Manipulatives

1. How do you hold every student accountable when using manipulatives for an activity?

I think that it is important that each student is held accountable when using manipulatives for an activity. It is important that the teacher assess each students' knowledge and ability to apply knowledge of a concept through manipulatives. One way that each student could be held accountable when using manipulatives for an activity is to have them each student record their thought process when using manipulatives in their math journal/notebook. This allows the teacher to individually assess each students' knowledge of a topic through the use of manipulatives. The activity we used in class was a great way to hold the students accountable. We each had to record the various activities and we had to work together to create the various activities that each type of manipulative could be used for.

2. Why do people say "hands-on, minds-on," instead of just "hands-on"?

Hands-on, minds-on is referring to the use of manipulatives. When students are using manipulatives they are engaging the mind by using and manipulating different objects to strengthen and build of different math knowledge and concepts. Students' brains are engaged in the manipulatives that they are using and as a result they are stimulating their minds. Manipulative use in the classroom allows the students to apply their knowledge in a way of "doing." When students physically manipulate objects they are applying their knowledge of various mathematical concepts to real-life situations, and they are able to physically see the process of how some math concepts and carried out. When manipulatives are used correctly, paralleled with strong and effective instruction, they will begin to investigate, explore, and ask various questions about concepts.

3. How do the process standards fit with the manipulative activity?

The process standards relate closely with manipulative work. Students are using problem solving when working with manipulatives when they must find a way to use the manipulatives in a way to solve the given problem or situation. Reasoning and proof is incorporated in manipulative use because the students are using the manipulatives to reason and problem solve to provide proof for their solutions they find. Communication is related to manipulative use when they it is used to express the process of a mathematical concept. Students may also have to use the manipulatives in their justification for a solution that they have found. Connections can be made when students are using manipulatives because they are able to use the objects to help relate to two mathematical concept together with a visual and physical representation. Representation is always applied to manipulative work. Manipulatives are used so that representations can be formed. Manipulatives can be used to represent fractions, sorting, patterns, probability, and many other mathematical concepts.

MTMS - A Framework for Analyzing Geometric Pattern Tasks

Friel, S. N., & Markworth, K. A. (2009). A framework for analyzing geometric pattern tasks. Mathematics Teaching in the Middle School, 15(1), 24-33.

A Framework for Analyzing Geometric Pattern Tasks is an article that discusses how teachers can use geometric patterns to promote the understanding of functional relationships by students. The article focuses on figural reasoning during the process of inductive reasoning. The authors state "when students use figural reasoning, they are able to make sense of patterns, ..., by paying attention to visual cues that can be organized and translated to numeric sequences (Friel & Markworth, 2009). The article was organized into three sections. The first section discussed various problem solving processes that support the use of figural reasoning as a way to explore and interpret geometric pattern tasks and generalize function rules. The second section discussed a framework for distinguishing the complexity of geometric pattern tasks. The framework may be used as applied contexts for figural reasoning. The third section provides a summary of various ways for how the long-term and extended use of geometric pattern tasks contribute to an overall development of students' functional thinking.

I thought that this article was good, but could be confusing at times. There were many instances that I found myself having to go back and re-read multiple sections. I liked that it explained and directly stated how the topic discussed in the article relates to the NCTM Principles and Standards for School Mathematics. I really liked that the there were many examples provided in the article. Having those tables and figures helped apply what was being said in the article with real world student work. I thought that this could be helpful to teachers. Sometimes it is difficult to fully understand how something can be applied to in the classroom. Providing the examples helps teachers have a greater understanding on the techniques explained and discussed. Although I found the tables and figures provided in the article to be very beneficial, there were times when it was very confusing and irritating to flip back and forth between the pages to try and see what was being discussed and referenced in the article. I liked that the authors included a brief paragraph explaining that geometric pattern tasks, although a valuable way to promote figural reasoning and develop a rich conceptual understanding of functions, it has its limitations as well. This information provides teachers with a realistic view of the topic. That realistic view allows teachers to be prepared for the limitations they will reach, and allows them the opportunity to be prepared for dealing with those limitations.

