Chapters 4 and 8

Chapter 4 has a lot of connections to the things that I am seeing in my practicum setting. I am in Mrs. Phillips' first grade classroom. The section of the chapter about addition and subtraction facts are very pertinent to what my students have been learning in math. The students have spent most of the year developing their number sense. They spent a day on each number. They would find ways to use other numbers to "build" each number. The students would talk about the way in which all of the numbers related to one another. The students have also spent time learning about how numbers can be related to five and 10. For example, the students spent several days learning how to count on from 5 to make other numbers. The students would write the number five and then draw how many more dots were needed to create a specific number. The students have also had a lot of exposure to the tens frame. The students fill in a tens frame each day as part of their calendar time. The class talks about the different ways that the frames can be counted.. They look at what each tens-frame looks life for each number.

Along with number sense, the students have also spent a lot of time with beginning addition and subtraction. The students have learned a wide variety of strategies to solve addition problems. The students are discouraged from using their fingers and are encouraged to use other methods to solve addition problem. The students are beginning to make these connections to subtraction problems as well. These strategies are the same ones that were mentioned in chapter 4. These include doubles plus one or two, using five or ten, and counting on. The students have also worked with one more and two more problem.

At this point in the year, the students have been working with addition facts through ten. I believe that they will begin to extend these facts. The students are becoming very fluent with strategies for solving their problems. They will be at the stage to be able to drill for some of these strategies in the near future. I can see from the lessons and from the students achievement that this is where the math curriculum is leading.

The multiplication and division part of this chapter along with the chapter on measurement has little to do with what I have seen in my field placement. The entirety of the math lessons up to this point have involved number sense and beginning addition and subtraction. Some measurement has been added in to the calendar curriculum for this month. At this point, the only thing that the students do with measurement is to compare two objects. The students are instructed to pick an object, and then find one that is either longer or shorter. The students have learned that the objects need to be lined up in a way that they have the same orientation and start at the same place to make it easier to "measure" or compare the two objects.

Chapter 11 and 12

1. How does the task presented in class (examining fair tests) compare to the content covered in chapter 11?

Chapter 11 covers the process of collecting data, categorizing and displaying the data, and how to use that data to answer our questions. This is what we did with the task that was presented in class. Chapter 11 says that the first goal in Data Analysis and Probability is to "formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them." This is exactly what we did in class. We came up with a list of questions about our wrists. This is an appropriate topic because chapter 11 talks about how children are the most interested in themselves. Once we came up with the list of questions, we found questions that we could answer with data. We chose to find out what the sizes of our wrists are. We then found the measurements and listed them in a chart. From that chart we were able to see what was wrong with the way we found our data. The children would need to be able to come up with these conditions like we did as a class. This would make their data meaningful and help find the answer to the question. The students would have to understand the concept of classifying to be able to organize and compare the data that they found. We may compare the sizes of our wrists among individual, among classes, or among grade levels. This would require that the students have certain criterion set for their data collection to get an accurate picture of the data. Chapter 11 says that children should decide on the classification. In our example the class decided on the questions and the criterion. We talked about how the measurements compared to one another. We had our information organized in a chart. We could have taken this information and used other visual representations. The students in the class would have decided what the best way to represent the data would have been. At the last part of our task, we found the measures of central tendency. We used these terms to talk about our data and to help answer our question.

2. What are you seeing related to data analysis and probability in your own classroom settings?

In my classroom, that children have started doing a weather graph as part of their calendar math in the morning. The students collect data by charting the weather each day in a bar graph. The students talk about which type of weather they have had the most of, least of, or which types of weather have had the same number of days. Today was the last day of the month so the students talked about the graph as a whole. The class also talked about how our graph might compare to one from another part of the country. For example, places are having snow now. Their graph would be different from our class' weather graph. This is all of the data analysis that the students have done in my class. There has been very little probability done in my class. At the beginning of the year, the students would pull a coin each day from the mystery cup. The coin would be placed on the magnetic tray whenever it was pulled out. The students would use the pan to predict which coin was the most likely to come out based on what was on the pan. For example, the students thought that it was more likely to pull a penny because there were many more pennies left on the board.

3. Examining the SC early childhood content standards (K-3) for data analysis and probability. How do the state standards compare to chapters 11 and 12?

Each year the South Carolina builds upon the standard for Data analysis and Probability beginning with the kindergarten standard until the third grand standard "The student will demonstrate through the mathematical processes an understanding of organizing, interpreting, analyzing and making predictions about data, the benefits of multiple representations of a data set, and the basic concepts of probability."Chapter 11 covers the first part of this standard with data analysis. The chapter talks about how children collect, organize, and interpret data. This is based on their questions and how they choose to classify the data. This again goes back to their ability to categorize information. Chapter 11 also discusses the various forms that the children can choose to represent their data. The students must understand the benefits of each type of representation in order to correctly decide which is the best way to represent their data. The last part of the standard- knowing the basic concepts of probability, is discussed in chapter 12. The students begin to understand that some events are more likely to happen than others. The students start to work with the continuum of probability as well. The continuum is a great way to help students visualize the concepts of probability.

