Student Reflections on my Class

So the students had to fill out this survey of questions to help the teachers to get  an idea of what to write about in the letters of recommendation. I found the surveys very helpful for some students. Other students did not understand the question or the point of the question. Some students were too vague or didn't give really good specifics.

However I wanted to share with you some of the comments students made about my class and my teaching. Reading some of these made me feel like I was doing something right in the classroom. These are responses to why calculus is their favorite class (I did not edit the students answer so that you can see what level of English my students have).

Student 1: Seriously, I am very interested in Calculus. The teacher is very good at teaching and always gives us a lot of examples to practice which improves our knowledge of Calculus a lot. Miss Lomax is very interesting and we can play little games in her class. I like the challenging questions the occurring in Calculus which let me always want to do my best in this subject.

Student 2: Calculus is very interesting and can be connected with other courses such as physics. Calculus always brings up several methods for one question. Some methods used to solve the questions are miraculous. The solving process provides the opportunity of thinking. The in-class group discussions and fellow cooperation are quite excellent.

I put a lot of emphasis on solving problems or explaining problems in different ways which is what I think student 2 is referring to. I make sure students who solve problems in a different or unique way share their method with the class. I make sure which discuss the methods and compare them as well.

this student also enjoys solving problems different ways as well

Student 3: When solving one difficult question , I will have a feeling of fulfillment. Also, I love the approaches to working Calculus are cool and fantastic.

Student 4: Because the knowledge of calculus are practical, which can be used in many spheres of our everyday life to solve some common but cumbersome problems. Also the calculus teacher is really skillful who can keep attracting students’ attentions in class.

Oh here is one of my favorites

Student 5: Mathematics has been my defect for my grades for more than 6 years. However, when I first turn to the calculus courses, I was awakened. I never thought that mathematic problems could be solved in such amazing and interesting way. I have no interests in Math simply because I was not able to find the applications in real world. Derivatives and Integral opened a new gate for me. I then fell in love with this charming course.

Student 6: Because our calculus teacher is very humorous and kind. She always explain my questions patiently before making sure I truly understand the knowledge point. Also, I think Calculus is very important for my college study and even useful for my future work.

These are just a few that I read and enjoyed. I will read through more of the surveys and add more student comments.

Update on Pre-Calculus Course

So the meeting was super informative and helpful this weekend. It gave me a really good idea about what to cover. We met with a retired Chinese math teacher who works for the publishing arm of our company. He went over how the Chinese math curriculum was organized and designed. This helped show the areas in which are students were strong and others in which they were weak. We went through our current textbook and looked over what section was needed for AP Calculus AB, AP Calculus BC, AP Statistics, SAT II subject tests, and the Chinese graduation exams. So I will focus on the topics they will need for the AP tests. The Chinese math teachers prepare them for the SAT II subject test and I have a feeling my students don't take the Chinese graduation exam.

We are still discussing which chapter is important prep for some of those classes, but I have a basic idea about where to start.

I will still be flipping the classroom and got the resources from the other teacher who is currently doing that. This will make it so I don't have to use the 90 minutes I see the students each week on lecture. I will use this series of pre-calculus videos to support my teaching. He does not have videos for every chapter so that may guide me in what I teach and what I don't.

I will start with chapter 4 which is over trig. The Chinese math curriculum focuses a lot on trig, but does not focus on all six trig functions. They do not spend much time with cotangent, cosecant, and secant. The students easily understand and pick up those functions, but they need to be discussed. The Chinese math curriculum also does not spend any time on inverse trig functions. So starting with the trig chapter will help ease them into the class so that some of the stuff they are seeing they understand and some is new. This chapter comes with notes to take as the students are watching the videos. However does not come with worksheets to do in class. I will have to pull problems from the book or elsewhere.

I will then move onto chapter 3 which is over exponential functions and logarithms. The Chinese curriculum does not talk about these functions a lot or rather uses different notation than we do for these functions. So I will spend some time on that. This chapter does not come with any handouts. I will have to make my own following his format. I may make a very simple one for these videos just to check that they watched it. Then have them hand those in and grade them pretty easily.

Now I may pick and chose what I do in the beginning chapters. I know students are really strong in those beginning chapters. I don't think it will hurt to go over earlier stuff just because a lot of them know the math, but don't know it in English. I will make sure to put emphasis on the early stuff in lots of different ways as well. The students need to understand everything graphically, algebraically, and numerically. So I will make sure to include a lesson based on this video.

My colleagues say that the students are really weak in statistics. The Chinese math system does not do much with probability and statistics. The students are also not asked to explain, describe, analyze, or justify their answer in the Chinese math system. So statistics is about doing just that analyzing and describing data. So I will include a stats unit. I am afraid to do it, but it will help me get over my fear of teaching statistics. My colleague at the meeting was really trying to encourage me and give me advice about how to advance my career. So I will include a unit on Chapter 9 which is an overview of stats and probability. This unit has worksheets and notes which will help immensely since I don't know much about stats. I am thinking of having the students design or complete an experiment. I think I will work with the 10th grade science teacher to see if we can take that on together.

Now that content will probably only take me a semester and maybe just a little bit over. I doubt I can stretch that into a year. I will look at what other chapters are suggested, but the problem is I won't have pre-made videos to use. I think it would be weird to switch back into a normal class and I don't want to make the videos. I will look around to see if there are other videos I can use or at least good power points I can use. The text book came with a set of power points to use. They look like pretty good ones with examples and things. So I can make the videos my self using those power points. It will probably not be as good as the other videos since I won't have the nice set up that he does and I will probably be making them must faster. But then I won't have to spend my time in class going through the power points and just focus on the students understanding of the concepts.

If I make it so that I have 6 or 7 groups that have to present then I can stretch this small part out. I want to keep the group size small and want to make one person from each group present. So I think I will still have it so first period students work in groups to solve a problem and then work on how to present the solution. I will have as many groups present in the second period. I will have a minimum of four groups present in a period. Then have 3 or 4 groups present the following week. Then have some time to review or ask questions. Then have a quiz over the content that was just presented on. We may not get through as much content, but it will help their English abilites and presentation abilities so much more.

I think each week I will give them a handout to fill out while watching the video. We may not talk about the content in the video each week, but that gets them working ahead. It allows them two weeks to watch the video, rewatch the video, and ask questions about the video.

So the break down of the class will look like this:

Final: 20%
Tests/Midterms: 20% (Tests every month)
Quizzes: 15% (One every other week)
Presentations: 15% (Each student will present every two months graded on these rubrics)
In-class: 15% (presentation exit slips, class dojo, any other activities)
Homework: 15% (filling out notes worksheet on video every week)

If you want more resources/information going to this post about my pre-calculus course or this post about my calculus course. 

Pre-Calculus Course

Grade breakdown:
Final: 20%
Tests/Midterms: 20% (Tests every month)
Quizzes: 15% (One every other week)
Presentations: 15% (Each student will present every two months)
In-class: 15% (presentation exit slips and class dojo)
Homework: 15% (taking notes on video every other week and short assignment from book the other week)

Alright so my bosses trust me and my knowledge of the system to teach pre-calculus. They don't want to give pre-calculus to the new teacher coming in because they don't really know what they are getting. This would allow the other teacher time to get their footing at the beginning of the year and teach classes that have a specific outline of topics given by the collegeboard. I will take this as a sign that my bosses think I am capable and trust that I will do a good job. So this is my idea for the class right now. I will get more specific and add more resources after I go to the mathematics company department meeting on Friday.

