Manipulative Activities Reflection

By doing the manipulative activity in class, I now have a better understanding of the types of manipulatives as well as their uses. After examining several different types of manipulatives, I now have a better idea of the kind of manipulatives I would like to have in my classroom. I saw some manipulatives that we used today, such as pattern blocks and snap blocks that I would like to use in my classroom. Other types of manipulatives I saw were not so versitille and I would not nesecarily want to use them in my classroom. I believe that manipulatives are essential to use in math classes, as well as other subjects. I believe that all students can learn but they all learn differently, manipulatives are a great teaching tool and great for visual learners. When students use manipulatives, they deepen their understanding by being able to physically use something to represent a mathematical concept. Students deepen their understanding by connecting a math problem to something real, they can use the manipulatives and mess around with them to understand the mathematical problem. In order to determine if students can transfer their understanding from manipulatives to other situation you must gently ease them away from using manipulatives. By having the students use manipulatives on a regular basis to solve math problems, you slowly ease them away from using the manipulatives until they say they do not need them anymore. Once the students stop using them and realize they do not need them anymore, that is how you can assess their understanding or growth.

Technology Reflection

Throughout the course of the semester we have used several different types of technology. A lot of the technology we have used I was completely unaware of or have never used before in my life. Some technology such as email and Sakai I use on a regular basis for my other courses and general Bradley information. However, most of the technology used within this course was completely unfamiliar to me and I learned a great deal from. First, we used Wiki, which is something I had never heard of before in my life. Since we did use it towards the beginning of the semester, and only a couple times, I still need to become familiar with it but it will definitely be something I will further investigate. We also used Prezis and Jing videos throughout the semester, which I had never heard of or used before either. I will be using one of these forms to make a video for my portfolio which will be the true test if I indeed learned how to make a video properly. I had never made a video for a class before, and I found them to be extremely beneficial. They re-iterate the purpose of the assignment in a visual way. Videos are important to make when we are teaching if we have students that are visual learners. I can also see it being beneficial as a teacher if we are absent and can put our lessons onto a video, or if a student is absent we can do the same thing. Throughout the semester, we have been using Blogger to write blogs about various things. I had never had a blog before, but I found it extremely easy to create and use throughout the semester, with a few mishaps here and there. I think blogs can be useful within the classroom to keep students and parents updated. Now that I know how easy it is to make and use, I can see myself using a blog to keeps parents informed of classroom activities. I think using Blogger is more appropriate to communicate through versus a social website such as Facebook. We also used Smart Board every day, almost, to sign into class. I have heard of teachers using SmartBoard for this purpose but have never seen it done, I was glad to get the experience in using SmartBoard for this purpose. Overall, I think we used more types of technology within this class that I have in all of my college courses combined. Although at times it was frustrating, since technology can be somewhat unreliable, I found it extremely beneficial. By being experienced with all of these various types of technology, I will be able to use them within my own classroom. It is important as teachers to remain current in the education trends, which will always be involved with technology. Technology is a major part of education today and it is going to be essential as a teacher to remain current and know how to incorporate technology within the classroom.

Video Analysis 3

"Looking Behind the Numbers"

