• Student's name: May 

• Major: Food Tech (UPH)

• What does she like about math? She likes about math because the result itself is exact, precise, and detailed. Furthermore, through the use of math people can prove scientific findings. 

• What she does not like about math? She does not like math because the formulas of it are too complicated. 

As a teacher, what should I do to change her negative perspective about math?

1. She has a good claim that math is useful for scientific research. Thus, as a teacher, I can use this perspective to decrease her negative perspective about math. I will start introducing that math is beneficial in our daily life: it is not only for scientific findings  but also it is part of our daily lives. A practical thing that I can do is to be contextual in my teaching. If I do story problems, the situation should be understood by the students: real-life situation. So that the students can understand the essential of math in our daily lives.

2. In my opinion, she has that negative view in math because the way she learned math in her school was directly to the formulas without knowing how does the formulas exist and the meaning of it. To help her, I want to implement the 8 mathematical practices: let the students be involved in learning actively. These MPs are very helpful to see students' thinking and equip them to be effective in learning math. Furthermore, as she mentioned that the formulas is too complicated.  Thus, by implementing the 8MPs, it is can be very helpful for her to understand the use of the formulas as tools and models to solve problems: not just as unknown tools. 

3. Lastly, since she thinks that math is precise and detail, she might feel overwhelm to be able to find the correct answer. Thus,  I will encourage her that in doing math need a lot of practice. It is okay to have many trials because that is the beauty of math. A thing that I can do is to give my students different questions but in the same topic. I will allow them to have practice, to reason, to critic, and to reflect. 

According to my observations, here is the list of how math is being implemented in our daily lives in the campus:

1.      Grouping lists of students: division

2.      Parking lot: geometry, distance estimation, and proportions.

3.      Cars and motor bikes: how many transportations each day? Counting and estimating.

4.      Buyers in the food junction: statistics.

5.      The cheapest food in the campus: Money management implementation

6.      Several buildings with different styles and shapes: geometry.

7.      Elevators and escalators: angle and multiplication

8.      Distance between the dorm and building B: estimation of time.

9.      The depth of the swimming pool: measurement

10.  The number of scholarships and regular students: statistics.

11.  Grand chapel: counting the number of the seats: addition and multiplication.

12.  Library settings: height of the bookcases and estimation of the amount of the books.

13.  Schedules in the service desk, library, building B, mail room, etc: calculations of time

14.  Affordable food: comparison of price in several places (food junction, kopkar, etc)

15.  Stairs in every building: addition and count-on

16.  Design of the UPH fest: pattern.

17.  30 anniversary of UPH: place value (tens, ones)

18.  Faster pathways: estimation of time and distance.

According to Piaget’s stage of development, elementary grades are in concrete operational stage (Hallahan et. al, 2014). I need to understand that children learn math concept by using materials and analyzing what is happening. Furthermore, I also need to be aware the need of my students, for instance visual learners. All of this means that implementing concrete things such as mathematical tools (base-ten blocks, pattern block, rulers, etc)/manipulatives in crucial in primary classes. Lets take a look how am I going to teach number, measurement, and geometry in the following paragraph. 

First, in teaching number, students need to see the object and count it. For instance, they need to use blocks to count. Based on the video, students mentioned and understood what they see. For instance, in the mountain diorama, the teacher asks the student what he sees, the students points out to the things that he sees. This is what happen to math concepts also, we need to facilitate their learning with things that they can see and touch so that they can individually construct the meaning through the usage of concrete things. In addition, we need to scaffold the learning according to their age development, so that they can construct the knowledge accordingly.

Furthermore, in grade 2 they will start to measure length of objects using measurement tools and start to understand the value of units. Thus, they need to use the actual tools so that they can understand the unit. In elementary higher grade levels (grade 3-6), they are in transition to the abstract concept. It does not mean that the teacher will totally discard the usage of concrete things. It is a very crucial time for the students to have a strong concept of how math works. Thus, as the students understand volume and area measurement, they need to actually see it before they move to the implementation of the formulas. Students can use mathematical tools such as conservation experiments, puzzles games, and unit blocks. 