First off let me say that there will be three sections of Pre-calc and each section will meet once a week for 90 minutes. There should be about 24 or 25 students in each section of the class.

I am thinking of flipping the classroom and putting all the learning/teaching on them. I know a colleague of mine is doing this with the pre-calculus class here in China. I want to do it too and will talk to him about how he does when I see him at the math meeting on Friday.

My idea is that each week the students watch a video over content. Then they come to class and discuss it in small groups. My hope then is to assign a problem that they should know how to do after watching the video to each small group. Then have one person in each group present the problem to the class.

My plan is that the first 35 minutes to 40 minutes of class the students will be in groups of four discussing what they learned from the video and solving an assigned problem. They will also work on preparing a presentation for the class for the next period. I think I will make each group mixed ability as much as I can. I will try to make the problems fairly challenging so that even the smarter students have to work at it a little bit. I am not sure where I will pull the questions from. I might modify free response questions from the AP site or use AMC questions or problem of the week questions. I am worried I will be searching around forever for the questions though. I will also look in the textbook to see what questions I could use, but I find that those are usually not challenging enough. I will have to decide on what content I am teaching and which videos I will be using before deciding on questions. I will talk to the other teacher who flips the classroom about the videos he uses. Also at the meeting we will be discussing and writing the syllabus for pre-calculus. So I will do a lot more planning when I get back from Nanjing on Friday.

My idea is after the students have discussed and prepared then they will present in the second period of class. So I am thinking that there will be 8 groups of 3 or 7 groups of 3 with one group of 4. I will ask one student from each group to present. I am thinking that the presentations should be at least 5 minutes long so we can get through several in a period. If they go over 5 minutes I think I will just cut them off at 7 minutes. So we will try to get through as many as possible and the ones we don't get to will present the next week. The following week the remaining students will present.

I will have students fill in an exit slip during the presentations. My principal suggested I implement something like this to help keep students engaged during presentations. This will be collected at the end of each day that there are presentations and will be included in the in-class category. It will be their responsibility to turn it in at the end of the day and if they don't they will get a zero. I will read the exit slips and then use the answers to plan review on the days where only a few students are presenting. Then after my guided review I will give the students a quiz.

So that means that every other week the students will be getting guided review and a quiz. The quiz will be multiple choice and be very short. I am thinking only 5 questions.

I hope to have longer multiple choice exams periodically through out the year. I don't know when I will have them. I hope to have them every month or every two months. Hopefully get through three rounds of presentations, which will take six weeks, before giving a big exam. So big test in mid October and then the midterm a month later in mid November. Then have a test at the end of December and the final at the end of January. That means two tests, midterm, and final. So that means I will have two tests and a midterm in that category. The midterm will be worth more points in that category, but there will be three exam scores in the category. Which will make so each exam is around 7% of their grade. If I weight the midterm it will mean the midterm is 10% of their grade while the other tests are around 5% each. That will even out nicely. However it will be hard fitting in more content in between some of those tests. I might need to put new questions over old material on the tests to make them long enough. This will help show students improvement and make sure they are not forgetting anything.

I did the math and this means students will be presenting about every six weeks. I will ask for volunteers in the beginning, but then I will choose the students who will present to ensure everyone presents once before beginning their second presentation.

I will grade the students presentations on this rubric. It is the rubric I use with my calculus class I just took out the focus on the AP test.

General Rubric
Detailed Rubric

I wanted to grade the students notebooks in this class, but I think I will be able to do that better in Calculus since I see them more often. I will teach them how to take notes using the cornell or two column note taking system that I will expect them to use next year. I will have to model how to watch a video and take notes in this format. I will have them take notes on the video and take notes during class presentations following the two column format. I could periodically check that they are taking notes and include this as a homework grade. I could assign the video and expect that they take notes on the video following that format. Then when they will turn in their notebooks. I won't grade the notes as closely as I do with my calculus students. I will just check that they toke notes and hand them back.

Then to prepare them for the quiz that is coming I will assign 10 problems from the book that will be due the following week.

That means every other week I am either checking notebooks really fast or I am grading short assignments from the book. I will try to make the book assignments easy to grade so I can get them done really fast. I want to make this not add too much to the amount of grading I have to do. I want to focus on my calculus class more because the students and company put a lot of pressure on getting good results.

I will use class dojo with this class as well. I will use it more during that period where students are discussing the video and preparing the presentation. This are the criteria I will be using:

Positive:

Different solution method

Critical thinking

Questions

Helps others

Negative:

Unprepared

Speaking in Chinese

Off task

If the students use a unique solution method or solve the problem using multiple methods in their presentation I will give points to the group. I will have to listen to students discussion to catch their critical thinking. If the students ask questions during presentations then they will get points and if a discussion using critical thinking skills starts after a question has been asked I will give the appropriate student points. I will watch how the groups work together and give the points to those who help their group members. This will encourage the students to work together as a group and really collaborate. I will have to switch groups every two weeks so they work with different people. 

Test Driven Education

From Teachers with a Sense of Humor.

I think this sums up a lot of my students in China. Many of them are just good at taking tests and don't really have much to offer.

The program and company I work for hires foreign teachers who teach following a Western pedagogy. So students debate, discuss, do group work, and inquiry based learning. This way the students are exposed to this type of teaching style they will experience when they go to colleges abroad. Most of my students go to America for college, but some attend schools in Canada.

However after going through the program and applying to foreign colleges some students still believe the Chinese model is better. The Chinese education puts an amazing emphasis on tests. You have to take an exam at the end of middle school to determine if you go to high school and what high school you attend. The best schools (usually with the best resources) take the highest test scores and then lower schools take the worst. Schools are numbered in order of their rank. My school is one of the top schools in the region and its name means first under heaven. Some of the top students with the top scores come to Tianyi and then only the best from that group of best becomes a part of the AP program. Although they usually leave spots open for people with low test scores that just happen to be friends with the principal or is the son of a politician  Many things are done by nepotism or by who you know in China. Then there is Wuxi #1 school which is also a part of the company that I work for that offers A-level (British curriculum) and IB curriculum. Then there is Wuxi #2, #3, etc. They all get students with lower and lower scores. Some students do not attend high school at all.

Then at the end of high school your entrance into college is not based on grades or letters of recommendation, but on a score. The students spend their entire senior year studying for an exam. This exam takes three days to take and rules are strictly enforced. You can not be one minute late for the exam and you will be turned away. Female students must have their hair tied up and many consider taking birth control so they don't have their period during the exam. Days before the exam students have been known to study while connected to an IV full of vitamins to ensure that they don't get sick. These IV's are often paid for or provided by the government. Your score on this one exam determines which school you will get into. You must have a good score to get into a good school.

So many of the students that come into this program with the idea that education should be heavily exam based. Most of their schooling before this has been based on exams and they are good at taking them. They study for the SAT, SAT II, TOEFL, and AP tests constantly. One student took days off from school to study for an SAT II subject test over American History. They had never really formally studied it and probably had some misconceptions about it from their Chinese teachers. However after a few days studying they did well on the exam. This student in particular seems to excel at everything, but not to enjoy any of it. The student can ace an exam and write pretty good papers; however sees no value in the holistic and inquisitive nature of American education. The student I am discussing will be attending Notre Dame in the fall and I hope will find something to be passionate about besides taking exams. I had several students that were similar to this one who were not passionate about anything and had no extra curriculars. They did not really participate in class, but did well on every assignment or exam. I had to write them letters of recommendation that we pretty short and didn't have much substance because all there was to them was the ability to take exams. If these students don't learn that there is more than just exams they won't have much to put on her resume when applying for jobs like the character in the cartoon.

