Unit Plan / Lesson Plan / Project Plan

Doing, assessing and designing a math project

Group Member: Darshan, Prem, Rong The project we have chosen from the “Mathematical Magic” book is a variation of the Fibonacci sequence. As we all know, Fibonacci sequence has a lot of amazing properties. This game is a generalization of one of Fibonacci sequence’s property. In this game, some seemly unrelated numbers are actually related. By using mathematical analysis, we can find out the reason behind the magic. Students will be surprised that we can use just one number to predict the sum of the ten numbers. Students will benefit from the game by witness the power of math. However, this game has a lot of addition calculation, which we think is both strength and a weakness. We say it is strength because this gives students a lot of practice in doing addition fast and accurately. The weakness is that if students make only one small mistake, the result will not match the magician’s prediction. Therefore, we think it might be better for the students to play in groups of two. Each student picks one number to play. Luckily, this game can be modified for two students to play at the same time as it needs two numbers to start playing. When they are doing the calculation, one can do the addition and the other can check the calculation result. This will reduce the calculation mistakes to minimum and ensure the game to play smoothly. Another problem to this game is that the calculation time and accuracy will highly depend on the students’ math calculation skills. Therefore, for lower grade or less competent students, it might be better to allow them to use calculators if the game takes too long. The “Read Your Mind” Game The project we have created is for Grade 9 students. A major problem for many grade 9 students is that they regard math as a hard and boring subject. The project is intended to arouse students’ interest in learning math. Math can be challenging. Math can also be fun. In order to understand the theory behind the game, they have to understand polynomial addition, which is closely related to the topic of Ch 5 in “Math Links 9”. Therefore, this game is also a good activity for students to get a deeper understanding of polynomial addition. The detailed activities are described step by step on the poster. We create this game by ourselves. We also reference to the “Math Links 9” textbook and “Mathematical Magic”. The length of the game will take about 10-15 minutes. Students are required to create a key from the chosen number. During the game, the students are needed to take out their pens and paper. They need follow the magician’s guide to produce a key from each chosen secret number. Because this is supposed to be a fun activity for the students to participate, the students will not be marked on their performance. However, for those students who are actively participating in the game and solving the mystery behind it, some bonus marks will be given as a reward.

Solving the Black Friday Problem

My Two Most Memorable Practicum Experience

Biog 11: I did my short practicum at Langley Secondary School. The teachers were very friendly and helpful even if they were not my sponsor teachers. I had a chance to listen to various classes such as drama, home economy, etc. On class that I was especially impressed with was Biology 11. I thought it might be a boring class because students just needed to memorize a lot of stuff to do well. The class was talking about mitosis. After the initial introduction to the topic, the teacher told one interesting story after another in between the lecture. Some of them were related to the Pro-D day he just attended. He talked about the new biological discovery and researches at UBC. He also talked about H1N1 pandemic which was the hot topic. When he went to the sideway of the class, every student turned around. Clearly, everyone was interested. The class finished before I realized it. Math 9: I taught one Math 9 lesson on polynomial addition and subtraction. As I observed the same class before, I notice that the young kids were very active and talkative. How can I use this high energy to my favour? I decided to use a game to gain students interest. The game was called “Guess your Secret Number”, which could be explained by polynomial addition. It asked each student wrote down a number that no one else knew. Then after a few calculations on the secret number, students would tell me their calculation result. By pondering over the result, I would guess what their secret numbers were. After the game started, the students actively participated. At one period of time, the classroom was so quiet because the students were all doing the calculation and trying to play the game.

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.