Furthermore, in learning geometry students need to see the shapes in the real life. The usage of concrete things is to have the idea of students to visualize what the shapes look like. It is crucial for lower grade to see the shapes (triangle, square, etc) so that they also can see that they can find geometry in their daily lives. 

All in all, as a future primary teacher I need to be aware the emphasis of implementing physical/concrete objects to grasp the idea (to make sense) and make it relatable to the students’ lives.

References: 
Hallahan, D. P., Kauffman, J. M., & Pullen, P. C. (2014).Exceptional learners: An introduction to special education.(12th ed.). Pearson.

Billstein, R., Libeskind, S., & Lott, J. W. (2016). A problem solving approach to mathematics for elementary school teachers (12th ed.). Boston, MA: Pearson/Addison-Wesley. ISBN-13: 9780321987297. 

Missmith891 (2011). Piaget's stage of development. https://youtu.be/TRF27F2bn-A?si=D02p9LK8qB3oX7vM. 

1. Anticipating likely student responses to cognitively demanding mathematical tasks

In my prior lesson plan, I planned a lesson plan that was applicable and relatable for grade 2 students. I considered their prior knowledge and their needs (differentiation). For instance, I understood that they have the idea of measurement in their prior classes. They understand that objects can be measured (longer, shorter, bigger, wider, etc) but they do not have the concept of units yet. Thus, considering this, I wanted to introduce the measurement tools (rules, yardsticks, and measuring tapes) and the idea of units (centimeters and inches). Through this observation of students prior knowledge and needs, I implemented centers to review their prior knowledge, and I made a contextual problem that they needed to solve: measuring objects for easter decoration day. By considering the students, I wanted them to have a really deep understanding in using right measurement tools. 

2. Monitoring students’ responses to the tasks during the explore phase

In the class, after I allowed them to do centers, I explained the main topic for the day. I informed them that they need to measure at least 2 objects in the class. During the exploration phase, some of the students were trying to choose the correct tools to measure their chosen objects. Sometimes the came to me and asked why it was hard for them to measure the object. Then, I monitored and redirected their thinking by asking them question. For instance, the student found it hard to measure the whiteboard with ruler, then I asked them, is it easy for you to use this ruler? what if you try another tool? Such simple questions were somehow beneficial to support them in their learning to try again. 

3. Selecting particular students to present their mathematical responses during the discuss-and-summarize phase

 I chose two students, Jasmine and Joy as the representatives of the class. They explained their findings and I let the rest of the class to give comment of their result. Some of the students agreed but some gave another suggestion on how to use the tools to measure objects. This is very helpful for me to summarize the main claim of the lesson of the day, choose appropriate tools to measure the chosen objects. 

4. Purposefully sequencing the student response that will be displayed

I purposefully chose the students to present their result, that can lead to the topic that I will speak about. I chose Joy’s result because she was using a yardstick to measure the height of the door meanwhile other use a measuring tape to measure the same object. I purposefully chose hers, and asked other students about the tool that Joy used. Other students gave suggestion about is as well and it led to the idea of how to use the tools appropriately. 

5. Connecting different students’ response and between students’ response and key ideas.

As I chose 2 students to present, I connect their ideas to the mathematics topic that I want them to know about. I connected the idea of measuring from zero mark on the ruler. I also connected the idea of how to measure object with the right measurement tools. For instance, they can use measuring tapes to measure length that are not straight or flat and they can use a ruler to measure shorter objects.

B. Six guidelines:

1. Begin with concrete representations

I used measuring tape, rules, and yardsticks. I also provided the pictures of those tools on my PPT.

2. Develop understanding

The main purpose in my lesson plan was choosing the right tools (rulers, yardsticks, or measuring tapes) to measure objects in centimeters unit. Thus, to develop students' understanding about measurement I incorporate relatable context in the class. I tried to make the activities in a known context which is easter day. I wanted them to know how and why they measure the objects in the classroom. Prior to this, we also had 3 centers as a review and scaffolding from the previous topic.