Letters of Recommendation

I just want to share my process for writing letters of recommendation.

I try to do my best to scribble down events or things students say in the classroom. I will also try to sit down and type up notes on students after class in a google document. I take notes in a google doc so they are all in the same place and to ensure I will not lose them. I have had the computer I use at school break on several occasions and the IT guys fix for that is to wipe the computer by re-installing the operating system. I think I need to try and make sure I do this more often. I also need a more organized system because I found notes on students on tons of different papers and scribbled everywhere. So I need a more organized system.

When my students give their long presentation at the end of the year I keep a copy of my comments and feedback on the presentation. This is a major source for ideas about what to talk about.

This year I had students write an essay comparing separable differential equations and non-separable differential equations. I made sure to save this writing sample to help me write letters of recommendation. Next year I think I will have to make a copy of each students journal entry so that I have some evidence of their writing and ability to explain math concepts in English.

I will also have to write down more comments when I grade notebooks. I am considering whether I should type comments in my google doc about the students note taking skills or I should give more written feedback on the actual rubric. I give some written feedback in their actual notes to show them exactly where and what to improve. I just don't want to transfer that to the rubric. I also may be able to easily making comments about their note taking skills this time next year because I will be grading their notes way more often next year.

Since I make sure to take notes on students participation, have detailed comments from assignments, and spend a lot of time getting to know my students I am able to include very detailed examples of how they act as students. I was told by someone who used to work as a college counselor that is what made my letters great. So I need to work on gathering more assignments and observations to include in my letters. I think that might help make some of these medium tough letters to write even easier to write.

Discussions in Calculus

I stumbled across this book called Paradoxes and Sophisms in Calculus when looking through my newsfeed on facebook. One of my college math professors had shared it from Mathematical Association of America website.

It includes difficult and interesting problems in Calculus. Many of the problems look false, but are in fact true. Although some may look true, but have some minor flaw. This makes students focus on common misconceptions and help them eventually deal with them. Many of the problems were historically important to the development of calculus. I think these problems can help students have a deeper understanding of Calculus.

Some teachers commented on the post saying they thought this would just make an already difficult subject more confusing. I could see this really confusing and not correcting misconceptions of American students. However these problems are just the challenging questions my Chinese students need. Now I think they may confuse some of the lower students that struggle, but I think the majority of my students will benefit from these questions. The top students that are constantly asking questions that go beyond the course basics would extremely benefit from these questions.

I hope to buy this when I am home in the States and bring it back to China. I am thinking of using the questions as a discussion piece. Where students will do a think, pair, share over the question. I think there are plenty of questions to fit in the curriculum. I may have them write down their answer to the question after the discussion and include it their in-class grade. They will be definitely good time killers and a good resource for in-class activities. I would like to maybe add them to the homework load, but I think I already have enough to grade. I may add them as homework in the second semester where I usually don't have enough things to grade because most of second semester is spent reviewing. I might have to look over the questions to see if one would make a good journal entry and make them journal entry questions. I could also periodically put them on quizzes or tests as extra credit.

I could use them in a multitude of ways, but I think it could really enhance the curriculum. It would allow the course to go beyond the AP test, but still give them a solid foundation of calculus that will only help them when the test comes.

I got invited to be a part of the curricular development team that  supports the math teachers that work for our company this June. I will share this resource with the team and try to make this available to the other calculus teachers that work for our company. I am planning to apply to work with the curricular development team next year and hope that I can use that time to make resources like this book available to my fellow coworkers.

Continue to make a difference in a child's life!

A colleague that I work with here in China shared this with me recently. I watched the video and thought of all the amazing teachers I had as a student and all the amazing teachers I am friends with or even related to. I thought I would share this with everyone in hopes to encourage them to continue doing the amazing job they are doing. I also want to share a few of my thoughts I had while watching the video.

What struck me was that in America when working with low achieving students or students with self esteem issues you have to address their confidence first. You have to build a relationship with the student so that they will work for you. You have to make them feel like they deserve to succeed. I have known some incredible teachers who have been able to inspire some of the lowest students. I have watched them or heard of their success and I just don't know how they do it. I taught some remedial and low level math classes with students who had really given up. It was a big struggle teaching them. However I enjoyed the content and I liked those kids, but I don't know if I was able to really get through to them. After teaching Calculus for two years now I have realized what I am best at and enjoy the most is upper level mathematics. I have always enjoyed the really advanced math topics and can see those topics in creative ways. I think I am meant to teach the advanced students and the advanced classes. I think this may stem from my own math background. I was bored in math class and needed more of a challenge in middle school. No adult championed my cause, so I took it upon myself and fought to get put in a higher math class. I also think I meant to teach those upper level classes because I can see how to connect my mathematics education in college to activities in high school. I respect those who can teach and inspire those lower level students. Those students need champions to fight for them, but I think the high level students need a champion as well.

Also what I found interesting is that her co-worker had the view that teachers were there to teach and the students were there to learn. I think that perspective on education is true in China. I feel like that mentality is ground into the kids and is expected in the culture. There is more emphasis and pressure on getting a good education. The Chinese education system does not teach the whole student. They only really look at test scores and grades. We have to work closely with parents and students to help them understand that to get into American universities they need more than just test scores and grades. This pressure and focus on scores often causes students to take drastic measures. However since teachers are not teaching the whole student they do not see the warning signs of a student about to snap, hurt themselves, or hurt others. I think that the American education system tries to teach the whole student. You are a mandatory reporter and you are encouraged to build relationships. Many schools have homeroom or advising classes that focus on supporting the students and helping them succeed. The educational philosophy and approach is much more about teaching the whole student and building relationships.

I also loved the part about how she taught a math lesson wrong. The students all realized she was teaching it wrong, but didn't tell her because she was just so excited and engaging. My first thought was that will never happen in China. The top students will call you out if you are wrong. It took me a while to adjust to this and be able to handle this. I have learned to turn those moments into teaching moments. We have a discussion about why it is wrong or how it is wrong. I have them solve the solution in another way to see how it is wrong. I also use it as a moment to highlight how you will loose points on the AP test. I have also learned how to laugh it off if need be.

Before I go I just wanted to share a story about how my math teacher made a relationship with me and helped encourage me to keep studying math. I knew I wanted to be a math teacher in 8th grade and in high school I shared that with each of my math teachers. So I already wanted to be a math teacher, but the math teachers I had my last three years of high school just made me want it even more. I can think back to one moment that I will never forget. It was sophomore year in Algebra/Geometry 2. I forget what the lesson was about really. However the teacher was trying to explain a concept and some students were struggling to understand. They were asking a bunch of questions. So I raised my hand and then got called on. I explained the concept in another way that I thought made it easier to understand. I always enjoyed helping others understand math. However one student made fun of me for being a know it all. I didn't really pay attention to the comment or didn't really even register it. However after the class was over the teacher pulled me aside and said that I should not listen to that student. She made sure that this incident didn't discourage me from continuing to study math. I loved math at that point in my life and nothing was going to stop from studying it, but the fact that my teacher cared enough to make sure that was true was extremely touching. I have stayed in touch with many of my teachers. They seem happy to hear from me and whenever I talk to them I hope that one day I will be on the other side. I hope to make a difference and hear that from a student one day. I think any teacher wants that.

