Math Interviews: Reflections

I found the responses from all the teachers are varied. The teacher I interviewed was my grade 4, 6, and 8 teacher. I remember him most for caring about the students and his discipline. Since he taught in the middle school, he would teach math and science to the students. Thus, he uses games as an incentive for students to keep on task and for classroom management. I now see that Bus Driver is a game which engages the whole class, even the struggling students. It helps the struggling students practice their multiplication tables and gives them an achievable challenge or goal. Bus Driver also makes math fun.

For the students we interviewed, I found it interesting the math and logic puzzles and challenges were the highlights of the students instead of any particular curriculum subject. They seemed have problems with concepts that were similar. For example, after plotting out y = x^2 for the student, we asked the student to plot y = 1/(x^2). He plotted out 1/x. He seemed to mix up the concept of reciprocal and inverse also. I would be clear on definitions and make sure I do not use the wrong terminology. I would not use the term, take the inverse of 6 when I mean take the reciprocal of 6.

Math Interviews: Group Summary

We interviewed a grade 8 mathematics teacher from a middle school in the Coquitlam school district via e-mail. He had expressed the biggest challenge was trying to teach the basic operations such as addition, subtraction, multiplication, and division to the struggling students such that the students would not give up. Consequently, he would also have to keep the top students from being bored. For the top students, he allows them to be peer tutors and provides math challenges or puzzles for them. The math class schedules goes as: class game of Bus Driver, presentation of the math challenge, lecture (25 minutes) and assigned questions (30 minutes). Bus driver is a game with multiplication flash cards. These cards are never shuffled. The bus driver is the winner from the last game. They start off facing off another student in the class. Whoever is the fastest at answering the next flipped up card is the winner for that round. If it's a tie, the teacher keeps flipping the cards (sometimes many at a time) to see who is the fastest. The winner is the bus driver and goes to the next student. The game ends when everyone has a turn and is proclaimed the bus driver and has their name written on the board as the bus driver. Even the struggling students love participating and sometimes may guess the answer ahead of time to win.

By interviewing two students in different grades, we were able to observe some interesting similarities and differences in their responses. First half of the interview questions dealt with what the students like and not like about mathematics. When we asked them what their favourite parts in mathematics were, the grade 9 student responded that he likes to work with integers because they are straight-forward and it is easy to remember the rules. The grade 11 student responded that he likes algebra for a similar reason, but he also likes riddles and logic puzzles. It is interesting how the grade 9 student likes the straightforward and easy concept, while the grade 11 student likes the challenging puzzles and riddles that allow him to think beyond the simple rules and concept.

We then asked them to share some of the challenges they encounter in mathematics classes. The grade 9 student responded that one thing he finds really difficult is translating the word problems into the equations. Also, he is confused when the same symbols are used to represent different things. Similarly, the grade 11 student had a problem with understanding the idea behind the concepts and rules. For example, he has difficulties with understanding the differences between the inverse function and the reciprocal function. Some of these areas of difficulty in math reminded us of the articles that we have read by Skemp (1976) and Robinson (2006) stressing the idea of relational understanding instead of instrumental understanding. We thought that the students are having difficulties in those areas, because they lack the relational understanding of the concepts.

They also expressed an interest in class when the teacher used different media for explanations (that is, anything but the chalkboard). The use of geometric shape blocks on the projector depicting larger shapes and ideas was found as interesting. Presenting students with “challenge-of-the-week” (COW) puzzles and problems got the students interested in math beyond the daily lessons. With the grade 11 student, humour made the classroom more relaxed and the math lessons more interesting–but we can’t all be comedians! In the students’ opinions, logic puzzles would be an interesting addition to math classes. Real-life applications of math would also make the classes more practical. When asked about group work, the reactions were mixed. The grade 9 student did not prefer group work too much because of the added distractions from getting the work done. The grade 11 student has had math projects, such as designing a water slide using cubic functions, and enjoyed the idea of working with others, finding value in comparing answers and thought-processes with others.

Memorable Teachers

Self / Parents: For most of elementary school and up to grade 9, I learned mathematics on my own (besides using compasses in geometry). Since my teachers would not challenge me enough, I would learn multiplication, division from my parents and exercise books they bought me. I would understand the mathematics instrumentally and relationally. The only part of the problem would be I wouldn't be doing math as fast as my friends who were in Kumon. School was for refining my math speed / skills. If I didn't know how to do something, I would ask my parents. The only difficult concept I could not learn on my own was long division in grade 2 or 3 and the base concept in grade 5.

Grade 10/11 teacher: I really enjoyed my grade 10 and 11 IB math teacher. He was very fun and funny. He would promote that math is fun and had a big smile on his face. Sometimes he would make some jokes and tell stories of his past. Sometimes he would make math jokes and math cheers. He's memorable for bringing the parabola dance into the classroom. Most of the students are comfortable with him.

http://www.youtube.com/watch?v=URziKVxG0kA

Response to article

It should be every teacher's desire for the students to understand mathematics at a higher level. Students should not only be able to compute answer but also use manipulate math concepts they learned to do other questions. For this reason, I thought it was interesting that the students did not have a good grasp of the higher mathematics with the lecture style than the hands on approach. Our class really prefers to teach instrumentally. Perhaps this is not so teacher centered teaching but rather student centered learning. This suggests I should let students have more time to learn the material rather than just lecturing a lot. Once when I was doing my volunteering, I had to teach a whole class on grade 10 math. I did the think pair share with the class as an activity during the lecture. The students had to multiply polynomial fractions and simplify. I had the students talk to their neighbours, share and change their answers. When they were done sharing, I asked for different answers and polled how many got each of those answers. Then I told them showed them how I would do the problem and where it could have gone wrong to show the other answers. I do question this method in whether it would help them have a higher level of understanding.

