Teaching at Concordia International School Shanghai
I taught two weeks of Honors Pre-Calculus to three classes of students as well as two weeks of Aerospace to one class of students, as the substitute teacher for Dr. Tong, one of my most important mentors. While teaching aerospace did not amount to anything meaningful to myself because it was mainly supervising their project time, I learned several valuable lessons when teaching the Honors Pre-Calculus students. I taught Polar Equations and Limits, as well as a brief lesson on the Fundamental Theorem of Calculus. I also assisted with Mr. Klammer's AP Physics and AP Computer Science classes, by giving the occasional guest lecture and helping with grading assignments and exams.
This was one of the happiest times of my life and was my first encounter to the many benefits of a synergetic environment of learning and teaching. I found that the more I taught the material, the more I was able to reinforce my own understanding of related topics, which then enabled me to better share my understanding with my students and my peers. This teaching experience became one of the most important lessons that contributed to my success at Cornell University.
I had written some notes in 2018 about some lessons I learned when it comes to teaching in STEM, and I would like to share that with you here.
Misunderstanding Course Standards
The first realization that occurred to me during the initial phase of teaching these students was the disparity between their personal standards and the standards of Dr. Peter Tong. Specifically, this disparity was most clearly seen through the grading of their tests. Rather than having points subtracted for not understanding the concepts that the tests assess, Dr. Tong's students were receiving low scores because they do not meet the standards of the answers that are expected. For example, students would receive only partial credit for a correct ellipse graph because they were missing axes labels, graph scale and coordinate labels for vertices, foci and the directrix.
To remedy this issue and help increase the scores of the students, I drafted a short, one page document detailing the expected standards of Honors Pre-Calculus for Dr. Tong's perusal and approval. These standards specified what format answers should be in, and what is expected for a full point answer on a test or quiz. Upon Dr. Tong's approval, I distributed this Guide of Standards to the students via e-mail. I heard the students were very grateful and I observed that a number of them got higher quiz and test scores after the standards were made clearer.
Lecturing Techniques - Applications
For the Honors Pre-Calculus lectures, I sought to review the previous day's lectures by applying them to real-life scenarios. I have all too often heard students challenge teachers on how the coursework they're learning will be beneficial or even applicable to the real world, so I aimed to silence these challenges before even a word was uttered. For polar equations, I decided to challenge them with a tank gunnery problem, something from my national service that would garner their interest. The range and r value of the polar coordinate would be solved by a system of equations (one for projectile x position and one for projectile y position), and the angle of fire would be determined by a simple Pythagorean problem (one leg of the triangle given as the distance between own tank and spotter which sights the projectile landing in front, the hypotenuse of the triangle as the range of the projectile). To reinforce the need for the students to always consider units and ensure that they write units in for quizzes, tests and exams, I gave the spotter's distance from own tank in meters, but I demanded the r value in km and the angle in Swiss military mils system (6400 mils in 2pi radians). I gave the fastest problem solver a small chocolate bar or snicker as a reward.
I have learned that it is prudent to involve real world scenarios in review problems. Students would understand the mathematics behind the concept after finishing the homework and be prepared to apply it. Not only would the students see the value and application of the concept they just encountered, but they would also be prompted to use their increased math knowledge in conjunction with critical thinking skills to solve a problem encountered in the real world.
Lecturing Techniques - Visualization
In my experience learning polynomial, ellipsis and hyperbola graphs, I had major difficulty in recognizing the type of graph from looking at an equation. I was unable to determine the characteristics of the graph simply by analyzing the different parts of an equation. I sought to prevent this issue from happening with Honors Pre-Calculus students by utilizing Desmos and its unique oscillating slider tool to demonstrate how graphs change as certain numbers within their equations change. I used this to maximum effect during the polar graphing lecture, as I was able to demonstrate with 4 variables of a, b, c, and d in different parts of the polar equation how each component alters the characteristics of a limacon from inner loop, to cardioid, to dimpled, to imperfect circle. I was able to demonstrate the unique characteristics of circles written in the form of polar graphs, as well as explain why rose curves follow an odd-even rule of determining the number of petals. I later used the same technique to explain how eccentricity changes the characteristics of a polar graph. By allowing them to visually identify the gradual change in the graph in conjunction with the manipulation of a few numbers, I believe I have successfully taught my students the different components that changes a graph's appearance without it being a set of rules and variables they will forget by the next week.
