Journal Article: Water Bottle Designs and Measures

Carmody, H.G. (2010). Water bottle designs and measures. Mathematics teaching in the middle school. 16(5), 272-279

This was a wonderful article, there were so many neat ideas and concepts introduced throughout the article! In the beginning of the article the author mentioned how students are excited about the unit of finding surface area and volume because of the authentic and hands on activities. Students will work ahead on homework because they actually enjoy what they are learning, and believe it or not, they discuss math concepts outside of the classroom. The article suggested having an engaging classroom and hands on activities that have open ended questions. I thought this was such a helpful article for future and new teachers because it breaks everything down. The article has a list of questions to make the lesson open ended and ways to incorporate higher order thinking. The is also tips to ensure that students fully comprehend the project or assignment. All the ideas listed are ones we discussed throughout the semester. (Which makes me super happy!) Ideas include having peer collaboration, written reflections, using manipulatives or visuals, project proposal, and algebraic notation.

The activity that is discussed in this article is actually a 19 day project, and there is an outline of the activity included in the article. This is yet another reason why this is such a fantastic journal. This shows that this project is a success and really helps students further their understanding. The water bottle activity is where students need to create an object that holds a specific amount of water. They need to sketch three drawings where they using mathematical terminology and explain which drawing is their favorite and why. The article concludes by introducing other ways to incorporate surface area and volume into the classroom, such as making race cars.

I thought this was a very useful article and had many interesting lessons throughout. My favorite part of the article was the beginning where the author gives the reader a schedule of the timeline of the unit. I thought this was very useful and laid how much time to spend introducing and reviewing the topic and then how much time should be spend creating the project. As a future teacher I have no idea how quickly or slowly my future students will learn, so it is nice to have a sample timeline to base lessons off of. I will most definitely use this lesson, because it seems super exciting!

Technology

After completing blogs this semester I have realized how valuable internet usage is in the classroom. Having students reflect on their discoveries and what they liked and disliked about the activity helps students deepen their learning. Since their are so many professions that are based off use of technology, incorporating technology at an early age will help students prepare for the future and enrich the learning process.

What I most enjoyed about completing the blog this semester was reading the monthly journals from the NCTM website. I thought they were meaningful and easy to adapt to any grade level. I like how they were already tried out in an actual classroom and reading it in the journal meant that the lesson had positive outcomes and results.

Looking back at the semester I also realized how much technology was used, besides just for research purposes. From day one I was introduced to a variety of different ways of technology. For example the Wiki page, the math applets, and the videos on the NCTM website, using the SMART board, and practice using the manipulatives. All of these programs will help me further deepen my learning which will help me in my future classroom. Technology is changing rapidly so as teachers we need to stay on top of it, because tomorrows leaders our in our hands.

Reflection on Manipulatives

After last class period and throughout the semester I have come to realize how valuable manipulatives are in a classroom, specifically a mathematics classroom. They benefit students in a variety of ways. After Wednesday's class period I learned something very valuable: quality over quantity. There are a lot of different types of manipulatives out there, but that doesn't necessarily mean it is a good idea to purchase them all. Some manipulatives could be used for several different content areas, whereas others were created for one specific content area. I also learned that as a teacher we need to be resourceful. There are a lot of manipulatives that can be hand-made, so if there is a budget you need to stay within, consider making a manipulative. Some manipulatives are targeted for a specific grade level, so as a teacher I need to be aware of this. I enjoyed the base ten logs, because they can be used at practically any level, and a ride range of topics. They can help students from the basics such as part and whole, all they way up to complex fractions.

Manipulatives can be used to help describe a concept through using visual representation. This way students can actually see how formula, pattern, etc. works. Manipulatives are extremely beneficial for students with special needs. They help students connect what is on the board or in their books to something real, something they can touch or hold. Manipulatives can also help students who are unable to see, figure out a problem. Manipulatives could also be used as a stress relief or sign to just relax. Mathematics is often a subject that students struggle with, so having a manipulative help them understand the topic of discussion can be a sign for the student to just breathe and relax. It reminds them that they already know how do this type of study. This would be great for students with Autism or Asperger's Syndrome.

Error Analysis

I am really glad we spent so much time looking at errors from samples of students' work. This helps me, as a future teacher see what errors students make. This also helps because its a good way to see what needs to be explained more thoroughly and how to present the material so it makes sense to the students. I felt like some of the errors were easy to spot, while others were more tricky to spot; so I'm glad there was a variety of error problems presented to class.

This helped me because it showed me what students really struggle with. Being a special education major, I have no doubt that more likely than not several of my future students will struggle with mathematics. Doing the errors, will help me because it is a nice introduction of common errors students may make. This way, when I am teaching math and a student makes a mistake, it won't be my first time correcting math errors.

I thought the most beneficial part of the error problems, was learning how to help students learn the material and to prevent them from making the same error in the future. More likely than not, students forgot to do a step in solving the problem or didn't do the step correctly (ex: instead of adding, they multiplied) so when we were reviewing the error problems, we discussed how we can teach them why its one way instead of the other. The reason discussed in class were more than "just because" or memorize how to do a certain type of problem. I thought this was interesting because, when I was going through grade school, I didn't learn the reason behind why you did specific types of problems, I just memorized the formulas. I believe if I was taught the reasons why we do something, I would have understood more and actually enjoyed going to math class. Overall, I believe this was a very beneficial activity because we can directly relate it to the real life classroom.

