Math Manipulatives

I enjoy using manipulatives during math lessons. The students are engaged and think it is fun. They do not realize how many connections they are making and that they are developing a deeper understanding of math. Unfortunately, the problem comes when there are not enough manipulatives in the classrom. When I taught elementary school, this was a problem for me. I had to use manipulatives in center activities, or I had to have students work with partners. That was fine; however, there are some students who will sit idle while the other partner does all of the work. When I use manipulatives in middle school, I have not encountered the problem of not having enough. We have large manipulative kits with 20-25 sets in each; however, most of the teachers do not touch these kits, so I have two in my classroom. So far this year, I have used fraction bars, unifix cubes, and base-10 blocks from the kit. I have upcoming units planned that use geoboards and tangrams. There is a set of algeblocks for each student in my kit, but I am not familiar enough with these to use them. I personally have a set of pattern blocks at home; I enjoy making designs on the coffee table with them.

As for the second abstract, I will be comparing Mac versus PC laptops. This should yeild much more information than software. It is also a topic that will be useful to all of us for school and home purposes.

My Math Technology Experience

I would have to say that I am not very tech savvy. I am one of those individuals who can do something technology related if shown how with step-by-step instructions; however, I am also the type of individual that believes turning off a computer will fix anything that may be wrong. And, if it doesn't fix it, it's time for a new computer. Being a grad student, I am proficient with Microsoft Word and Powerpoint, have some experience with Excel, and am great at downloading music and uploading photos. I have a Starboard in my classroom, which is a very cheap version of a Smartboard. I do use the basics, but have not taken the time to explore all that it is capable of.

This is my seventh year teaching, all in Clark County. I taught 1st grade for one year, 3rd grade for five years, and now I am teaching 6th grade math at Creative Arts Magnet School. I have to say that this is my favorite year yet!

I am looking forward to this course because there are many aspects of technology that I am interested in. As I mentioned, I have an interactive white board that I enjoy using; however, I also have a slate and responders that I would like to try. I have not devoted any time to trying to figure out how to use it because I am sure it will lead to pure frustration. So, a training, or more information, would be of great benefit to my students and myself. Teaching middle school math, students have access to graphing calculators. I have not used graphing calculators since taking Calculus 10 years ago, so I need a refresher. I've set up a blog page, but would like to learn how to publish a webpage that my students may use. Math software is something I am unfamiliar with. I have had limited experience with Math Type, but would like more exposure to math games and graphing programs other than Excel. Although it doesn't have much to do with math, I would like to gain some knowledge of setting up and navigating through computer networks. So much of school communication revolves around networks, and I would like a deeper understanding of how it works.

Those Pesky Codes

When faced with the task to develop codes for the concept of Area, I went first to my textbook. I know the power standards well, and knew that area is specifically mentioned in one or two standards depending on the grade level. However, when develpoing a coding scheme for a textbook, I realized that area is embedded in many different standards. So, I thought it would be best to go page by page through my Holt Course 1 textbook and make note of the different types of problems involving area. I did not write down page numbers, but I developed very task specific codes. At this point in time, I thought I had a good jump start on the codes and would only need to refine a few. However, when I got to class, it became obvious that I was no where near done. In fact, the codes were so specific that I had a far larger number of codes than necessary. In essence, I created more work for myself. It has been much more difficult for me take specific codes and generalize them then to make general codes more specific.

Working with my peers, we went back to NCTM's Principles and Standards as well as the Common Core Standards. After reading through the documents, I pulled out standards that may relate to area and formulas to find area. I am now in the process of seperating my specific standards into the different standards, all the while trying to make them a bit less specific. I can honestly say, this is an exhausting and frustrating task. It would be beneficial to have written steps for us to follow in order to develop the codes instead of general guidelines. More examples would be useful too. I have never been exposed to a task like this or even the idea of coding a textbook. It is quite an undertaking and I am not sure of the ultimate goal. I hope to develop an appreciation for authors of textbooks if they go through the painstaking task of coding each standard in their text. I have definitely come to realize that I have no desire to be responsible for writing textbook curriculum.

