CGI Chapters 4-6

I have really been enjoying the CGI text just because of all of the examples it gives of the many ways that students approach various math problems. Chapter 4 is a continuation of the principles established in the previous chapters but it applies to multiplication and division instead. I remember when I first came across partitive and measurement division I was an adult and I had difficulty figuring out what the difference between the two of them were and why it was even important because with my background the math in them seemed to be the same. Since this is my second experience learning about the differences I can now grasp that partitive and measurement division are uniquely different for children learning about math and that needs to be kept in mind. If you try to solve the problem by using counters or drawing pictures the difference becomes immediately obvious. I also liked the idea that the text raised about introducing simple multiplication and division problems in as low as K-1 so that students get used to the idea of using math in this way.

I also also surprised when the book addressed the common misconception that students should have a good grasp of base-ten in order to be able to work with two and three digit numbers. They raised the point that as long as students can count they can solve problems with two digit numbers. Of course, this is sped up immensely once students realize they can count in groups of 5 or 10, etc. Many of the students in my second grade class are in the process of refining their knowledge of the base ten system. They spend a lot of time with a chart they have that lists ones, tens, and hundreds and they have cubes they use to model problems that are two digit addition and subtraction. Most of the students can make blocks of 10 cubes and use them that way while other students still prefer to count out each block.

I also was interested in reading about the various strategies that students use when working with multi-digit numbers such as incrementing, combining tens and ones, and compensating. These are all still strategies that I use in mental math but I've grown so used to it that I don't spend much time thinking about what exactly goes into solving a math problem anymore. When working with students to have an awareness of all the different ways students can do something is helpful so you can help a student find the strategy that works the best for them.

CGI Ch. 1-3

I think of the initial ideas that I took away from these chapters was the statement that children learn in their own way that is uniquely different from adults. One way this was demonstrated was by giving examples of three different ways to look at the same math problem that was set up differently. Since I have had a lot of experience with math I see all of the problems as essentially the same whether I am adding 3 plus 5 or 5 plus 3. Every child can approach the math problem differently and teachers must be sensitive to that. In chapter 3 when the book detailed some of the various strategies that students use to solve problems I was a little bit surprised that the book said that children pick up on most of these strategies naturally and without being taught them. In my placement class my students are working on regrouping and a lot of them use the direct modeling strategy. They have a ones/tens/hundreds chart that they use with cubes and they will count each cube to come up with the answer. I remember in MTH 201 learning about partitive and measurement division (I think those were the correct terms) so some of that was coming back to me while I was reading about the difference that students perceive when they are splitting things into groups or when they have groups and they are trying to find the whole. There is so much more that goes into selecting appropriate math problems for the students in your class that it appears at first. In the video that we watched in class last week the teacher purposely picked problems where it was times 5 each time to see if the students noticed the pattern. As teachers it is important that we think about the different ways that students can approach a math problem when we create it so that it is easier to anticipate how they might go about solving it. The idea of having discussions in math seems so reasonable I'm amazed that I was never exposed to it really at all K-12. I hope to spend more time learning about how to implement discussions into my math classroom to further enrich my students learning.

CGI Chapters 1-3 Ideas

After reading the first three chapters of "Children's Mathematics" I already know that I will like this book. It's so interesting to watch a child count on their fingers for a particular problem and then go back and understand why this was helpful to them. Throughout the three chapters, I found multiple ideas and strategies that stood out but I found two of them most helpful and interesting. The idea of direct multiple modeling strategies seemed to be most helpful for younger and less advanced students. Since I am placed in a kindergarten class at this time, I was able to relate to these strategies better. Although the student's rarely do addition or subtraction, they are currently working on counting. I can see using physical objects being the best way to instruct them with join problems. The other theory I found really interesting was the whole idea of the join, separate, part-part-whole and compare problems. I never really realized there were so many ways to write simple math and addition problems. It is helpful to know these ways though so we can have variation in our work. It also helps us as teachers know that our students truly understand the multiple ways of performing addition and subtraction problems. Through the three chapters I only really came across one question. On page 22, we see two students performing the "counting down to" strategy. Both of the children were sure to either not count the first one or not count the last one to reach the correct answer. It had me wondering if this was a practice taught to the students or whether it was something the realized through trial and error. If it was taught, how would you teach something like that?

CGI Ch.1-3 Reflection

After completing this weeks selected readings from Cognitively Guided Instruction I found many concepts to be both helpful and extremely interesting. One of my greatest fears as a future educator has been my own confidence and knowledge of strategies to teach to children to utilize when solving mathematical equations. According to Ch. 2 of our text, however, it states, "All of the strategies we have described come naturally to young children. Children do not have to be taught that a particular strategy goes with a particular type of problem. With opportunity and encouragement, children construct for themselves strategies that model the action or relationship in a problem." While I questioned whether or not this could really be true, I reflected back upon experiences I have had within my own classroom. While I have never explicitly seen my CT teach the children any one strategy to use when solving mathematical equations, there still seems to be a wide array of techniques put to use by the children. For example, on one particular day there was a story problem written on the overhead projector that read " Jill has eight library books. She returns 5 of them to the library. Now how many library books does Jill have?" As I walked around the room to observe how each child chose to solve the problem, I was amazed at the wide array strategies that were being used. While some children obtained physical objects, or manipulatives, from the shelves, others were drawing pictures, writing equations, counting on their fingers, etc. As our textbook suggests, this is because children are intuitive and hard wired to develop a plethora of problem solving strategies independently. I find this to be both interesting and exciting because it enables children to become invaluable resources for each other.