Journal Article: Put the Right Spin on Student Work

Cohen, J.S. & Ely, R.E. (2010). Put the right spin on student work. Mathematics teaching in the middle school. 16(4), 208-215.

This article was about teaching probability using a spinner. The article first introduced the activity and had the directions of how the play the spinner game. Students were paired up, and together they needed to create a spinner. The spinner can have as many sectors or portions of the spinner twice. Each partner spins the wheel and records their data. The winner is the first person who has the sum of: 2, 3, 4, 5, 6, 7, and 8. The next round of the game the partners need to alter their spinner. In the final round students need to alter their spinner again and put their findings in a blank spinner. The students then need to describe and justify why the layout of the spinner is vital and why the final spinner is more compatible than the first spinner. The article concludes by describing that fact that having very specific predetermined goals will help the teacher pick out the important realizations when choosing students work. (Cohen & Ely). The idea of probability wasn't fully understood by all the students but the vital main points of probability were introduced with the spinner activity.

I really enjoyed this article. I thought it was neat of how the game alone does a nice job explaining the concepts of probability. Even though this activity wasn't the easiest to follow I thought it was a nice way for some students to grasp it. Some students thrive on very original games and this may just be the game for them. When I was in 7th grade we did an activity where we needed to sew intrinsic shapes with colored string on black paper. We then had to calculate the angles of of the shape. Most students were very confused and thought this activity was weird because we were sewing in mathematics class. Personally, I loved the activity. I liked how we learned something new without using paper and a pencil. Knowing that an unusual activity works for others is great, but a general rule of thumb is too keep a record of which students are grasping the material and which ones the teacher needs to give extra attention to.

I thought the spinner game was fantastic! I will most definitely adapt this into my classroom. I really enjoy the fact that students need to work together to figure out how to make their spinner have the sum of each number. Being a special education major, I need to have a list of creative, authentic, and hands on ways to teach a variety of topics. Mathematics is usually a content area that students struggle with, so having creative lesson plans will hopefully help students stay intrigued throughout the lesson and comprehend the task at hand.

Journal Article: What Math Knowledge Does Teaching Require?

Ball, D.L. & Thames, M.H. (2010). What math knowledge does teaching require? Teaching children mathematics, 17(4), 220-229.

This article was mainly about different ways and techniques to implement in the mathematical classroom. The article has little blurbs that describe how teachers can incorporate visual aids in the classroom. One teacher has a chart that has the numbers 1-100. This helps students with the concept of base ten and finding patterns. Patterns include adding quickly getting the numbers close to the 10's number. The authors also touch on the concept of asking questions to students that really make them think why their answer may or may not be correct. This helps students to fully understand and retain that specific concept of mathematics. One of the toughest concepts for mathematics teachers is figuring if students fully understand the concepts or if the concepts are being missed by students. The article concludes by having the authors explain that teaching mathematics may be a difficult task, but having visuals to help students fully understand the concepts helps and deepens the knowledge of mathematics.

I thought this article was interesting, but very repetitive. All the information was important I thought, but I feel like there have been previous journal articles that have touched on these concepts. What I really enjoyed about this article was that the authors listed frequent tasks of teachers and defined each task. The definitions were short and too the point. After reviewing the tasks, I thought that this was a nice rule of thumb to incorporate in all classes, not just a mathematics classroom. The task that I thought was a great one, but is often overlooked is appraising students' unconventional ideas. Looking back to when I was a student, I think more teachers solely focused on getting the right answer. If you give positive feedback to students who may not have gotten the correct answer, they more likely remember this event and how to correct their errors. This also will make students more comfortable approaching you if they have a question.

After reading this article a way to incorporate some of these tips into my own classroom is to constantly incorporate visual aids into my classroom. More and more students show that they have better learned the material when there are visual aids. I enjoyed how the elementary teacher used the chart to teach base ten numbers and patterns. The numbers are laid out nicely and it is clear to see. I also need to remember to take a break from constantly teaching and make sure each student fully understands one concept before moving onto the next mathematical concept. Overall, even though this was targeted towards a mathematical classroom, I think many of the tips and concepts could be adapted for any type of classroom; which is great because I do not have a job yet, so I don't know what level/content area I am going to teach.

Video Blog Three

This set of videos main focus was on solving and understanding algebra and understanding the formulas and the reason why the formulas work. The videos gave three examples of lessons used to teach algebra. All the lessons used manipulatives to help students understand the concept of algebra. Also, the lessons given could have be adapted to any age/grade/ability level, which was great. The first lesson was having students look at the V-pattern of how geese fly and having them come up with a way of figuring out how many geese there were besides counting each individual goose. The students were given manipulatives to solve this problem. Students were able to bounce ideas off of each other, so communication was also a big help in solving this problem. The second lesson was having the students view beams used for construction and and figure out the pattern of how each beam was laid and how it differed from level to level of beams. Once again, students used manipulatives to help them solve this problem. Students were given tooth picks to make their own beams and help them discover a formula for this type of problem. This lesson was very authentic and holistic for each student because each student had their very own lesson. The whole class had different formulas. The teacher took advantage of this situation and discussed with the students how even all of these equations look different, some of them are the same. The third and final lesson displayed in the videos was very interesting and had the students keep track of the rate of hair growth. This portion of the videos was interesting because the teacher had the students relate this to real life, but also had them create a formula. Throughout the videos, both NCTM and the CCSSI Standards were practiced.