Does Common Core Match NCTM Standards?

As a sixth grade math teacher who relies heavily on NCTM standards and lessons, I wanted to see how much Common Core State Standards aligned with NCTM's Principles and Standards for School Mathematics. The comparison is not as easy to make, because the CCSS are not divided between the same strands as NCTM. NCTM divides their content between between 5 content standards: Number & Operations, Algebra, Geometry, Measurement, and Data Analysis & Probability. CCSS content is divided between Ratio & Proportional Relationships, The Number System, Expressions & Equations, Geometry, and Statistics and Probability. This leads to some deficiencies and confusion.

For example, standards from the measurement strand for NCTM have been placed in other strands or deleted altogether in CCSS because there is no measurement strand in CCSS. Standards covered under Measurement for NCTM include conversions between metric and customary systems, area, perimeter, surface area and volume, formulas for finding areas of geometric shapes, and using scales to develop ratios and proportions. In CCSS, converting measurements is covered under the Ratio & Proportional Relationships Strand number 3,  Use ratio and rate reasoning to solve real-world and mathematical problems.  Area and volume are covered under the Geometry strand in CCSS.

However, the data Analysis and Probability strand in NCTM is fairly close to the Statistics and Probabilty srand of the CCSS. NCTM's standards are designed around collecting data and describing it as well as discussing probabilty of events. CCSS discuss the same items, yet it seems to be a bit more vague. NCTM provideds specific content to teach (ie. - formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population). Compared with the standard in CCSS (recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers), teacher will find that the NCTM standards are more specific and easier to follow. CCSS is too vague leaving room for error.

Overall, within the middle grades, I would say that because the CCSS are more vaguely written than the NCTM standards, many of NCTM standards can fit within the CCSS, just under a different strand than the strands all educators are used to.

Professional Development with the CCSS

Professional Development will be vital to the life and success of the Common Core Standards. Teachers who are exposed and trained in the right way will be more willing to implement the standards and be supportive. If teachers are not trained right, or the standards are just pushed upon them, there may be some harsh backlash.

Dream Scenario:

I am currently working in the middle school setting, where professional development may look very different than at the elementary school setting. But, my underlying point remains the same - teachers need time to absorb these standards. The CCSS should not be thown out to them the few days before school starts or at the end of the year with the expectation that teachers will implement them immediately.

I would recommend having someone from the State Department of Education of CCSD present an introduction of the CCSS standards and reasons for the switch in a staff meeting at the beginning of the school year, one year BEFORE the standards are implemented. With the idea of professional development for a middle school math department, the CCSS should be rolled out one strand at a time. For example, the 5 strands of standards for 6th grade math are Ratios & Proportional Relationships, The Number System, Expressions & Equations, Geometry, and Statistics and Probabilty. The math department should be presented with the Ratios & Proportional Relationships strand in January of the year before implementation. The department should meet and discuss the standards, look for similarities in the current standards(if possible, align them with the current standards), and look for where that standard is supported in the current curriculum. In February, teachers would receive the Number System strand, and so on. By May, teachers will have thoroughly investigated the new standards and will hopefully feel comfortable implementing them in the coming fall.  Ideally, teachers should be given staff development time to review the standards, and also be paid for afterschool or Saturday meetings. Also, teachers need to be able to ask questions about the standards with people in the Department of Education directly. In turn, the State Department of Ed should also be asking for feedback and willing to take suggestions about implementation.

Reality:

CCSD will inform the administration that they are working on a professional development plan and are working with the State Department of Education to roll one into place. About 4-6 months before the implementation date, administrators will be informed that it will be their responsibility to inform the staff and to provide professional development how they see fit. Administration will go back and forth trying to develop a plan. But ultimately, teachers will receive a fullday training of all of the standards all at once on the first staff development day of the new school year. (One of those 3 days before students arrive.) The next day, teachers will be able to develop lesson plans with these standards in grade-level or subject-level teams. The third day will be more of the same, and the teachers will be unhappy about the lack of time they have for setting up their room. Teachers will have questions that administration won't be prepared to answer. But administration will gladly do some research to answer those questions or concerns during the first staff development meeting after the students arrive.