Chapter 3: Developing Meaning for the Operations and solving Story Problems

Chapter 3 focuses on the four operations-addition, subtraction, multiplication, and division. The main idea of this chapter is that the four operations are very connected and should not be taught as separate ideas.

This chapter talks about the traditional definitions of the operations. It stresses that teachers should not teach that the operations have these definitions because they are too strict and limited. Instead, addition and subtraction problems should be taught according to the four structures. The four structures are join, separate, part-part-whole and compare. If these structures are used over the traditional "put together" or "take away" methods, the children will have more fluency within their mathematical reasoning. This is important to remember when introducing the symbols to children as well. Children should know that the equal sign is not just a symbol for the answer. The should realize that it means that the things on each side are the same. The minus sign should be seen as minus or subtract and not as take away. The children should learn the cumulative property and the zero property with addition. The cumulative property can be taught by pairing problems with the same addends in different orders.

The same concepts hold true for multiplication and division. There are structures for these problems. Two of these structures are equal groups and multiplicative comparison. The multiplication and division properties can be very helpful for students. Again, the name of the property is not as important as the concepts themselves.

In teaching mathematics, Model based problems are good for teaching the four operations. I will use model-based problems in my classroom. I will also use story problems that are meaningful for the children. For example, I will use the students names and experiences that they can relate to. I could use their recent field trip to find the mice on main in a story problem to help the children anchor their learning to a meaningful experience. Also, as the text stresses, I will stay away from the key word methods. These can be confusing and lead a student to use the wrong operations. The students should have to explain how they solved problems and got their answers. This will make their learning more meaningful.

Week 3: Chapter 2

The information in chapter 2 is directly related to the Patterning and Partitioning videos. In this unit, the students are exploring the number relationships that will be the foundation for all of their future work with numbers. The students are focusing on the four types of relationships for numbers 1-10. The students work on spatial relationships, one and two more, on and two less relationships, anchors to 5 and 10, and part-part-whole relationships. The teachers use many of the tools mentioned in Chapter 2 to teach these relationships. For example, the dot patterns are used for spatial relationships. The 10 frames and double 10 frames are used to anchor the numbers to 5 and 10. The monkeys in a tree and double decker bus lessons are used to teach part-part-whole relationships. The double tens frame lesson can help extend these relationship concepts past the number ten.

My favorite activities in this chapter were the tow activities that were based on the game of "War." This is a game that many children know and will be able to connect these new number concepts to the game. I like that this game can utilize many of the number relationships to find out what number each child has. This allows the children to come up with their own method and gives them some freedom to problem solve. I also think that the children will have fun with this activity. This will make them more motivated to work with the numbers and their relationships.

Challenge 5: Role of Mathematical Tools

I have always been a supporter of the use of manipulatives and other concrete tools in mathematics education. However, I realize now that these tools are often overused and misused in the classroom. Students should not be force to use tools or be told to use them in a certain way. Students should be allowed to use these tools in a way that make sense to them. This will promote better problem solving and more creative thinking among our students. Concrete tools are often used to teach students how to do something rather than help them understand concepts. These concepts are the most integral part of math. Students need to be given the chance to figure out what each concrete representation means and stands for in their own way.

Symbols are "tools" in math that communicate ideas. I have learned that there are often misconceptions about the symbols in math. Most often, students learn a definition for each symbol.. This limits what a symbol can communicate. A definition is not the same as learning what concept the symbol stands for. The example of the equal sign in the article is very illuminating. If a student learns that the equal sign means the answer, they will not understand that it has more to do with a relationship between different ideas. Students will become more "symbol literate" if they see the symbols actually used in different ways rather than learning a definition for each symbol.

Challenge 4: Video Response

Derek: Derek has a much more sophisticated understanding of numbers after the instruction. He recognizes how numbers relate to one another and how you can use these relationships to make addition easier. For example, Derek is asked what is 5 +4. Derek realizes that 4 is one away from 5. Derek then adds 5 plus 5 to get 10, but then takes away the 1 that he added in to get the correct answer of 9. Derek uses the same process for the subtraction and word problems.

Elizabeth: Elizabeth also uses the relationships between numbers to understand addition facts now. When given the problem of 7+6, Elizabeth uses her knowledge of 7+7=14 to get to the correct answer of 13 because it is 6 is one less than 7. Elizabeth counts the cows when she they are shown to her. Elizabeth uses number relationships again to solve the problem about how many cows were hiding.