3. Encourage communication

In my microteaching, I did classroom discourse, where the students needed to explain their thinking in an interactive way. For instance, when I chose Joy to elaborate her result in measuring, she found out that her yardstick is not enough to measure the length of the door. Then other friends gave her suggestion that she can measure it twice. It means through communications, students understanding can be solidified.

4. Make connections

In making this lesson plan, I focus to make it relatable to the students lives. Thus, I decided to choose easter day as the context. Easter day is very famous in our society, where we celebrate it. Thus, I used this context in math class especially in measurement topic.

5. Motivate children.

In the class, I found out that some of the students were struggling to use the measuring tape. They complained that it was hard to measurement tape. I encouraged them to try again  by using other tools and be patient, because practice makes perfect.

6. Provide opportunities for practice

In my microteaching, since I ran out of time, I could not create more practice. Thus, in this part I want to scaffold the measuring ideas. I want my students to understand what to do if their measuring tools is shorter than the object that they need to measure. I want them to practice more on it so that they can find a strategy that they actually can measure it twice using the same appropriate tools. 

In my microteaching, I create environment where all the students have equal opportunity to learn. For instance, I let them to choose the object that they want to measure, then I provide them the tools that they need to use. For instance, I provide them yardsticks, rulers, and measuring tapes. They can choose to use any of these tools to measure the objects in the class. I give them opportunity to talk, give ideas, and give comments to their friends' work. Another example, I focus on the second language learners. I want them to have equal access in learning based on their need through helping them in explaining their thoughts (follow up questions), simplifying my wording and explanation, using visual aid, and introducing the term. 

2. Focusing on a balance of conceptual understanding and procedural fluency

In my microteaching, I do not really reach this balance. I realize that I lack a deep conceptual understanding and I do not have enough time for the procedural fluency practicing. In my opinion, it is because this is the first topic. Thus, for the future consideration. I would like to make this balance happens. I want my students to grasp the conceptual understanding by expressing the length of an object as a whole number when they measure it using non-standard unit.  I want them to know how to measure by laying multiple copies of a shorter object (the length unit) end to end (iterating) with no gaps or overlaps. I would do it by giving them opportunity to measure the object again. Then, try to discuss it. Furthermore, I would like to scaffold the problems in different context and I want my students to do it accurately, flexibly, and efficiently. Thus, giving them more time to practice is beneficial for their procedural fluency. Practical thing that I want them to do is to solve story problems about measurement as well (in other context)

3. Ensuring activity student engagement in the NCTM process standards (problem solving, reasoning, communication, connections, and representation)

In my microteaching, the engagement of the students in learning process is somehow fulfilled. I give them problem to solve, which is Easter decoration day: measure to decorate. Then, I continue ask them to reason about their result, they even present their worksheet and show the class how did they do the measurement. Furthermore, I allow the class to have whole-class discussion, where we can give critique/comments about our friends work. In the center, I also encourage my students to communicate their thinking and they can write it on the worksheet, they can explain how they do the measurement. For representation part, I use the real measurement tools, I put the picture on the PPT, and I bring the object that I measure and decorate to be shown for the class.

4. Using technology to enhance understanding

In my microteaching, I do not use technology in the class, since I use more practical activities: measuring object in the class. However, in giving instruction I use computer to present the content and instructions. In further consideration, I might use document camera to support my students in their explanation to the whole class. 

5. Incorporating multiple assessments aligned with instructional goals and mathematical practices

I do centers (to assess their prior knowledge), discuss what the problem is about (MP1) do worksheet (as written result of their measurement activity) (MP 4), do whole-class discussion to assess their participation (MP 2), do peer-share time to know students' thinking (MP3), and I allow them to choose the tools to measure (MP 5),

6. Helping student recognize the power of sound reasoning and mathematical integrity

I encourage this to my students through follow up questions, where my students can think more of what they are saying. I also encourage my students to proof their thinking through showing the class of how they get it. For instance, I would ask them how do you do your measurement. 

