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Cathcart, W. G., Pothier, Y. M., Vance, J. H., & Bezuk, N. S. (2011). Learning mathematics in elementary and middle schools: A learner-centered approach. Pearson.

Oregon Department of Education : Mathematics Standards : Mathematics : State of Oregon. (n.d.). Www.oregon.gov.https://www.oregon.gov/ode/educator resources/standards/mathematics/Pages/MathStandards.aspx

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics.

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Interview report

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Before Class: Standard Comparison

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Before Class, Video-How do Children Learn Math

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Lesson Plan Analysis 1

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Lesson Plan Analysis 2

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Types of Problem and Example:

Process Problem: Grade level: 7

Standard:

7. AEE.B: Solve mathematical problems in authentic contexts using numerical and algebraic expressions and equations

Question:

This one is for all you pet lovers out there! A hog, a dog, and a frog all want to cross a bog. The hog slogs across at a speed of five feet per second. The dog logs a time of 20 minutes for its crossing. The frog jogs across twice as fast as the hog but half as fast as the dog. What is the distance in feet across the bog?

How to answer:

First, let's convert the dog's crossing time to seconds. Since there are 60 seconds in a minute, 20 minutes is equal to 20 * 60 = 1200 seconds.

The hog crosses at a speed of five feet per second. We don't know the time it takes the hog to cross directly, so we'll find it later using the relationship between the speeds.

The frog jogs twice as fast as the hog, so the frog's speed is 2 * 5 = 10 feet per second. The frog also jogs half as fast as the dog. The dog's speed is distance/time. We don't know the distance yet, but we know the dog's time is 1200 seconds. Let's call the distance across the bog "d". So the dog's speed is d/1200 feet per second. The frog's speed is half of the dog's speed, so the frog's speed is also d/2400 feet per second. But we already found that the frog's speed is 10 feet per second. So we can set up the equation d/2400 = 10. Solving for d, we multiply both sides by 2400 to get d = 24000 feet.

Therefore, the distance across the bog is 24000 feet.

Translation Problem: Grade level: 7

Standard: 7.AEE.B - Solve mathematical problems in authentic contexts using numerical and algebraic expressions and equations.

Question: With cell phones being so common these days, the phone companies are all competing to earn business by offering various calling plans.

One of them, Horizon, offers 128 GB of data per month for $45.99, and additional data is charged at 6 cents per each GB used.

Another company, Stingular, offers 128 GB of data for $29.99 per month, and each additional GB used is 35 cents each.
For how many total GB per month is Horizon's plan a better deal?

How to answer: see the picture.

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Mathematics Assessment and Unit Plan Writing

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Mathematics Assessment and Unit Plan Writing

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Analyze Students' work

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Exploring Number an Operation Sense and Developing Basic Fact Fluency

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Exploring Number an Operation Sense and Developing Basic Fact Fluency

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Center Activity:

Center 3: Thinking Block Addition: I realized that students could use this activity to learn using addition to solve subtraction activities and use subtraction to solve addition problems.

Center 7: Broken Calculator: Can you find three ways to solve 61 divides 14 on a calculator without using the divide key?: I realized that there are many methods to solve one problem. I need to teacher students methods as many as possible.

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Exploring Fraction Concept and Computation

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Exploring Fraction Concept and Computation

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Activity 8.21 Hexagon Whole

Activity8.23 Quilting Pieces

Multiplication Activities:
1. Activity 8.21 - Hexagon Wholes:
- Problem: Designate the yellow hexagon as the whole. Find the fractional value of other pieces and find how many wholes for different quantities. For example, find how many wholes are there for 5 blue pieces if the blue piece is 1/2 of the whole.
- Solution: Each blue piece is 1/2 of the whole. For 5 blue pieces, we calculate 5 ✖️1/2=5/2=2 1/2 wholes
2. Activity 8.23 - Quilting Pieces:
- Problem: A quilt is 8 feet by 6 feet. Each group prepares a picture that is 3 feet by 2 feet. Find what fraction of the quilt each group will provide.
- Solution: The area of the whole quilt is 8 ✖️6 = 48 square feet. The area of each group's portion is 3✖️2=6 square feet. The fraction of the quilt each group provides is 6/48=1/8
We can also look at it as 1/2 of the length 3/8 and 1/2) of the width 2/6, and 1/2✖️1/2=1/4 for the area fraction considering the scaled dimensions. But when considering the whole quilt area, it is 1/8

