Wednesday, September 30, 2009

Relational Understanding and Instrumental Understanding

I think that the author makes a very interesting point in debating the pros and cons from both views of understanding, more precisely on understanding mathematics. I still think that relational understanding is a more valid goal for a mathematics teacher but apparently relational is not independent from instrumental understanding, as one is helped by the other. Relational understanding seems to promote the independence of students when it comes to showing their mathematical knowledge. On the other hand, instrumental understanding leads to too many rules, that once broken, e.g. making a mistake, it is hard to find the right way.

The idea of seeing math as a town is a new, yet sounded somehow familiar to me. Understanding mathematics relationally is like knowing the city so that one can go from one point to the other, and understanding mathematics instrumentally is like knowing a number of routes between a number of points.

An idea I got after reading this was to maybe ask my students to draw their own "mathematics" maps, to see how they can connect their knowledge. And why limit it to mathematics only? I would like to see how math connects within itself and also how all those concepts connect with other areas like art, science, sports, business...

Monday, September 28, 2009

Interview with Students and Teachers

Interview with the Student
So I interviewed a student from high school, "J." We spend some time together and I taped his answers. I transcribed the interview.

How would you answer the question "why learn math?"

To make you think. There are some jobs that do not really need math, but math just really makes you think. It makes you see a problem in many different ways. They train you to not be closed minded.

How do you best learn math?

Practice. Not just reading. You have to do it.

Do you enjoy learning math? Do you find it interesting? Do you think it is being taught well?

I can't say I always enjoy learning math, I think it is taught pretty well. Math is a hard subject. I don't enjoy it always. It has nothing to do with the way it is being taught that I don't like it; it is a pretty hard subject. Sometimes I don't like it. Sometimes I find it interesting. Sometimes I find it pretty boring.

Which assessment methods are you best at?

I would say on presentations. I just don't have stage freight. I don't really have a problem talking to people.

How would you define mathematics?

Numbers. Lots of numbers. I don't know. . . adding, subtracting, multiplying.

Interview with the teacher:

How much homework do you assign and why?

Just enough to feel over whelmed but they don't break down. Keep them out of my hair during class time, keep them occupied.

How do you deal with difficult students

Use of humor, have the students see you as a person.

How would you improve the teacher education program?

UBC program, shortened to 6 month practicum, focus on practical. No wishy washy theoretical crap, total lack of practicality. Social Justice, Special Ed good.

Boys more willing to experiment and try different things, girls require a lot more structure, less likely to take risks. However they put in more effort.

more hands on work not the answer, math work not being done, not enough theory, more crunching numbers. Overly calculator dependent.


Conversation with a math teacher
Q1: How much homework, do you assign, if any, and how often? Why?
In the lower grades (9 and 8) homework is primarily work not completed in class.
For grades 10 about half and hour per class and grades 11 and 12 about 45 min to 1 hour per class.
Q2: How do you get difficult students or those who just don't show any interest and do not participate
involved?
I try to engage them by doing a wide variety of activities and showing how it applies in their lives. I am finding
the new curriculum (grade 8 last year, grade 9 next year) to be excellent in this regard. The new textbooks are
amazing. When I was a new teacher I thought if I just explained things well enough and loved the students, they
would all succeed. I've learned that the students must bring their own energy to the class in order for them to be
successful.
Q3: What kind of training did you go through before started teaching? Do you feel
it was adequate?
I am an engineer by training and worked in the field for over 15 years. I find that this experience helps
tremendously specifically in making the lessons more engaging. I have a Ph.D. in Chemical Engineering. To
qualify as a teacher, I took the 11 month program at SFU.
Q4: Do you see a different between how and the way boys and girls learn math?
As a profession, teacher are working at incorporating boy/girl learning patterns. More specifically, we are trying
to make lessons more boyfriendly
by allowing a lot of partner work and movement in the classroom.
Q5: What recommendations would you have for mathematics education reform?
The only recommendation that comes to mind may surprise you we
need to make it easier to fire ineffective
teachers. The number one factor for a child's success is the teacher. We need to stop blaming outside factors and
critically look at ourselves in terms of delivering the best lessons in an environment that allows individual students to embrace their own style of learning and thrive.

