Comment on Freewriting

By writing on a set period of time freely and continuously, free writing allows me to break all the rules and thinks outside the box. Sometimes, I am amazed at what I have written down, which normally I would not think about or would not dare to try. My conscious mind always guides me “This is not right. That is not the way to go”. It shoots down many ideas before new thoughts being fully developed. Especially during the brain storming stage, I am stuck with ideas I don’t like too much. Hence, free writing is a very useful tool for me to use. However, free writing will inevitably produce a lot of useless information. Sometimes, after pages of writing, I realized I cannot use any of them. Therefore, we need to balance between these two sides of free writing in order to make good use of it.

Division by Zero

Math is The art of rigorous prove The practice of predictability Zero is The origin of undefined outcomes The start of abstraction Ancient people Have one finger to represent 1 Have nine fingers to represent 9 Yet No finger to represent 0 We cannot show 0 by any object in real world Furthermore Any number times 0 The result is an ambiguous 0 Therefore Division by 0 Any result is possible Zero is The defiance of math’s predictability Result? Math cannot let zero do whatever it wants Hence the rule No division by zero

Micro Teaching Reflection

For our teaching of the Pythagorean Theorem, I feel it goes on well over all. It also needs improvement on many areas. We start with a real life application (ladder problem) and ask the students to think about how to solve it. Most feedbacks show that they like the intro/bridge. Some students hope the topic to be more clear and suggested writing the topic on the board after the introduction. Good idea! Then we use visual prop to prove the theorem geometrically. Students, especially visual learners, like it this way because they can see why this theorem is true by themselves. Some students feel the proving part is a little bit rushed. They wish it was discussed more thoroughly. Most students like the game we provided. But some students do not have a chance to fully participate in because we only have one sheet for each group. Next time we need to provide more. Some students notice the discrepancy of policies among three of us: allow using the ruler to measure the triangle or not. We need more communication among us beforehand. We wrap up the teaching with the solution to the ladder problem and summary. Students like the way we finish it. However, they do mention that terms like “orthogonal” needs some explanations.

MicroTeaching: Pythagoras Theorem

Bridge/Intro: Real life problem related to Pythagoras Theorem: Ladder problem For a ladder to reach second floor window(13ft high) and away from the wall(3ft).What's the length of the ladder? Teaching Object: teach them the proof of the theorem by geometric proof with props Learning Object: the theorem, the principle and the usage of this theorem Pre-Test: Start by asking if students are aware of the content about pythagorean theorem Yes? allow them to come up and write down what they know No? simple! let's start with our lecture Participatory Activities: divide students into 3 groups and hand out the game sheet. Each group of students will work together to solve while teachers will join them to help out when needed. Post-Test: might take the unfinished game home as homework. Solve the ladder problem together Summary: briefing of this theorem and introducing new topics for students to think about which makes connection to the next class RUN-DOWN Greeting | Problem for students to think(Rong) | Learning objective(Paul) | Pretest(Paul) | Proof of Theorem(Stan) | Game(Stan) | Solve the Problem together(Rong) | Summary(Paul)

“What-If-Not” strategy

“What-If-Not” strategy from the book “The Art of Problem Posing” is a very good math method in many ways. First, it opens up the door of opportunities for people to investigate many different paths to find some more useful corollaries or attributes. Sometime, it can even lead to another important theorem. Second, by thoroughly investigating many possible variations, we can get a deeper insight as to what this theorem can do, where this theorem can be applied and how this theorem is used. These ideas will insure that the theorem can reach its full potential and be applied in as many areas as it can. However, the major problem for this method is that by creating so many different cases, most of them will be proved to be dead end and useless. The procedure can be very time consuming, although prudential choice of some related attributes might significantly reduce the number of different attributes created and thus reduce the investigation time. For our micro-teaching, we will be working on Pythagorean Theorem, one of the most well known and widely used theorems. After presenting a^2 + b^2=c^2, we will ask the students to link the formula to the graph. We will ask students to think what if the triangle is not a right triangle. The angle facing c side is smaller or bigger than the right angle. How does the formula change intuitively just by looking at the graph? By investigating these variations, students will have a better understanding that a^2 + b^2=c^2 is not only a necessary condition of the right triangle, but also a sufficient condition of the right triangle. By checking the 3 sides of a triangle fit the formula, we can be sure the angle c side faces is right angle. This seemly easy understanding has extensive use to check or to get a right angle from an existing line in all civil engineering areas.

10 Questions/Comments on “The Art of Problem Posing”:

1. Do you think it is a good idea to teach students problem posing skills at all? 2. If yes to question 1, at what stage of teaching do you think we can encourage students to pose questions? 3. Which one do you think it is better for students: limit and guide students the kinds of questions they pose, or let students ask any questions they come up with? 4. For x^2+ y^2=z^2, you have actually come up with many questions I did not expect, do you think it is a good idea to raise so many questions to ask the students? 5. Do you feel that too much broad questions will distract students instead of keeping them focused? 6. Which types of questions do you think teachers should focus on to present to students? 7. At the beginning stage of learning of a topic, I would let students focus more on internal exploration instead of external exploration, do you agree with my idea? 8. I feel approximation instead of exact answer is a very important technique which most high school students lack. What types of questions are you going to use to introduce approximation ideas to students? 9. Comment: Historical questions are actually good group projects for students. I feel students should explore the history of math and math stories by themselves. 10. Comment: You have offered some very effective problem posing techniques. I am especially impressed with the idea of “challenging the given”.

As a Good Teacher,or a Bad Teacher

Dear teacher, I have been in your math class from G8 to G12. Now I have graduated from high school for 2 years and am taking some math classes in UBC computer science department. I have learned tons of useful skills from your class and understood the solid math ideas. Now I am doing so well in the math class. I am one of the top students in my class right now. While most other students are struggling through the course, I am feeling everything is familiar. It sounds to me that you have gave me those ideas beforehand. For example, in class of linear algebra, most students cannot grab the idea why those matrix are coming from. For me, it is just so natural because you have introduced to us the ideas of matrix from our G12 algebra class. While other students were still wondering why they need to learn linear algebra, I have already known that linear algebra is an essential part of computer graphic. I need to thank you for your wonderful job in teaching not only what math is, but also how math is used in real world. Danny A UBC Undergraduate Student Hi, I must say I have been through a hard time to learn math. I hope math is easier. A student My Hope: I not only teach students math knowledge and skills, but also motivate the students. I let students understand how math is used in the real world. My Fear: The students might be frighten when they realize the real application of math is so much complicated.

Video Review

In the video, the teacher used some simple yet effective techniques for an effective teaching. Interactive: By tapping at the wall and asking the students what number it is, he successfully grab the attention of every student in the class and let all students participate in the activity. Through this interactive activity, the students will get the feedback from the teacher right away. If a student makes a mistake somewhere, he can adjust his understanding and catch up to the rest. Repetitive: This activity lasted for a while during the class, which builds up the fluency for the students. As students are still at young ages, they need the repetition to memorize the techniques. Sometimes the teacher think students have got the idea and quickly move onto another topic, just because a few students have answered the questions correctly. But in reality, many students have not understood what is going on. Even those students who answered the questions correctly, they have not built up the fluency to do it quickly. Therefore, repetition is an effective technique for the kids to build up their math skills. Visualization: When introducing the variable x, he drew a simple diagram to visualize why x+3-2=x+1. As a visual learner myself, I like to see how things change right before my eyes. As the old saying says “seeing is believing”, most students, including me, are convinced.