Division Activities:
1. Activity 8.26 - Fractions Divided by Whole Number Stories - Garden Plots:
- Problem* Three gardeners are equally sharing 1/4 of an acre for their plots. What part of an acre is each gardener's plot?
- Solution: I divide the fraction of the acre by the number of gardeners. So, 1/4 ➗3=1/4 ✖️1/3=1/12 of an acre for each gardener.

2. Activity 8.28 - How Much in One Whole Set?:
- Problem: If 8 counters represent 1/4 of the whole, find how much is one whole set.
- Solution: We know that 8 is 1/4 of the whole. To find the whole, we divide the number of counters by the fraction it represents. So, 8➗1/4=8✖️4 = 32 counters in one whole set.

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Decimal: what I prepared for presentation and micro-teaching

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Investigating with Concepts of Probability

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Investigating with Concepts of Probability

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I have designed a math unit plan centered around rational numbers for grade 7 students. This plan was crafted with a meticulous consideration of students' cognitive, social, and math developmental stages. In the microteaching session for lesson 2, which focused on the conversion between fractions, decimals, and percents, I aimed to foster students' understanding of rational number operations through visual models and strategic use of tools.

The unit plan that I designed showcases a comprehensive approach. By integrating the Common Core and NCTM standards, it ensures a solid curriculum foundation. The use of diverse formative assessments, such as quizzes, journals, and class discussions, provides a multi-faceted view of student progress. These are matched with my teaching philosophy as well. Students can engage in the classroom and be supported by the teacher to fulfill their different needs. The usage of visual models like number lines and area models serve as powerful tools for concept explanation. Students can easily feel concrete to formal operational thinking.

My unit plan actively engages students through various hands-on activities and real-life problem-solving tasks. For instance, in the lesson on applying rational number operations to real-life problems, students are required to plan a school event budget, which makes abstract math concepts tangible and relevant.

Also, my unit plan incorporates various formative assessments like journal entries and class discussions. These allow students to reflect on their learning process and receive timely feedback. For example, after the lesson on integer division rules, students write in their journals about their understanding and challenges, enabling the teacher to identify areas where additional support is needed and offer personalized guidance.

If implementing this unit in China, several adjustments would be beneficial. Firstly, considering the longer class periods, more time could be allocated for in-depth practice. After introducing new concepts like the conversion between fractions, decimals, and percents in lesson 2, additional complex problems could be provided. For instance, instead of just a short quiz, there could be a series of exercises with increasing difficulty levels, allowing students to master the skills thoroughly. This extended practice time would enhance students' proficiency and confidence in handling these mathematical operations. Secondly, group work could be further enhanced. In larger classes, dividing students into small groups for problem-solving activities would not only manage the workload effectively but also foster peer learning. For example, in the lesson on simplifying rational number expressions, groups could be assigned different sets of expressions to solve and then present their solutions and reasoning to the class. This would improve students' communication skills and deepen their understanding through collaborative discussions. These changes are crucial as the Chinese educational context often emphasizes thorough practice and collaborative learning to ensure students' solid grasp of mathematical knowledge and the development of their comprehensive abilities.

The overall performance that I took in this class can be included in many parts, which are the class activity I took, the presentation and microteaching for the concepts of decimals, and the making of lesson plans and unit plans. Whatever the activities before or during the class, I can engage and finish them positively. For the presentation about the decimal, I made seven centers and presented them during the class. My students can follow it, and I give guidance while they are in the centers. Different centers focused on the different ways, which all focus on the decimal. For my lesson plan, I revised it based on the teacher's and friends' feedback. My micro-teaching used a lot of time on reviewing, and then I revised my reviewing method from teaching to center activities. I think it will be more effective to allow students to learn and review by themselves what they have learned in the previous grades. I realized that only teaching in the class cannot make students really engage in the classroom. I need to find more teaching strategies to provide an engaging and supportive environment. Therefore, the key takeaways from my experience are the importance of real-world connections and visual aids effectiveness, such as Number lines, area models, and online manipulatives significantly aid in concept comprehension. I have become more adept at designing engaging lessons and integrating technology. I have also improved my ability to analyze student responses and adapt instruction accordingly. For example, through the misunderstanding that students show during the class, I know how to teach them in another way that they can easily understand.

