Reflection Question: Week 4

What are the key ideas presented in chapter 3?

Chapter 3 presented a range of important ideas that contribute to developing meaning for operations and solving story problems. Meanings, interpretations, and relationships to the four operations of addition, subtraction, mulitiplication, and division are each presented throughout chapter 3 and connected to help children establish an understanding of how they can use such operations in real-world settings. One important concept presented is that both addition and subtraction are connected. Another key idea within chapter 3 is that both addition and subtraction involve an action. One is a joining action of putting together where as the other is a separting action or taking away. Subtraction is the operation that names a missing part and addition is the operation that names the whole in terms of the parts. Important ideas regarding multiplication that were presented throughout the chapter include: multiplication involves counting groups of like size and figuring out how many are in all or the process of mulitplicative thinking, multiplication and division are connected/related. Division names a missing factor. In addition, models can be used to solve problems for all operations. They are also used to help figure out the operation involved in a problem despite number size and can be used to help number sentences have meaning.


How do these ideas inform your understanding of teaching numbers and operations?

The key ideas presented throughout the chapter help to establish meaning and purpose behind how to teach numbers and operations effectively to students. It is very important that as a teacher, I look at more than just the answer a student gets but instead, look at the techniques the child used to solve the problem. I should show students how operations are connected and help students develop the strategies needed to work through a problem and give it meaning. I realize that it is important to pay attention to how a student works through a problem or the strategies used so that it will give me the information that I need to understand their number development and the strategies being used. I also realize from the ideas presented that it is essential to pay attention to how a child solves a problem or the methods being used. This will help me grasp and understand as to what numbers should be used in problems for the next day.

From the points established throughout the chapter, I have become more aware of the connection and meaning between specific operations. I have also learned that it is important to encourage the use of problem analysis and avoid the use of key words such as "altogether" and "fewer" whem dealing with adddition or subtraction. Being able to provide good explanations behind specific mathematical operations is essential in the learning process. Rather than simply looking at specific "key terms" used throughout a problem, it is important to actually break the problem down and figure it out mathematically. This will help the child become more aware of how he/she was able to solve the problem and help give meaning to how the answer was found.

Reflection Question: Week 3

How does the information and the tasks presented in chapter two connect to the videos of lessons you viewed as part of challenge 5?

Chapter two presented the task of using dot plates. As I recall, this task was presented by one of the first students in the videos. The child used the concept of counting dots to help them work through a counting task. Also, the concept of "one and two more, one and two less" is presented and used within the videos as an additional method when reflecting back on the way one number is related to another. This task is used through the activity "dot plate flash" that was mentioned throughout chapter two.

An additional concept that was used throughout the videos was the task of using missing-part activities. The activity "covered parts" that is metioned throughout chapter two was also seen throughout the videos using counting cows and a barn. The cows were hidden underneath the barn and either pulled out or taken away to provide the child with the task of being able to focus on a "single designated quantity as the whole".

As mentioned, there were several pieces of information and tasks presented throughout chapter two that connected to the videos viewed as part of challenge 5. It is evident that both the videos and the information presented are somehow connected, using the same concepts and providing students with similar tasks in order to understand the information being presented.


What task (activity) in chapter two was most interesting to you? Why?

I really liked the activity, "Fill the Chutes" that was presented to help children develop their understanding of counting by engaging each child in a game that involved both counts and comparisons. This is an activity that seems to be a fun learning activity that would provide children with a fun approach to counting. The game is very similar to the game, "Chutes and Ladders" that many children may already be familiar with. This is a way to get their attention and provide a fun alternative to worksheets. I found it both interesting and something that children would enjoy. Its a fun way to get children excited about counting!

Reflection Question- Week Two

How did each article help further your understanding for your topic area Mathematical Tools?


Thompson's article, "Concrete Tools and Teaching for Mathematical Understanding" allowed me look further into the importance of concrete tools and what teachers should be asking, "What, in principle, do I want my students to understand?" I also found that concrete materials are used appropriately for two purposes:

• "they enable the teacher and the students to have grounded conversations about something concrete"
• they furnish something on which students can act

From this article, I learned that for concrete materials to be an effective aid to students' thinking and to successful teaching, "the effectiveness is contingent on what one is trying to achieve" and reflects upon the teachers ultimate goal. Meaning, it reflects on whether the teacher is asking, "What do I want my students to understand" rather than "What do I want my students to do". After reading Witherspoon's article, "And the Answer is Symbolic" I was able to grasp a better understanding behind the misconception of what a symbol means in terms of mathematics. I was also able to see the importance of fostering mathematical literacy and the communicative role of mathematical symbols in elementary school.

The use of concrete materials in early childhood education is essential. However, I have learned that it is very easy to misuse concrete materials and to use them incorrectly. As mentioned in the article, using concrete materials doesn't mean that a child will automatically grasp the understanding behind the mathematical meaning of the concrete object. While we must use concrete materials to teach math, it is also important to understand that children must know and understand the meaning and representation of the concrete materials being used.

We cannot assume that each child knows the meanings of symbols and concrete objects but we must make sure that each child understands what the symbols represent in a mathematical expression.

Overall, the articles provided me with a better understanding on the role of concrete materials when teaching mathematics and the importance of fostering symbolic literacy.