Process problem

Grade level: Grade 4

Standard:

4.GM.B: Solve problems involving measurement and conversion of measurements.
4.GM.B.5

Apply knowledge of the four operations and relative size of measurement units to solve

problems in authentic contexts that include familiar fractions or decimals

Question:

Mrs. Lampita is preparing her garden for the spring season. She has a rectangular plot of land that measures 10 meters by 5 meters. She wants to divide this land into smaller square sections to plant different types of flowers. If she needs to plot four kinds of flowers on four equal-sized land, how large is each land of flowers?

How to answer:

The process:

Start from the size of the whole land then get the size of each land.

The size of the whole land is obtained from the length of two sides.

Area of the plot=Length × Width

By multiplying them we get 50 meter squares of the size of the land.

Then divide the land into four sizes. This means that the 50-meter squares are divided by four.

Area of each section= Area of the plot/Number of flower types

Which is 50/4. In the end, we get 12.5 meter squares of each plot for each flower.

TRANSLATION PROBLEM

Grade level: 4

Standard:

4.OA.A.3 Solve multistep problems in authentic contexts using whole numbers and having whole number answers using the four operations, including problems in which remainders must be interpreted.

Question:

The bus driver starts her route with no passengers. At the first stop, 12 people get on the bus. At the second stop, seven people get on. At the third stop, five people get off the bus and three people get on. At the fourth stop, four people get on and six people get off. At the fifth stop, three people get off the bus and no one gets on.
How many passengers are on the bus after the fifth stop?

How to answer: (attached)

APPLICATION PROBLEM

Standard:

4.OA.A.2 Multiply or divide to solve problems in authentic contexts involving multiplicative comparison, distinguishing multiplicative comparison from additive comparison.

Problem:

One person from your table can take 3 weeks to drink 1 Liter of Orange Juice. But because there are 4 people in your group, how long will it take for each person in your table to drink 1 Liter of Orange Juice.

For Context:

From this problem, students are encouraged to learn how to estimate and how to make an educated guess from doing this problem.

Students will also be given 1 Liter of Orange Juice in each table and they will receive cups. Students should predict how long it will take for them to drink the orange juice.

How to Solve:

The estimated guess should also be 3 weeks for one person. If the students did not guess 3 weeks then they should try guessing how much orange juice each person can drink in one week. Then, they should try guessing how much orange juice a person can drink within two weeks.

Work sample 11 "Statistics: Data" (Satisfactory) 

Analysis on the standard and sample: 

Standard: 

• Students describe different methods for data collection and representation, and evaluate their effectiveness. 

Work sample: 
In the work sample, the student is able to choose and explain which question is best to be asked for the survey. For instance, the students chooses a survey question that provides a lot of different answers (checklist) for the respondents to answer effectively.  In this work sample, the student meets the standard especially in collecting data (asking and choosing best question is important for collecting data): the student chooses the effective question to collect the data and give a reason for it. 

Standards: 

• Students describe different methods for data collection and representation, and evaluate their effectiveness. 

• Students construct data displays from given or collected data 

the student able to construct the date in a table and make it into a column graph appropriately and concisely. Additionally, in the column graph the student is able to choose appropriate scale (put the number as well) and label the appropriate data on the axes. Lastly, the student colors the graph to show the difference of the data. However, in the table, the student does not accumulate the total number of devices after recording the actual data. In my opinion, this part is quite acceptable since the student accumulates the prediction part but not in the actual result. 

Standard:

• Students describe different methods for data collection and representation, and evaluate their effectiveness. 

Work sample:
According to the work sample, the student predicts first before collecting the actual data. Prediction is an important process in students' thinking that lead them to reflect on their own thinking with the actual situation. Prediction will allow them to evaluate their effectiveness in collecting data. 

How would you assess the children’s understanding based on that written work?