Cool Math Art Project Idea

http://math.slu.edu/escher/index.php/Tessellation_Art_Project

I really like some of the examples they gave.

Sending Secret Messages with Light (Microteaching)

1) Things I thought went well:

• asking for verification of understanding of the subject
• would refresh what needed to be known

• interactive
• drew analogies with already known concepts

2) Ways to improve:
I have taught this lesson a few times with my friends. I may have "lost" a student but I wouldn't know because the students kept nodding their head. Asking specific questions on the facts that needed to be known would have allowed me to see whether they knew the subject or not.

3) Reflection on peers feedback:
One peer enjoyed the participatory activity of roleplaying Alice and Bob and thought the involvement was great. However, another peer suggested more participation. The visuals helped to explain the lesson. Another peer thought the material and presentation was very suitable for the topic. All the peers would have liked more theory or notes on the topic. They agreed that the objectives were clear. Quantum cryptography is a difficult topic to understand. I thought this lesson and activity was appropriate for the time and the topic. I prepare some resources and notes for the students after if this was the sole lesson. However, it would be a great introductory lesson or a good activity to reinforce quantum mechanics.

BLOOOPPS

1) BRIDGE: “hook” or introductory activity
Ask students, what can you do with light?
You can send secret messages with light!
2) Teaching OBJECTIVES:
Have everyone involve in the active learning / the game.
3) Learning OBJECTIVES: “SWBAT” 
Students will be able to relate eigenvectors to the polarization vectors of light and the implications.
Students will be able to relate measurement as a matrix acting on a vector.
SWABAT explain that measurement on light will change the polarization unless it is already an eigenvector.
4) PRETEST: Ask orally about understanding of light and quantum mechanics.
5) PARTICIPATORY
Lay down ground work rules on light.
Explain how Alice and Bob (Quantum Cryptographic System) work.
Get students to pretend / role play Alice and Bob.
Ask what information is being sent classically vs what information is kept as the message.
6) POST-TEST
Ask students one by one what they learned.
Key questions:
What about measurement allows for light to be a medium for sending secret messages?
What is the classical (verbal) information that is communicated through between Alice and Bob?
What is this information used for?
What part of light does the message information come from?
7) SUMMARY or conclusion
Alice and Bob are actual machines of a quantum cryptographic system. Such systems exist and were used in the Swiss 2007 election to encrypt the ballots.

Reflections on Skemp

Skemp presents the idea of faux amis to introduce the problem with different meanings of understanding and extends it to that of mathematics. I would relate the “instrumental understanding” to knowing simply which rule to use and how to compute the answer. “Relational understanding” is knowing the concept which leads to explanation with more than just the equation. One of the examples given was a “teacher reminding a class that the area of a rectangle is given by the area being A = L * B” (area is length times the breadth) (1). I had difficulty with how an elementary school teacher would teach the relational understanding of the concept of area. The only way I could think of was a picture of the rectangle with a grid drawn over it. If one was to be very thorough about it one would have to remind the class that length was a measure of how much space a line took up and that area was like the same concept but in two dimensions. One would also have to help the students understand why one has to use multiplication. One of the gems of Skemp's article is the quote: “Unless the two sides (who have a different interpretation on what was going on) stop and talk about what game they think they are playing at, long enough to gain some mutual understanding, the game will break up in disorder and the two teams will never want to meet again”(2). I would agree it would be quite important for the student and teacher discuss the end goal of the lesson if they are different. The frustration that would mount up for the student would cause them to have problems each lesson and may cause them to steer away from the subject in the future. It would help the student to understand why they were learning such things and for the teacher to understand the needs of the student. As a teacher candidate for physics, I would hope to teach the student relational understanding of the subject. This would help them have the skills to analyse and tackle new problems. If they only understood the subject instrumentally, they would only be able to take the same type of questions they learned and look for some equations that would give them the answer; they would not be able to teach a peer how the equations applied to the physical situation. As a music enthusiast, music taught “as a pencil-and-paper subject” compared with students “taught to associate certain sounds with these marks on paper” (3) was a very strong example depicting the importance relational understanding. The first group would miss the idea of music completely and miss the goal of learning music. They would not know music at all. Skemp states, “it is nice to get a page of right answers, and we must not underrate the importance of the feeling of success which pupils get from this” (4). As a new teacher, I am more enthusiastic on students learning which would leave me not remembering students want to feel some sort of accomplishment. In this education system, marks are perhaps used for students to judge each other and a source of pride and self-confidence. However, I would like to at least empower the students to be at a point where they can say “I can do this”. Skemp states that the difficulty with teaching relational mathematics is “the backwash effect of examinations” (5). In the current education system, measurements of success stem mostly from examinations. This results in students just wanting to cut corners and learn mathematics the quick and fool-proof way (instrumental learning). Physics education groups have been developing questions testing the relational understanding of physics instead of computations. I think students would switch from just learning instrumental mathematics to relational mathematics if relational mathematics was tested. Overall, I learned mathematics and physics relationally and would be frustrated at just an instrumental understanding of these subjects. However, I would have to be mindful and plan carefully how to teach the concepts and problem solving without the students just filtering all the information and learning just the computational part of the lesson.

Welcome!

Welcome world to my MAED 314 blog.