventually were able to learn more complex ideas such as dividing fractions. Through problem solving groups, class discussions, and reflections, students were able to understand the flexibility of a unit. These sixth grade students showed the capacity to learn complex ideas on which more abstract concepts could be built.

I believe that having students investigate the different ways to represent fractions is crucial to their understanding of the basic concept. As a student, I learned to jump right into symbolic work and never learned the basis for my concepts. Upon reading this article and seeing different student representations of their use of fractions and fraction divisions, I was able to visualize and understand the part-whole concept. Having the students take the different colored circles--all the same size--and fold them into different fraction representations really shows the importance of understanding a unit and how different fractions make up a unit. As a future teacher, it is crucial that we let our students learn the importance behind the symbolic work that they eventually do. I liked the way that the teachers let the students learn through discussions and reflections. Learning how to problem solve through picture representations of fractions before learning symbolic work was something that I really admired. We should not underestimate our students and should give them the opportunity to figure out the concepts so that they can build upon that knowledge in order to use symbols as representations.

PBL Comparisons

#1 Summary of PBL--Lounging Around (grades 7-8)

In this PBL assignment, students in grades 7-8 are to decorate a lounge area given a set budget by the school board. The students in this PBL use different skills such as measurement, prior knowledge about geometry, and budgeting skills. This problem that was assigned to the students relates to real-life skills as well as applies to a topic area that may be of interest to most students. This learning takes place over a time span of 16 days and involves the use of communication in groups. During these 16 days, students learn about different math concepts through worksheets in order to understand the meaning of budgeting. Students also make 3-D representations, listen to a guest speaker, and present their budget plan for their lounge area. All areas of curriculum are presented in the PBL and student uses a variety of different mathematical concepts to complete the PBL.

# 2 and 4/Comments About Strengths and Weaknesses/Critiques & Examples/Changes I would make

There are many strengths to this PBL but I also believe there are many weaknesses. When reading over this PBL, I can clearly see how the principles and process standards are incorporated into the overall PBL. Students are presented with a difficult problem where they must develop a budget plan for a lounge. This lounge area would be something that students in 7th and 8th grade would find interesting and relatable to their real life. As I looked read through the overall document, I felt that the PBL concept was good but the organization and ideas behind the concept did not flow together smoothly. The PBL was student centered but did not give each student an individual part to be accountable for. By working together, the PBL was student-centered and collaborative, but lacked individual accountability. Overall, I did not feel like the problem at hand was the difficult part to solve, but more so the many different tasks that did not seem to flow together. By completing a variety of assignments and homework papers, I feel that the students would get lost in paper work and methods for trying to solve the problem. Even working together, the project seemed completely overwhelming without the established roles given to each specific student. After re-reading over what makes a good PBL and the steps that makes a good model for a PBL, I think that this version would be overwhelming and confusing. However, the concept behind the PBL started off in a positive way, but became too divergent and messy toward the middle of the project. Students did not even start working on their budget plan until approximately day 14. This did not give the students much time to complete their actual budget plan. If I were to change anything about this PBL, I would make it more organized. I think that by giving each student a role to complete, students would better be able to research a specific part to contribute to the group. I also believe that the guided questions could be more focused and less divergent. Students working together on this project might get lost in the overwhelming amount of worksheets and assessments. I think that by creating a more authentic assessment relating to journal work or a portfolio, students would be able to show their individual learning process while working together. I would make my mini-lessons more incorporative into the actual PBL assignment rather than just basic activities that teach concepts of proportions and fractions. I feel like the mini-lessons given in this PBL are examples of the way that most every day math teachers teach. Instead of exploring a concept, they are given a worksheet in order to find answers. These are just a few ways I would change this PBL assignment.

# 1 Continued

Summary of PBL--Operation "Redo the Zoo" (grades 5-6)

Students get the opportunity to take a field trip to the zoo and create their own design and budget plan for an exhibit they'd like to see at the zoo. There are four groups of students who each are given a specific role and task to contribute to their overall problem. These students will become an expert on their role and will research information and bring their thoughts back to the remaining group members who will also share their information. Over a 15 day period, students will design an exhibit and a budget plan for a specific zoo area. Students will keep math journals to document information and their learning process. Through mini-lessons, students will develop a portfolio. At the end of the PBL, students will present their design and budget plan to the rest of the class.

