Visualization 

 Challenge & Selection   The challenge for that I used Visualization , or Visual Thinking , was to achieve a deep understanding   of advanced analysis for approximately 200 students in a university course . I was teaching a huge   tutorial lesson for them in a German university . The actual lectures were done by a professor who   was very abstract and explained math very high level , because from his viewpoint , the intuitive part   about it was anyways clear . He did not notice that we were facing a bunch of undergrads who had   learned at school and in previous university courses that math mainly was about juggling with   formulas . 

 My aim was that as many students as possible gain a deeper understanding of mathematics plus to   enable them to travel through these abstract worlds laying in front of them with as less formula   luggage as possible . But I only had 90 minutes each week to achieve this . 

 That is why I decided for the tool of visualization . For me , even if math gets more abstract , it seems   like a big landscape that one can discover . And instead of explaining to the students how to move   from A to B in this landscape , I tried to give them a mental map that enables them to go from any   point they want to any other point they want . 

 For this , I decided to break down connections they seemed to find difficult into pictures , sketches ,   that they can remember quickly . I used mathlab to generate some of these pictures and I drew many   pictures on the board – very classically with several chalk colors – and tried to connect these pictures   for them with the formulas and theorems they described . 

 Application   Whenever I had to explain another theorem or formula to the students , I asked them why somebody   invented it . I tried to break things down to easy connections they could see in the world , like corn   fields in the wind to describe vector fields . I encouraged them to forget about the severness they had   learned to apply for their studies and to go back to their childhood and try to imagine . I wanted them   to feel what we are doing . 

 If I was explaining to them what different differential operators were deduced from , I had prepared   pictures within mathematica that explained the connection easily . If it was about less complex topics ,   I plotted the easiest example I could think of to the board and tried to make the students come up   with the right idea themselves by asking them questions . Whenever they got a step closer , I plotted   that part of the answer to the board . If plotting was too static , I tried to use my body language to   make them see the context . 

 A nice example of this was the proof of the hairy ball theorem . It says that – if the world was a ball –   there is always at least one point on the world where there is no wind blowing . For that , I visualized   different proofs and methods for proving , e.g. we cut balls open into two halves , then pressed them   flat to gain something like a round plate and then we walked around that plate . This one worked out   so well that it was even possible to explain it to school students who had just turned 12 years old . 

 This kind of visualization motivated most of the students to learn the formulas belonging to the   picture as well . And even more , they did n’t have to learn them by heart anymore , because they could   remember the visualizations of each formula element and the full picture and could puzzle it   together . 

 Insight & Approach   What I have learned from conducting this way of teaching is that it is very important to have a close   look especially at the weaker students whether they get the connection between the visualizations in   my class and the abstract formulas in the professor ’s class . But I also learned that it helped the   average students to understand the content of the abstract class much better and that it encouraged   the strong students to learn even more , to take their phantasy into their studies more than before   and to evolve to – in total – much stronger math students . 

 I also learned that this way of teaching brought me much closer to the students then teaching them   in formulas . Enabling them to “ see ” what they are actually doing seemed to take the fear away to ask   me questions . The nicest moments were moments when students with questions got up in this huge   hall , walked to the board and towards me , asked me for the chalk and dared to plot their question to   the board . Even if they could not express themselves so quickly in the formula language , the fact that   we communicated with visualization enabled them to express their ideas and uncertainty much   faster . It also helped me to understand with which parts they actually had trouble understanding and   what I could explain better or more closely to enable them to get the full picture and a much better   insight . 

 Meanwhile I am working in economics and not at university anymore . Also here , I use the tool of   visualization frequently to solve more complex problems and to enable a group of people to gain a   common understanding of what it is we are actually talking about . 

 If I should ever teach high level math again , I would always choose the tool of visualization again to   explain the connections students have to discover . I would even go so far that if math was only   teached in this way , most students who failed in math at school would not have had any trouble and   I can only advice this tool to anyone who has to explain more abstract content to a group of people .   It prevents frustration of not understanding in the audience , helps people to remember and makes   talking about actual understanding deviations much easier . 