Error Analysis

I believe that the error analysis activities were the most beneficial thing we have done in this class. Prior to taking this course, I would have checked the answers to see if they were either right or wrong. It had never occurred to me that there are patterns in the errors that are made. I never realized how much I could learn about students in my class just by analyzing the errors they made and determining where they did not understand the concept. It is important for teachers to spend time and consider the educational needs of each student. I also learned that I am able to make notes on different concepts that I may need to spend more time on the following year.

I thought it was beneficial to me, as a future educator to know different strategies for solving addition, subtraction, multiplication, and division problems so that I am able to instruct students with various techniques depending on their knowledge and competency with each mathematical skill. Before the error analysis activity, I hadn't really realized how often students can trip up on multiplication problems that I find to be pretty simple.

Technology

Technology has been used throughout this entire course in many different ways. We discovered and used a Wiki to collaborate as a group and complete projects electronically. We then learned how to use a GoogleDoc when working on a group project. We also used math applets and learned how they can benefit students. The SmartBoard and computers were used throughout the course as well.

I thought that the use of the different types of technology throughout the semester was very beneficial. I am naturally comfortable with using technology. I enjoy knowing how technology works, and how to fix or resolve any problems or questions someone is having with technology. I thought it was interesting to use a new technology for an assignment. Some may have been frustrated when learning to use new technology. I feel that the use of technology in this class is beneficial to me as a future educator. It is important to incorporate technology in the classroom very often. I believe that it is important for teachers to have a full understanding of the technology they are assigning their students to use. After having the opportunity to use Wikis, GoogleDocs, math applets, a SmartBoard, and other technology used in the class, I have more of an understanding of what technology tools should be used in certain situations.

Prior to this class I had used a SmartBoard a few times in high school. I always thought that a SmartBoard was an awesome tool to have in the classroom. Throughout the semester, we used the SmartBoard in almost every class session. I learned so many things about what a SmartBoard can do, and how they can be used by the teacher and students to aid in the students' learning. Before I began this class I would have never thought of using a blog, Wiki, GoogleDoc in my classroom. I didn't really know what a math applet even was. This class has taught me of many different ways that technology can be incorporated in the classroom to aid the students.

"Graphing Calculators as Tools" - Mathematics Teaching in the Middle School

Browning, C. A., & Garza-Kling, G. (2010). Graphing calculators as tools. Mathematics Teaching in the Middle School, 15(8), 480-485.

Graphing Calculators as Tools includes several graphing calculator activities and focus on what we believe are four powerful ways to use this technology as a tool to explore mathematical idea:

1. Collecting or generating raw data
2. Examining multiple cases
3. Providing immediate feedback
4. Showing graphical and numerical displays

Teachers can use these ideas effectively through TPACK (technology, pedagogy, and content knowledge). The interconnection and intersection technology, pedagogy, and content knowledge. The article discussing different ways of investigating algebra through the use of graphing calculators and technological tools. Students use graphing calculators and stand alone motion detectors to investigate linear relationships and other functions. Students can make connections to positive and negative slope. The accessories available for graphing calculators allow students to physically investigate linear relationships and functions with motion sensors. The calculator they creates a graph from their movement. Students can then analyze the graph of their movement. Students investigate geometry and measurement using the SMILE application. SMILE has the potential to strengthen students' understanding of angle and angle measure. Students also use graphing calculators to investigate data analysis and probability. Students can use the graphing calculators to use probability simulations where they develop estimates of experimental probabilities for the problems they are interested in investigating in a more efficient way that using traditional tools.

I thought that this article was very helpful and useful for classroom teachers. I liked that the article provided a multitude of activities that use graphing calculators. I also liked that the article included activities that utilized different graphing calculator accessories with the calculator. The accessories available for graphing calculators allow students to strengthen prior math skills/knowledge, as well as learn new mathematical skills and concepts. The article is useful to classroom teachers because it specifically explains how certain applications on the calculator can help students grasp a deeper understanding of mathematical concepts and skills. I thought that this article presented very interesting and engaging activities, but it is very expensive to purchase enough graphing calculators for the class. Even if only a few calculators were purchased, the cost of the applications and accessories used in the activities are also costly. I liked that the article addressed how the activities and math skills explained in the article to the NCTM Content Standards.

"Supporting Language Learners" - Teaching Children Mathematics

Brown, C. L., Cady, J. A., & Hodges, T. E. (2010) Supporting language learners. Teaching Children Mathematics, 16(8), 476-483.