1.) Purpose of Activities:
The videos were taken in an 8th grade math classroom. The students were examining mean, median, mode, and range by using real life statistics. The students first were to find the statistics of a players on a basketball team and were to determine the MVP. They determined the MVP based off of the mean, median, mode, and range they calculate with their classmates. The students then wrote a letter to the basketball coach explaining statistically why that player deserved to be MVP. The second task the students were given was to calculate the mean, median, mode, and range of the hours spent watching the Olympics. Once the students completed that, they were given mystery sets of data to analyze and figure out various means, medians, modes, and ranges.
2.) Connections to Process Standards:
Throughout these videos, I saw several process standards being demonstrated. One standard that was used from the beginning task was reasoning and proof. The students has to mathematically reason why one player should be the MVP of the basketball team. The students has to use their statistics that they figured, to make a viable argument why that person should become MVP. Going along with the letters, another standard I saw being used was communication. Communication was demonstrated by the students writing letters to the coach to mathematically reason why the player should be MVP, as well as the students communicating by working together in pairs and as a class to construct the mean, median, mode, and ranges for various sets of data. The students had to communicate with one another, as well as the teacher, in order to solve the problems and communicate their individual data results. The students also were problem solving throughout the videos. They were given problems with sets of data and were required to solve the problem using various methods. During the ending tasks, the students used methods such as "guess and check" to solve their mystery problems. A main standard I saw being used during these videos was connections. The entire concept of this lesson was making outside connections  to the real world. By using real world data from a real basketball team and from Olympic data, the students could connect to the mathematics they were using. Also, by writing the letter to the coach and doing peer editing, the students were making interdisciplinary connections as well.
Connections to Standards of Mathematical Practice:
There were also quite a few standards of mathematical practice being demonstrated within these videos. One standard that is easy to see being used was constructing viable arguments and critique the reasoning of others. The students demonstrated this standard by writing the letter to the coach about which player should be MVP and having to reasoning and construct a viable argument as to why. The students also peer edited each others letters, therefore they were critiquing one another. Another standard I saw being used was making sense of problems and persevering in solving them. Students demonstrated this when they were trying to solve the last task by using various mathematical techniques such as "guess and check" to solve the mystery problem. A third standard I saw being used was using tools appropriately, this was demonstrated when the students used their calculators to figure out the mean, median, mode, and range easier.
3.) Reflection:
I greatly enjoyed these videos. I thought making the connections to the real world by using basketball player statistics and using the Olympic data was a terrific idea. I can really see students being engaged and wanting to solve problems using this type of data. I also thought writing a letter to the coach and making interdisciplinary connections was fabulous, not only did they write the letter but the students peer edited it also. I particularly liked the Olympics activity, I am doing an Olympic unit in my novice teaching classroom. Although second graders cannot do this type of math, this would be wonderful for a middle school classroom. This would connect wonderfully to an Olympic unit, especially if the Winter Olympics were currently on.

Error Analysis Reflection

After spending several weeks throughout the semester analyzing errors, I feel I have gathered enough data to provide an in depth reflection on the error analysis process. First, I enjoyed looking at common errors children make and further analyzing them to discover not just what their mistake is, but why they're making the mistake. I think it is extremely beneficial as a teacher to take the time to analyze the errors our students make. I can honestly say, I know that as an elementary student I made several of the same type of errors that we analyzed. I am certain if my teachers took the time to analyze my errors, the problem could have easily been corrected. If my problems were corrected rather than just marked wrong, I might not have gotten so discouraged and developed my dislike of mathematics. I also thought by working separately within our tables to discuss what the errors were and then why they were made was helpful. I then thought by sharing our ideas with the entire class to see what we thought of was beneficial as well. There were several times in which my group had thought we solved the error work and analyzed it correctly, but when discussing with the entire class, we were in fact incorrect in our analysis. I learned many things after analyzing the errors, the main thing I learned is that there is always a reason a student answers the way they do. Many of the errors were made from a simple mistake, or misunderstanding, which led to wrong answers. These problems and answers, however, made complete sense to the students. From my experience within this classroom working with the error work, I have been able to use my gained knowledge within my novice teaching classroom. Some of the examples of the error work were mistakes made when doing two digit subtraction, the student did the problem right to left instead of left to right. After analyzing the error work in class and discussing the reasoning behind this type of errors, I was able to notice it right away with my second graders during novicing.These types of errors are extremely common and must be addressed right away in order to succeed with other mathematical operations to come. Luckily, my cooperating teacher is familiar with these types of errors and addresses them immediately. I can see first-hand the importance of analyzing student's mathematical errors in order to help the students be successful down the road.

Journal Summaries

1. Fair Shares, Matey, or Walk the Plank

Wilson, P., Myers, M., Edgington, C., & Confrey, J. (2012). Fair shares, matey, or walk the plank. Teaching Children Mathematics, 18(8), 482.