I was also very interested in reading about the variety of ways to word a story problem in order to truly capture a child's cognitive ability. While we discussed in class how important the wording of a story problem is and the various benefits that differentiated wording ensues, I couldn't help but wonder HOW to do this. The reading, as well as the charts and graphs provided within the text, discussed in detail each specific type of problem and what that problem was testing. On page 12, figure 2.2. we are given a chart inclusive of eleven diverse ways to present the same exact story problem! Personally, I find this to be an excellent point of reference for me to continuously refer to in the future and fully intend on taking advantage of this information.

Lastly, though I do not have any questions regarding the reading, I do have a few questions regarding our class discussion about standards presented in Thursdays class. One of my classmates brought up the issue of private schools and wondered what standards they were required to adhere to. Having gone to private catholic school from k-12th grade myself, this was a topic I have also been curious about. Do national and state standards only apply to those schools which are governmentally funded? And if that is the case, then who creates and enforces standards within private schools? Also, there was never a special education program present within my schooling but it seems that nearly all public schools are inclusive of this. Is this also something that is mandated by the state for public schools? These are just a few questions I have had for some time now and our discussion on Thursday seemed like a perfect segue for me to present these ideas and gain some insight!

My Learning Goals

My learning goals for this class are fairly simple but perfect for me. I have never been very good at math which causes me to be nervous about teaching it. In this class, I would like to be given the knowledge and tools to do well and teach math to my students. I want to learn how to be a GOOD math teacher. I know it may seem as though this is a given, but in my experience I never really had math teachers that helped me understand math. I want to know how to help my students really understand math rather than just memorizing how to do it. I want to be able to know when they need help and how to help them. I also would like to be knowledgeable across all grade levels. In the past, our TE classes seem to be very specific to younger or older grades. Since I am not that good at math, I would like to be prepared no matter what grade I end up teaching. Along with math, I would like to continue my education on teaching in general. I always enjoy being exposed to fun activities or new ways to explain something to children. I believe a lot of this can happen in the field placement, but I would like to enhance that exposure in TE 402. I look forward to working with the other students in order to become a well educated math teacher.

Learning Goals

I'm coming into this course with high hopes for myself and the advances that I hope to make in my own teaching of math. While I felt that I was always proficient in math in elementary school, it was never a subject that really got me excited. I hope that is something that I can change for my students in the future. In this course I want to explore math in ways that I have not done before. In my placement my CT integrates partner and group work into math on a regular basis. As a student of math, I always viewed it as a solitary activity, which perhaps contributed to less enjoyment on my part. Integrating math into all subjects is important to me because I never want my students to have to ask, "How are we ever going to use this?" I definately want to spend some time focusing on how to reach students with varying levels of interest/ability in math. Even though I was a good math student in elementary school my lack of interest in math correlated to poorer math grades by high school. I have to admit I do occasionally worry about teaching math, not because I'm intimidated by the material itself, but that I will inadvertently resort to some of the same techniques used by teachers that did not peak my interest in the past. We briefly discussed the ineffectiveness of worksheets that asked students to repeat the same problems over and over in class last week and that just reminded me of so many experiences that I have had in math classes in the past. Above all, a goal that I really have for myself for this semester is to more deeply explore ways to captivate student's interest in math. I want to do this by reflecting on the types of math assessments that I can use to really determine a student's learning and how to tailor my teaching to reach the desired effect.

Learning Goals

My primary learning goal is to develop a sense of excitement and enthusiasm with regards to the subject of mathematics and try to lessen and/or eliminate the negative feelings that I currently have associated with this topic. For this evolution to occur, I think I need to begin by findings ways to motivate and encourage myself to become a logical thinker and learner. Though this is not my area of expertise, it will be important for me to constantly establish positives to this learning approach and consider the multitude of ways that this learning style will be helpful to me in the future. In order to change my thinking, I need to change my approach. I hope to discover some of these new approaches and strategies that I can utilize when performing mathematics and incorporate those strategies into the lesson plan that I will be teaching to my first grade students within my field placement. For example, when we were asked in class to comprise a list of the ways in which the number 16 can be reached, I was astounded at the variety of approaches and strategies that could be used in completing this problem. Not only were numbers incorporated, but graphs, charts, and drawings as well. These are the types of strategies that would have made my childhood experiences with mathematics more positive. I hope to continue to learn new approaches to mathematical equations and, as a result, be able to transform my attitude towards math in the process.