Chapter 4 & 8

How does the information presented in chapter four or in chapter eight connect to what you have seen in your practicum placement or how does it compare to what you have experienced in the past?

In my practicum placement, I am seeing numerous things that are mentioned in chapters 4 and 8. My teacher has been using the "think addition" strategy to teach subtraction to the class, demonstrating that you can add on from the smaller number instead of counting backwards to subtract and find the answer. She teaches them how addition and subtraction are related by helping them understand and acknowledge that 13 minus 7 is the same as "7 and what makes 13." This concept builds upon the commutative property for addition, in which the children understand that the order of an addition problem does not matter because it still has the same answer (5+3 is the same as 3+5). She allows the students to have individual time to work on and practice solving problems, sometimes allowing them to choose which method they want to solve them with. If they are learning a new strategy, then they must use that specific strategy to practice and learn it, but sometimes they are given options for adding and subtracting on worksheets or during their math game, for instance. For measurement, she has used non-standard units to have the children measure length, width, and weight of objects, such as blocks and plastic bears, demonstrating that "to measure" means that the attribute being measured is "filled" or "covered" or "matched" with a unit of measure with the same attribute. She uses multiple unit models rather than a single copy of a unit in an iteration process in order to properly show students how many blocks some object is in length, or how many bears the pumpkin weighs, which a balance scale is used for. In teaching time, the students are starting to learn that minutes are smaller than hours and that it takes 60 minutes to make one hour. Also, they are learning the purpose of the "big and little hands" in context of which to use when counting the minutes.

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.

Week 3 Reflection

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

There were numerous tasks that were presented in chapter 2 that related to the videos from challenge 5. Both talked about ten-frames and using "quick images," meaning the teacher flashed the ten-frame for a few seconds and asked the children what they saw. Using the ten-frame strategy, both also talked about the importance of having the children explain how they found their answers and allowing them to discuss different ways to reach the number that was shown on the ten-frame so they can learn from each other. Another thing that was similar between the videos and the chapter in the book was part-part-whole relationships. In the book, the activities had to do with creating 2 parts (groups) of a certain number, meaning that the children would be breaking a whole number into parts. In the video, the teacher was playing a version of "Simon Says" with the children and had them put up a certain number of fingers on each hand and then think about how many fingers they were holding up altogether. This showed parts because the children had separate numbers on each hand (2 numbers, 2 sets) even though they were adding them rather than splitting them. The Double Decker Bus video also demonstrated part-part-relationships in this same way. This video discussed more the process of changing the parts to equal the same whole number, changing between 3 & 0, 2 & 1, 1 & 2, and 0 & 3.

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

From chapter 2, “Build it In Parts,” “Patterns on the Hundreds Chart” and “Is It Reasonable?” were 3 activities that I really liked.

• “Build It In Parts” allows children to use a hands-on problem solving approach and helps them move towards an understanding of numbers as multiple parts instead of a whole. They love using manipulatives so it will be a fun activity for them and will allow them to figure out parts on their own or talk it through with a classmate. There are numerous possibilities of parts to split numbers into so there really is no wrong answer as long as everything equals the main focus number, so this activity is very encouraging for children because they have so many opportunities to get a correct answer.
• Patterns on the Hundreds Chart” is an important activity to do with the class because it helps them recognize patterns while they are learning and practicing their numbers from 1 to 100. They also get a chance to explain their findings, telling what their pattern is, how they found it, and why they think it is important. They will be able to listen to their classmates’ findings as well and learn about other patterns in the chart that they might not have seen, which are all beneficial to helping them subconsciously (or maybe even consciously) learn and retain their numbers between 1 and 100.
• "Is It Reasonable?" allows children to relate and make personal connections with math, which we know will encourage and help develop a better understanding of math. This activity lets the students be silly and have fun while still making them think logically. For example, asking the class, "Could a puppy dog be 10 feet tall?" is obviously not reasonable and the students would most likely laugh at this question, but following it with, "Could a school be 10 feet tall?" would be relevant to them and would be true, allowing them to get an idea of how big/how much ten feet is.

Week 2 Reflection

How did each article help further your understanding of your topic area (student thinking)?

In the first article, “Understanding Children’s Reasoning,” Bright tells us that although observing is a great way to see what children are doing, the best way to understand what children are actually thinking is to simply ask them questions about their answers and how they got them. This is important because performance is basically a symptom of mathematical understanding, so it is more beneficial to figure out the child’s understanding than to base judgments on mere performance. When asking children for their explanations, there are various types of questions that teachers can ask in order to receive varied responses and thought-out explanations, rather than “I don’t know” or “I guessed.” Questions should be specific to the problems being solved, or else the children may begin to see the questions as a routine instead of understanding that the teacher is genuinely interested in their reasoning. Once the teacher has a grasp on the children’s understanding of a certain concept, it is imperative that he or she does something with this knowledge. There are a variety of methods that are available to use in order to assist children individually, as well as a whole class, but that does not mean that it is a simple task to decide which strategy will best fit each student’s learning abilities. Some of these strategies include adjusting numbers for individual instruction, presenting problems based on information about what types of thinking are demonstrated, small group instruction, suggesting alternative strategies in order to expose the students to a wider range of techniques, and monitoring the effectiveness of instruction.