So the video starts by saying that everyone has been affected by a teacher or an adult in their life. So I leave you with some food for thought. I want you to think of all of the teachers and adults that have made you become the person you are today. Think about the ways they have shaped you and how that has made you become the person you are today. If you feel so moved I encourage to reach out to them and thank them for everything they have done.

I know that I learned something from each and every teacher I had. They all made a difference, but I want to
dedicate this post to the following people who have made some of the biggest impacts on my life. So this post is dedicated to: my Mom (had to start the list with mom since it is mother's day), my Dad, Granny, Bestie, Mo, my aunt Shirley, Mrs. Brooks, Mrs. Patterson, and Carol.

Inquiry based Calculus Lesson

I try to make my lessons where the students have to participate and interact with the material. I can not always achieve that, but I was able to create a lesson where the students really got to explore and make predictions in my class.

So to start off the unit on applications of derivatives I use a directed reading and thinking activity(DRTA). I have students look at the title of the chapter, the subtitle, the example problem, and more. After each I have them write on their personal whiteboards about what they think they will learn and how it relates to what we have learned so far. I keep track of predictions up front on the whiteboard. I do this at the beginning of each new chapter or each new unit. Then I read the overview at the beginning of the chapter and I pause to recognize the student that had the correct prediction. So then I begin to define extreme values and the different terms for these. Then I have students find the extreme values of x^2 when the domain is all real numbers, from [0,2], from (0,2], [0,2), and (0,2). This way students can see the function may have one extreme value in most of those intervals, but only has two extreme values in the closed interval. However I don't simply tell them this. After the students have filled out a table with their answers I put the correct answers on the whiteboard up front. Then I have the students look at the table and make up their own theorem based on the table. They have look at the data and come to a conclusion on their own. So after letting them have time to think and write on their whiteboards. I call on several students. I usually start with someone who was able to make a really general statement about the data and move on to students who got more specific. What I am looking for the students to do is to come up with the Extreme Value Theorem on their own. After we have talked about the students conclusions I tell them that they just came up with this theorem or came close to getting the theorem. I give the formal definition and explain when it will be used. I often get really good discussions about the theorem even after I have revealed. Several students had to give examples of when it worked and counter examples to some students predictions about the theorem. With almost every class it has sparked really good discussions all done in English. I also tell them in this lesson that this is what real math is. It is more than just memorizing formulas and solution processes. It is looking at data and information, then drawing your own conclusions. Then later discussing them with your fellow mathematicians.

I then move on to discuss and define local extreme values or relative extreme values. Then I put up a graph and have them copy the graph onto their personal whiteboards. I have them label the points as local maximums and local minimums on their whiteboards. Then I have them hold them up so I can see the students responses. I call on the students to go over the right answers and reveal the answers on the power point. However once this is done I have them write about what those points all have in common and how they could find those points without looking at the graph. I give them time to think and write on their whiteboards. Some students respond with that is where the first derivative is zero or where the first derivative does not exist, which is what I am looking for. However I have also had students notice that the function switches from increasing to decreasing or decreasing to increasing at those points. So the students come up with the definition of critical points and the idea behind the first derivative test on their own. Then after we are discussing that and student response I put up the formal definition of critical points.

I wish I could make all of my lessons this inquiry based, but I have yet to be inspired with how to do so. However as I teach more I think I can come up with more lesson plans like this. I have had my supervisors observe this lesson several times because it is a good sample of my teaching and I am really proud of this lesson. I got really good feedback and impressed them with this lesson. I think this lesson sums up my view on mathematics education. I want to engage students and make them think. I want them to be involved and to challenge them. This incorporates a lot of things that I learned about in a lot of my classes for my Masters.

Lesson Plan and Materials

Introducing Integral Approximation Methods

So I wanted to share with you one of my favorite lessons that I came up with a year ago and have done several times now with several Calculus classes.

So I wanted a clever way to introduce the Right Riemann Approximation Method, Left Riemann Approximation Method, and Trapezoid Approximation Method.

So I went over an example of how to use each method. I ask students to make predictions about which method comes next and about whether the approximations are an overestimate or an underestimate. I make sure they understand why the approximation is an overestimate or an underestimate. This where I ask them to connect to the graphs behavior and whether the graph is increasing or decreasing (a major point on the AP test). So after introducing the method I have the students practice one of the methods. I split the class into either groups of three or groups of four. Each person in the group picks a different method to use and so in each group you have one expert on each method. However I make the groups according to ability. I do this because each group has to approximate the integral using a different number of subintervals. The smarter students get a greater number of subintervals. So the lowest students approximate the integral using three subintervals and then the subintervals increase by one as the students ability increases.

I have the groups put up their answers in a table on the white board. Once everyone is done we talk about what patterns they see. They talk about how the numbers are increasing and decreasing. They compare the four methods and talk about which one they think is more accurate. I put up the approximations for the integral using 25, 50, 100, and 1,000 subintervals which are listed in the textbook I use. We talk about how little the numbers change the more subintervals use. I finally put up the exact answer and see how close the groups got to that answer and which number of subintervals got the closest. Analyzing this table easily leads into how if we let the number of subintervals go to infinity we get the actual integral. So in the next class I talk about integrals and show how Riemann sums lead to integrals.

Last years 11th graders groaned and moaned about using that many subintervals. Then with this years 11th graders wanted to go above what I assigned them. I gave them a certain number of subintervals and they were like no we want to use even more than that. Then I did this same activity with this years 10th graders and had an interesting response. In one of the classes the entire class wanted to pick on the very top students and make them use 100 subintervals. I talked the class down to making them use 15 subintervals. I checked with the group if it was ok and they ended up not having a problem with it. It really isn't that difficult using more it just means smaller and decimals and more numbers to keep track of. When showing examples of the different methods in one of this years 10th grade classes a really good discussion started about which method was more accurate and whether the approximation was an underestimate or overestimate. They really analyzed the function, its behavior, and the methods. They were doing most of the discussion in English which was great to hear. So I am looking forward to teaching them more and having lots of in-class discussions with them.

Materials for lesson. Please look in the notes section of the powerpoint for more details.

Women in STEM Fields in China

So first off let me say this the stereotype that Chinese students are good at math is mostly true. Most of my class are great mathematicians. The top students in my class ask some of the most difficult questions which I have learned how to field. I often have to say what a good question why don't we discuss or let me get back to you. When I went to a comedy show recently and told the comedian I taught math he just laughed. The comedian could not get over the fact that I taught math to Asians and told me I should brag about it. Well I don't know about bragging, but it is difficult and rewarding work. I am spoiled and think that these will be the best math students I ever had.

Let me tell you though this ability to excel at math occurs in both girls and boys. In China I don't think there is a gender bias if you are interested in STEM (Science Technology Engineering Mathematics) fields because the school system crams math and science down everyone's throat. You have no choice, but to be good at math and you must take tons of high level courses. However recently I was talking to a couple of girls about women in STEM fields and was intrigued with their responses.