Supporting Language Learners is an article that describes many different instructional practices that will help support English learners. The article begins with a heading stating "incorporating these instructional practices for two math tasks into lessons to support English learners to an excellent, equitable program." The strategies described in the article are not exclusively for English learners, and regardless of their cultural background the strategies can provide support for all students. The strategies are grouped in two main categories: strategies to support English language acquisition and strategies to promote low-anxiety classrooms. Linking new concepts to past experiences is beneficial to students. There are advanced organizers that help language learners make that connection. Expository and comparative are the two forms of advanced organizers. Developing students' vocabulary by identifying words that might cause confusion for the students and address the words during the lesson. Teachers should use visual cues to help students develop their vocabulary. Visual cues include diagrams, props, photographs, drawings, models, and real-world objects. Make sure to adjust your teacher talk. Repeat important explanations and directions to improve ELL's comprehension. Slow down when speaking, but not so much that it sounds unnatural. This is the most simple and most helpful strategy for ELL students. Simplify sentence structure, be consistent with words and phrases when giving directions. The article then provides an example of two tasks that utilize the first group of strategies. The second category is strategies that promote low-anxiety learning environments. Teachers should use predictable routines. Use signals to indicate the close of one activity and the beginning of another to aid students in understanding expectations. Contextualize when giving directions. Simultaneously using physical demonstration with verbal explanations help language learners understand. Language learners appreciate peer help and support, and the volunteers feel a sense of satisfaction in helping. The teacher should provide guidance to the volunteers and distribute responsibility in the group.

I thought this article presented very helpful information that would be useful to a classroom teacher. I thought that this article presented information that was very general and could be applied to all subjects and in a lot of different situations in the classroom. I was somewhat disappointed that this article did not relate to mathematics instruction more specifically. I have learned about these strategies in prior classes. I liked that the article included a list of words, relating to mathematics skills and operations, that cause confusion in ELL students. It was helpful to see the strategies used in the examples provided in the article, even though they were very basic.

Interviews and Conferences

Anhalt, C. O., Civil, M., & Fernandes, A. (2009). Mathematical interviews to assess Latino students. Teaching Children Mathematics, 16(3), 162-169.

This article discusses the use of mathematical interviews to assess Latino students. The article explains how using interviews illustrates the correlation between language and math. English Language Learners must describe their understanding of math in one-on-one interviews. In this article the teacher is using these one-on-one interviews to assess the students' understanding of measurement. 15 working-class Latino fourth through sixth grade students were interviewed. Each student was asked to solve three NAEP tasks on measurement.

One example given was dealing with language issues that ELL students face with interpreting or solving problems. The student interpreted the question "if both the square and triangle above have the same perimeter, what is the length of each side of the square?" The student interpreted the word if differently because of the different interpretations of the work if between the student and instructor. The instructor asked many probing questions of the student so that they can understand what the student believes the question is saying, and they can also understand how to assist the student with interpreting and answering the question and others they may be faced with in the future. The instructor learned that omitting words that cause confusion to the ELL students will help them succeed in the math classroom. Language has a significant role in the performance of students. Using interviews allow the teacher to ask appropriate questions so the student can form a better understanding and strengthen their language skills. Teachers were able to use gestures the students made to interpret the explanation the student was giving.

The article then addressed how students struggle with applying math knowledge to problems because the questions are lacking context. Interviews allow for the teacher to realize this problem, and address it by providing a context that is meaningful to the student.

Teaching Children Mathematics March Article

McMillen, B., McMillen, S. (2010). My bar graph tells a story. Teaching Children Mathematics, 16(7), 430-436.

This article discusses the effective instructional strategy of connecting stories to qualitative coordinate graphs. The article includes a lesson plan that incorporates story telling with mathematics. Students are encouraged to create a story about the graph. In order for students to properly tell a story about a graph, they must be able to describe the relationship between two concepts, events, or ideas. Students are able to reinforce the relationships within mathematics. They are also able to better understand the connections to other subjects outside of mathematics.