This article discusses the importance of teaching young children the value of equal-sized groups. By educating children at a young age about equal-sized groups, they will be more likely to excel when dealing with fractions, ratios, and multiplicative operations. This article focuses on a study done on children of all ages that shares a pirate treasure and later shares a cake. The focus of these studies is to see the logic and reasoning behind the way they are sharing the pieces into parts, equal or non-equal. The term equipartitioning is mentioned quite a bit throughout the article, this term refers to creating equal-sized groups or parts of collections or wholes. When children are creating fair shares there three pieces of criteria that should be met: 1. creating the correct number of groups or parts 2. generating equal-sized groups or parts 3. exhausting the collection or whole. When the students shares the pirate treasure they were given different problems, first there were only two pirates and they had to share the treasure. Next, there were more pirates to share the treasure with and the students had to explain their rational. This type of problem and questioning was also in the form of a pirate birthday party and they had a rectangle cake with n number of pirates, then a round cake with n number of pirates to share. The way a student divided the whole demonstrates different knowledge, as well as their reasoning behind it. By justifying their answers, teachers can make connections between fair-share experiences and other mathematical concepts.

I really enjoyed this article and have found several interesting and useful strategies within it. The information given was especially pertinent to me in that I want to teach younger elementary grades. By knowing that teaching fair share problems to young children enables them to be more successful in mathematics down the road, I will be sure to remember this and emphasize this technique within my classroom. I liked the use of pirate treasure and a pirate birthday party, this is something young children would enjoy and can easily be incorporated into a multidisciplinary unit. I also liked that the article gave a couple different books to incorporate when teaching fair share problems, I think using books with all subjects is extremely beneficial and math books are more difficult to find.


2. Putting Mathematical Discourse in Writing

Bolyard, J., Lynch, S. (2012). Putting mathematical discourse in writing. Mathematics

Teaching in the Middle School, 17(8), 486-492. 

This article describes a research project in which sixth graders write about their problem solving efforts. A term used frequently in this article, and one of the main foci of the research, is metacognition. Metacognition is being aware of, evaluating, and regulating their own mathematical thinking. The research project was designed by a sixth grader math teacher, she started a pen pal correspondence between her sixth graders and preservice teachers enrolled in a math methods course. The sixth graders responded to a main question and additional open-ended questions by completing a graphic organizer with the following information: 1. the problem's main question 2. potential solution methods 3. the student's work 4. the student's final response. Then, the students wrote a letter to the preservice teachers describing their understanding of the problem, their methods, their reasoning, and their answer. The teacher then sent the preservice teachers the letters and the graphic organizers. The preservice teachers responded with feedback on the student's work and explanations and questions further explaining their efforts. For students that understood the problems, the preservice teachers asked higher-order thinking questions to further enhance their understanding. The last step in this cycle was for the students to respond to the preservice teachers questions. Based off of this correspondence, the teacher analyzed the areas in which she needed to work on with her students. She found out the students got more out of discussing their reasoning with groups, rather than writing about it. She also discovered that by only doing one graphic organizer per group, instead of per person, this allowed for more discussion which enhanced their problem solving skills. This is one example of how written discourse can provide information for instruction while developing student's thinking.

I thought this article was interesting in several ways. I really liked the idea of having the students write out their reasoning and logic behind problem solving, I think this would be a great way to get students to think about it. I also thought having the students write their reasoning in a letter form to preservice teachers was fabulous. I think this would have been beneficial to both parties and I wish we were doing something like this in our course! This provides better communication skills, especially mathematics communication skills. The students are putting their logic into words and giving reasons behind why they answered the way they did, which further enhances thinking. I found it interesting that the students found it more beneficial to discuss within their groups their reasoning rather than write out their thoughts. I also thought making a graphic organizer with the information was a great idea. There were several ideas from this article that I think would be extremely beneficial to do in a middle school classroom.

Conferences and Interviews: Students lead parent-teacher conferences

This article focuses on student-led parent conferences. It discusses the importance of focusing on a student portfolio during the conference. It is important to have the student chose what to put in his or her portfolio to share with their parents. It is also essential to make a schedule and stick to it. By having a schedule it allows for little off topic conversation and gives the students something to go off of while speaking. Evaluating right away is also important, there are only four main questions needed for a good evaluation of the conference. Another beneficial idea when doing a student-led conference is for the parents to fill out a questionnaire afterwards to get immediate feedback for the teacher and the student. A few things the students did not like was the time constraint and they were unable to previously view their end of quarter reports. Although there are a couple negatives, the majority of the students thoroughly enjoyed student-led conferences. It gives the students, parents, and teachers a better picture of who the student is, who he/she has achieved, and what the students future goals may be.