One of my junior Calculus students asked me about careers in the STEM field. She asked me about actuarial work and about accounting. She asked me which was better for women. Apparently her mom thought accounting was more appropriate for a girl. I said that you should not let anyone's opinion about what is appropriate or not appropriate based on your gender stop you. I told if you should just do what you think you are best at and will enjoy. I told her that it was comments like those that often stop girl (or boys for that matter) for entering certain fields. Although what you have to remember is the parents have a lot of say in what their children study and my students for the most part listen. Almost all of my students parents want them to become engineers, doctors, or go into some other STEM field. Because of the one child policy in China the parents depend on their children to get good jobs that will allow their children to support them in their old age. It is a very different reality from American students that are taught to speak their mind and discuss decisions with their parents. I vividly remember my father talking to me about career options before I went to college and about everything I could do with a math degree. However I knew my passion was for teaching and I knew I was going to be a math teacher. I had the next five years planned out by myself and knew that path ended with me teaching high school math.

Then recently another incident involving girls in STEM fields came up. I overheard a student from the engineering club tell the clubs adviser that one of the students did not want to attend the STEM fair because she would be the only girl going to the event. I knew I had to talk to her and make sure that she did not let that stop her from going. I have often had to be the only girl or be in the minority at different STEM related events throughout high school and college. I thought the student should go because it was a good experience and it would look good on a college application. I talked her and she told me this wasn't the real reason it was just because she could not think of a good reason to say no. It sounded like she thought the project she had been working on with other students was not really ready to be showcased in a fair. However I was glad that I took the opportunity to at least check in and make sure that she felt comfortable as a girl interested in STEM projects.

I just thought these were interesting events related to the classroom, math, and culture that I wanted to share with you.

Math Competition Class

So since they are cutting my calculus class down to 5 periods a week to make room for a humanities class that means I would only teach 15 periods (periods are 45 mins.). My company requires that I teach at least 20 periods. So next year I will be teaching a math competition class. I will have the students solve problems from previous AMC tests to prepare for the AMC test in February. I will most likely meet each class for one period a week. So that will mean I am teaching 18 periods a week teaching math and then add the two periods a week that I am glee club adviser and I will be at 20 periods a week. I am hoping that it will be an optional course for my top students. The lower students will really struggle with the content and I will have trouble ensuring that they succeed.

I plan on having the students prepare for the AMC first semester and then prepare for the Euclid contest in March. I know students can get involved with writing the exams. I am not sure how that is done and will have to look into it. If so I will add that as part of my class and make an assignment to be write a question for the exam. I may still include the assignment of writing your own question even if I can't arrange to have the students take part in the writing of the exam.

Grade break down:
Final: 20%
Quizzes/Midterm: 30%
Presentations: 30%
Homework: 10%
In-class: 10%

For the final exam and midterm exam I will have to find some full exam of the test. I don't want to cobble together questions and want to save the sample questions for in-class use. I will have to look for an exam to use and write it after the AP test.

I was going to have one quiz a week so the students get practice on the questions. I will be using the Mathematical Association of America's archive of questions to make the quizzes. I was planning on using the sample brochure questions to make the quiz. They have about five questions that gives a good overview of the test. Here is one example. I will have to look at the scores at first, but I will have to find a curve to use. I assume that the students will struggle with these questions. I could give them the same amount of time on the quiz as they will on the test which is three minutes per question and then curve the quiz like crazy since they most likely not finish. That means about 15 minutes for the quiz which I can fit in each week. However I could give them 45 minutes to finish and might not have to curve it as much. So I could do it so I have a quiz every other week.

I was going to make this class presentation heavy since the class is just about solving questions to prepare for the exam. I thought about lecturing some myself, but since the topics that are presented are so wide and cover a lot of different things I wouldn't know where to start. I will use wolfram alpha articles to supplement students understanding. However I need something that can define advanced math topics in simple language since my students are in high school and second language learners. I might look into getting a copy of the textbook my professors use for Contemporary mathematics since that course covers a lot of random advanced math topics. So I think one week I will assign problems to pairs and let them work on it. They can discuss with me as they solve it. I will have students sign up for days they want to present at the end of calss. I will monitor how much work they get done and maybe give them another day in class to work on the problem. Then the next week I will have a quiz. Then the following week have presentations start. Then rotate between presentations and quizzes until we finish the presentations.

So 1-2 weeks spent preparing, then rotate the weeks between quiz and presentation for 7 weeks (based on having 28 students in a class). If I have a smaller class I will rotate between quizzes and presentations until presentations are done, then restart the cycle.

I was going to use the worksheet set of problems for presentations, for example this one. The questions are labeled with their difficulty at the bottom. If the class is required for all of my students then I will give the weaker students the easier problems and my stronger students the more difficult problems. If it is an optional course I may keep a record of who has presented what level of question. I will make sure students get chances at each type. If I can't find a full exam somewhere to use for my final and midterm then I will have to use these questions, but I really want to save these for presentations.

During presentations I will grade the group/individuals using the following rubrics.

General Rubric
Detailed Rubric

If I have a class of 28 I will do groups of two and I will grade each person individually doing my best going between the two different rubrics. If we make it optional and the class is really small I might have individual students present one problem. However those in the class will grade one of the individuals in the group using the rubric. I might also have some people in the audience circle one of the ten areas to improve from this article. Then I think I will make some sort of exit slip they have to fill out. I will check that they are giving good feedback and filling out all of these forms. They will get a score for the feedback that they provide and will be a part of their in-class grade.

I was not going to assign homework for this class, because I did not know what I would use for homework since I was using the worksheets for presentations. However someone from the Center for Education in Mathematics and Computing (CEMC) at the University of Waterloo spoke to my students. He mentioned a problem of the week that you can subscribe to. So I think I will assign the students the problem of the week as homework. I will require them to show any work that they can on the problems and that they don't have to get the right answer, but just try the problem. So to grade them I will quickly scan them to see that they have tried. I think it will be easy to see who tried and who didn't. For those who tried they will get a 5 and those who didn't will get a 4.

I am not sure whether I will give them the 9/10 or 11/12 question. I want to keep my grading load low so I don't want to assign two questions. I may just decide which question is more interesting or more linked with the AMC. This will be a course for 11th graders, but I worry about giving them that question because of their English level.

For the Euclid Contest/Second Semester:
I think for this there is enough questions in the archive to pull from for my midterm, final, quizzes, and presentations. Although there is a resource manual that I can buy to supplement the class and most likely use questions from it for quizzes if need be. I will have to talk to my school/company about getting that paid for by them. I could also buy it with my own money so the resource is mine to continue using. It is not too expensive.

I would still continue giving the students the problem of the week as homework in the second semester.

We are required to give the students homework over the Chinese new years vacation which is three weeks long. I may give them two problems to solve over that break both the problem of the week for the 9/10 grade and the problem of the week for the 11/12 grade.

Also the Euclid contest is in March. So I could include more Euclid stuff until the exam in March. I will have to give a full Euclid test before the actual one and will most likely have to schedule that in my own class time. I will have to talk to my principal after that. The Chinese system is still very test driven and my students may not enjoy the math unless there is a test waiting. We could work on writing and solving our own questions after the tests are over. The midterm and finals will land after the competition tests are over in the second semester. So for the midterm and final I could give them another test based off of questions from the archive even though they are done taking it. That will really effect the course.