I know that I still need to improve in the future, so I made two goals for myself. I think I need to enhance my ability to respond flexibly. Students may think and give many unexpected questions that, beyond my lesson plan, I should be familiar with the topic and content I am going to teach as well as possible. Then, I can answer their questions and will not feel anxious. The method that can enhance my flexibility is to read more sources and prepare more before the class. Another goal that I want to achieve is the differentiation for students who have special needs. Even though it may be heavy work to do, I need to create more tiered assignments for students with different ability levels. For example, in a lesson on rational number operations, design three levels of worksheets: basic, intermediate, and advanced.

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As I took Advanced Primary Math, I think a lot of my teaching philosophy has changed. In the Chinese educational environment, students need to master concepts very effectively. As a result, teachers do not use many interesting methods when teaching, but only talk about concepts and do exercises. However, now my teaching philosophy has changed, and I think the atmosphere in the classroom is very important for students to learn math well. Only an interesting classroom atmosphere can arouse students' interest in learning mathematics. If I use my previous experience of learning mathematics to teach, my students can easily develop an aversion to mathematics. Therefore, my teaching philosophy is to create an engaging and supportive classroom for students. I believe that students should be actively involved in the learning process. This means providing opportunities for hands-on activities, group work, and real-world applications. I also emphasize the importance of communication, reasoning, and representation in mathematics, as these skills are essential for understanding and applying mathematical concepts.

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In my engaging and supportive classroom, I will use problem-based learning to engage students and develop their problem-solving skills to help them make sense of problems and persevere in solving them(Cathcart et al., 2011). By presenting students with mathematical tasks connected to real-life situations which is also connect with Problem-Solving Principle, they can utilize their mathematical knowledge and skills to address these challenges. In doing so, students will also enhance their problem-solving abilities. This helps them see the practical applications of mathematics and increases their motivation to solve the problems. I have incorporated this belief into my micro-teaching about the conversion between decimals and fractions. Students need to convert them in a shopping list. I think this connection with real life also stir their interest in learning math.

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Incorporating NCTM principles into the classroom can make it more engaging and supportive. NCTM principles include problem-solving, communication, reasoning, connections, and representations(National Council of Teachers of Mathematics, 2000). Problem-solving should be presented as a real-life problem. This helps students see the practical applications of mathematics and increases their motivation to solve problems. For communication, I think there is a learning theory called social constructivism, which is suitable to be implemented here.  It allows students to learn the concept by communicating and cooperating with each other(Santrock, 2011). To implement the "communication" principle, I will organize class discussions where students can share their problem-solving strategies and solutions. This helps them clarify their thinking and learn from others. Then, students will get the chance to present their work in class; this is also another way to build their confidence and communication(expression) skills. As the teacher, the way that I communicate with them is also important. I can give them in-time feedback and guide them once they need help. In this way, I can ensure that every students can join into the classroom and engage in the class activity within the group.

Another principle that I will implement is connection. It needs teacher to show students how different mathematical topics are related and how they can use knowledge from one area to solve problems in another. For example, show how geometry can be used to solve problems in algebra. additionally, connecting mathematics to real-world situations and other subjects can improve students' engagement into the class. For example, using examples from science, history, or art to show how mathematics is used in different fields, they will find the interrelationship between different disciplines.

The usage of diagrams, graphs, models, and other visual aids can help students understand abstract mathematical concepts in a concrete way. For example, use a number line to show addition and subtraction or a graph to show a relationship between two variables. If teacher just speak and give instruction, students will not fully engage in the classroom, Math will become boring for them because there are only abstract concepts need them to remember. However, the representation will allow them think the math concepts in a concrete way. When they are trying to use the representation to build their understanding, they will understand more through the hand on activity.