I enjoyed this article and I think student-led conferences are a great idea. I would love to use student-led conferences in my classroom and I think this was a great article to display ideas about it. The article also gave an example of a schedule for a student-led conference as well as an example of a parent questionnaire. This article displayed many benefits for not only the students, but the teachers and parents as well.

Journal Summaries

1. CCSSM: Getting Started in K-Grade 2

This is the second article in the series which focuses on each grade band. The article first focuses on characteristics in K-grade 2. Next it discusses why teachers may have to change their thinking of mathematics in the classroom. Last, the article offers suggestions that allow teacher to reflect on their implementing CCSSM standards into the classroom. The main domains of the standards that relate to kindergarten, first, and second grade are: operations and algebraic thinking, number and operations in base ten, measurement and data, and geometry. In the article there is a great table that displays CCSSM wording and translates it to student friendly wording. There are some great tips on how to reflect and implement CCSSM into the classroom. The following ways are what is suggested: familiarize, reflect, identify, make, keep, have, at...and several paragraphs are attached to each word with additional suggestions.

This is the second article in this series in which I also read the first article. I am glad the articles are going to focus on each specific grade band. I think these articles are extremely helpful to educators and future educators that want tips on how to successfully implement CCSSM into the classroom. Most districts do not give suggestions or ways to use these standards but simply tell you to implement them. As teachers, I believe examples and strategies are beneficial to look at. I think these articles are great for teachers that want help and need help in implementing and better implementing CCSSM into their classrooms.

2. From Artithmetic Sequences to Linear Equations

Video Analysis 2

1) In the lesson "Looking at an Angle", Mrs. Sanchez teaches her 7th graders about angles. She begins the lesson by taking the students outside to look at the angles a ladder makes. This type of teaching connects to the real world and allows the students to relate. Back in the classroom, the students do activities that relate to angles, height and distance ratios. Students work together in small groups to determine angle measurements and height to distance ratios as well as plotting a graph. The lesson is all connected to the ladder and what angle is needed in order to it to be safe to use.

2) I saw many process standards demonstrated throughout this lesson. The process standards that stood out to me were communication, connections, and problem solving. Communication was demonstrated by having the students work in groups and with partners to solve problems and work together. The students then had to communicate as a whole class in order to plot the graph with all of the angle measurements. Making connections was demonstrated throughout the entire lesson because the concept of measuring the angle of a ladder automatically made a connection with the students. The students were using real life angles and real life scenerios to solve problems. The students felt connected to this math problems because they were determining if the ladder was safe to use. The students also demonstrated problem solving when they worked in partners and small groups in order to figure out different measurements. The students were only given certain numbers, either the angle or the height and distance, and had to figure out the opposite by problem solving. The students then had to determine if these measurements would be safe for the ladder.
I saw many of the Standards of Mathematical Practice being demonstrated throughout these video clips. The three standards that stook out most to me were make sense of problems and persevere in solving them, use appropriate tools strategically and model with mathematics. The students make sense of problems and persevere in solving them by working together to solve the problems, understand the different approaches to the problem, and understand the effects that are involved. The students use appropriate tools strategically when using their protractors and compasses to measure angles. The students model with mathematics when connecting the measurement of angles to the placement of ladders and using real life scenerios.

3) I thought the lessons in the video clips was very engaging and well planned. The concept of the lesson I thought was a great idea and really got the students interested in the topic of angles. Since they know a purpose for measuring angles they got more excited about it. I liked how the students worked with partners and small groups to continue measuring angles and height and distances. I also thought the teacher was well organized and liked her direct instructions. She had guided questions that helped the students along, yet it was no obvious. I also liked that the videos also gave a script of what the teacher and students were saying, it was easy to follow.

Review of Math Applets

Grades 3-5

5.2 Understanding distance, speed, and time relationships using simulation software
http://www.nctm.org/standards/content.aspx?id=25037