So there is still a lot up in the air about this course. However I wanted to brain storm and get an idea for the general structure and get my resources for the course gathered together. A lot will have to be passed by my principal and the Chinese staff here at the school before anything is final. I am excited about it because I think it is going to be very interesting. I think for my stronger math students it might be the first time they are really and truly challenged. However they always seem to rise to the occasion and excel at any question I give them.

Math Notebooks

I have thought of trying to incorporate writing and notebooks for a long time in math. I thought this was a way to make math more fun. I also thought this was a way to engage students in math using another type of intelligence. I hoped to engage the eloquent writers to express their understanding of math using the medium they are most comfortable with.

This was a major focus during my work towards my masters. I have been able to include more activities in my math class where students write down responses on their individual whiteboards. I still want to include more writing assignments though, but have trouble balancing computation problems with writing problems. I still want to find a way to work more writing assignments into my curriculum, but I have focused on keeping good class notes in the students notebooks.

Although after writing this blog post a thought occurred to me about how to incorporate more writing (ah the power of writing/blogging). When I start a new chapter with the students I use a directed reading thinking activity (DRTA). I ask the students what they think the title means. I ask them what they think they will learn based on the title. Then I have them look at the picture at the beginning of the chapter and the example problem that relates to it. I ask them how the two are related. I ask them to think about how you would solve the example problem. Some students get really carried away and want to know how to solve the problem that second. I then ask them how the example problem relates to the title and what we will learn in this chapter. I then have them read the title of the first section of the chapter and ask them how it relates to the title of the chapter. Then I ask them what they think they will learn in the chapter. I have them read and write their answers on their personal whiteboards. Sometimes I do a think, pair, share to answer these questions. However I ask students to show their whiteboards to the class and call on students to share with the class. I keep a list of ideas after each round on the chalkboard at the front of the classroom. As a class we compare answers and as we get more information we can see if our predictions were right. Often to wrap up this activity I read the summary of the chapter to the class which confirms and denies the predictions made. I often will read it and say this is just what student X said. This activity helps them learn to make predictions by looking at titles, pictures, and other text. I encourage students to use the strategy in other classes. It also helps build background for the students and helps me understand how much they have studied. It also helps make connections from past material to the new material. I think to build even more background I think I will do a writing activity after the DRTA. I will have the students answer a question on their whiteboard. The question will be the big idea of the chapter. I think I will then have students respond to the question again on the weekly quiz. This way they have several drafts to use before the final and they will have another chance at answering once they have learned some of the content. I will provide feedback on the written part. They should have two quizzes before the end of the chapter so they will have two chances to practice their answer. Also if for some reason a lesson runs short and I have extra time then I can have students write and/or discuss that chapters prompt. Then once we have finished the chapter they will write their final response on the page right after that chapters notes in their notebooks. I will have to instruct them carefully to save a page or two for that chapters journal entry. Although as long as they include it somewhere in their notebook and list the journal entry in their table of contents then I guess it won't matter. It just would be nice to have all the notes on that chapter followed by the journal entry over that material.

I hope that this helps the students express themselves better on the free response questions of the AP test. They are putting more and more emphasis on explaining the answers with words. The questions are really looking for calculus with words. This has proven difficult for my students since English is their second language. I also hope if they can explain a process or concept in general using the words they can solve any problem.

I will need the AP scores to back this up. The company and the Chinese administration put a lot of pressure on getting high scores. This is doubly true in math since all of the students excel in math, so they except a 5 out of 5 for all the students.

Here are the prompts I am thinking of using. I got most of these from the assessment resource book that accompanies the calculus book by Finney, Demana, Waits, and Kennedy (3rd edition) (citing my source). Some are the writing to learn question in the books exercises. Some I have tweaked or added to.

Chapter 1 (Pre-Calculus and Review of all Types of Functions): Explain why a function that is not one-to-one does not have an inverse function.
Chapter 2 (Limits and Continuity): Explain the importance of continuity in discussing limits. Also discuss the value of determining limits as x approaches infinity. For example in economic problems, the limit of a function can give important information. If a company has an increasing revenue or profit function, what does the limit as x approaches infinity tell us?
Chapter 3 (Derivatives and Differentiability): What does the derivative represent? Use the concept of the derivative to define what it might mean for two parabolas to be parallel. Construct equations for two such parallel parabolas and graph them. Are the parabolas "everywhere equidistant", and if so, in what sense?
Chapter 4 (More complicated Derivatives): How does the chain rule help us find the derivatives of almost any function? Why is implicit differentiation useful in examining the derivatives of curves that are not functions?
Chapter 5 (Applications of Derivatives): Think about the importance of the second derivative test. In economic applications, what does the second derivative test reveal about cost functions and revenue functions? And why is that information important?
Chapter 6 (Integration): Let f be a positive continuous function that is concave up. If the trapezoidal rule is used to estimate an area between f and the x-axis, will the result be an overestimate or an underestimate? What if the midpoint rule is used instead? Explain how you know if it is an overestimate or underestimate. (often a question on the AP test)
Chapter 7 (Integration Techniques and Differential Equations): Compare and contrast non-separable differential equations with separable differential equations. Analyze the following: equations, solution method/process, how to use initial conditions, and slope fields. Use the following vocab words: general solution, particular solution, initial condition, non-separable differential equation, separable differential equation.
Chapter 8 (Applications of Integration): Describe in what situations should you use the disk method, washer method, and shell method? Think about rotating the same curve about the y-axis and the x-axis. What similarities could be seen about the volumes of these solids? Are they the same? Would you use the same method to find the volume?
Review: The students will write about the mathematicians we have studied going to a dinner party and talking about the topics we have studied so far. The assignment will let them be creative and help them review concepts. I will give them guidance on what to write about, but still the ability to choose and be creative. I am really looking forward to reading these.

I came up with the review entry after taking a ESOL class online from Willamette. The example came up in the textbook we were using for the class and was tailored to an English class. The example was student writing about author's from a certain time period at a dinner party. I read that and thought why can't it be about mathematicians. The textbook I use for calculus has some really interesting comments about the different mathematicians for example it talks about how Michel Rolle spent a long time making fun of Calculus, but then ended up having contributing a theorem (Calculus, Finney, Demana, Waits, and Kennedy, 3rd edition). So I flipped through the book to see who else was mentioned and for other paragraphs that mentioned the mathematicians contributions. I had fun actually reading my Calculus textbook and finding little gems in it that I had previously skimmed over.

The one for Chapter 7 I came up with by myself. I noticed last year that my students could not put the equations with the vocab. They could solve a separable differential equation no problem if you put it in front of them, but if you asked them to describe what they did using the above vocab then they couldn't. The differential equations questions on the AP test are loaded with the vocab I listed. I put more emphasis on the vocab this year and had students write "a comparison essay" as homework comparing non-separable differential equations and separable differential equations. I got some really creative pieces. I think it helped solidify the vocab so when they saw it on the test they knew what the question was talking about. Most of the students could describe how to solve a separable differential equation at the end of this year, but there was still a few that did not know. They were the few that struggle normally though and was not shocking to see that they did not know, but it was disappointing  Although hopefully now after reviewing what the words mean before the AP test they will do well on the free response question that is over separable differential equations. It just shows that some students need constant reminder and review for the words to sink in.

I suggest that if you want to do something similar look through the assessment resources that come with your textbook. Hopefully I can continue this if I go on to teach other subjects and out of other textbooks. I would need some ideas for writing prompts at least as a start. So hopefully other books come with alternate assessment ideas in the resource books that accompany it.