Jim: After instruction, Jim now understands how he comes to an answer. Jim understands that he can break apart numbers to make addition easier. Jim understands that you count down in subtraction. Jim also uses his understanding of how numbers relate in subtraction. Jim uses more strategies in his problem solving after the instruction.

Lauren: Lauren also uses the relationships between numbers to solve problems. She uses what she already knows and related this knowledge to the problems she was given. Lauren also breaks down numbers to make addition and subtraction easier.

Challenge 3: Video Responses

Derek: When solving the question Derek is very thoughtful. Derek gets the right answer for this question, but he explains it by saying that he multiplied it all together and then added one. For the next question, Derek gets the wrong answer. He understands that you start with the first number and then add up. For the subtraction problem, Derek understands that you start with the larger number and take the smaller number away from that. He uses his fingers to count down. Derek used the same method for the word problem, but he was unable to solve the problem. Derek seems to understand the general processes of addition and subtraction. Derek also understands number concepts regarding which numbers are larger or smaller.

Elizabeth: Elizabeth knows simple addition facts from memory. Elizabeth also has an understanding of how numbers relate to one another. She shows this when she uses the first answer of 6 to “guess” the answer of 7 for the question about how many apples would there would be if she had 3 and her instructor had four. When using the cows in the video, Elizabeth shows that she is able to count and corrects her mistake when explaining her answer. Elizabeth says that she counts in her mind, so she understands that you take one number and count up to add. When there were 6 cows out and 4 in the barn, Elizabeth gets the wrong answer but she shows that she understands the problem when she says that she did not count to 6 but started at 7 and counted up. Elizabeth also shows that she understands that subtraction is taking away with the last problems.

Jim: Jim understands that subtraction is taking away numbers. He says that he “took away 6 and 5” to get the answer of 4 apples. Jim tries to do all of the problems in his head without using his fingers. For the addition problems, Jim does not understand how to get the answer and either guesses or says that he has not been taught that. Jim has trouble explaining how he gets his answers. He says that he just thinks about it. Jim does understand how numbers are related. When asked how he got the answer 3 for the last problem, Jim says that 3 comes before 4. This makes sense since he was asked to take away 1 from 4.

Lauren: Lauren was taught the number and dot method, and she uses this in addition problems. She understands that she counts the “dots” that are on the numbers and add them together. Lauren has these dot methods memorized. Lauren understands that subtraction is taking away numbers. She is able to count backwards in her mind. Lauren understands that you start with the high number and take away the smaller number. Lauren is very good at problem solving and uses her fingers to count up the difference between two numbers and finds how many pieces of candy a boy should put back for a word problem.

Week One Reflections

What does the term early childhood mathematics mean to you?

To me, early childhood mathematics is the part of a young child's education that builds a foundation for all later logical and mathematical reasoning. Early childhood mathematics involve students interacting with numbers and their relationships. It also involves learning to analyze the relationships among other entities. This includes comparing, organizing, and categorizing by different qualities. This foundation is essential for a child''s future ability in mathematics. It is imperative that children build this foundation along with confidence in this area.

Key Points from Chapter 1

Chapter One discusses the best strategies to teaching early childhood education. Children learn by construction their own knowledge, therefore it is important to provide opportunities for students to build this knowledge. The more connections that children can make with their existing knowledge, the more meaningful the new learning will be. This is an important concept for teachers to remember when planning activities. Connecting new information to something that is important to the child will make it easier for them to learn and understand the concept. What the word "understanding" means in mathematics is also an important thing for teachers to keep in mind. Understanding does not mean that a child must be able to always come to the correct answer. Understanding means that the child knows the process of how to solve a problem and why this process works.

The classroom setting is very important in providing the optimal learning experience for our students. Children need to have the opportunity to work with one another and they need to be active in their learning. This should be taken into consideration when planning. Math should be taught by way of problem-based tasks. These types of problems allow students to use their own knowledge to problem solve and work on finding a solution. These types of lessons promote the process of learning math over the correct answer. Problem-Based tasks also allow for authentic assessment. The teacher can observe, interview, and collect data from these problems. The students should be encouraged to write, draw, and explain each step. This is much deeper thinking than simple skill practice.

Problem-based tasks allow for variations to meet the diverse needs of the learners in our classrooms. They can have multiple solutions, different levels of sophistication, and different questions for different learners. These problem-based tasks also allow teachers to write problems that suit exactly what they want to address. Again, these assessments are a great way to provide fair grading and feedback. Teachers can see what the children have learned and base grades off of rubrics that show progress.