NCTM Process Standards:

• Communication: Throughout all the videos, communication was heavily displayed. In all three lessons, the students worked in small groups to create a formula to solve the problems. Then the students and teacher would have a whole class discussion, where they students will ask any questions they may have and discuss their findings. Having students work in small groups and then the whole class works on different types of communication.
• Connections: All three lesson plans did an excellent job connecting the math concepts to real life. In the first lesson, students connecting the pattern of how birds fly to mathematical formulas. In the second lesson, the teacher connected formulating formulas to construction beams, and in the third lesson the rate of hair growth was used. The variety of ways mathematics was connected to the real world is beneficial because it will reach out to more students.
• Problem Solving: Problem Solving was displayed in all three of these lesson plans. Students needed to formulate a formula and equation to solve each problem. They were given manipulatives to help solve the problem.
• Reasoning and Proof: Students were working in small groups and explaining the formulas and equations they created. They each needed to give reasons how they came up with their equation. During the whole class discussion, the teacher made sure to ask higher order thinking questions pertaining their equations.
• Representation: Students were given manipulatives to help them solve the problems. Manipulatives brings the problem to live and helps the students really picture the problem. An example includes using toothpicks to represent the number of construction beams and to figure out a pattern of how each beam is laid at each level.

CCSSI Standards:

• Construct Viable Arguments and Critique the Reasoning of Others: Students worked together in a variety of group sizes (partners, small group, and whole class) to decide on one way to form the equation. In order for students to agree on one form of an equation, students needed to justify to their peers their reasons behind their equation.
• Make Sense of Problems and Persevere in Solving Them: Students were able to use manipulatives to help them find the answer and create a formula. Once the students found a way to formulate a formula they were also able to use the manipulatives to solve the formulas. The manipulatives also helped the students communicate and tell their classmates why this formula would work.
• Model with Mathematics: Throughout each lesson, mathematics was displayed. Students used manipulatives to help solve mathematical equations and formulas.

I thought these videos were great, I really enjoyed seeing the students engaged during these lessons. Typically, math is a subject not many students enjoy, but these lessons definitely make math a more enjoyable subject. The class didn't have textbook work and handouts, which is awesome, because that is how I remember most of my math class career. Having such hands-on lessons makes the learning process more authentic, and helps students retain information for longer periods of time. Throughout each lesson, a variety of NCTM Process Standards and CCSSI Standards were displayed which is fantastic, and helps develop multiple strengths not only in mathematics but other subject areas. Using manipulatives also helps students who learn best by visual aids, and students learn best when they can relate something to their outside world, and all these lesson did exactly that.

Journal Article: Portfolios and a Backward Approach to Assessment

Britton, K. & Johannes, J. (2003). Portfolios and a backward approach to assessment. Mathematic teaching in the Middle School School, 9(2), 70-76.

This article was pretty interesting and it did a great job reviewing the strengths and reasons why teachers should adopt the concept of Portfolios. The authors thoroughly explained what exactly is a portfolio and then had different teachers write about how they use portfolios in their classroom. After reading this journal article, I thought the main reason why these authors published this article was to explain the idea that our society needs to start looking at the students growth and what they learned, struggled with, and enjoyed during a unit or span of time. According to our authors, students and parents look so deeply into grades, but in reality what does that A mean? The authors concluded the article by explaining how portfolios help students have pride in their work and show them their learning process from the most basic concepts to the most abstract concepts.

I really enjoyed this article, but I thought a lot of of it was common sense. I am most definitely going to adopt the concept of portfolios in my classrooms, because there are so many positive outcomes from using portfolios. Not only do portfolios help students see their growth, allow the teacher to see what the student has pride in, and shows parents/guardians how their child is doing in school; it also allows students to have pride in all their hard work. There are so many students who dislike school, but having a portfolio to visually show students all their hard work will boost their confidence and hopefully have them enjoy school a little bit more.

ICTM Mathematics Conference

On Friday, October 15, 2010, I was able to attend the ICTM Mathematics Conference in Springfield, Illinois. I wasn't exactly sure what to expect when arriving at the conference. After signing in, my job was standing at the top of the stairs in the Lincoln Hotel and direct people where to go. This job wasn't horribly difficult, and it was really nice because I was able to talk to a high school math teacher and a math professor who works at Eastern Illinois University. They were both speakers and were waiting for their room to open up, so we chatted while they were waiting. It was really great to talk to the high school math teacher because she was explaining an activity she uses when teaching algebra. She used manipulativs and she said it was easy to manipulate the activity to the ability of her students.

After I completed my volunteer shift, I decided to go to a lecture on SmartBoards. The reason why I chose to attend this lecture is because I know it will be really useful because SmartBoards are becoming a "staple" in more and more classrooms, and I need to become more familiar with them. I liked this workshop because the speaker went through the basics and then introduced us to some more original options that the SmartBoard provides. My favorite 'app' that was presented was setting up math problems with your voice. The speaker programmed math problems in the SmartBoard and once she told the board to begin, it would solve the problem, showing all of the teacher's work. This would be useful because as you are presenting the topic you can have eye contact with students and have them see the work that goes along the problem. Overall, it was a great experience, and it was nice to connect with other mathematics teachers, of all levels. I felt like I learned a lot, and will be able to use this new information later in my teaching career.