I think the dinner party review assignment can be done in any math class at any level. Most math textbooks include short biographies on mathematicians. You could also turn it into a sort of mini research project. You just have to suggest topics or people that relate or connect to your content. So I think the hard part would be to come up with suggestions for students to write about because that would take some time on your part to research the background of the math you are teaching.

I plan on reading these and checking these journal entries when I collect the notebooks. I will just have to figure out how to grade them. I think I can use part of the rubric I use for mathematical presentations and then I studied rubrics to use with ESL students in some of the ESOL classes I took online. I will have to look back through the textbook I have to find a rubric that will work. I want it to be something quick and easy to use. Any suggestions or links to rubrics would also be greatly appreciated.

Let me talk about what I have done with the notebooks so far and what I plan to do.

During my student teaching I had students keep a table of contents for their notebooks and number the pages in their notebooks. I thought this kept things organized and if their was something specific they needed to study they could easily find their notes on that subject. It also makes grading them easy because then you know which page to turn to when you are grading something specific. I continued that work with my Chinese students. Some students are good about doing this and others have not. I just have the students label what part of the textbook we are working on and use the title of the section from the textbook. I always have this displayed on the first slide of that day's powerpoint. Some students leave an empty line between each entry and others leave extra space when they start a new chapter. I also have a student that put in tiny stick notes on the pages where each new chapter starts in her notes.

I started working with my students on using an organized system for their notes during my student teaching and I tried a couple of things. This year I focused on using a Cornell notes like set up. The students put the main idea of the content or summarize what the notes on the right are about. So they could put the name of the theorem in a small column on the left of notes and in the large section on the right they could have the theorem written out. I gave students this main ideas at the beginning and then wanted students to come up with them on their own. I wanted them to either come up with them in class so they are processing information during class. The other option being they review their notes and add the main ideas in the left. I noticed once I was not giving them specific main ideas students were either not including them or giving really general main ideas. Some students would just write example, but not what it was about or why it was different. Some would just write equations and I really wanted them to use words to summarize the notes. Last year I found when I handed out study guide sheets that listed the objectives to be tested on the exam the students did not know the English words connected to the math that they knew. I could put a problem in front of them and they could solve, but if I asked them to describe what type of problem it was using words they couldn't. I hoped to solve this problem by having students include some of those words in the main idea column.

I don't want to feed them the main ideas in class either on the whiteboard or on the poweroint because I think they gain a lot more learning coming up with them on their own. I have one student who always asks how to classify what we are learning so she can label it in her main idea column, but only a few students listen to my answer. I am not sure how to help them develop more specific ideas. I wonder if I could stop and ask them what we are learning on this slide and how it relates to what we have been doing? It could be a way to check in with students and the class about their understanding and help them come up with main ideas.

Then they use the bottom five lines of the page to make notes about tips and strategies for the AP test. I often remind them or give suggestions of things to do on the AP test. I began writing these tips on the board and then expected them to listen for them. I noticed a lot of people stopped writing them down. I need to get the kids to recognize that when I say AP test that should be a cue to them to use that bottom part of the page.

I have a part on my rubric that talks about underlining, highlighting/color coding, starring, boxing, or using other techniques to make information stand out. However still were not make the big key concepts like formulas,derivative rules, vocab, or even titles stand out. So I think I need to mention that and go over that more. Although when I first started in Chapter 1 and 2 I talked about it a lot and modeled it some. However maybe I need to give more reminders. The students can do this during class or after class when they are reviewing their notes. Some of my students have really found a way to make color really effective and organize their notes.

I also wanted students to relate or connect ideas in their notes. For example how derivative equations of different trig functions relate. They could also relate to things they learned in physics. They can connect definitions to examples. I don't think I gave them enough examples of this. I think next year I will bring some of the best notebooks from this year to illustrate to the next years class what I am talking about. I think I will have to make photo copies so I will always have examples to go along with my rubric.

I just recently finished grading the notebooks and was sick of finding handouts shoved willy nilly in them. One student had English handouts in their notebook. I made a comment about that, but no one listened. I think next year I will take away one point from the format category for having loose papers or handout in their notebook. I have had some students who taped or glued my study guide handouts in their notebooks after that chapters notes, but they listed it in their table of contents and it was placed in a logical place.

Rubric

I took most of the rubric from the AVID programs rubric, but I added some of my own ideas that tailored to math and what I wanted. I also got rid of categories that I thought didn't really lend itself to math. I told the students the AVID program uses a similar rubric and those are some of the best students who are working hard to go to college. I told them they start doing this in 7th grade and they were shocked. It showed them that other people in America felt you needed these tools in college. They seem shocked and impressed at first at least.

Now let me explain how I use the rubric. I circle each numeral in each category. So this means that students could get a 5 for using plenty of space between their notes, but a 3 because they are not using abbreviations. This way the students can see how well they are doing on each skill. Then what I do is I average the score in each category. I always round up to the nearest tenth. Then I added up the averaged score in each category and then find an average for the overall score. So it is only worth 5 points in the end. This way they get a wholistic score on the notes. I have debated about averaging in each category then making the notes 20 points (5 points for each category), but I am not sure if that is too much. Next year I hope to grade the notebooks every three weeks until I have looked at notes for each chapter. So that means that in total that will be 40 points for the notebook. Although next year I am putting more weight on the scores so I should probably keep the points low. I have also thought about finding the average score between all four categories then multiplying by ten so there are no decimals. Then each chapter would be worth 50 points, but that seems too much. I will have to think it over.

The scores are often low using the rubrics since I am holding them to high expectations. I often will curve the grades or give back points in the end. I often find to when I start grading them I start mean and become more lenient. So if I go back and give more points it evens things out. However some students rise to the high expectations and take amazing notes. I recently let those students have extra credit because they took such good notes. I am not giving too many extra credit opportunities this year so I thought this would be a good chance to get some extra credit.

After I graded their notebooks the first time I gave them some written feedback to improve which described some examples. I will have to give this type of feedback more often next year. I will probably give this to the students every three weeks or every month. We finish a chapter a month usually so that will help them refine their notes for the next chapter.

My Calculus Class Next Year

I have decided I am sick of grading homework and want to give more meaningful homework next year. So I am restructuring my class. I thought of this over vacation and wanted to get it down in front of me. I will work on creating and adjusting resources for next year after the AP test is over.