This applet simulates two runners running along a track. Students control the speed and the starting points, they can then replay it and see a graph viewing the time-distance relationship. The objective states that this will help students understand ideas about functions and representing change over time. This applet is pretty easy to use, with reading the directions first. I did not read the directions first and I was confused, after going back and reading the step by step directions given, it was a simple simulation to use. The visual presentation and graphics are not great, but they get the point across. This applet could have been a lot more visually pleasing by having better graphics and more detail.
I think this applet overall would be beneficial for students to learn from. Although, I did not think it is the more graphically pleasant applet, the content is good. This applet allows students to be the controller and change the speed, the way the runner is facing, the amount of steps the runner takes, the starting position the runner takes, and how fast or slow the runner goes. Automatically, all the information is relayed on a graph that displays both runners and whatever information the student enters. I think there are great follow up questions and tasks for the students. An example of an additional question, that can be an interdisciplinary lesson, is: Set the starting position and length of stride for both runners. Run the simulation. Now write a story that describes the trip. For example, "The girl is going really fast. She catches up to and passes the boy, who is going slow," or "The girl started way behind the boy, who was already halfway to the tree by the time she got going. She went really fast and caught up to him more and more. Finally, at 75 she passed him and kept going really fast and got to the tree first." Although there is no set assessment, you could use any of the questions as an assessment, as they all involve the applet and knowing how to use it.
There are also great follow up reflection questions for the teacher as well. Some of the reflections include: Do you think the students would enjoy using this activity? And what important ideas about functions and representing change over time will the students get out of this activity?


5.1 Communicating about mathematics using games
5.1.1 Playing Fraction Tracks
http://www.nctm.org/standards/content.aspx?id=26975

This applet supports students' learning about fractions. The objective stated is students have opportunities to think about fractions and how they relate to a unit whole, compare fractional parts of a whole, and find equivalent fractions. This applet is extremely easy to use and is self explanatory. This applet is designed to be done in partners, however, it could work with just one person as well. Each player gets a fraction and they have to find fractions that can make it, or find the fraction on the board. Since this is also from the NCTM website, it is made similarly to my previous applet. It is not visually pleasing either, and the graphics are not very good. Once again, while the graphics are not as pleasing as they could be, the content is still good. This would not be an applet to use for a lesson, but more so if the students have extra time or need additional help with fractions. Other than the actually game through the applet, there is no assessment. There are no additional tasks or questions other than: To extend this game, students could make their own boards with different fractions, with decimals, or with a combination of decimals and fractions.
There are reflective questions for the teacher once again, such as How can playing this sort of game help children build understanding of equivalent fractions?
Like I said, this would not be an applet I would introduce my class to for a lesson. This would be more of a "filler" or something for students to do if they have extra time or need additional help in fractions.

Journal Summary 2

The Value of Debts and Credits

Akyuz, D. (2012). The value of debts and credits. Mathematics Teaching in the Middle School, 17(5), 332-338.

Certain teaching practices can support students' mathematical reasoning. The topic in this article is integers, which are typically a challenge in the mathematical community. A persons financial net worth provided the context for the activities in this article. The integer operations used were addition, subtraction, and multiplication. This type of instruction is realistic mathematical education (RME). This type of instruction allows students to use their informal knowledge of mathematics and progress to move towards formal mathematical reasoning. There is a detailed chart in this article that explains how to foster mathematical reasoning. The article is based off of a study done over a five week period in a 7th grade classroom in Central Florida. There were 20 students, 13 boys and 7 girls, 3 students had mild learning disabilities. The data included teacher interviews, audio and video taped classroom sessions, field notes, teacher notes, research meetings, and a collection of student work. The most important practices that have been found to support student reasoning involved: encouraging students to give concrete explanations to think of effective solutions, to make conjectures and prove them, and to provide different and sophisticated solutions. The article demonstrates how an expert teacher incorporates these practices into her teaching. Although this lesson focuses on debts and assets it can be modified to fit other topics. The article then gives a description of the practices and provides classroom examples. Teachers need to know hot to support students mathematical reasoning from concrete to abstract, to motivate, support, and give opportunities to explore mathematical reasoning and concepts.

I thought this article had some really good ideas for including mathematical reasoning into the classroom. I do not recall doing anything like this when I was in middle school, however, times change. I think that including real life scenarios to teach a math lesson is a great idea and is extremely beneficial to the students. The students  feel that it is actually knowledge they need to know, therefore, they will be more enthusiastic and take an interest in the lesson. Personally, I thought thought the wording throughout the article was a little heavy. I am use to reading articles and journals geared toward primary grades, since that is what I would like to teach. This article, geared towards middle school grades, had a lot more wording and mathematical vocabulary that I am not use to. I was not familiar with these practices and so I found this article to be a learning experience for me.