As a teacher I thought this article was beneficial. I liked that it included a lesson plan on how to use stories to explain graphs. The lesson plan was very detailed so there was no confusion on what to do. I liked that this article included the worksheets that were used in the lesson. Having those handouts is very helpful to have as a new teacher. As a new teacher I am worried about confusing the students, and the detail about the lesson that was provided in the article helped me fell more comfortable with the activity and instructional strategy.

March MTMS Journal Article

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

This article discusses reasoning books. Reasoning books have been proven very beneficial to students that are learning and understanding mathematics. Mathematical reasoning books can be used as a tool that helps students develop their reasoning and proof, communication, and vocabulary skills. These books also create a place for students to create a forum to explore and answer the difficult questions. The article explains the reasoning books being used in each way. The article also includes examples of student responses in mathematical reasoning books.

As a teacher I though that this article investigated a great and different way to assess students in math. I liked the idea of using mathematical reasoning books discussed in the article. I liked this idea because it allows teachers to fully assess students in math. This is important because teachers are often found to be not asking the questions that are difficult to answer because they would have to "invest" time in the question by having a complete understanding of material and have explored the material enough to know everything about it. Another thing I liked about this article was that it included student responses and an analysis of each. It gave me a better understanding of how to asses each student's responses in the reasoning book.

Math Applet Review #2

"Cubes" Grade 3-5 http://illuminations.nctm.org/ActivityDetail.aspx?ID=6

The objective of this applet is for students to help students understand volume of a cube. Students will have to determine a rule for finding the volume of the box. Students are given a box, a cube, a row of cubes, and a layer of cubes. They use all of those things to fill the box and determine the volume. The students are able to fold the sides of the box up or down so they are able to visualize the box when it is filled. The dimensions of the box can easily be changed. The students can determine the width, depth, and height that they desire for their box. Students can easily add and take away cubes from the box. After the student fills the box with the appropriate number of cubes, then their is an exploration tab they can clock on. The exploration portion explains more about what volume is, and a very brief and vague explanation of volume. Students then use that explanation and the box they have filled with cubes to determine the volume and how it was discovered.

I did not find this applet to be very effective. I thought that it did not explain why they (the student) will be filling the box with cubes. I thought that the activity did not accomplish what it said it would they way I thought it should. The applet did not give the student any indication of a correct answer. I thought this could cause a lot of confusion among students. The applet also did not include a way for the student to check if the volume they found was correct. I think that a teacher could use this applet as an additional tool to represent how volume is found and what it is. A student who did not know anything about volume may struggle with this activity.

Math Applet Review #1

"Five Frame" Grades Pre-K - 2 http://illuminations.nctm.org/ActivityDetail.aspx?ID=74

Through the use of this activity, students will learn basic number facts. Specifically, students will develop their counting and addition skills. Students use pictures on a grid to determine how many of the object there are on the grid, or how many spaces on the grid are empty. If the student answers the question correctly a happy face appears on the screen. A score of how many they answered correctly is kept at the side. If a question is answered incorrectly, it does not count against them and they have the chance to answer the question again. There are a total of four different activities in this Applet. In the other activities, students must determine how many circles it will take to fill the grid, or place the appropriate amount of circles in the grid. The fourth activity has students using the cirlcles and the grid to represent addition problems. Students are able to easily manipulate the pieces on the screen. The students can continue with the game for as long as they would like to because there is no end.

I believe that this applet is a great way for younger students to become comfortable with counting and addition in an interactive and visual way. This applet quality information regarding directions. This applet would be most beneficial for beginner to intermediate learners. The applet did provide problems that were more challenging, therefore not promoting much growth in math content and process knowledge. This applet would be useful for integration in a beginning level lesson about addition and counting. This applet would be useful for teachers to use as supplementary activities in the classroom. Students could use this applet at school during free time, or at home for additional practice. This applet does not provide the teacher with information regarding how well the student did completing the problems. That does not allow the teacher to assess how well students understand the concept from this applet.

Rubrics at Play by McGatha and Darcy - MTMS

Darcy, P., & McGartha, M. B. (2010). Rubrics at play. Mathematics Teaching in the Middle School, 15(6), 328-336.

