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.