Journal Summary 1

The Big Picture: Examine the Structure of CCSSM  and Consider Major Ideas That Will Influnce Their Implementation.

This article is an introduction to a series of five articles to support implementing the CCSSM. Future articles will address additional topics, ideas, and different grade bands.
The article introduction the idea of the standards and gave some background information about the CCSSM. This will be the first time in US history that a set of standards is established and taught in nearly every state, 44 states have currently adopted the standards. These standards introduce a significant change in the way we teach mathematics. This article provides an overview of CCSSM by addressing four main parts: 1.) Describe the standards which apply to K-12. 2.) Describe the parts of the grade level standards. 3.) Explain how the standards develop across grade levels 4.) Discuss the intersection of the domains of the CCSSM.
The article further explains what the eight standards are, and then explains the grade-level standards. The grade-level standards include a 2 page introduction describing the critical areas and broad clusters for each grade, supplying teachers with essential ideas to focus on throughout the year. The teachers can sort the different standards into categories such as: standards  they are ready to implement, standards they have some ability to implement, and those that need the most preparation to implement. Also provided in the article is a list of questions the educators and districts get to begin conversations that lead to strong advocacy of the CCSSM. Throughout the article there are examples, tables, and pictures explaining and describing how to read and understand the different sections of the CCSSM.


As a future educator, I found this article incredibly helpful. This article is specifically designed to instruct educators on how to use the CCSSM in their classrooms and how to understand the standards  themselves. Since this is the first of a series of five, I'm sure the other articles will go more into specific examples with different grade levels and so forth. Personally, this is an article that I would like to read especially if I was confused or overwhelmed with not knowing how to incorporate the standards into my lessons. It gives step by step instructions and definitions on how to use the standards and what the standards are. This would be an article, or series of articles, to refer back to once I have a classroom of my own.

Dacey, L., & Polly, D. (2012). Common core standards for mathematics: The big picture. Teaching Children Mathematics, 18(6), 378-383.

PBL Review part 3

Building a Safe Place vs. Technology Grant

The PBL about building a safe place is for 7th and 8th grade students to come up with a solution to present to the Peoria community board. The students will come up with an idea for a community center that includes a budget, educational centers, fundraiser ideas and more. This PBL includes numbers and operation, algebra, geometry, measurement and more. I think this PBL activity involves several types of math and also connects with many other subjects as well. This PBL is a great example to me of what an effective problem based learning activity would look like. The PBL about the technology grant is about a teacher that wants to apply for a $50,000 grant for the low-income school he teaches at. He assigns his 7th and 8th grade students the task of coming up with a grant proposal to present to the board. The focus of the math in this problem is budgeting, I think this activity could have and should have incorporated more math concepts into the equation.
When comparing and contrasting this two different PBL's, each has things I like and dislike. Overall, I much prefer the set up of the "Building" PBL. I think it's much more organized, easier to read and more professional looking. I also think it incorporates more math into the activity, the mini lessons are easier to follow and more detailed as well. In "Technology", I do like that there are several questions for the teacher to use to prompt the students as well as specific examples for adaptations. The strengths of the first PBL in my opinion are the mini-lessons, the assessments, and the way the activity is written in general. The strengths of the second activity are the adaptions for special needs and gifted students as well as the prompts and role of the teacher. While I think math is the main focus of both PBL's, I once again do not think there are enough math concepts in the "technology" PBL. I like that there are different layers to the "building" PBL and therefore different types of math required.

PBL Review part 2

Problem Based Learning Faculty Institute

The website I found is from the University of California- Irvine, and it's a faculty institute about PBL. This website explains what PBL is and that it's a student-centered approach. It then goes into the criticisms and benefits of PBL and what the instructor's role is. The instructor is not to be passive but should model different problem-solving strategies and should ask the students higher-order thinking questions. The section on this website that I really like and thought was helpful is a chart of characteristics of PBL. There are 3 columns, WHAT, HOW, and WHY? An example is WHAT: Student-centered and experimental HOW: Select authentic assignments from the discipline, preferably those that would be relevant and meaningful to student interests.  Students are also responsible for locating and evaluating various resources in the field. WHY: Relevance is one of the primary student motivators to be a more self-directed learner.
Other categories within the WHAT section are: inductive, build on/challenges prior learning, context-specific, problems are complex and ambiguous and require meta-cognitive thinking, creates cognitive conflict, and collaborative and interdependent.

