Unit Plan Final Version

In PDF version:

https://drive.google.com/file/d/1aCtag7yoVAF7GWE4Kp5deXF-QYwpnhSp/view?usp=sharing

In Microsoft Word version:

https://docs.google.com/document/d/1rwV6H-bOq2ZQndBZwUyOS6r7xddsNzrf/edit

Course Reflection

I enjoyed the course 342 very much. It is definitely the most interesting course I was taking this term. Every time when I was doing the homework for this course all my fatigue went away. I want to thank Professor Gerofsky for sharing so many interesting stories and problems and useful ideas and concepts with us. I am also interested in studying curriculum and planning lessons, so this course is really my favorite one. 

My favorite article is "Arbitrary Vs. Necessary." How we, as math teachers, make the knowledge more necessary to students rather than arbitrary is what we should consider when we are making lesson plans.  To use this concept correctly, I think I need to re-read it one or two more times. 

My favorite puzzle is… I like all of them. The locker problem and the scales problem are two puzzles that took me the longest time to solve. I found high school students love doing puzzles too. I think it is necessary to continuously collect and compile relevant (if possible) interesting puzzles for each lesson plan. It could help us to foster a love of mathematics in students. 

My favorite activity is " Teaching Perspectives Inventory." This activity helps me understand my own teaching perspectives better. I would like to do it again in the future. I believe Nurturing will always be my dominant perspective. I am not sure if the developmental perspective will change or not, but it is my recessive teaching perspective now. 

I hope one day I could also make my Math course interesting like EDCP 342. I would like to thank Professor Gerofsky for letting me have such an amazing learning experience. 

Let's party! Math Party of EDCP 342--My favorite Math games

 Game 1: Hua Rong Dao (华容道）

In China, we call this game Hua Rong Dao (华容道). It is also called Klotski, a sliding block puzzle. Today, I want to show you how to play it with cards. So you can play it at any time. You could play 3 x 3, 4 x 4, or 3 x 4, or any number of cards. 

Game 2： 24 game

I started to play this game when I was in grade 5. I found high school students like playing this game too, no matter which grade they are in.  Please help me to solve these two puzzles. You could use any operation.

Unit Plan

https://drive.google.com/drive/my-drive

Please click this link to review it on Google drive.

EDCP 342A Unit planning: Rationale and overview for planning a unit of work in secondary school mathematics

Created by: Cheryl Zhang

School:         Dr. Charles Best Secondary School 

Grade:         11

Subject:       Pre-calculus 

Unit Topic:  Quadratics  

Preplanning questions: 

(1) Why do we teach this unit to secondary school students? Grade 11 Pre-Calculus Mathematics builds on the topics studied in Grade 9 and Grade 10 and provides background knowledge and skills for Grade 12. In Grade 9 and Grade 10, students studied the basic algebra skills, such as simplifying algebraic expressions, the family of 1st degree polynomials, the linear relations and linear function, etc. Based on these skills and knowledge, students are able to explore the family of 2nd degree polynomials, the quadratic functions, in Grade 11. Quadratic functions are a slight advance in linear functions and provide a significant move away from attachment to straight lines. As the prerequisite course of Pre-Calculus 12 and Calculus 12, students should be comfortable solving quadratic equations.   Quadratics can solve many real-life problems involving area and projectile motion. It is not only the access to understanding calculus, but also the access to understanding science, such as gravity, and many other subjects. Quadratics allow us to analyze the relationships between variables. Using quadratic graphs people could estimate values for some given values.   By studying quadratic functions, students are able to apply their understanding of quadratics to variety of real-world, model real-world data with quadratics, graph them, find solutions for them.   Quadratics is primarily used to find the curve that objects take when they fly through the air. Today, we know quadratics can be used to find shapes, circles, ellipses, parabolas, and more. Quadratics is used in the design of products in almost every field, such as car industrial, entertainment, astronomy, etc.