# 2 and 4/Comments About Strengths and Weaknesses as well as Critiques and examples

I believe that overall this is a strong PBL for grades 5-6. The overall problem is relative to what the children know and adds on to their prior knowledge through an interesting topic. The subject of 'the zoo' is always very interesting at any grade level and designing your own exhibit and budget plan would be something that not every student gets to do in real-life. Although this is not directly related to an actual activity they could do in real-life, the concepts behind the PBL directly relate to real-life concepts that the students could connect to on an every day basis. The PBL is very organized and presents a a difficult problem in which students are each given the role of some sort of 'expert.' The students then are able to decipher the problem, make a plan, and establish methods for solving the plan--all essential components of a model PBL. Students then are able to take their 'expert' part and gather information that relates to their individual role for their group. When students come back together as a group they use communication skills to decipher the best method for designing and budgeting their problem. Students build upon their research throughout the project and complete graphs, charts, pictures, and representations. The problem is messy and seems overwhelming at first, but when divided among 4 other group members over a period of 15 days, students can realize that a big problem can be minimized through communication and individual accountability. Higher level thinking is evolved throughout the entire PBL and students keeps logs and journals of their learning process. The teacher guides the overall PBL by giving different challenges that can help break the overall problem down a little bit while also enhancing the learning process. Students seem to always be working hands-on or interactively through the use of applets and technology. They also get to listen to a speaker from a zoo and go on an actual field trip to the zoo. Cross-curricular needs are met through many of the standards and goals presented in other subject content areas. All of the objectives presented are well-established and are grade-related. I really do not see any weaknesses throughout this PBL. If I had more experience with creating a PBL or seeing one in use I might be able to pin-point specific weaknesses, but from the knowledge I have gained about PBL's I think this is a great example that models all of the necessary components of a challenging PBL that is grade appropriate. I don't think I would change a whole lot to the overall organization of this PBL. I might add some assessment ideas to go along with the actual PBL presented. By developing a portfolio along the way, students might find the learning process more valuable. If during the mini-lessons students were able to add to their portfolio and obtain valuable information guiding them back to their problem at hand, I think they would be encouraged to keep working toward better methods. This is the only part that I would change. I loved how the group incorporated different units for other subjects as well as incorporated grade-appropriate standards and goals.

# 3 Compare and Contrasting

PBL: Lounging Around VS. PBL: Operation "Redo the Zoo"

The Lounging Around PBL seemed to be challenging but way too overwhelming. The Operation "Redo the Zoo" PBL was challenging but was guided and divided into a more reasonable problem for the grade level given. After reading the problem for the Zoo PBL it seemed overwhelming, but as the guided questions and division among research through 'expert members,' students would be able to work through the stress of the situation. I also believe that through working individually and together, it provides a better opportunity for the groups to discuss reasoning and proof behind their research as well as lots of communication skills. I believe that in the second PBL, the standards and objectives were more directly related to the problem compared to the Lounging Around PBL. Although both experiences related to real-life situations, the budget plans and the amount of student-centered learning was completely different. Students in the Lounging around PBL seemed to not have any rhyme or reason for completing their PBL. When reading the model steps for a good PBL, it seemed as if the Operation "Redo the Zoo" PBL was incredible. Not only did it make the problem student-centered, higher-level, and learner-centered, the PBL focused on a situation where students used guided questions and steps to create a plan to solve their problem. Overall, I believe the Zoo PBL was a much better representation of a model PBL than the Lounge PBL.