Rubrics at Play is an article about rubrics and how they should be used to be most beneficial. The authors stated that assessment must be formative and rubrics are often found to be useful as formative assessment tools. Holistic, analytic, specific, and general rubrics were all explained in the article. The authors explained that rubrics should be viewed as more than just an assessment tool. Rubrics can help provide the teacher with feedback, further students'understanding of the topic being learned, and they can facilitate the discussion of ideas. The process of creating rubrics using student work, and the assessment task are given. Student-created rubrics were also highlighted. The authors provide many examples of how students can be involved in the process of creating a rubric. Student involvement in rubric creation promotes students to take ownership of the assessment process. This strategy allows the students to understand what is expected of them from the assignment. When a rubric is created correctly it can be a beneficial formative assessment tool. Rubrics help teachers by acting as a guide for analyzing the work of students. Rubrics also provide the teacher with feedback regarding their instruction. Rubrics also enable to teacher to provide each student with feedback that is specific to them. The specific feedback allows the students to move to higher skill levels. This encourages students to become independent learners. The article included examples of different types of rubrics, a personal story of creating a rubric, and reflection and discussion questions for teacher to use.

I thought this article was very informative. I liked that the article provided examples of rubrics. It can be difficult to create rubrics and providing examples helps explain the proper way to create a rubric. The authors included a chart with the four categories, excellent, proficient, marginal, and satisfactory. I liked that the description of these would be useful to a teacher because it helps them create a clear line between each grade that could be given. The information in this article would be useful for a classroom teacher because it provides steps for creating a rubric that can be used as formative assessment. I thought that the incorporation of a teacher's story about creating a rubric, and creating a rubric with their class was useful, because it gave a real life example of how the process carried out. It can give a teacher a better understanding of how the rubric formation process will be done.

Classroom Characters Coach Students to Success -Edwards

Anderson, G., Edwards, S.A., & Maloy, R.W. (2010). Classroom characters coach students to success. Teaching Children Mathematics, 16(6), 342-349.

This article explains the instructional mode called "1-2-3 Time." This instructional mode divides the class into small groups, that rotate through different instructional activities. 1-2-3 Time allows for the integration of technology, children's literature, writing, and even board games into daily activities and lessons. While using this strategy half to 2/3 of the class are completing independent learning activities, and the remaining students are working with the teacher. The article continues by explaining how the entire instructional mode is considered teacher-led instruction. The system has four math coaches. These coaches offer different suggestions about problem solving from many differing viewpoints. These math coaches are represented by four different stuffed animals or dolls. Students have these objects to represent the four viewpoints and remind them to use them. The article then continues by explaining how math learning games and reading and writing activities are integrated into "1-2-3 Time."

I thought that this article was very interesting. The concept of having a stuffed animal or doll represent a problem-solving strategy was very novel. I think that the instructional mode that was outlined in this article would be very beneficial in the classroom. More and more often teachers are stressed to include visual representations in a lesson to strengthen students' learning. I liked that 1-2-3 Time is an instructional mode that is not specifically for math. The integration of the different content areas could make it more attractive to a teacher because they will not have to worry about switching to a different instructional mode.

Review of Two PBLs

PBL: ADOPT-A-LOT

Summary

The PBL called “Adopt-a-lot” is a about a school that was given a piece of land and $35,000. The 5^th grade class had to design a park that met the needs of the community. They are given the $35,000 to go towards the building of the park, but once they exceeded that amount they would have to decide how to fundraise. Throughout the PBL, the students will create and explain graphs, and surveys used to collect data about the environment. They would also gain experience of creating, prioritizing, and distributing a budget. Students will solve problems involving scale factors. Students will have to recognize and apply mathematics in situation outside of the mathematics classroom.

Strengths and Weaknesses

A strong point of the PBL was overall design of the PBL. This PBL did a great job on the overall design. The grade level given for the math being done was not appropriate, but it was organized well. A weakness was that the math level being used by students was not aligned with the grade level given for the PBL.. The work that the students would complete was at a 3^rd grade level. The overview of the PBL stated that it was designed for 5^th and 6^th graders.

Appropriate Grade Level?

This PBL did not have the appropriate grade level listed. The PBL was supposed to be used in a 5^th or 6^th grade classroom. The work that was asked of the students was at around the 3^rd grade level.

Assessment

The assessment of the math content and process was very vague. The rubric that was given gave too much slack in each point value. It allowed too much leniency.

