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.