#5. Thoughts on how math is or is not the main focus of the work/level expected

Lounging Around PBL

I think that math is definitely a strong focus in this PBL but I believe the objectives and standards behind the actual PBL are not strong. For example, I see that students are able to use transformations and flips and turns in order to create a lounge, but I do not think the actual concept is directly related to creating and building a lounge. When students would build the lounge they would primarily be focused on measurements and budgeting. The math concepts seem to diverge into other subject areas as the PBL continues. The overall PBL did not seem organized with math concepts or ideas. Doing worksheets and being assessed via worksheets does not seem authentic. Although math was a focus in the PBL, I think it was too unorganized to really see all the math concepts involved in the entire process.

Operation "Redo the Zoo" PBL

I could easily follow this PBL and was better able to see the actual math that went into the steps and model of the PBL. I liked that the students were able to take a field trip to the zoo to first being the process of visualization for the process. By planning out their math ideas first and then building on their prior knowledge they have a stronger basis to build from. Dividing each student up into an expert on part of the problem also gives the student an area of mathematics to focus on. When students come together they can learn about other ways that their group members used the same math concepts to relate to their area of expertise. When students use math journals and create diagrams, charts, graphs, etc. it really shows the use of math concepts and standards for this specific grade level as well as the authenticity that the teacher will be able to assess the students on. By keeping math logs of budget plans, measurements, time periods, students use many aspects of mathematics to explore concepts that they are familiar with. Students also use higher-level thinking in order to build upon their prior knowledge during the steps of this PBL.

# 6. Assessment

Students participating in the Lounging Around PBL were assessed based upon checklists and rubrics. I think that the checklists provide a valuable assessment for the teacher to see how the students are doing during specific assignments. The overall rubric assessment incorporated every activity and did not really show how math content or processes were evaluated. Their explanation of how math concepts and processes were detailed and understandable, but when it came down to the actual evaluation and assessment I think that the rubric did not do justice. I do not believe there was enough criteria for assessment for it to be an authentic assessment. I also believe that more math concepts needed to be incorporated into the assessment. In the PBL described below, students were able to use journals and portfolios to show their math concepts learning as well as their processes for completing the PBL. I think that this is a more authentic form of assessment based on the standards, objectives, and goals.

Students in the Zoo PBL were assessed both formally and informally. I liked how the students were assessed daily through student journals so the teacher could specifically which students may need more guiding throughout the process. This is where adaptations for some students could be realized and the teacher could better guide the student. Formal assessment thought a portfolio also shows a different type of assessment and quizzes were also given based on new knowledge learned. By incorporating both formal and informal assessments the PBL became more authentic. Students were also graded on their presentation of their overall PBL. Students are able to show their reasoning and proof through these types of assessment as well as show their communication and understanding through their presentation. Work in the portfolio also gives an example of how the students built upon prior knowledge and made connections to other subject areas. Overall I feel that the content standards and the processes were clearly shown through assessment in this PBL.

PBL Journal Article Review

The teacher in this article created a PBL based on the book, Harry Potter. What I really liked about this PBL assignment was that it was cross curricular. The teacher took their literature reading, Harry Potter, and turned the book into a real-life budgeting problem. The students were fascinated with the book and were excited to be assigned a character with a specific budget. Budgeting was a topic that was not completely unfamiliar with the students. The teacher introduced the budgeting problem and explained the term budget and how it applied directly to their real-life. The children were given a character with a specific budget amount and had to create a budget for their entire year worth of school supplies for Hogwarts. Students were given a price sheet showing the amount for specific items. Some of the items were required during the school year, and others were luxuries that the students could purchase if their budget allowed. This shows a great use of PBL in a classroom. The teacher took a real life problem and related to the world of her students through Harry Potter. In order to address the budgeting problem for the school year, the students had to use prior mathematical knowledge in order to break down the problem and start to find methods to solve the problem. The teacher noted that many students used different methods for solving their specific problem. One student decided to save some of her money and put it in the bank so that she could have access to more money throughout the school year. Other students wanted to buy luxury items and based their budget on trial and error. Students helped each other with their differing budgets and their methods for working on their problems. The teacher was more the facilitator while the students lead their own learning experience. Once the students had spent two class periods working on the PBL, the students discussed their methods to their specific budget plan with the class. In the debriefing period of the PBL assignment, the teacher stated that her class was able to make higher level connections about budgeting based on their own experiences working with their ‘imaginary’ Hogwarts budget. One child said that his parents had to budget a member in his family going to college while another student said that a previous teacher budgeted for school supplies for the entire classroom.