Lecturing Techniques - 3 Phase Problem Introduction
This next technique is one I gleaned from Mr. Joel Klammer, the chief physics teacher at Concordia. Some university students often complain that the lectures given by the professor are off-topic or do not assist with the completing the assigned work at all. This causes students to lose interest in what the professor has to offer and increases the temptation to default on classes. A 3 phase problem introduction provides a comprehensive and gradual process in introducing students to applying concepts they learned in the lecture to problem sets given as assignments and assessments.
The first phase of problem introduction involves the teacher introducing a standard difficulty problem that the class will encounter in assigned work and assessments. The teacher will employ "Deliberate Action" by spelling out the steps he/she is doing to solve the problem. By doing this, the teacher is able to show students how to approach a problem and what strategies to employ to get to the answer.
The second phase of problem introduction involves organizing the students into small groups that will tackle a similar problem with additional challenge or complexity. Not only does this encourage teamwork, communication and critical thinking in a group setting, it allows for two distinct effects to occur amongst the small groups. Firstly, the students who have a good grasp of the material are given a chance to reinforce their skills by teaching it to those having difficulty. Secondly, the students having difficulty with the problem are given a secondary perspective on how to tackle the problem. Occasionally, the teacher's approach and problem solving strategy may not make sense to some students or be "not the preferred style" of other students. The secondary perspective offered by their peers can help these students by providing an alternative strategy or a secondary explanation of the approach.
The third phase of problem introduction is simply giving the students a problem to solve individually. This either ensures that the students are able to solve the introduced problems individually by applying their newly learned concepts, or signals to the students that they need to invest extra time and effort outside of class time to familiarize themselves with the material.
Though this strategy takes considerable time to execute, I have found it to be the most reliable method to teach problem solving strategies to the students. In Honors Pre-Calculus, the first phase of problem introduction helped me identify certain heuristics that I possess in problem solving that they were never taught. For example, I am able to determine sine and cosine values of 30-, 45-, and 60-degree angles using the left-hand technique. According to some students (judging from gasps and dumbstruck faces), this technique was extremely useful and eliminated their difficulty in memorizing the correct values for the correct angles.
Grading
One of the unavoidable experiences of being a teaching assistant is the task of grading papers. While initially it may seem like an arduous chore, grading papers allows students to understand how best to present their answers on their own assignments and assessments in order to facilitate grading and evaluation. This minimizes miscommunication between teachers and students and lessens the chances that a student must consult a teacher for the reason behind grades given for a esoteric answer. Below are some techniques to facilitate effective, reliable and objective grading.
Grading Tactics - Parallels between Test Taking and Test Grading
Grading a test and taking a test have uncanny similarities in tactics. When a student takes a test, he/she should skip problems that are too difficult, too time consuming or requires more time to determine a problem solving approach. The student should return to these questions later to ensure that he/she has enough time to complete the test. Grading a test shares the same tactics. When a particular student answer is too complex, too unorthodox or too esoteric to evaluate quickly, the grader should skip grading that answer and leave it for the end. This allows the grader to "clear out" all the simply correct and simply wrong answers given by the rest of the class, and gives the opportunity to address the more complex answers by breaking them down into a set of standard common mistakes that can be graded fairly across the set of, unusual student solutions, shall we say.
Grading Tactics - Assembly Line Style Grading
In order to ensure that the points taken off for a particular mistake is constant across all tests, it is better to grade all the students on that particular problem before moving on to the next. By grading one problem or one page of problems at a time, teachers can identify commonly committed mistakes and clarify the correct problem solving strategy to the class for review later on. The downside to this "assembly line" approach is that students may receive points off for repeating commonly committed mistakes in multiple questions of the test. If "error carried forward" is to be considered, the questions requiring the same approach should all be graded together (for example, a question with multiple parts that depend on each other should be graded together instead of 1 part at a time).
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