Chapter 11 & 12

1. How does the task presented in class (examining fair tests) compare to the content covered in chapter 11?
In class, we completed a task on our own and then discussed as a class if it was a fair representation of the data, and we explored what makes data analysis fair and the procedures and rules that may need to be established. We discussed how if results are unfair, they alter the results and make them worthless, and we came up with ways to keep this from happening. I learned that it is important to teach the meaning of the data collection, analysis, and graph representation because otherwise it is not relevant to the students and is simply a monotonous task to be completed. Chapter 11 discusses these same principles, including also that a connection to the real world is beneficial in addition to teaching the reasons of importance when teaching graphs and data analysis. The chapter also discusses various data can be collected that is meaningful to children, such as their favorite things, numbers of different things (pets, siblings, etc.), and measurements. Just as we did in class, the chapter talked about having children classify their data, which is important so that they will learn to make sense of real-world data in a variety of ways.


2. What are you seeing related to data analysis and probability in your own classroom settings?
I am in a first grade classroom and I see a few aspects of data analysis and probability present. One of the students' favorite things is their tooth chart. Each time a student loses a tooth, he or she writes his/her name on a tooth shaped piece of paper and Mrs. Smith tapes it above the last one on the chart in the correct month's column. Each time this happens, she talks to the class about different things about the graph, such as how many more/less are in this month than last month, how many altogether, how many has a particular student lost, and other things as well. During calendar time, Mrs. Smith always includes the weather and she keeps tallies of how many days it has been sunny, rainy, cloudy, etc. She does the same thing with this chart as with the tooth chart and asks the students various questions about how many more days has it been cloudy than rainy, if any are the same, etc.


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?
The standards for data analysis and probability are closely intertwined with the content in chapters 11 and 12. In kindergarten, students are expected to organize and interpret data using drawings and pictures. Chapter 11 discusses collecting data from kindergarteners and teaching them to classify objects, as well as the introduction of picture graphs and classification loops. The standards specify expectations in the organization and interpretation of numerous types of graphs, as well as the use and creation of surveys for data collection and analysis. Chapter 11 discusses these graphs, but also talks about not having children make the graphs until later grades, which is represented by the standards of the early grades that show simple organization and interpretation only. The standards also specify expectations of children to learn probability on a continuum, just as described in Chapter 12. The chapter says that it is important to teach children probability on a continuum rather than using percentages and fractions, and the standards follow this by increasing from "likely or unlikely" to "more likely and less likely," and so on.

Chapter 3 Reflection

What are the key ideas presented in chapter 3? How do these ideas inform your understanding of teaching numbers and operations?

• The two word problems at the beginning of the chapter showed me that addition is not always putting together and subtraction is not always taking away. I realized that addition could be used for a problem in which “gave away” were the key words that I noticed, immediately thinking it would be a subtraction problem when in reality I was wrong. It is important to teach children numerous addition and subtraction structures because when children learn the limited definitions for addition (put together) and subtraction (take away), they will most likely have trouble later when solving problems that do not follow these simple structures.
• The difference between addition and subtraction is that addition is used to name the whole when the parts of the whole are known, and subtraction is used to name a part when the whole and the remaining part are known.
• Using context or story problems is more beneficial for children because they model the real world and help the students develop a more practical sense for math instead of simply solving problems to get the answer.
• It is important to teach students symbolism correctly, and I have seen numerous examples of the symbols taught incorrectly, which can confuse children later on. The minus sign should be read as “minus” or “subtract” but not as “take away,” and the plus sign should be and usually is substituted as “and.” He equal sign is usually thought of as “the answer is coming” and should be taught as “is the same as” because there can be two equations on each side, which doesn’t necessarily represent an answer.
• When using model-based problems for addition, I had never really thought about the importance of leaving the two parts separate, whether by keeping separate piles, using different colors, etc. The students need to be able to identify which ones they started with, what was added, and the whole amount after the action of adding was completed.
• Addition and subtraction are interrelated and should be taught together. When teaching these, teachers should use the same models for both types of problems in order to help the children begin to realize the connection between the two operations.
• When teaching multiplication and division, the big terms for the parts of the problem are not meaningful or beneficial to children. Instead, they should be taught as parts and whole, which is useful in helping the children make the connection to addition.
• Just as with teaching addition and subtraction, it is beneficial to teach multiplication and division at the same time or at least one soon after the other in order for children to learn how they are connected to one another.
• It is important to teach multiplication using contextual problems, just as with addition and subtraction. Students should use whatever techniques they feel comfortable with to solve problems but they need to be able to explain what they did and why it makes sense, preferably in writing which could be in the form of words, pictures, and numbers.
• In multiplication, the distinction between which number represents the sets is not necessarily that important, but it is important that the students understand the concept that 4x8 means there are 4 sets of 8 or 8 sets of 4. Students will use repeated addition equations to show their work for these problems before they learn and understand the symbolism for multiplication, so they will need repeated practice and explanations of it.
• As long as numbers are within the students' counting strategies, they can and should be used in multiplication and division problems. When children are given challenges involving larger numbers and have not learned a computation strategy yet, they will most likely develop one on their own.
• Students should never be taught to look for key words in a problem! Not only can they be misleading as to which computation to use to solve the problem, but some problems don't even have any key words and if a student only relies on looking for the key words, then he or she is left with no strategy to solve the problem. Also, students should be taught to solve contextual problems by analyzing and making sense of the problems rather than looking for an easy and quick way to solve the problem.
• As we have discussed before, asking children to explain their strategies and processes of solving the problems is extremely vital to gaining insight of their understanding and abilities. It is more beneficial when a student is able to describe how they came upon an answer and explain the way that they carried out certain operations, rather than just writing the equation and the answer on their paper.