The article is geared towards college professors using PBL strategies for college students. Although the context is a little more advance, the idea of PBL is the same. I did like the chart provided and thought it has good explanations of why and how to use PBL and why it works. Another reason I liked this website is that there are many examples provided with some student evaluations. I also liked that this website had some criticisms of PBL because I was interested in what they might be. As a future primary school teacher, I would try to find a different website when looking up information about PBL because this is not geared towards younger students. Although the ideas are the same, the language is more complex and the examples given do not apply to primary grade students.

http://www.pbl.uci.edu/whatispbl.html

PBL Review part 1

Problem Based Learning, or PBL, is a learner centered approach to teaching. PBL can be used in any grade level classroom, as long as the material is developmentally appropriate. PBL gives students complex, real-world problems to solve that do not have one simplistic answer. The teachers role is to be the facilitator and ask questions, allow plenty of time, and provide resources for the students to use. The students role is to, in small groups, figure out what they know and what they need to know to solve the problem. The students then apply that knowledge to come up with a solution or multiple solutions to the problem. Problem based learning allows the students to use higher-order thinking, problem solving skills, reasoning, communication, and many other skills to solve problems.

Developing Mathematical Reasoning Through Games

The article I choose was written by Jo Clay Olson, a 25 year teacher from pre-k to high school. She wrote about her experiences with mathematical games and the relationship to mathematical reasoning. Engaging math games encourages students to explore number combinations, place value, patterns, and many other mathematical concepts. She put the three P's in place- Plan, Play, and Please be patient. With the three P's in place it provides a framework for teachers to explore mathematical ideas and discussion. When choosing a game for your students to play you must: play it yourself to gain familiarity, discuss ideas and how they can emerge in class, determine the level of competitiveness for you class, anticipate responses and outcomes, and create a list of questions to prompt students' thinking. When you Plan you must decide how you will introduce the game to your students, how you will decide teams/partners, what materials you need, etc. When you Play you should walk around the classroom after you introduce the game and listen to the conversation with the students, take notes on strategies used, and think of discussion starters. In Please be patient it is important to provide repeated opportunities for you students to play the game, and watch as the mathematical strategies change. In this article she provide three games to use, two for primary grades and one for intermediate. Include in these games are question prompts, and answers from her students, and "cautions" she found during her experience. Games are fun and create a context for students' mathematical reasoning. Games encourage students to respond different using different strategies and enhance further mathematical reasoning.

I enjoyed this article very much. I enjoy playing games and would love to incorporate games into my classroom. Not only does this article provide several games and tips to go along with them for math lessons, it provides reasoning from an experienced teacher. I liked the three P's strategies of planning games and can see myself using several strategies I have obtained from this article. I also liked that within the different game sections in the article, she provided question prompts and tips to further enhance students' thinking. I would like to use these game ideas and strategies in my math classroom.

Olson, J. (2007). Developing mathematical reasoning through games. Teaching

Children Mathematics 13 (9), 464-471.

Reason abstractly and quantitatively

-Mathematically proficient students make sense of quantities and their relationship with problem situations

-Two abilities to solve problems involve quantitative relationships:
                  1. Decontextualize which is the ability to abstract a given situation and represent it symbolically and represent symbols as they have a life of their own
                  2. Contextualize which is the ability to pause in the process and think about the symbols involved

-Quantitative reasoning creates a representation of the problem, attends to the meaning of quantities, and knowing the appropriate properties of operations and objects

Video Analysis 1

The videos were about a fourth grade lesson plan on variables. The teacher, Mrs. Klein, introduced the concept of variables by assigning a number to each letter of the alphabet. The students were in groups and they were assigned various tasks to use the different variables to create words and assign numbers to the words. The tasks were designed to introduce variable and the concepts of the links between numbers and letters.


Questioning: Reflective task 1: Describe how the teachers questioning and the manner in which student responses are handled, contribute or do not contribute to a positive classroom learning environment.
The teacher uses prompts to get students to discuss and share their ideas throughout this lesson. By having the students explain and discuss with their peers and teachers it allows the students to bounce ideas off one another. Class discussion gets all of the students thinking and their creative juices flowing. This type of questioning and class discussion creates a positive classroom learning environment for the students to be successful.