EMAIL: 
ID_NUM: 
NAME_STUDENT: 
PHONE_NUM: 
STREET_ADDRESS: 
URL_PERSONAL: 
USERNAME: 

### Task: You are a PII (Personal Identity Information) Detection. Given a essays written by students, you have to identify ONLY the following:
NAME_STUDENT - The full or partial name of a student that is not necessarily the author of the essay. This excludes instructors, authors, and other person names.
EMAIL - A student’s email address.
USERNAME - A student's username on any platform.
ID_NUM - A number or sequence of characters that could be used to identify a student, such as a student ID or a social security number.
PHONE_NUM - A phone number associated with a student.
URL_PERSONAL - A URL that might be used to identify a student.
STREET_ADDRESS - A full or partial street address that is associated with the student, such as their home address.
### Input: Visualization 

 Challenge & Selection   The challenge for that I used Visualization , or Visual Thinking , was to achieve a deep understanding   of advanced analysis for approximately 200 students in a university course . I was teaching a huge   tutorial lesson for them in a German university . The actual lectures were done by a professor who   was very abstract and explained math very high level , because from his viewpoint , the intuitive part   about it was anyways clear . He did not notice that we were facing a bunch of undergrads who had   learned at school and in previous university courses that math mainly was about juggling with   formulas . 

 My aim was that as many students as possible gain a deeper understanding of mathematics plus to   enable them to travel through these abstract worlds laying in front of them with as less formula   luggage as possible . But I only had 90 minutes each week to achieve this . 

 That is why I decided for the tool of visualization . For me , even if math gets more abstract , it seems   like a big landscape that one can discover . And instead of explaining to the students how to move   from A to B in this landscape , I tried to give them a mental map that enables them to go from any   point they want to any other point they want . 

 For this , I decided to break down connections they seemed to find difficult into pictures , sketches ,   that they can remember quickly . I used mathlab to generate some of these pictures and I drew many   pictures on the board – very classically with several chalk colors – and tried to connect these pictures   for them with the formulas and theorems they described . 

 Application   Whenever I had to explain another theorem or formula to the students , I asked them why somebody   invented it . I tried to break things down to easy connections they could see in the world , like corn   fields in the wind to describe vector fields . I encouraged them to forget about the severness they had   learned to apply for their studies and to go back to their childhood and try to imagine . I wanted them   to feel what we are doing . 

 If I was explaining to them what different differential operators were deduced from , I had prepared   pictures within mathematica that explained the connection easily . If it was about less complex topics ,   I plotted the easiest example I could think of to the board and tried to make the students come up   with the right idea themselves by asking them questions . Whenever they got a step closer , I plotted   that part of the answer to the board . If plotting was too static , I tried to use my body language to   make them see the context . 

 A nice example of this was the proof of the hairy ball theorem . It says that – if the world was a ball –   there is always at least one point on the world where there is no wind blowing . For that , I visualized   different proofs and methods for proving , e.g. we cut balls open into two halves , then pressed them   flat to gain something like a round plate and then we walked around that plate . This one worked out   so well that it was even possible to explain it to school students who had just turned 12 years old . 

 This kind of visualization motivated most of the students to learn the formulas belonging to the   picture as well . And even more , they did n’t have to learn them by heart anymore , because they could   remember the visualizations of each formula element and the full picture and could puzzle it   together . 

 Insight & Approach   What I have learned from conducting this way of teaching is that it is very important to have a close   look especially at the weaker students whether they get the connection between the visualizations in   my class and the abstract formulas in the professor ’s class . But I also learned that it helped the   average students to understand the content of the abstract class much better and that it encouraged   the strong students to learn even more , to take their phantasy into their studies more than before   and to evolve to – in total – much stronger math students . 

 I also learned that this way of teaching brought me much closer to the students then teaching them   in formulas . Enabling them to “ see ” what they are actually doing seemed to take the fear away to ask   me questions . The nicest moments were moments when students with questions got up in this huge   hall , walked to the board and towards me , asked me for the chalk and dared to plot their question to   the board . Even if they could not express themselves so quickly in the formula language , the fact that   we communicated with visualization enabled them to express their ideas and uncertainty much   faster . It also helped me to understand with which parts they actually had trouble understanding and   what I could explain better or more closely to enable them to get the full picture and a much better   insight . 

 Meanwhile I am working in economics and not at university anymore . Also here , I use the tool of   visualization frequently to solve more complex problems and to enable a group of people to gain a   common understanding of what it is we are actually talking about . 

 If I should ever teach high level math again , I would always choose the tool of visualization again to   explain the connections students have to discover . I would even go so far that if math was only   teached in this way , most students who failed in math at school would not have had any trouble and   I can only advice this tool to anyone who has to explain more abstract content to a group of people .   It prevents frustration of not understanding in the audience , helps people to remember and makes   talking about actual understanding deviations much easier .
### Output: 