I really liked the entire PBL assignment in this article for numerous reasons. The students were already reading the book, Harry Potter, for their literature class. The teacher saw that the students were very interested in the subject of Harry Potter and incorporated it into a real-life budgeting problem. By using a higher level topic such as budgeting, the teacher was able to help the students understand the topic by bringing it down to their level. Through specific characters in the story, the teacher was able to assign each child a budget based on their character. I really liked how the students were all given a different budget so that they had to use their own method to buy supplies for school although the concept method for all the students could be similar. Students were able to discuss with each other the different methods and approaches they were using to buy their school supplies. By having required items and luxury items, students had to learn to manage their money appropriately. Students went about the problem in many different ways which was very interesting. Some students started with required items while others tried the method of trial and error. Allowing the students to discuss their methods showed other students in the class better methods for buying their school supplies based on their budget. The teacher really focused on the PBL being student centered while she took a step back and became the facilitator. Students learned the learning process through trying different methods for buying required items/luxury items with their given budget. In order to make the problem more difficult, the teacher stated that you could use ‘wizard’ currency which could be valued different than our American currency. This could be applied at a higher grade level as well as made cross curricular when studying different cultures and money. I did not believe that there were many weaknesses to this PBL method. I think that the teacher focused on the main criteria for a PBL assignment and did a great job relating the topic of budgeting to her students. The only weakness that I could really see was that the students did not really relate much to the subject of budgeting yet. However, I do not believe it is a bad idea to relate this subject to younger aged children if it can be done through their world. I think the teacher did a great job of relating a bigger concept to her students’ real world.



Beaton, T. (2004). Harry potter in the mathematics classroom. MathematicsTeaching in the Middle School 10(1), 23-25.

PBL Readings

Problem-based learning is where students are more involved through the learning experience while the teacher poses as the faciliator during the experience. This instructional strategy is student-centered and allows the investigation of challenging real-life problems. Students are presented with a difficult problem where they must work together to come to a solution. In order to do this, students must use communication skills to figure our what the exact problem is, what they already know based on prior knowledge/experience, and how they are going to get to where they need to end up. In the classroom, a basic KWL chart can be used to find out these basic questions and needs. The teacher helps guide through asking questions as well as directing students toward resources for finding the answers. Students will then use resources to collect information that can guide them toward finding methods for solving the challenging problem that they have been presented. However, the teacher allows the students to take the lead role in solving the problem. Each student uses communication in group work but also is responsible for their own part in the learning. These types of problems that students are faced with do not have a simple 'correct answer.' The students must work together to find potential solutions to the problem and then conclude on the best method for solving the problem. Ideas are then presented and the students get to pick the solution that best works with their problem. I liked in the readings that the teacher was preseented as the 'coach' in the teaching process. The teacher is there to guide and encourage the students while the students must actively find the solution. When the students come up with the best solution to their problem, it is important that they reflect on their overall problem and the solution that they fit to the problem. I like that during this process students are actually learning the actual learning process. PBL's can be used in any type of situation that involves a problem. Any type of career can use a PBL for a problem that they are facing. We work on problem-based learning on a day to day basis. We may not be writing down exact steps to how we are solving problems, but when we encounter a basic real-life problem, we being this process. For example, if you have no water pressure in your house, you must find the root cause of the problem. In order to do this, you need to develop a plan. This plan includes the process of figuring out what you know, what you need to know, and how are you going to get to a solution. There potentially could be many solutions for how to fix the problem, but the best method or approach is needed. I may gather background knowledge to try and save some money and then involve the use of other people, such as friends or family to try and solve the problem. Eventually, when I come to the solution for the best method, I am essentially using this same process of PBL that we are using with our students in education. PBL's are important in day-to-day life which is why it is critical that our students be prepared to use this type of learning daily.

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.