Master Geometry while Co-Teaching Journal Article

This article was in the Mathematics Teaching in the Middle School, October edition. It was written by Jessica H. Hunt, and she begun the journal article by writing about just a few benefits of co-teaching. Typically, co-teaching occurs with a general education teacher and a special education teacher. It benefits the general education teacher because it typically gives them more confidence when working with students with disabilities, because they see how special education teachers teach them new material. It helps the special education teachers because it gives them a better idea of what the general education curriculum is about. There are five types of co-teaching, they are: one supporting, one teaching, station teaching, alternative teaching, parallel teaching, and team teaching.

The article continued by explaining these five types of co-teaching. One supporting, one teaching is exactly what it sounds like. One teacher is teaching and the other is completing administrative tasks such as checking homework or managing behavior. (Hunt) The general education teacher doesn't always have to being teaching the whole class. Station teaching is when the teachers divide the chapter or content being taught and the students rotate around the classroom. Alternative teaching is where one teacher teaches the a big group and the other teacher teaches a smaller group of students. The smaller group doesn't necessarily have to be a group of students who need extra help. It could be a group of students who have a specific interest. Parallel teaching is when the teachers split the class in half, and teach the same information, but its just in smaller groups. The students join together at the end of the lesson. Team teaching is where one teacher may lead the lesson and the other teacher may model the lesson.

I thought this journal article was very beneficial! The reason behind this is because I am a special education major, and in a few of my classes we have been talking about the pros and cons of co-teaching. To be completely honest, it seems like such a great concept, but in reality it seems very difficult for the concept to work well. I've heard that the special education teacher is just like an aid and helps students who need the extra one-on-one help, and the general education teacher feels uncomfortable because there is constantly another teacher in their classroom. I really liked how this article broke down the job roles of the general education and special education teachers. I don't know where I will get my first job, but I will definitely keep co-teaching in mind, and if I do end up co-teaching with a general education teacher, I will definitely re-read this article because I feel like there were a lot of great points.

Computational Estimation Journal Article

This article was in the Teaching Children Mathematics, October edition journal. It was written by Maria G Fun and Christine L. Latulippe. The article began by having the authors explain how much we use estimation in our daily lives without even thinking about it. We use estimation when we are driving and figure out if we need to stop for gas or if we can make it to the next stop. We use estimation when figuring out a tip at a restaurant and when shopping and figuring out which store has the best deal. In order for individuals to develop these estimation skills, teachers need to teach students. There are several types of estimation, and the article begins by explaining the main two. Reformulation, which is used for rounding, truncating numbers, and substituting compatible numbers. The other one is Translations; which is used for changing the mathematical structure. The authors continued on giving examples of each way of estimating and gave lots of lesson plan ideas for teaching estimation.

Some of the lesson plan ideas that the authors included were games where students had to quickly guess the total, kits with a missing numerator in the set of the problem and a missing denominator in the second set of the problem, and bringing in newspapers where the headings used exact or estimated numbers in the headings. The great things about all of these lessons is that they were easily adapted to the desired ability level. I really liked the lesson on the newspaper because it was applying what you were learning in the classroom and relating it to real life. This was great, because students like to know how this relates and why it is important.

This article was a nice article to read, and I thought it was great how the authors put in ideas for lessons, but I felt like it was almost common sense. There have been a ton of studies done on how estimation is vital, and it needs to be taught in the classroom. It is great that all of our cell phones have calculators in our phones to do the math, but we do need to know what numbers to plug in. I also think it is important to teach the students what the calculator is doing for us. I will definitely teach estimation in my classroom. Something that I need to remember is to relate the problems to my students. So if all of my students are into baseball cards, make the questions based off of buying baseball cards.

Video Blogg Two

These videos was mainly showing the viewer how to connect mathematics to the outside world. The videos were based in a 7th grade classroom in Missouri. The classroom featured was learning how to find volume and surface areas of the three - dimensional shape, the prism. Instead of using more traditional manipulates such as the wooden shapes, the teacher used everyday items, like the cereal boxes. The videos stressed that having hands on activities keeps students intrigued and helps them stay connected. The concept of building knowledge from one lesson to another is also very valuable when in a mathematics classroom. In order for individuals to understand one concept they need to understand simpler concepts. Having hands on actives can help students learn and retain more information.

While reviewing these videos, all the process standards were used. Students needed to use their problem solving skills and investigation while figuring out or investigating the surface area and the volume of the prism. They also needed to communicate with their peers of how they needed to solve the problem. The students needed to use reasoning and proof to show their work and why they think they have to correct answer. This activity was really great because it related to the outside world in a creative manner which helps students connect this to the outside world. When I was a student I always wanted to know why we were learning what we were learning, and how does it apply to the real world. This teacher did just that. The students knew that learning this information was important and it connects to prior knowledge and helps students realize how important this concept is.

Overall, I thought this was a good video because the teacher showed his students a topic in a very hands on manner which helps them comprehend and improves test scores. I personally learn better when I visually see something in action; so seeing this video showed me that having creative lessons really do help students. This authentic learning style will help students comprehend information, and this was easy to spot in the classroom.

Math Applet Review Two

Shape Sorter (Grades 6-8)
http://illuminations.nctm.org/ActivityDetail.aspx?ID=34

The shape sorter was created for students in sixth through eighth grade. Students need to sort all the shapes in the Venn diagram. The Venn diagram is where the individual compares two things using two circles that overlap each other. The section where they overlap is where the put the characteristics that are the same for the two things that you are comparing. In this case, the student can choose two rules when working with shapes and then sort the shapes based upon the rules. This activity was very easy to understand because the directions were very clear and there was an example of how the program worked. This is an effective program because the students can choose the rules they want to sort the shapes. Overall, I thought this was an amazing Math Applet. I really liked it, and thought that it was very creative how the students choose what they work on. It was easy to understand and used visual presentation well.