PBL: LOUNGING AROUND

Summary

This PBL is for 8^th grade students where the students must design an addition to the school that will house more space for the 8^th grade because of the dramatic increase in the class population. The students will build on data analysis and probability, geometry, number and operations, algebra, and measurement. This PBL will take place over 22 days.

Strengths and Weaknesses

A strength of this PBL is how the objectives are presented in the daily plan and on the objectives page. One the objectives page, it was nice to have them organized by content areas. Having the objectives for each individual day plan was nice. It is important to have the specific objectives that should be reach that day stated directly on the page. A weakness of the PBL is how the daily plan is organized. Having to read through an entire paragraph to find information can be tedious. Using bullet points for the Daily plan is more effective. It is easier to find specific information when it is formatted as individual bullet points. Another weakness is that there was no PBL web. The group did not create a web for the PBL. The work had a weak overall format of the work. There was not an organized overview of the important points of the PBL. Listing grade level, duration, rationale, math focus, and higher level thinking in a list form, it allows the teacher to quickly see the important points.

Appropriate Grade Level

The focus of the PBL is half on math, and half on other subject areas. The math that is required by the students is repeated very often throughout the PBL. Students are not building on skills constantly, but they are just using the same ones more often.

Assessment

The assessment rubrics were not detailed enough. They left a lot of room for students to question what each grade required. Some of the areas being assessed on the rubrics repeat.

Compare and Contrast the 2 PBLS

These two PBL’s contrasted regarding the appropriate grade level given to it. The first PBL was not appropriate for the grade level stated, but the second PBL was. Both PBL’s had great organization, although one had a more organized overall layout of the PBL, and the other had better organization of the objectives. Both PBL groups had weak assessments. Both groups need to improve their rubrics.

PBL Article - "How to Buy a Car 101"

How to Buy a Car 101 is an article about the instructional strategy called problem-based learning, or PBL. PBL is based on the primary concept of relating the curriculum to an real-life situation. The explanation of the PBL design is given, beginning with the teacher asking the students an engaging question. The explanation continues to through the process as students are comparing possible solutions and researching the best option. The PBL that was described was about how to buy a car. The benefits of PBL were given. The benefits of PBL are numerous. There are also many adaptations that can be made to PBL for special needs learners and ESL students. Finally, the author gave tips for teachers that were attempting PBL for the first time.

In the article, under the adaptations section, the author stated that a way to adapt PBL for special needs kids is placing them in a group with students who do not have special needs. This could cause more problems for the special needs student. They may work slower and the other students may just do the work for them. This does not allow the special needs student to benefit from PBL. The article did a great job of maintaining a constant theme of buying a car throughout the entire article. I thought a strength of the article was that it included tips for teachers using PBL for the first time. Many teachers are intimidated by PBL. It is scary to think that you are only a guide and the students are making their own decisions. Another strength of the article is that it includes the rubric used for the PBL. It helps understand how PBL can be assessed.

Flores, C. A. (2006). How to buy a car 101. Mathematics Teaching in the Middle School, 12(3), 161-164.

PBL Readings

Problem-based learning helps students develop better problem solving skills and expand disciplinary knowledge bases and skills. Problem-based learning places students in an active role of problem solving. Students are faced with poorly phrased problems that tend to mimic real-life problems. PBL is a student-centered instructional strategy. Students must work together to find a solution to the problem they have been faced with. The teacher acts as a coach or facilitator to the students throughout the PBL process. Once the students have received their problem, they must define the challenge or problem. They must determine what they know from prior knowledge, what they need to know, and how they are going to get to the solution. The use of a chart similar to a KWL will help students answer these questions. The teacher should provide resource advice and ask questions throughout this process to act as a guide. Students should list many possible solutions and rank them according to how successful they will be. Once they determine which solution will be most successful, they begin research the knowledge and data that will support the solution. The group needs to discuss the possible resources they can utilize as well as assigning and scheduling research tasks for each member of the group. Deadlines should be established so students are completing work in a timely manner, but the student should set their own individual deadlines for each part of the PBL. The teacher gives the final deadline for the project. Students then organize the information they found through research and the solution and present their findings to the group. When students present their findings, they need to include the problem statement, questions they asked, the data that was gathered, an analysis of the data, and support for solutions or recommendations based on the data analysis. It is pertinent that students reflect on the overall problem and the work they conducted to solve the problem. This encourages the students to learn how to learn. When completed, the PBL would have promoted the integration of knowledge, critical thinking, problem solving, communication, reasoning, learner empowerment, interpersonal skills, and decision making.