Battleground Schools Review

The article categorized math education methods into 2 polarities: progressive and conservative. It then shows the different views each category has in many different areas. It also discussed many different factors that complicated the math education in North America. The article went on to explain three twentieth-century reform movements in math education. First, it introduced “Progressivist Reform” from 1910-1940. It explained John Dewey’s teaching techniques along with progressivist’s views of math teaching to give students “the challenge of doing and experimentation in mathematics, accompanied by the sense-making activities of reflective practice.” Second, it introduced the “New Math” during 1960s. The Cold War between the United States and Soviet Union pushed the reformation for more abstract and highly complicated math at high school. Led by the School mathematics Study Group (SMSG), they created mainly conservative and even “teacher-proof” curricular materials, aiming to educate “future elite scientists and mathematicians.” Third, it discussed the “Math Wars over the NCTM Standards” beginning from 1990s. NCTM standard was under the influence of progressive and constructivist approaches and focused on the back-to-basics curricula. However, the low ranking of American eighth-grade students in the world caused the traditionalists to fights back for more rigorous math in high school. Based on the views of the different groups, I stand on the progressive side. However, on many areas, it sounds to me both conservative and progressive have valid points of view. For example, on “Goal of math learning”, I would think fluency is as important as understanding. I remember I once did a complicated computation during a math exam at G12. When I reached a step which needs the square root of 49, I simply stopped there and thought I must have done something wrong. I simply forgot square root of 49 equals 7. Less of fluency stopped me from doing the calculation further. Looking back from this experience, without fluency, the result would still be frustrating even if you think you understand the materials perfectly well. Furthermore, math abilities are usually evaluated by exams in high school. How can a person say he’s good at math if he cannot finish the math exam in time and constantly make basic mistakes because of fluency problems? On the other hand, if a person reaches a certain level of fluency at math, it also helps him to understand the materials better. Therefore, I feel fluency and understanding are two inseparable sides of math education.

Interview Review

Based on our interviews, I have found that a student teacher needs to have extended knowledge not only on math but also on other science courses. That’s the reality of a substitute teacher. We need to prepare for the unexpected classes to teach. For math teaching, I not only need to know how to solve the problem, but also need to know other different ways to solve it and understand it. As a teacher, we also need to deal with other issues such as grading on students, which sometimes is the greatest challenge even for seasoned teachers. Also, classroom management skill is the key to student's success. Especially for lower grades, effective teaching also means to manage the students effectively. For students’ interviews, I notice that the likes and dislikes of math are closely related to how well the students feel they can do on math subject. It’s more likely that the students will study hard on math and do well if the teachers encourage them and help them do well at the beginning. Most students agree that math is useful, but only on the basic calculation level. If we can provide more useful (not make up unrealistic ones) real life and work examples, students will gain further understanding of math’s role in our society and have better motives (other than go to the University) to study well in math. For example, how do experienced carpenters find a perpendicular line to the existing line?

Interview with Stduents and Teachers

Interview with Mr. Lucas:
1. What’s the greatest challenge you or new student teacher will face?
• For student teachers, you can’t get all classes in math. Sometimes you need to teach other courses you are not well prepared for.
• Even for seasoned teachers, the greatest challenge would be the constant arguments among students, parents and teachers whether the students should get the mark they desired.

2. How well prepared were you for your teaching?
• I am very well prepared
• This includes mastery of the curriculum, background concepts, and extended fields

3. How about the use of lesson plan? Do you think it would be useful?
• I have taught the course for many years. I don’t need a complete lesson plan.
• I use assignments and tests to evaluate the students, and I know very well how the class will go on.
• A checklist of what to do in class is good enough.

4. Do you approve variable standards in your math class?
• No. Students should meet certain standards in order to continue studying in math.
• It doesn’t have to be the common standard set by the school, but when the standard is set, students should work to meet the standard.

5. How different is it to teach different grades and which grade do you prefer?
• For different grades, different standards are required and different focus is concerned, for example, calculation skill based or concept based
• Higher level of math teaching(such as calculus) is preferred for me because students will focus more on the problems and then develop on the problem solving and critical thinking skills
• Teaching students at lower grades often need to deal with behaviour issues.