Grade break down:

Final: 20%

Tests/Midterm: 30%

Quizzes: 15%

Projects: 15%

Notebooks: 10%

In-class: 10%

This year I tried doing a project that helped students prepare for a free response question as well as develop their ability to explain their reasoning using different methods. The AP test emphasizes understanding the material using a graph, algebra, tables of values, and words. I started off with a project at the beginning of the year to help students work on justifying their answers using those methods. The project covered concepts that we had recently covered, but it also included concepts they need to review because we were going to explore them deeper. I enjoyed the sort of review and preview of material this year. So I will do the project again this year. However the students procrastinated on it. So I need to break it down some more. So each week they will have a small piece of the project to work on. So they will have to find the domain, range, x-intercepts, and y-intercepts of their given function. Then support their answer using a graph, table of values, algebra, or words. The problem I had this year was I couldn't grade the homework as well as the projects. Now the projects will be my main assignment to grade. I think this may be harder to grade than the homework out of the textbook I have been giving, but I think it will help them prepare for the AP test better. They will be preparing for the free response questions that are what most of my Chinese students do poorly on. They will also be developing analytic skills that will help them in other disciplines as well as their future math skills. Some of my students really wanted to attack the problem using each and every method. So they were developing the ability to view the problem in different ways and using different information. I will collect these each week and try to get them handed back in a week. Then at the end of the project I will take time in class to have students peer review the projects. Each student in one section of the class will have a different equation to analyze. The more advanced students will be given more complex equations and the weaker students given a less complex equation. So then I will take the projects of the students who studied the same equation in the different classes and have them grade each others paper. I hope the students will learn a lot from grading the other students paper and give good feedback. I think I will then take a quick glance at the projects and the scores. However I think I will just average the scores that were assigned by the students. I will include in the in-class category points for giving good feedback. So if a student did not provide enough comments, reasons, or the scored was not at all accurate then they will loose points. I will have to outline what good feedback is and have the students take it seriously. 

Project 1: General Properties of a Function

I plan at the beginning of the year reviewing basic properties of functions: domain, range, intercepts. Then I will have them apply the knowledge they just learned: continuity and using limits to find asymptotes. Then I will have the preview/review material we will learn about in more depth later: increasing, decreasing, concave up, and concave down. The resources include handout describing the task, detailed rubrics outlining how to get the points, and a rubric grading sheet for each small piece of the project as well as the final project. 

So the students will work on this project in September and October. It may even extend into November. I will see how small I can chunck up the project. 

Project 2: Connecting a Function to its Derivatives
Sometime in November I should wrap up teaching about derivatives. Once that is over I am going to have the students start the next projcet. The students will have to find the derivative and second derivative of the function then apply them. They will have to find the local extreme values using both the first derivative test and the second derivative test. They will have to find the absolute extreme values. They will have to find the point of inflection and justify it. They will have to find where the function is concave up and concave down. I will require them to use the words to support their algebra. I will make sure they show all their algebraic work as well. These sort of questions are always in the free response section of the exam. I will give a lot of feedback about if they have justified their answer enough for the AP test on this project. I know that my students can determine where the function has these propeties, but they do not know how to support their answer using words or showing enough work. The resources include handout describing the task, detailed rubrics outlining how to get the points, and a rubric grading sheet for each small piece of the project as well as the final project.

Project 3: Fundamental Theorem of Calculus

The following semester I will have the students work on a similar project but the equation that they will be given will be the equation from the previous semester, but with an integral. They will have to use the fundamental theorem of calculus as well as what they learned in previous units. The resources include handout describing the task, detailed rubrics outlining how to get the points, and a rubric grading sheet for each small piece of the project as well as the final project.

The equations I plan to use are listed in the assignment table chart that you will find with each project. I decided to have students focus on intervals when doing the second project so that they would have to check the endpoints when looking for extreme values. The classifications column is where I labeled the equation as easy(E) or hard(H). The other numbers and letters refer to groupings. The numbers all refer to equations of the same family. The letters that accompany them match together equations that are similar and often differ by a sign change. Each student in the class will get a different equation but this chart will help keep track of the students in the other classes that have the same function. When I assign functions I will type in their names.  

The other project that will go into the project grade will be the review project I started doing last year. I started it to help students review for the test and to help students work on their English presentation skills. I think I will give them a copy of this article and discuss it before presentations begin so they can focus on how to improve their presentation skills. I will pass out a packet of old free response questions from college boards website. I give this to them a month or so in advance. I remind them to keep working on them. I also give them some time in class to work on them. Then around the end of march or beginning of April I have students pick which problem they want to present. This allows them to choose a problem they are comfortable explaining to the class. Then they have to give a ten to fifteen minute explanation of the problem to the class. This means they have to solve the problem using multiple methods, explain every step, explain how to solve on the calculator, or explain key things the college board is looking for when grading to fill the time limit. I grade them both on content as well as presentation skills. I want them to review the material, but I also want them to work on their presentation skills. I had to give presentations in several math classes and as a part of my major. They will have to present to future colleagues as well. I made a rubric to grade this by combining a lot of different rubrics I found online. Feel free to use in your classroom or adjust what I have made to use in your classroom. I will be trying out a new rubric next year. It is based on the one I used before however I added more emphasis on the math content area. So I increased the point value in the content area. I ran into a problem this year where students solved the problem using a second method, but that method was not really appropriate for the AP test. I didn't mind them including those solution methods, but they should have included information about which method to use on the test. I added that to compare methods as well. I also added a category about making sure the students explain why they can use a certain theorem. The students need to know that they can only use theorems when the function is continuous. The AP test often has questions where you can't use a certain theorem and must explain that you can't because the function is not continuous. So I hope by including this category it will prepare them for those type of questions. 

General Rubric
Detailed Rubric

This year I am going to get the audience more involved. Three students will grade the students using this rubric. Then three other students will grade the solution of the problem using the AP rubric. I will also require students to ask one question during the course of the presentations. I wanted to make it more than one, but was not sure if I made it more that there would be time for all the students. I was going to check this off and put it in the grade book. 

I will look at their feedback and check that they are doing it to give them a feedback grade that will go in the in-class category. I will check off who has asked a question and give them points for that in the in-class category. 

I will average the scores given by the students using the AP rubric and put that score in the homework category. 

I have also considered averaging the students scores on the presentations with mine. 

So next year I will be teaching 5 periods of class a week instead of 6. So two days a week I will see a class for 90 minutes and one day a week I will see the class for 45 minutes. So during the 90 minutes I will teach one section of the textbook. So I will teach two sections of the textbook a week. Then on the day I have the students for 45 mins I will have the students take a short quiz. Then once they are done with the quiz they can work on their project. Depending on the schedule I will probably have them turn that weeks part of the project in at the end of the period. 

So instead of homework I will have a weekly quiz. I am think the quiz will have five multiple choice questions on it and the students will have 15 minutes to answer the questions. That is the amount of time they will get on the AP test. The remaining class time I will check in with students on their projects and give them time to work on it. If a student is done I can let them get started on the next part. I will have to have the handouts ready far in advance though. I am planning to use AP questions on the quiz. 

I made a rubric to grade students notes and taught the students how to take good notes this year. I wanted to grade and collect the notebooks a lot, but did not have time since I was spending it grading everything else. I have also learned that I can only grade one class set of notebooks in one week. So I think each week I will collect the notebooks of one class. That way I am grading and giving feedback on their notes every three weeks. I will talk about how I grade notebooks and what I have discovered so far this year in another post.

For the in-class portion of the grade I will grade any activities that are done in class. However next year I am planning on using an app called class dojo to monitor student behavior and participation. One of my colleagues here in China uses it with his classes. He gives points for things like critical thinking and takes away points for speaking Chinese. At the end of the six weeks he looks at the summary points and gives them the grade according to that. I have to decide what criteria I will be looking for. I will also have to make sure I can use the app in the classroom since their is no wifi and the internet. I know it can be done, but I will have to test it out before I use it next year.

Here is my criteria:
Positive:
Different solution method
Critical thinking
Questions
Helps others

Negative:
Unprepared
Speaking in Chinese
Off task

I want to make these categories meaningful. I am try to think about what behavior in class makes a good math student. I would love more suggestions of categories to add.

I will try to keep you updated on how using class dojo impacts my classroom.