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.