Since this work sample is being implemented as an end of unit assessment, it means I will assess what they learn throughout the unit. I can assess their process of thinking and reasoning. I would like to assess them based on these criteria (I will provide a rubric as a tool for me in assessing): 
- how they select appropriate questions. 
- how they collect data 
- how they represent and construct the collected data 
- how they give explanation based on their selected answer. 
- How they use the strategy appropriately. 

• Probability is a comparison of the desired outcomes to the accumulated possible outcomes. Basically it is about how likely events are around us (It somehow connects to smart guessing). 

• Probability can be represented on a continuum or number line that has 0 as impossible and 1 as certain. This continuum can visualize of how an event likely can be. For instance, if an event is in 0 it means the event will never happen. But, if it is in 1, it will certainly happen.  So, just imagine a number line that only has 0-1 (0%-100%)

2. Types of Probability:

• the difference between theoretical and experimental probability is in the result determination. In theoretical probability grounded on our thinking/logic in analyzing the experiment, meanwhile the experimental probability is determined only by collecting the empirical data or on the experiment results. 

• Simulation is a very effective method to have to answer real-world problems or questions especially when it comes to changes of chances. Thus, a model or simulation is helpful to visualize the real-world situation and come up with solutions of the probability. 

3. Define the terms below and provide examples from the textbook:
Sample spaces 

• Definition: its a set of all possible outcomes for an experiment. 

• Example: A bag contains: 2 red, 3 yellow, and 5 blue tiles. So the sample space is all the ten tiles. 

Independent event 

• Definition: an event or situation has nothing to do on the other events. In other words, it does not affect other event since it can stand by itself. 

• Example: Tossing a coin twice. We can see the probability only in this event. 

Dependent event 

• Definition: the result of a second event depends on the result of a first event. 

• Example: 2 boxes, One box contains one genuine dollar bill and two counterfeit bills, and the other box contains one genuine and one counterfeit bill (you do not know which box is which). You may choose one box and from that box select one bill without looking. What are your chances of getting a genuine dollar bill? Here, there are two events: selecting a box and selecting a bill. The probability of getting a dollar in the second event depends on which box is chosen in the first event.

Area representation/model 

• Definition: its a tool to list all possible outcomes of simple and compound events. 

• Example: are you spring a dog? This activity helpful for students to arrange the outcomes of the probability in a complex context. 

4. Activities and Experiments:
1. Get all 6

• Purpose: to understand that the larger the size of the data set, the more representative the sample is of the population. 

• I learned: This activity is helpful for me to attempt for determining the probability of an event through data collection. 

2. Drop it 

• Purpose: Students get to understand that some probabilities cannot grounded on the logic analysis only, but must through gathering empirical data. 

• I learned: I learn there are 2 types of probability: theoretical and experimental. This activity helps me to understand experimental probability better. 

3. Keys to a new car 

• Purpose: Students get to know that dependent events depends on the other event. 

• I learned: this activity is an engaging way to explore probability of dependent events. 

Math Teaching Statement

Teaching mathematics is a challenging journey where we find ourselves most of the time in “aha! Found it” or “aghh, what’s the answer” moments. It implies that learning math needs effort and consistency. My teaching philosophy of mathematics stems from the beliefs that science is doable and applicable in lives as long as we are striving in solving the problems. The core of my teaching philosophy is the constructivist theory, that claims learning math must be centered on students, as the main characters in learning. Constructivist theory implies that all people create or construct meaning of anything around them (Van de Walle et al., 2020). I believe that every student can construct their understanding of math through hands-on activities, discourse, and collaboration. It means that students are not only passively receiving information from the outsiders, but themselves are the ones who make meaning of learning. The lessons and core idea of those resources is students as active learners, they are the ones who construct the meaning through their understanding in the exploration. For example, allowing students to have class discourse, centers, math talks, and 3Acts activities.