The Progression of Standards in Common Core

As a sixth grade math teacher, I have been spending my time familiarizing myself with the sixth grade Common Core standards. I  wanted to see what the standards looked like in fifth grade, and also, what will they look like in seventh grade. With Common Core, it is not as easy to see a progression between the elementary and middle schools - it is a bit of a jump. For example, in elementary, standards are given for the following five areas: Operations & Algebraic Thinking, Number & Operations in Base Ten, Number & Operations - Fractions, Measurement & Data, and Geometry. However, in middle school, the standards are presented in the following five areas: Ratios & Proportional Relationships, The Number System, Expressions & Equations, Geometry, and Statistics & Probability. With the different categories, it is more difficult to find the progression. Since Geometry is presented in both elementary and middle school, I will evaluate the progression of standards within the Geometry strand.

In fifth grade for Geometry, students should be able to recognize an x- and y-axis and plot points on the first quadrant of a coordinate plane, identify attributes of two-dimensional figures, as well as identify the hierarchy of two-dimensional figures. I am surprised that there is no mention of three-dimensional figures in fifth grade. I taught three-dimensional shapes when I was teaching third grade.

When students get to sixth grade for Geometry, students are responsible for finding the areas of polygons using other shapes, use formulas to find the volume of given three-dimensional shapes, draw two-dimensional shapes on a coordinate plane, and use nets to find the surface area of three-dimensional shapes. I am surprized that students are expected to use formulas to find volume and surface area of three-dimensional shapes when this is the first exposure to 3-D in the Common Core standards. There also is no mention of whether students are only responsible for plotting points in the first quadrant of the coordinate plane or all four quadrants. In seventh grade, students will be working with finding actual lengths from scale drawings and vice versa, draw triangles with given angle measures, describe relationships between two- and three-dimensional figures, use circumference to solve problems, find volume and area using formulas, and use facts about angles to solve equations to find unknown angles. Using formulas to find volume progressed nicely from the students' exposure in sixth grade. I did notice that there is no mention of angles in sixth grade, so the curriculum in seveth grade must be filled with angles if students are expected to be able to find measurements of unknown angles.

Overall, there does not seem to be nearly enough geometry in the elementary grades. In middle school, the Common Core standards emphasis on Geometry increases each year. Some standards are progress from previous standards and some are brand new concepts.

Common Core Standards

As an educator, I have had experience with different state standards. I received my teaching credential from California and taught lessons on the California State Standards. I have been teaching in Las Vegas for seven years developing lessons for the Nevada State Standards. Wanting to move from the area, I have taken exams for licensure based on Minnesota State Standards. I have first handedly seen the difference of standards between these three states, and there are 47 other states and standards. Common Core standards will make learning expecations, assessment, and mobility for teachers and students much easier.

The common core standards for math seem to address the "mile wide and inch deep" curricula we have in place today. Some opposers argue that the common core are too general and may "weaken" some state standards. Looking through the 3-6th grade core standards, I noticed that many of our current, more specific, power standards can easily be covered by the common core. The common core standards are not as specific, but they facilitate a deeper understanding of mathematical concepts. The standards also stress relationships between concepts, something current standards do not address. For example, the 3rd grade standard of measurement included finding area, but it also emphasizes relating area to multiplication in the numbers and operation standard.

Although not stated as NCTM standards, the common core standards also address the 5 process standards NCTM present in Principals and Standards for School Mathematics: Problem Solving, Reasoning and Proof, Communication, Connections, and Representation. These process standards are what provide the depth of understanding in mathematics. Unlike NCTM, these process standards are embedded within content standards.

Not as many new specific standards are introduced each year, rather previous learned concepts are further developed in subsequent years. The standards are also written so that they require higher level thinking skills. This again, adds to a deeper understanding of important mathematical concepts.