Problem-based learning is used in many ways in the classroom, but it is also used outside of the classroom. Problem-based learning can be used in the classroom with a mini-lecture. The students work in small groups and they material resources are provided for them. The students then solve the problem in that class period. We solve problems in our day-to-day lives. When solving the problems we face, we determine the best and most efficient and effective way of solving the problem. We research how to reach the solution and then we execute it. PBL is an important strategy in a student's learning.

Communication Article - "You Had to Be There!"

"You Had to Be There" is an article discussing the use of a demonstration classroom to teach other teachers about effective mathematics education. This article gives us an insight into the classroom of Duane Heide, a mathematics teacher in Canada. Mr. Heide answers questions about what attracted him to becoming a teacher. The article continues as Mr. Heide describes a typical situation a visiting teacher might see in the demonstration classroom. Heide explains in the article how he opens and closes the lesson in a way that allows students to works with each other and discuss how to address and solve the mathematical problem they have been given. After the class is dismissed, Heide holds a debriefing session for the visiting teachers in the classroom to ask questions and discuss what they just observed. The article is concluded with Heide answering questions about the experience of being the teacher in a demonstration classroom and how the students react to effective teaching of mathematics.

I thought that this article was very interesting to read. I had never heard of a demonstration classroom. As I read the article, I was intrigued by the amount of information that a teacher can learn from visiting a demonstration classroom. I thought this article was beneficial because it taught me that communicating with fellow teachers can enable a positive learning environment in the mathematics classroom. I liked when the article discussed the use of discussion among students during the lesson. This was beneficial to read about because it taught me that through communication with others in the classroom, students are able to better understand mathematical concepts. Discussion among students allows them to organize their thoughts and ask questions for clarification.

This article would be beneficial for classroom teachers. Through this article, teachers are reminded that lifelong learning is important to student's success in the classroom. Teachers can learn from this article that collaboration with other teachers can help improve their teaching skills. The article includes a list of questions that a teacher can ask themselves and discuss with other teachers to improve their teaching. These questions are very beneficial and useful to teachers. Reflective teaching by asking these questions can improve classroom practice. This article would be beneficial to first time teachers to help them begin feeling comfortable teaching mathematics.

Kotsopoulos, D. and Heide, D. (2009). You had to be there! Teaching Children Mathematics 15 (7), 410-415.

Communication Process Standard

The communication process standard is vital to the learning and understanding mathematics. As students ask questions and justify their reasoning behind the solution they found, they are communicating. Students must be able to communicate what they are thinking to others. In order to do this successfully, the student spends time organizing their thoughts and reflecting on what they have done.

The organization of the student's thinking helps them communicate clearly to others. It is important that students are encouraged to communicate their mathematical thinking to others in a clear way. Often, we find that students are intimidated by math, so the constant encouragement to communicate mathematical thinking will help students improve their skills. Teachers should encourage appropriate communication of mathematics as students become older. Students must also learn to communicate in written form.

Communication among peers about mathematics is very important. Discussion among students allows for them to see other perspectives and methods they can use to solve mathematical problems. Discussion of different solutions teaches students to become critical thinkers about mathematics. Students are able to form these critical thinking skills through analysis and evaluation of other students strategies.

Lessons on Variables - Lesson in Grade Four

The fourth grade videos are about a lesson covering variables. The students will be building a "variable machine." Prior to this lesson, the students did not have a good understanding of variables. The teacher taught the concept of variables through the use of the variable machine that each student made. The variable machine was made up of two strips of paper, one with the alphabet on it, and the other strip with the numbers 0-25 on it. The students took the two strips of paper and placed them next to each other to create a ring. The two strips of paper placed next to each other matched a number with a letter. The letter acted as a variable. The students worked together in groups and determined the value of the names of all members of their group using the variable machine. The students had to work together to figure out how they could increase the value of their names using the machine.