Interview with students:
1. What is Math to you? Why are you taking math? Do you like it?
2. What is the greatest difficulty or barrier you face in learning math?
3. Has it ever come across your mind you feel that math is not important?
Why or why not?
4. Do you think that your math teacher actually care about your grades? Or
do you think he only pays attention to those who are good at math?
5. Math is only for smart people. True or False? Why or why not?

Berta
1. Math is I don't know what math is to me but I think it is important and that’s why I’m taking it. I like math when I understand it but it can be really difficult if I don’t.
2. Most of the difficulties I have in learning math is when I don’t understand the question like if it is a long question with hard words I have to read it at least 5 times to understand it.
3. Somehow. Because there are something that we learn in math class that I think we won’t use in the future but math is important.
4. NO I don't think my teacher cares. But he is funny in class
5. False. Math is for everyone.

Winie
1. Math is a course that is required for me to graduate high school. I do like math as it teaches me how to apply different formulas to different equations.
2. Trigonometry was the biggest barrier I had encountered when learning math in school as I had a difficult time trying to understand it.
3. To be honest, unless one wishes to pursue a career that involves serious math skills, I don't find that learning more complicated math is very useful in everyday life.
4. My math teacher does care about my grades and everyone else's in class. He tries different methods to help everyone with different learning paces to understand the material.
5. False, as everyone can do it, but only if they are motivated and willing to do so.


Summary:
1. For new teachers, the challenge is not only from math teaching, but also from administration and management jobs.
2. We need to have extended knowledge on math, not just knowing how to solve a question.
3. A student teacher need to have a good and detailed lesson plan to guide us. As we become more experienced, we can just use checklists.
4. Variable standard might be Ok for some other courses. But it is not a good idea for math because new math skills are built on previously learnt skills.
5. Classroom management is very important in lower grades.
6. Most students think math is useful, but only at basic calculation level.
7. Most Students like math only when they can do well on math. The reason they want to learn math is because they need math to go to university.

Article Review: Changes in Instruction

Mr. Robinson’s article illustrated a period of time that he changed his teaching method from a “controller” of the classroom to a “facilitator” of the classroom. Instead of making the class his one man show, he changed his teaching style by letting students actively participate in the math topics. The classroom has shifted to become student-centered environment. As a math tutor for many years, I know the importance to get students involved. Active participation means better understanding of the materials and better results. Furthermore, by letting students discuss with each other. Students will begin to help each other. I remember one seasoned teacher told me: “Students usually have the same age and similar living experience, one students’ explanation to another student is usually better than teacher’s. An example obvious to the teacher is not necessarily being so obvious to the student. “ Another important change Mr. Robinson mentioned was his questioning method. He changed his assignments from procedural questions to the ones that need deeper thoughts. I totally applaud his changes. Procedural will not help students understand the materials better. That’s why some of the students did very good at every assignment for each chapter, yet they failed the final exam which needs the students to combine the knowledge from different chapters to solve the problems. For many students I tutored, if I just review the contents of the textbook, the students will get bored. They have already spent so much time in class to understand the materials. Therefore, questioning skill is the one I use most often. Through insightful questions and listening to student’s answers, I will get an idea of how the student understands the materials. Then I will either use examples or exercises to help them correct the misunderstandings. That way, my students were doing very well in the school.

Two of my most memorable teachers

One of my most memorable teachers is my math teacher from G4 to G6. Although she was very strict with students, she encouraged discussion during the class. Everyone was involved in her classroom. Even those girls who hated math got involved and raised their hands to answer the questions. She aroused me of great interest in math. When I told her my thoughts about a problem, even if it was not perfect, she would encourage me to continue and help me grasp the ideas through her guidance. The other one was the math teacher from G7 to G9. She was memorable to me because she brought me great pain in learning math. I was so into the teaching methods of my previous teacher that it was a torture for me to listen to her classes. During all her classes, she was copying things from the books and explained it very very slowly. She hoped that by going slow, everyone would understand it. Her classes were all about writing nicely and drawing the perfect circles. Every time she drew a perfect circle, students cheered. She was enjoying that. She also gave out tons of assignments which were just a repetition of basic questions. By comparing different teachers' teaching methods, I realized the teacher who encourages discussion will encourage students to learn more actively. The teacher who can arouse greatest interest of math from the students will make students learn better. As a teacher, one not only need to teach the materials, but also need to motivate the students to study and make the most out of the learning.