Assess: Personal Reflection 1: What techniques do you use to determine whether students have learned the materials you are teaching?
Assessment is essential to determine whether students have truly learned the material you are teaching. Assessment can be informal or formal. Mrs. Klein used informal assessment for this particular lesson by listening to students conversations for understanding. She analyzed what the students could pick up on, what they were looking for, what they could see within the problems, and how they solved the problems. Another way to assess informally is through participation. Formal assessment is another technique to determine if the students have learned the material. Formal assessment could be a test, a worksheet, a project, etc. Through assessment, informal or formal, the teacher can determine whether students have learned the material.

Responding: Reflective task 1: Describe the student-teacher interactions during the task debriefing discussions and assess the effectiveness of these interactions.
During the debriefing of each task, Mrs. Klein asked the students to reflect on the work and share some examples with the class. Mrs. Klein prompted questions which allowed the students to share their ideas and theories about the tasks. Mrs. Klein gave positive feedback to the students and encouraged their theories and ideas. This type of discussion creates a positive learning environment and a positive student-teacher relationship. By having the students share their ideas and how they solved the problems, it allows for other students to learn from the discussion.

I enjoyed watching these videos, I think watching a teacher teach a lesson in an actual classroom setting is very beneficial. Not only is it beneficial to see how she teaches a math lesson, but how she conducts her class in general. There were several interesting classroom management strategies that she used throughout this lesson, clapping for example. Another reason I enjoyed watching these videos is to get ideas for my future classroom and how I will conduct my class. I liked the round group tables with supplies in the middle, that is not something you see often in classrooms and I liked the idea.

"Communication Speaks"

This article was written by a fourth grade teacher who changed the way she taught math to align with the NCTM communication standard. She began the change by simply starting to ask the students to explain their thinking- which began to make the class more interesting and engaging for both the students and the teacher. However, the students needed an opportunity to verbalize their answers and they needed prompts from the teacher. Some examples given of the types of questions asked were: Why is this so? Explain your thinking. Show me. How do you know? The next area of communication she worked on was listening, which is essential to communication. She started to stand back and allow the students to share with one another and determine different ways to solve problems. This forced the students to listen to each other, make connections, and think creatively. The next area of communication addressed was writing. She had her students create posters and their peers evaluated them. This demonstrated not only the importance of words but examples, the sequence of the way things are written, and the preciseness of vocabulary. After adding communication into the curriculum she asked herself , "So what are my students gaining from this added communication in math?" She could tell her students were gaining knowledge and benefiting from the discussion and sharing of thinking. As asking questions became a daily part of class, not only was the teacher asking questions but the students began to ask questions as well and wanted explanations. She saw her students take ownership and became invested in learning. As the article concluded, Robin stated that by asking one small question it can have a huge impact on teaching and learning. The students must be taught how to speak, listen, question, and write, this all takes time- time well spent according to Robin.

I enjoyed this article and thought it gave great ideas of how to incorporate communication into a math classroom. I liked that it was written by a teacher who has tried these methods and saw success with them. I would like to take these ideas into my own classroom because they are not only found to be effective but simple to incorporate. This strategies are beneficial in any subject, not only math. I think by adding communication to all subjects would be extremely beneficial to the students.
 I know after reading this article and reflecting on my own math experiences, I do not remember using communication within the math classroom. I think it would have benefited me especially to answer and ask questions such as "why" and "how". I think by communicating in math, more students will like math and understand it better.

Kinman, R. L. (2010). Communication speaks. Teaching children mathematics 17 (1), 22-30.

NCTM Process Standard: Communication

- Organize and consolidate mathematical thinking through communication: formulate a question, explain answers and reasoning, diagrams, mathematical symbols, writing to reflect on their work

- Communicate their mathematical thinking coherently and clearly to peers, teachers, and others: students need to test their ideas in the math community within the classroom, students will benefit from being a part of discussion

-Analyze and evaluate the mathematical thinking and strategies of others: students working on problems with other students

- Use the language of mathematics to express mathematical ideas precisely: gaining knowledge of mathematical language, students use of mathematical language, comparing mathematical expressions with technology tools