In the second article, “Teaching Strategies: Teaching Arithmetic through Problem Solving,” I really liked that they discussed the fact that it’s most important for the student to understand the concept and the process and that the method is not the most important part of mathematics. Children learn in many different ways and use a variety of methods depending on their conceptual level, which is perfectly fine as long as they understand the concept and can solve the problem in a way that makes sense to them. Also, they pointed out that teachers should allow students to explain their solutions because it helps them verbalize their thought processes and it shows that there are multiple ways to solve one problem. This article also stresses that working in partners is beneficial to the students because it allows them to work things out on their own instead of having the teacher show or tell them what to do and they are able to help their classmates if someone is having a more difficult time on a certain problem.

In the last article, “Kindergarten is More Than Counting,” I learned that fluency with small numbers is crucial to children’s learning in mathematics. Fluency means being able to put together and take apart numbers without even thinking about it. Manipulatives are very helpful in teaching children numbers and number operations, but it is important that they do not become dependant on them. Children must develop number fluency and a mental imagery of numbers, and this is usually accomplished pretty easily through the use of quick images. Quick images promote flexible thinking and encourage sharing of ideas. Being flexible while working with numbers will provide a foundation for students in relation to mental mathematics with larger numbers and in developing computation strategies. The students should be encouraged to move past counting one-by-one and move towards using mental imagery (mental math) because it will be very beneficial to them when solving problems once they are fluent in the basic numbers.

Challenge 4

Derek: He adds and subtracts numbers using a strategy that allows him to do the math in his head. He breaks the numbers down in order to make it easier to figure out in his head. The word problem was more difficult for him because he had to think about what numbers to use and what operation to do instead of solving a simple problem (8+7). Once he figured that out, he came across the answer the same way as before.

Elizabeth: She understands adding doubles and then subtracting the extra to get the answer. She uses her fingers to figure out answers and pretends that there are extra fingers when the number is more than 10 to help her find the answer. She uses sequences of 5 and 10, such as when she is adding and subtracting the cows, which shows that she understands number concepts as well as the operations.

Jim: He has some trouble counting backwards for subtraction, but he is able to use sequences of 5 and 10 for both addition and subtraction (12-5, 5+5 is 10 and 2 more is 12 so the answer is 7). He He can add doubles and then subtract and/or add numbers from the answer in order to get the answer to the problem (7+8, 7+7 is 14 so one more is 15).

Lauren: She understands adding doubles and then subtracting the extra to get the answer. She uses her fingers when explaining how she got her answer to the teacher but does not need to use her fingers to count while she is adding or subtracting. She easily solves problems, whether they are simple operation problems or word/story problems, showing that she understands number concepts well.

Challenge 3

Derek: He has an understanding of addition and subtraction, that adding numbers makes a bigger number and you count up and that subtracting numbers makes a smaller number and you count down. However, he has trouble because he tries to do the math in his head or on his fingers but he seems to have trouble sometimes when he reaches 10 since he can't count any more fingers.

Elizabeth: She says that she guesses to get the answer and cannot explain how or why she came up with the number that she did. She understands number concepts and sequences but she tries to do the math in her head and gets a different answer once she is asked to explain it and tries to count it out on her fingers. She can do subtraction with the cows in front of her, but with small numbers (5-2).

Jim: He somewhat understands the concept of adding and subtracting but cannot perform the operations with numbers that are higher than 6 because it becomes too difficult and he guesses answers.

Lauren: She adds by starting with the first number and counting the dots on the dominos until she arrives at her answer. For subtraction, she counts backwards in her head when she can see the manipulatives in front of her but uses her fingers to count down/backwards when trying to solve and understand the word problem about the boys with the candy.

Chapter 1 Reflection Questions

What does the term early childhood mathematics mean to you?

To me, early childhood mathematics means teaching children the basics and fundamentals of math and helping them develop foundational problem-solving skills that they will be able to carry on and build upon in their higher education. Children learn from their prior knowledge and from others' knowledge in a variety of methods, whether it is by counting on their fingers, computing in their head, or drawing pictures to figure out and understand a problem. Early childhood math is crucial because it teaches important concepts that children need to fully understand in order to move forward in mathematics and it will shape the way that children view math as they grow.

What key points did you take from chapter one that inform your understanding of how to teach mathematics for young children?

From chapter 1, the most important things that I learned were that understanding is more important that knowledge and that focus should always be on the student rather than having teacher-centered lessons. Students need to be the focus of lessons and learning, and the teacher should provide an atmosphere where students feel encouraged and excited to learn and explore the world of mathematics. They need to be able to explain their reasons for solving a problem a certain way and they need to know why certain things happen in mathematics, such as why you carry the 1 above the tens place when adding two-digit numbers. In order to understand a concept, there are many connections that need to be made within the knowledge that you already possess. Understanding is a process and learning cannot occur by simply memorizing information or procedures and then reciting or reproducing them later.