Self Assessment on Microteaching

1. I thought these things went well in my lesson: * For introduction, I have made a good introduction so that everyone knows that fast and neat is the characteristic of this new folding method. * For pretest, I have asked some girls and guys to show their ways of folding before I showed mine. Through comparison, people can really see the difference in speed and quality. * For activity, everyone participated and enjoyed the learning. I have received seven feedback sheets in total. Thank you guys!

2. If I were to teach this lesson again, I would work to improve it in these ways: * I wish I had brought more T-shirt so that everyone can practice on their own pace instead of being watched by everyone. * I wish I had brought long sleeved shirts so that people can see this method also works on long sleeved ones.

3. Here are some things I reflected on based on my peer's * People will get interested if the topic you do is useful and your teaching method is easy to follow. * People all want to participate in a fun activity. * I should have made a folding competition to get everybody activated.

[PLAN] Fast and Easy way to fold a T-shirt

1. Bridge: As Education students, we are all have a busy schedule and little time. Can we finish our household chore fast and easy? Today, I am going to introduce a fast and easy way to fold a T-shirt. 2. Teaching Objectives: Learn a good life skill in 10 minutes. Save your time for the rest of your life. 3. Learning Objectives: Everyone in the group will have a firm grasp of how to fold a T-shirt fast and easy. 4. Pretest: Let a guy try to fold a T-shirt. Keep a timing of how long does he take to fold the T-shirt. Do he do a good job? 5. Activities: A: Show how to fold a T-shirt step by step, with detailed explanation. Hold Q&A sessions. B: Let everyone try to fold the T-shirt. We will all comment on the speed and looking of the folded T-shirt. 6. Posttest: See who fold the best looking T-shirt and who fold the fastest! 7. Summary: If everyone of us can pay attention to little things in our life, we can save a lot of valuable time to enjoy the best of life.

Relational Understanding vs. Instrumental Understanding

After reading Mr. Skemp’s article, I have got mixed feelings towards both approaches. Through many years’ of study, I have got math teachers try to explain things in a marvellous (relational) way. But after the whole class, we are still at a loss regarding steps to solve the questions. By saying “Relational knowledge can be effective as a goal in itself”, Mr. Skemp have admitted that teaching in the relational way is a daunting task to accomplish by itself. Especially for students at young ages, their understandings and experiences are very limited. One relational explanation, which the teacher feels to be excellent, might be harder to understand than the original question for students. Thus, we need an explanation for the explanation. For example, on page 6, Mr. Skemp used the music teaching as an analogy to prove his point. For those readers who are familiar with music, they might feel the analogy is excellent. But for those readers who aren’t music fans, the words such as “stave, minim, crotchets, and quaver” took a whole course to explain well. Not to say let the readers understand what the author is trying to prove. Nor do I agree that math be taught purely instrumentally. For this part, I feel Mr. Skemp has done a good job to prove the point. See quote 4 and 5. In summary, the approach I suggest is: First, teach the students instrumentally. Give them a crutch to help them walk on their own. Second, when the students get an idea of what’s going on but still confused why they do this, give them relational explanation to clarify. The students will suddenly realize what they have done make sense and they will have a firm understanding of how to do it. [ Quotes ] 1: knowledge can be effective as a goal in itself 2: Imagine that two groups of children are taught music as a pencil-and paper subject. 3: Within its own context, instrumental mathematics is usually easier to understand; sometimes much easier. 4: It (Relational understanding) is more adaptable to new tasks. 5: It (Relational understanding) is easier to remember.