Test Driven Education

From Teachers with a Sense of Humor.

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

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

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

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

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

Letters of Recommendation

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

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

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

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

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

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

Discussions in Calculus

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Inquiry based Calculus Lesson

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

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

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

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

Lesson Plan and Materials

Introducing Integral Approximation Methods

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

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

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

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

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

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

Women in STEM Fields in China

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

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

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

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

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