Furthermore, I believe that learning mathematics is not only about one exact correct answer, but more than that, the process that the students go through is the important part. During the process, they learn something, they attempt to solve a problem, give reasoning, and display their work. According to NCTM, values problem solving processes and solution for learners (Van de Walle et al., 2020). It gives the sense of perseverance in solving the problem. This greatly important for students to maintain and practice their conceptual and procedural knowledge (Van de Walle et al., 2020). The core idea I believe in teaching is to help students to develop their critical thinking ability and apply their understanding in the real-world contexts.

Moreover, my strategy in teaching should reflect the idea of constructivist. Therefore, I focused more on multiple explorations and various types of problems. Multiple exploration can be through math talks, centers, read aloud, 3Acts, and using manipulatives. For the type of problems, I can implement the application, puzzle, process, and translation problems (Cathcart et al., 2014). In my lesson, I must be always put relevant and thought-provoking questions for all students to answer and ponder. Additionally, it is important to consider the real-world application of learning math. It is because all of us need to know the reasons of learning math, which is I believe to be able to solve daily life problems and to have more critical thinking in making decisions. Furthermore, through math we can be more appreciative and reflective, we can see the beauty of God’s creation in it. I believe that the complexity and solution of math point us to the Creator himself. Furthermore, in my assessment, I integrate classroom discourse because it is beneficial for students to communicate their thoughts and listen to others for more learning strategies. Teaching mathematics should enhance reasoning and speaking ability (Van de Walle et al., 2020). Additionally, I should address the misconceptions and different learning styles of students, consider differentiation. I believe that all students can make sense of math according to their learning styles. It means I need to embrace and address their learning styles so that they can enjoy and feel motivated in learning.

Class discourse such as math talks or number talks enhances students’ communication of math and their ability in computation will be seen in their justification (National Council Of Teachers Of Mathematics, 2000). It encourages to practice their inquiry and reasoning abilities. Furthermore, technology integration is highly valued (National Council Of Teachers Of Mathematics, 2000). Thus, letting students to experience technology as tools are important for them to be able to survive and bound with the modernity. It is important to be noted here that using technology is not only about following the modernity or trend, but also to prepare students for wider knowledge as global citizens. As a teacher, incorporating tools like online manipulatives, online apps such as khan academy and math is fun is highly recommended for my class.

In my classroom, teaching students is not just letting them explore and construct meaning by themselves without any guidance and standard that I want them to reach. Therefore, as a teacher, prepare ways to assess or to measure and know students understanding though assessing them (Cathcart et al., 2014). Assessments are beneficial for me to evaluate the students’ understanding and revise my instructions. For example, in my unit, I teach about unit conversions of length. First, I start with concrete example of how we determine something is longer or shorter. Then, I allow them to have centers as the pre-assessment. Then, I scaffold my assessment through as worksheet that contains real-word story problem as my formative. Through this assessment, I can evaluate whether students reach the objective or not.

Being a lifelong learner has been the motivation for me to see other people’s potential. I believe that everyone regardless their background or age, they can still learn in their own portion and ways. Thus, this idea is applicable for me as well, especially for teaching mathematics. As part of lifelong learning beliefs, I want to keep on updated of math developments through practicing my skills in analysing real-world problem around me. I want to know more how math is applicable in daily life. Also, I will try to practice my skill by solving problems in a math app. So that I can be more familiar with math.

In conclusion, my teaching philosophy rooted in constructivist theory where students actively engaged in exploration and discussion. All students have access to learn math and make connection to the real-world problems. My goal is to help students to have more critical thinking and problem-solving skills. Therefore, aiming consistent growth in learning math is crucial for me as a future teacher.

References

Cathcart, G. S., Pothier, Y. M., Vance, J. H., & Bezuk, N. S. (2014a). Learning Mathematics in Elementary and Middle School. Pearson Higher Ed.

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

Van de Walle, J., Karp, K., & Bay-Williams, J. (2020) Elementary and middle school mathematics: Teaching developmentally (10th ed.). Boston: Pearson Education, Inc.