(2) A mathematics project connected to this unit: Topic: Parabola Selfie Poster          Students will find parabolas in the real world. For example, the shape of a banana, the shape of a bridge, the shape of the brand of McDonald's, etc. This project may be done alone or in pairs.   Aims:  Students will know that there are many things in the real world designed using quadratics, and the graph of quadratic functions can be found throughout nature. Students will find out that a parabola is a curve, a conic section, and a graph of the quadratic function. Students will learn how to use the graphing calculator to generate an equation for their parabola. They will know how to analyze the graph by finding characteristics.   Process: Step 1: Modeling: All students will study a sample poster with the teacher. They will learn how to use Desmos to generate an equation for a parabola on a picture. They will get to know what they are supposed to do. Step 2: Find two parabolas in the real world. One is found in nature, and another one is found in manmade items. Take a photo with each of them. Printed out and put on the poster. It can be in black and white. Step 3: Students will upload that picture into Desmos. They will adjust the scale of the graph to match the dimensions of the parabola in the picture. Finally, they will generate two quadratic equations for both of their parabolas. Printed out the graphs and put on the poster. Step 4: Students will analyze the parabolas in terms of the axis of symmetry, vertex, domain, range, max value, min value, etc. Students will put all of these analyses on the poster.  Step 5: Students will write any funny things and ideas that they encountered or found out when doing this project. They will put all of them on the poster.   Timing: They have 4 weeks to do this project. Normally, I will ask them to submit their      poster two weeks before the end of the quarter. Then during the last two weeks,      students could exchange ideas based on the posters on the wall.   Assessment: On the poster, students must show all the items required in the rubric. I will remind them to check each item when doing the poster. The poster should be made neat and organized and submitted on time. This project will be 20% of the total grade.

(3) Assessment and evaluation:  Formative assessment: (1) Each class, students will work on small group activities and finish a short worksheet. I will assess students' level of understanding based on their works on these worksheets. (2) Students will have homework for each section. I will grade only for homework completion. Every two weeks I will offer a one-on-one conversation with each student to check their homework completion and address any issue they encountered. (5%) (3) Every week, students will do a homework quiz. I will not grade on these quizzes.   Summative assessment: There will be a unit test (15%) and a project (20%) to do for each student. This test and the project will be used to test students' comprehension of materials.

Elements of your unit plan:

a)  Give a numbered list of the topics of the 10-12 lessons in this unit in the order you would teach them. 

Lesson	Topic
1	Investigating Quadratic Functions in Vertex Form
2	Investigating Quadratic Functions in Standard Form
3	Completing the Square
4	Graphical Solutions of Quadratic Equations
5	Factoring Quadratic Equations
6	Solving Quadratic Equations by Completing the Square
7	The Quadratic Formula
8	Transforming Quadratic functions
9	Modeling with Quadratic data with Real-World Problems
10	Review and Unit Test

Article response: Dave Hewitt on Arbitrary vs. Necessary

This article inspires me from the following points. I will rethink how to question my students, how to explain curriculum content, and how to plan my lesson.  

1. "However, these are stories invented by teachers mainly as memory aids rather than justifications." (p.3)

If students feel it is hard to memorize names, I think inventing stories as memory aids is a great idea. This is what I am always doing to help my kids and myself memorize arbitrary things. When I was teaching my children Chinese characters and ten digits (0, 1, 2…9), I often invented some stories related to their structures or shapes. These stories work as labels that are attached to those arbitrary things. When we want to recall some arbitrary things, our brain could find them by their labels. However, I think most labels and symbols are not arbitrary. Most of them make sense, and it is easy to explain their meanings. For example, the symbols of "greater than >" and " less than< ", "perpendicular ⊥" and "parallel ∥",  "angle ∠ " and "triangle Δ", and so on. For those arbitrary labels and symbols in the mathematics curriculum, such as "division sign ÷", I will encourage students to invent labels by themselves to help memorize. I personally don't suggest teachers invent stories for students, because the brains of different people work in different ways. Students, especially high school students, need methods more than to be informed of some things.

2. " Those things which are necessary can be worked out: it is only a matter of whether particular students have the awareness required to do so. If not, then maybe it is not the best choice of topic to be taught at that moment in time." (p.4)

This explains what an invalid lecture looks like. Invalid means that the content of the lecture doesn't match for students' level of understanding. For example, if I taught five laws of exponents, I asked students to calculate (3/n^2)^4. To solve this expression, students needed to use two laws to solve this single problem. The majority of students should have the awareness required to do so, otherwise, I think I need to add extra lessons to teach them some basic mathematical knowledge before teaching laws of exponents. If I did not notice this circumstance and not add any extra lesson, students would feel confused when they doing practice. 