I thought this was a great program because it can be used during any part of the unit when learning about shapes and different characteristics of shapes. This could be an introductory lesson or a lesson used at the end of the unit. This would also be great if there is a class where students are at different levels of understanding. The reason behind this is the fact that the students get to choose the rule they want to work on and sort the shapes. Due to this, it allows a lot of growth for students. Students can have a variety, so students won't feel like they are doing the same activity over and over.

Math Applet Review One

Cubes (Grades 3-5)
http://illuminations.nctm.org/ActivityDetail.aspx?ID=6

This applet is mainly aimed for third to fifth graders learning about geometry and how to calculate volume of a cube, using mathematical terms such as width, depth, and height. The students first start off by having a five 2-D squares that are different colors and three different size cubes, these 2-D squares are 5x3. There is a single cube, a rod of five cubes, and a flat that covers the surface of the 5x3 squares. At the bottom of the screen there is a chart that lists the measurements whenever you add a cube, rod, or a flat. Using these three cube sizes the students would need to determine what the volume is using these cubes. Once the students finish this box, they can click, "change box" and begin working on another box pattern. I don't think this applet was that effective, and I also thought it was more confusing. The directions given were somewhat vague, and didn't really express how to figure out what the volume was. If a student didn't know how to calculate volume, I can see students becoming frustrated with not exactly knowing how to do this problem.

Something I do like while using this applet, is that each side of the square is color coded, and when you add the cubes, they are automatically match the colors alike. This distinction helps students realize how tall each layer is, and how it matches up to a one of the 2-D sides. I also like how there is a chart that is keeping track of how many cubes are the student has added. Another pro, is that there are many different examples to work on, so this can reinforce the idea of finding volume in a third to fifth grade classroom.

Overall, I believe this is a good start to a nice program, but the directions need to be clarified, because they are unclear of what the student needs to do. Once the directions are written more clear with possibly an example or two, I believe students will get more out of this activity because they will be able to understand the purpose of it.

Mathematics Teaching in the Middle School Journal Article

Moyer, T.O. (2010). Keeping all the trains on the tracks. Mathematics Teaching in the Middle School, (16)2, 116-121.

Keeping All the Trains on the Tracks
This article begins by explaining how a popular formula, distance=rate x time is commonly taught (and hopefully learned) in an eighth grade mathematics classroom. The author, a teacher explains how the approaches from the textbook typically do not "hook" or get the students attention right away, which there for is a struggle throughout the whole unit. The teacher then explains to the students how this works and gives students "real life" examples. She then sets up the equations and as a class they solve the problems. At this point, the teacher is still explaining the purpose of this and how the equation works. The article continues explaining how having a visual helps many students, and a way to adapt this into a math classroom is to design a chart, or program the equation into a graphing calculator, plug in the data, and then see the line graph that is created from the data. This is a great tool to adopt into a math classroom because it benefits the "visual," "listening," and "do-er" learning. The student has to listen to hear the instructions from the teacher, the students are physically putting in the data, and then the visual of the chart quickly appears on the screen.

I really like this article because it was so "user friendly." The author gave out very clear directions of how to teach this lesson. I also liked how the author included two activities directly related to this topic. This definitely helps teachers, especially new ones to get ideas of how to teach this in a creative and fun way. One thing I know I need to be cautious about if I ever do teach mathematics is to not rely too much on the calculator. It is an excellent tool to have and students should become familiar with a graphing calculator, the only issue is that I would not want students to become so dependent on the calculator that they can only complete math problems using a calculator.

Teaching Children Mathematics Journal Article

Young, E. (2010). Probability: a whale of a tale. Teaching Children Mathematics, 17(2), 106-112.

Probablilty: A Whale of a Tale
This article was a breeze to read due to the creative lessons described in this article. This study took place in two Title I third grade classrooms. The study consisted of introducing probability in both classrooms in a very nontraditional way, and seeing if it was more effective than the a more traditional way of practicing with endless amount of problems. The article first began by explaining that both classrooms introduced the topic of probability by reading a novel typically used in an English class. The title of the book is "Dear Mr. Blueberry." The reason this book was chosen was because of the vocabulary that was introduced and how it led to such a great discussion on how likely it was; based on the terms: likely, highly likely, not very likely, or impossible. The teacher then made a line chart where the students would have to put their term deciding if having a whale in the backyard would be likely or impossible. After a discussion, students were given new sentences and had to place their probability sentences next to the words that best described them. Students also had to draw pictures and write one sentence that were likely, certian, unlikely or impossible. Teachers would then display the pictures around the classroom after the group discussion. Another activity is having students match the sentences and the pictures together. Having creative activities help students learn the material in a fun and non stressful environment. The author concluded the article with another lesson plan with dice and having students roll the dice and predicting the sum of the dice.

I really enjoyed this article a lot because I thought the lesson plans were very orginial and authentic. Learning probability can be an intimidating topic because it can be difficult judging if something is likely or highly likely to happen. Having many ways of introducing this article will help students realize what the vocabulary terms mean and become more comfortable with using the terms. I also really enjoyed how the math classroom integrated a novel into the classroom. As a student, I always wanted to know how something applies to my life, and bringing in a book and having the teacher explain that todays lesson will directly correlate with this novel would be such a neat way to introduce a new math topic.