• Reflective Task 1: Describe how the teacher’s questioning, and the manner in which student responses are handled, contribute or do not contribute to a positive classroom learning environment. During the lesson, the teacher would ask the students many questions about what they had found. The way the teacher would ask the questions enabled the students to explain what they had found and how they had found it. The questions that were asked did not allow the students to fail. She would use answers that were not correct to teach the students the correct answer. The teacher created a more relaxed and friendly environment with the students in the classroom so when they were answering the questions they would feel comfortable, and not intimidated. I believe that the questioning contributed to a positive classroom environment. It allowed the students to feel confident in the work they had done, as well as allowing them to talk through the difficult parts of the work as a group to fully understand every part of what was being done.
• Lesson Analysis 1: Identify several strategies the teacher used to orchestrate and promote student discourse in this lesson. As the students are working in groups, the teacher is walking around to each group and asking them questions. She asked the question "what are you finding out?" After the student answered that question she said "you need to think about when you put one on 25 what that does to others." She says these things so that the students in the group would begin thinking of different ways to achieve a higher value. She also mentioned other ideas that might enable them to form a higher value. The students would then begin discussing what would make the value the highest. They would begin forming connections between the proximity of the letters and the value of the numbers. The teacher also promotes discussion in the lesson by continually asking the students what each other them are finding and having them explain how they found that result.
• Lesson Analysis 2: Describe what the teacher does to support learning while students are working in groups. As the students are working in groups the teacher is walking around and listening to the discussion each group is having. As she walks around, she stops at each table. She does many things while speaking with the groups. She asks them how they are achieving each answer. When the teacher asks that question it is supporting what the student learns because it makes them explain how they found the answer. When they have to explain how they found the answer it gives them a better understanding of the concept they are learning. The teacher also is watching what each group is doing to ensure the activity is being conducted properly. This supports learning because it is keeping the activity consistent. The students are supposed to discuss different ways to raise the value of their names without adding any letters to them. When the teacher clarifies what is being done, the students will be able to understand the concept better.

I found that watching these video clips were very reassuring to me. I have always felt intimidated

about teaching mathematics, especially those topics that can be confusing to students. When I

watched the experienced teacher in the videos teach the students about variables I felt more calm

about teaching mathematics. While watching the teacher in the videos, I noticed many of the

principles that were discussed in class being used. I liked being able to see the principles being

used effectively. I really liked the activity used in the lesson, and I liked being able to hear the

teachers justification of how certain things were done.

Teaching Principle

Important points of teaching pronciple:

1. A teacher must fully understand the mathematics they are teaching and be fully committed to applying the mathematics to experiences that will allow students to learn mathematics. Students learn best through different experiences the teacher provides.

2. The teacher must adapt the classroom to the proper environment needed for teaching math. Organization of the lesson is key to its success.

3. Improvement is important when teaching. A teacher must stay committed to learning and improving their lessons and teaching strategies. Continual learning is important because teachers need to increase their knowledge of mathematics and pedagogy.

What Values Do You Teach When You Teach Mathematics? - Alan J. Bishop

The article "What Values Do You Teach When You Teach Mathematics?" looks into the values that students are learning through mathematics. How a teacher allows their students answer a question can teach students certain values. The way the teacher responds to a student after they answer a question also teaches students about values. The article examines the connection between mathematics and culture, focusing specifically on values that are portrayed when teaching math and how we teach it.

Teaching values is not like teaching fractions. There is no right answer when teaching values. "The choices that you make depend on your values, which in turn influence your students' values. Understanding more about values is, in my view, essential to improving mathematics teaching" (Bishop, 2001). The article examines the different sociocultural values that are taught through mathematics. The different values that are taught through mathematics are rationalism, objectism, control, progress, openness, and mystery. The article gives different ways to explore the idea of values in mathematics teaching. There are different ways to examine lessons to see the values in each.

I thought this article was very interesting. The concept that values are being taught in a math lesson shed a new light on the subject area that I had never considered before. I liked that the article gave background information about values. I also liked the end of the article when the author listed different ways you can examine lessons to see the values being taught. I also liked that the article listed many questions that a teacher can ask themselves about a lesson to see how a value is taught by the way they respond to a certain situation.

Bishop, A. (2001). What values do you teach when you teach mathematics?. Teaching Children Mathematics, 7(6), 346-349. Retrieved January 22, 2010, from http://www.nctm.org/eresources/view_media.asp?article_id=766