3. "Inviting students to 'think about it' is appropriate for what is necessary, but not for what is arbitrary." (p.5)

Questioning is an important assessment method. If we ask students to think about something that is arbitrary, we are just asking students to recall in their memories for something they cannot remember. When we ask students questions, we need to make sure that "it is possible for students to use their awareness to try to work out for themselves" (p.5). These questions are necessary and in the realm of awareness. For example, if I ask "what are the four basic arithmetic operations," this is a question for something is arbitrary. If I ask " which one of the four arithmetic operations is the inverse operation of addition," this is a question for something is necessary. I think the second question is the one that students need to work out.

 

TPI test results and reflection

 

My TPI results

My TPI test result shows that my dominant perspective is Nurturing. Apprenticeship, Transmission, and  Social Reform are my Back-up teaching perspectives. Developmental is my recessive teaching perspective. My profile is a bit flat and it has a standard deviation of 4.24.

When I look at this result, I think it does show what I look like right now --- a beginner. I have the highest intention score in the Nurturing perspective. I feel comfortable being a nurturer. As a teacher, I think my priorities are providing students support and encouraging them. I have the lowest action score in the developmental perspective. This one is funny.  I am not surprised for this result because I believe in natural development. If students feel safe, encouraged, and are placed in a positive learning environment, development is inevitable. I will not let students know my expectations of them. Instead of that, I hope they can feel that I firmly trust them, and believe they can achieve their goals sooner or later. Everyone is living life at their own pace and vibrating at their own frequency. Expectations, especially from a teacher, may cause pressure since it sounds like a task. Oppositely, trust is an attitude and it sounds like encouragement. 

For the question "In my teaching, building self-confidence in learners is a priority," I strongly agree. For questions starting with "I expect people," I mostly give them "rarely" or "sometimes." Again, I just don't want to give others any pressure. For the question "  I share my own feelings and expect my learners to do the same," I don't know if it is a good idea to share teachers' own feelings with their students, while definitely, I will not expect my learners to do the same thing as I do. I will share some positive emotions with my learners. I think it is not appropriate to show students teachers' negative emotions.

Thinking about math textbooks

As a teacher, when I observe the image of Figure 1: A hand doing mathematics (Lappan et al., 1998, p. 6), I think the disembodied, generic hand is simple enough to help students understand the process. There is no need to draw a person in that image. The unnecessary part of the graph can confuse the readers.  This image is quite detailed, and it positions the students to recall their real-life experience that adding coins to the cup will eventually break the bridge. 

As former students, the figure shows a very interesting scenario that someone is trying to test how many coins a paper bridge can hold when each side of the paper bridge has 1 inch on the books. The ”1-inch“ label is a clue. It will make me eager to figure out the paper bridge's holding capacity. 

No matter as a teacher or as a former student, I think the textbook is necessary for learning mathematics systematically. Not like some other subjects, such as physical education and language, mathematics involves many concepts, theories, theorems, propositions, etc. We need a book to present them systematically. With a textbook, students could learn mathematics in a rigorous and systematic way. I always think learning Math is like building a skyscraper. The final height of the skyscraper will usually depend on how strong and deep the foundation must be built. To build a skyscraper, we need a guide book (Technical Drawings). Moreover, for Euclid's "Elements" and Euler's "Introduction to the analysis of the infinite", we all know how important these textbooks are. 

There is one issue that I concern about. Math textbooks are too expensive to have for most students and schools. Some schools do not have sufficient funds to buy Math textbooks for students. And Math textbooks change versions every several years, so it is another difficulty for school. I observed students in my practicum school are still using Math textbooks in old the version. 

Some people suggest digital textbooks or e-textbooks. I think students need to avoid spending too much time with electronic devices. There are many Apps on their electronic devices, so students cannot fully concentrate on the textbook and learning.

In conclusion, I think textbooks should be essential for Math learning, and they still play a fundamental role in schools.