Sample PBL Reviews

1. Lounging Around: A PBL Unit for Students in Grades 7-8
This PBL unit was mainly focused on working on geometry, algebra, measurement, data analysis and probability, numbers and operations. The scenario was that, at West Wood Middle School the principal wants to create a lounge for the 7th and 8th graders. The principal has allowed all the students to make a layout, and he will choose one of the designs. The principal has given limitations to what needs to the lounge, the only thing the students need to be conscientious about is staying within the budget. Some of the strengths for this PBL is that it is very organized and the standards are grade appropriate and match up to the objectives. I also thought the mini lessons were detailed and were easy to follow. Something I thought that could have been improved was having a few of the activities relate to the general idea of creating a lounge or something that is important to 7th and 8th graders. A specific example of this is Exploring Proportions Activity with the beans. It was a pretty general activity so I thought they group could have done something more interesting to middle school students, such as the ratio between different brands of shoes. Overall, I thought this was a fabulous PBL because of the creative lessons with clear directions and how this lesson was based upon the math standards and objectives.
2.Operation "Redo the Zoo" A PBL Unit for Students in Grades 5th-6th
This PBL unit was directed for students in the 5th and 6th grade. The students were going to take a field trip to the zoo, and observe the strengths and weaknesses of the zoo and things they wanted to change or add onto the zoo. They are then suppose to design a floor plan of different exhibits they wanted at the newly modified zoo. Something I really enjoyed about this PBL is that they within the groups of four students they are assigned a job role. This is a nice benefit because this way each student feels needed and a vital part of the group. During the adolescent years it is an important job for teachers to ensure that all of the students feel needed. Something that I thought this group could have improved on was having more detailed explanations. For example when it was the Higher Order Thinking portion, the group could have given specific examples of how students were going to improve their skills of analysis, synthesis, evaluation, and reflection. Overall, I thought this was a creative concept and if I was assigned this activity as a 5th or 6th grader I would have stayed intrigued in the topic.
3. Compare and Contrast the two PBLs
Both of the PBLs were interesting and it was evident that both groups spent lots of time completing this assignment. I personally liked the first PBL I reviewed because I thought the directions were more clear and there were more examples. Having said this I think the first PBL I reviewed is a better representation of what a PBL is suppose to look like. However, it was great how both PBLs incorporated technology into their unit plans, and being a Special Education major I really enjoyed reading the creative accommodations both PBLs had.

PBL Journal Review

This article was introducing what a Problem Based Learning assignment is, and the benefits of such an assignment. The author stressed that these types of problems are important because they improve students problem solving skills and challenges their higher order thinking, while combining previous knowledge while working on a long term project. The author believes that the teacher needs to teach the material in the PBL beforehand, and the PBL should then be a creative approach to ensure that the students know the material and can apply it to real world situations or challenges. The author then continued on to say that PBL assignments improve test grades because while completing a PBL, the teacher should be more focused on the process of getting the answer and how the students came to find the answer. While completing a more typical assignment;such as an exam, students are more focused on finding the correct answer, and may not know exactly how to do that. Studies show that students educated in the traditional content-based learning environments exhibit lower achievement both on standardized tests and on project tests dealing with realistic situations than students who learn through a project-based approach. (Roh, 2003) The article concludes by explaining the teachers role, and how the teacher needs to guide students in their knowledge and need to encourage students to use their problem solving skills to discover the answer.
I thought this was a very beneficial article; because the author explained what the expectations are of a PBL and backed up their main points with specific results from previous studies. My favorite part of the journal is how the author connected the PBL experience with the NCTM Process Standards. This helps the reader justify why this is an important teaching strategy and why they should implement this into their classroom. The author believed the main point or lesson that should be learned through completing a PBL in a mathematics classroom is giving students a new challenge, while connecting the real world to the mathematics classroom. One of the weaknesses that I picked up from introducing the PBL into your classroom is the change of character the role is of the teacher. Instead of introducing or teaching, the teacher now needs to play more of encouraging role in the classroom. The author recommends that the teacher does not give out any knowledge while completing the PBL, instead the teacher needs to keep the students encouraged and focused. This new job in the classroom can be very difficult for some teachers, and may be the reason some teachers may not try this in their classroom.

Roh, K.H. (2003). Problem-based learning in mathematics. (Eric's Digest).
http://www.ericdigests.org/2004-3/math.html

*There wasn't a volume number or issue number for this journal article.

Problem Based Learning (PBL) Readings

According to the readings, a Problem Based Learning experience, or commonly referred as a PBL is a hands on, in depth group project where students are given a challenging question that has many parts to solve. The question requires students to use past knowledge and use resources to solve the question, and the answer doesn't align just an equation. The question needs to be based on the real world, developmentally appropriate, and needs to be able to relate to the students interests. The teacher needs to give students time to work on the PBL collaboratively, but parts of the PBL need to be split up so students work independently. While students are working on a PBL, the teacher is suppose to have more of a mentor or coach role. The teacher is there to answer questions students may have, but not lead a group. The teacher needs to remember to give students time to work on this project and suppose to take this project very seriously and present the assignment as a document. Giving the students this much freedom sends the message that they are controlling their learning, while providing them of higher order thinking problems. With a PBL students can use a variety of resources (books, internet, interviews, going out in the community, etc.) to find their answers. At the end of this assignment, teachers should use a typical rubric to assess the assignment. The criteria should be clearly explained in the beginning of the assignment, so there aren't any mix ups. The criteria should include how smoothly and effectively the group worked together, how each individual worked on the assignment and how much the contributed, the solution of the problem, and how the group came to the solution.