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I think the Common Core standards are organized based on different level and grades of students. I can make my unit plan and lesson plans based on it. Then my teaching procedure will be organized. The MP encourage students to be active participants in the learning process. For example, when students make sense of problems and persevere in solving them, they are fully engaged in the task at hand. They are challenged to think deeply, ask questions, and explore multiple solutions, which keeps them interested and motivated. Reasoning abstractly and quantitatively and constructing viable arguments also promote engagement as students are required to think critically and communicate their ideas(Cathcart et al., 2011). This leads to lively class discussions. Then students can develop deeper understanding by engaging into the class activities.

The diversity of investigation includes making sense, impact in the future, and prediction(Cathcart et al., 2011).The "make sense" driver presents problems that are meaningful and relevant to student's lives, which immediately grabs their attention. They are more likely to be engaged when they can see the real-world connections and the importance of solving the problem. The impact of the future can allow students to know the importance of these math concepts. For example, the conversion of fractions and decimals can help them understand rational numbers better, and they can use this skill in future work. Therefore, they will be motivated by what achievements they will achieve in the future. The "prediction" driver adds an element of excitement and curiosity. Students are eager to test their predictions and see how accurate they are, which keeps them engaged throughout the learning process. Teachers can support students by guiding them in making reasonable predictions and providing feedback on their thought processes.

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To assess students, I must think about how to make all students engage in the assessment and get support for their improvement. There are many ways that I will use to assess my students. Actually, in China, the most common assessment method for math classes is exams and quizzes. However, after the learning of this course, I realized that this assessment method is too monotonous to assess all students in the class. It is obvious that some of the students may have special needs in learning math, especially for students with disabilities. Therefore, I will try to implement various assessment methods in formative and summative assessment. Firstly, I think regular class discussions allow me to observe students' thought processes and understanding of concepts. I can ask probing questions and encourage students to share their ideas more after they present. This helps me identify areas where students may be struggling and address those issues immediately. While students are working on group activities or individual assignments, I observe their behavior, problem-solving strategies, and communication with peers. This helps me assess their understanding of the concepts. Additionally, the periodic tests and quizzes measure students' knowledge and understanding of specific mathematical concepts. These assessments help me determine if students have mastered the material and identify areas where further instruction is needed. The self-assessment and peer assessment are necessary as well.  I encourage students to reflect on their own learning by asking them to write about what they have learned, what challenges they faced, and how they can improve. This helps them develop metacognitive skills and take ownership of their learning. Moreover, having students assess each other's work promotes collaboration and helps them develop critical thinking skills. They learn from each other's strengths and weaknesses and provide valuable feedback.

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I am committed to continuous learning and improvement in my teaching practice. I think implementing NCTM principles and MP into my classroom. For example, during the class, I should pose more questions that require students to justify their answers and explain their reasoning. This can help them develop deeper understanding of mathematical concepts and improve their critical thinking skills. This can develop their reasoning progress. Another improvement I think I need to have is the representation that can be used in the class. One way is to encourage students to create their own representations of mathematical concepts. This can help them develop creativity and ownership of their learning and deepen their understanding of the subject.

In conclusion, my teaching philosophy is centered around creating an engaging and supportive classroom environment so that they may have interest in learning Mathematics. In terms of methods, I incorporate problem-based learning, real-world connections, and collaborative activities to make mathematics more engaging for them. I can implement NCTM principles, Mathematical Practices (MP), Drivers of Investigation, and Content Connections to guide my instruction. This includes emphasizing problem-solving, communication, reasoning, connections, and representations. I provide students with opportunities to solve real-life mathematical problems using different representations and encourage them to reason through these problems and think about what knowledge they can apply. The group discussions and presentations can promote them to share ideas and build their own understanding.

My goals for students are aim to help students build problem-solving skills that they can apply in various contexts. I want students to see the relevance of mathematics in their lives and be able to make connections between different mathematical topics and other subjects. Additionally, through continuous assessment to students and the feedback from them, I can work towards ensuring that every student makes progress in learning math.