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
About Me
- DominiqueL
- I love to teach and I love math. Teaching has always been a passion since I was in 5th grade. I gained a love of math later in eighth grade. I have been told that I always have a smile on my face and a song in my heart which is the best description of me.
Friday, May 10, 2013
Thursday, May 9, 2013
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.
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.
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.
Saturday, April 6, 2013
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.
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.
Friday, March 8, 2013
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.
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.
Tuesday, March 5, 2013
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.
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.
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.
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.
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 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.
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.
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.
Thursday, December 1, 2011
Campus Photos
So I thought you guys would like to see more of the campus. So here are some photos from around the campus.
Here is the view of the school when you walk in the front gate. They are some great panorama photos.
The brick building on the far left is the buiding I teach in and where the AP center is.
The building in the middle with the halo ring is the information building. It has the library, auditorium, and the music classes are held there. Glee club meets on the 6th floor in the choir room. The IT department is also located in this building. I am told it is mostly empty though. I have not really explored though. I want to see the library though. I am told it is pretty empty and does not have many research books. None of the students use it anyways.
The brick buildings at the right are more classroom buildings I believe.
That white building on the far right I believe is ours and empty. The principal says they are going to tear it down and that will be our new building. Right now the TAG middle school kids are on the first and second floor of our building. We have classes and offices on the third and fourth floor of our building. So we have been promised our own building to ourselves.
This is another statue on campus. It is located near the building I teach in. This is called a ding and many were made in Ancient China. The Chinese staff were surprised that I knew the name of the statue. However I studied these and other types of common artifacts from Ancient China in the art history class I took that focused on the art of China. When I saw it I fell in love with it and was like oh look it is a ding. I got strange looks I think. I had to wait to take pictures of it though. The bamboo was growing everywhere and at the beginning of school they were on trimming it all back.
Here are the roses that grow in the garden area next to the building I teach in. At first when they were not in bloom they were quite ugly. However the first time I saw them in bloom it was calming. We usually walk through this garden area on our leisurely stroll back from the cafeteria after lunch. I hope the photo bugs are ok with my close up of the rose. I felt like my Dad or sister would have done it better or spent more time getting the right picture. I just took a quick shot.
Now here is a panoramic view of the lake from inside the pagoda. You can see the information building with the halo on top. You can see the dorm buildings across the way. The brick buildings on the far right of the second photo are the staff dorms.
So that is more views and sites from around Tianyi campus. I hope you all enjoyed seeing where I work. I am told the campus is even more beautiful in Spring. So I will have to post more photos when spring comes.
Here is the view of the school when you walk in the front gate. They are some great panorama photos.
The building in the middle with the halo ring is the information building. It has the library, auditorium, and the music classes are held there. Glee club meets on the 6th floor in the choir room. The IT department is also located in this building. I am told it is mostly empty though. I have not really explored though. I want to see the library though. I am told it is pretty empty and does not have many research books. None of the students use it anyways.
The brick buildings at the right are more classroom buildings I believe.
That white building on the far right I believe is ours and empty. The principal says they are going to tear it down and that will be our new building. Right now the TAG middle school kids are on the first and second floor of our building. We have classes and offices on the third and fourth floor of our building. So we have been promised our own building to ourselves.
This golden apple is a statue that you couldn't see in the panorama. I think the golden apple is supposed to represent wisdom and the color gold is supposed to be one of the best. You can see the fountain behind the golden apple. Then I included a close up of the fountain. It is only turned on one day of the year and that is open house day. I didn't get a picture of it because I was busy prepping for the big open house day lesson.
I will try to get a picture of it next year.
So this is the bridge around the lake on the far side of the lake behind the information building. This is where I took my school photo that you may have already seen. You see some of the lake and the area around it. The brick buildings in the back are teacher and staff dorms. The tall buildings in the very back are apartment buildings in the apartment complex I live in.
So these are panorama shots of the school around the lake. You can see the information building in this picture in the forefront of this picture on the very left. Across the lake is the cafeteria and the building I work in. The students dorms are also in the very back of this photo on the far side of the lake.
This is the pagoda on the side of the lake. They started building this at the beginning of the year. It was done sometime in October. We came back from our one week holiday and it was suddenly done. It didn't seem like it was going to get done that quickly. I think it is a very pretty addition though.
Now here is a panoramic view of the lake from inside the pagoda. You can see the information building with the halo on top. You can see the dorm buildings across the way. The brick buildings on the far right of the second photo are the staff dorms.
So that is more views and sites from around Tianyi campus. I hope you all enjoyed seeing where I work. I am told the campus is even more beautiful in Spring. So I will have to post more photos when spring comes.
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