Enhancing Think-Pair-Share Journal Article

Dubon, L.P. and Shafer, K.G. (2010). Enhanciing think-pair-share. Teaching children mathematics 16(8), 451-455.



The article I read was called Enhancing Think-Pair-Share. The article was based on a study done in a kindergarten classroom. Three researchers (Tyminski, Richardson, and Winarski) developed and decided to test their more detailed "think-pair-share" teaching strategy in a kindergarten math classroom. One of the reasons these three scientists decided to complete their study in such a young classroom was because of the high energy and the passion for learning was so evident. Having so much energy brings a new level of communication in a classroom, and having a classroom based upon the standards is so much more effective than having a classroom based on the information that is being taught. At such a young age, teachers need to teach students how to think at a deeper level, by "conjecturing, reasoning, representing, and communicating mathematics." (NCTM).



The authors and researchers really encouraged student-to-student verbal interaction. They liked the think-pair-share teaching theory, but they added a few more steps to make it more effective. They also added a protocol, to help students organize their thinking and to improve on their listening skills. When this occurs it strengthens their problem solving skills. (NCTM). The protocol begins by expressing the problem by stating, "My problem is.." then responding with, "I heard you say...is that right?" The first person then responds by saying, "My suggestion is..." and then finishing the conversation with, "I can agree to.." (NCTM). Having students practicing this will help them solve problems more effectively and listen to their classmates. For the modified think-pair-share teaching strategy there are five phases. The phases are: orientation, play-investigate, share-reorient, pair-play pair-listening, and whole group share. Having these phases breaks down what exactly needs to happen when practicing the think-pair-share strategy. Students need to know what is expected out of them and what the directions are. Then they need to investigate/play the lesson. Once they become familiar with the task they can then share knowledge with their peers during the share-reorient phase. Afterwards, students can play and explain to each other what they have noticed. At this time the researchers went around and asked students questions, when they phrased the questions differently some students had a difficult time answering them. To conclude the lesson, the whole class should have a class discussion where all the students where they discuss their discoveries and different ways of completing the assignment. Overall, this modified teaching strategy really flourishes when students use each other to help develop new ideas and concepts. The researches new method helps the students discover these new ideas through communication and having students practicing how to listen to each other. This strategy is not only great for a math classroom, but any classroom.

NCTM Process Standard: Communication

According to the NCTM article on Communication process standard in a grades 3-5 classroom, "students should be able to use communication as a tool for understanding and generating solution strategies." (NCTM). After reviewing the information about communication in a classroom, I realized that without communication, a classroom would not be able to run efficiently. Communication isn't just oral instructions from the teacher to the students, it also includes communication from students to students. According to NCTM, students learn best when they are involved in a hands on activity where they can discuss their observations and conclusions with their peers. (NCTM). Communication in a classroom is essential, because this is how teachers can evaluate where their students stand, and see if they can move on with new material.

For the Communication standard for grades 3-5, the students should be able to use math language more often and use this type of language correctly. When students write their writing should be well developed and explain thoroughly what they discovered and learned. Teachers need to remember to model the use and the importance of communication in the classroom. To begin a lesson the teacher needs to have a discussion with the class and go through what steps need to be taken when evaluating a new math problem. In the middle school years students feel satisfied when they come to the answer themselves, so knowing this teachers need to remember to guide students through communication to come to the solution.

When discussing how communication and the standards are related, you can connect communication with all of them. The one standard that took out more than any other standard was making sense of the problem and perceive when solving it. The reason why is in order for somebody to make sense of anything, communication is key. Understanding math terms is a challenge, so when students attempt to make sense of the problem, communication is used. Another example is when students use reasoning to solve math problems. When using constructable arguments and critique communicating is being used. In order to develop anything, communication is used. Overall, communication may not always be verbally discussing something, it could be in a written reflection or working on a problem and showing every step.

Video Blog and Analysis

The video blog I did was on transitions in an elementary classroom. There were three videos, and the main objective of these videos was to show the viewer how to make transitions while checking for comprehension from the students. The way the teacher did this was to give clear directions, and ask the students what is next. For example, the teacher was going over the proper heading on the paper and instead of telling the students what she wanted she asked the students what they thought what needed to be on the paper header. One of the students said the date, and instead of answering with just a 'yes', she said, "Yes, but what is the date?" I thought this was great because she took it to the next level of comprehension. In another video she called on one student to answer the question, and since the student answered correctly, she then announced tat if you are sitting next to that student, he can be a leader and show the other student that they can really rely on him and other classmates, instead of always asking the teacher. The their video reinforced the concept of giving positive reinforcement. The teacher gave praise when a student was following directions. This is crucial to students' success, because at the age, most students want to feel well liked by their teacher. The reason why I chose this topic was because this summer I worked at an overnight summer camp for children with disabilities, and I quickly realized that when we made transitions we needed to give warnings to the campers and the directions given needed to be very clear and precise. after knowing this, I wanted to explore and observe how teachers make transitions in classrooms without stressing out their students.
Lesson Analysis 1: Identify three transitions that were made in the lesson.
Three transitions that were designed to make the lessons run smoother were when the student went back to their tables to work on their assignments. The students were able to work together and ask each other for help if it was needed. In the third video the teacher had the students line up to leave the classroom, and instead of just asking the students to do that, she counted allowing students time to finish their activity and get into line. The third transition I noticed while watching these videos was when the teacher dismissed the students to their tables in groups. She checked for understanding from one student and if the student answered correctly they can be the "table leader" during the activity and dismissed the students table by table. This way an interesting strategy because the teacher now has a chance to talk to a smaller group of students giving them more attention, but allowing freedom to the students. This gives the class a sense of community instead of just having the teacher make all the decisions.
Lesson Analysis 2: Describe the strategies used by the teacher to make transition smoothly.
Strategies the teacher used to make the transitions smoothly was asking questions to the whole group and individual students. The teacher also counted down from a specific number to give the students time and preparation of when the next activity is about to begin. If a student does not complete their work the teacher does not rush them, instead she had the student circle the problem they were working on. The teacher gave positive feedback to students, so every time the teacher waked around the room and looked at their work, it did not necessarily mean they were doing an inadequate job. The teacher often gave positive feedback, which is definitely needed at this age level.
Reflective Task 1: Describe specific actions the teacher could have taken to improve the quality of the transitions within the lesson.
Something the teacher could have improved on was possibly at the beginning of every lesson explain to all the students how the period or lesson is going to run, and what you(the teacher) expects from the students. Telling the students what will happen while they are at that period will make transitions easier. The teacher then has to check for understanding by answering any questions the students may have.
Personal Reflection: Describe how you identify the transitions within a lesson and what strategies you employ to make these transitions effectively.
I believe transitions are most effective when students fully understand what they need to do and what is expected from them. I also believe transitions should be fun and unique, and in order to have the students feel like they have power or choices, the teacher needs to ask them what they think is next. This also boosts students self esteem when they answer correctly and get positive reinforcement from their teacher. When I am making transitions in my classroom I will have timer or bell that goes off 10 minutes, 5 minutes, and 2 minutes before the transitions begins. I will also verbally tell the students. This way there will be a verbal cue and an audio cue. Most importantly, I will always be consistent and keep my classroom well structured. A way to do this is to have a schedule clearly posted in the classroom where all the students can see it.

NCTM Principles on Technology

After reading the NCTM Principles on Technology, I know realized that technology is a very important part of the mathematics classroom. One of the parts that I thought was most interesting was that technology can help students stay focused for longer periods of time. Technology does not replace the teacher in the classroom, instead it aids the teacher and is another avenue of learning. According to the article, technology helps students process information in a logical and well organized manner. Technology often times helps as a visual aid, especially when students are learning algebra or graphing. After reading the article, the main point I got from it, was that technology is improving daily and in order to have a successful classroom, as teachers we need to keep our classrooms up to date which results in more comprehension from the students.
After reading the stardards, I thought Standard Five: Use Appropriate Tools Strategically best relates to using technology in the classroom. At first, you may not think a calculator, ruler, or protractor as technology but these tools are considered technology. Students should be using technology at all levels. In this standard, the author mentioned that all students at all age levels should be using technology. What I thought was interesting was that there are computer programs that help students better understand the concepts. Computer programs is a great way to introduce a new topic or review an already learned topic. These programs can be used by all ages.
In elementary classrooms, students should be exposed to simplier forms of technology such as completing simple math using calculators, graphing equations using rulers and using protractors to design geometric shapes. In high school classrooms, students should be able to make graphs on calculators and check for errors and using computer programs as a way to better understand different types of math.

Enhancing Mathematical Learning in a Technology- Rich Environment

Suh, J.M., Johnston, C.J., and Douglas, J. (2008). Enhancing mathematical learning in a technology-rich environment. Teaching Mathematics (15)4, 235-241.

I really liked this journal because I thought the authors gave really creative ways to incorporated technology into an allready technlolgy "savvy" classroom. The idea I liked most was creating a story map for fractions and money because it was connecting math with a Language Arts class. In Language Arts classes, students make story maps to review and fully grasp the concept of characters are and their charactistics. I beleive it is very important to incorporate similiar learning strategies into a variety of classes. I also enjoyed hearing that the authors really emphasized on interactive programs to help students unterstand that the information. This article was great becuase, there was a section where the authors explain for teachers of how to slowly but surely incorporate creative technology in the classroom.

Technology and Mathematics in the Middle Grades

Hollenbeck, R. and Fey,J. (2009). Technology and mathematics in the middle grades. Mathematics Teaching in the Middle School 14(7), 430-435.

This journal entry was about incorporating technology into the matematics classroom. Technlogy has made huge strides in the education world, and schools now have so many options of how to incorporate technology in a math classroom; resulting in an enriching classroom experience. Students learn more efficientlyu and retain information for long periods of time when informations is presented visually. The technology discussed in the article really focuses on how there are so many programs that are available for teachers to use and benefit from teaching using multiple sesnses, which helps students retain information.

An example would be the interactive computer programs. These programs help students becuase the student physically doing work, has a visula. and is heariing the directions. The article also discusses the Smartboard and how teachers can use these boards to visually explain a math problem, and have the students anwer questions. The article continues on discussing how each type or genre of mathematics coud adapt to technology and to make the lessson more beneficial to students.

This arcticle further explained the idea that having the technolog in the classroom really does improve students' comprehension. One of the main reasons is due to the fact that when using the programs discussed in the article, the teahcer is applying the learning process to multiple types of learning.