Conversation with two math students
Q1: How would you define "mathematics"?
Both Isabelle and Gabrielle: Adding, subtracting, multiplying, dividing, fractions.
Q2. If you were the teacher, how would you improve your math class?
Gabrielle:
1. take 5 minute break in between the classs. Her classes are 75 minutes.
2. have students get involved by asking them how to do a step/do the problem before explaining the process.
Q3. How do you best learn math?
Gabrielle: having a quiz after learning a new process.
Q4. How would you answer the question "why learn math?"
Gabrielle: budget, learn about money, taxes, buying things.
Isabelle: we don't need to learn half the stuff we do. We don't need to learn geometry.
Q5. Which assessment methods are you best at (if applicapable): quizzes, in­class assignments, home work, open
book exams, final exams, provincial exams, group projects/activities, in­class assignments, research projects,
open­book exams
Both Isabelle and Gabrielle do best with in­class assignments.
Isabelle says because she is more focused in­class.
Gabrielle ­ home work and in­class assignments are about even re successful learning/marks.
Q6. Do you enjoy learning math? Pls explain your answer.
Gabrielle ­ depends on teacher ­ if the teacher explains and proceeds to new problems at the students' pace and
gives students lots of examples so we understand.
Isabelle ­ no. I don't know why.
Q7. Do you find MATH interesting? Pls explain your answer.
Gabrielle ­ yes, because a specific rule will apply to all numbers. For eg, two negatives multiplied are always a
positive. But it's not interesting like social studies, where you learn something new all the time.
Isabelle ­ it's fun when it's easy. Otherwise it's not interesting.
Q8. Do you think it is being taught well? (comment on both the course materials and the teacher)
Gabrielle ­ the course materials are good this year. My teacher this year is good because he gives us a break to
relax our mind after 35 minutes, he takes time to make sure that he isn't going too fast and he checks to make
sure we are following his steps, he gives a lot of examples, and he tests us as we go along to make sure that we
understand. He also makes sure we are prepared for the chapter tests/quizzes by giving us assignments on the
new material before the quizz.
Isabelle ­ No because we spend too much time marking our homework and tests. We spend about 1/3 of the class
marking each other's work. The class is boring. The teacher teaches straight out of the text and she just tells us to
do the work. If we need help we have to ask.

Summary of "Battleground Schools" & Reflection

Mathematics Education is a very hotly debated issue in North America. On one side you have people who think that math should be taught in a strict teacher-proof way with clear objectives which entail fluency and lead towards easy-to-read results, such as multiple choice tests. This approach towards teaching math is referred to as "Conservative."

On the other hand you have people who support teaching with deep understanding of mathematical concepts as the ultimate and most important goal. This point of view towards teaching math is referred to as "Progressivist." Under the progressivist attitude, the teacher needs to become an expert in assessing student progress by diagnosing the student level of knowledge and skills. Progressivist argument also calls for the teacher to have strong background in mathematics, and in mathematics teaching and learning.

In the last century, mathematics education has passed through these points of view, and spending a little time in the middle. There are three periods: Progressivist, New Math, and the Math wars.

Around 1910 the progressivists answered to the then current way of teaching math of simply memorizing procedures and facts by calling for a deeper understanding of mathematics rather than a repetition of statements and the use of formulas to solve standard word problems.

In the 1960's the space race called for a reform in math education, since there was paranoia in the Western world, particularly in the United States, that the space race was being lost due to a lack of scientists. It was thought that mathematics programs in high schools were not keeping up with top research universities.The response to this problem became known as the New Math movement and it called for a reform of all math curriculum and even other subjects to make all students turn into prospective scientists, especially rocket scientists. The new curriculum included topics that neither teachers nor the parents of the students were familiar with. Hence many complications arose when it was tried to be implemented ata schools.

In the 1980's there was a tendency in the western countries to fight and break unions, including teacher unions. In an attempt from being left out of the drafting of curriculum in schools, the National Council of Teachers of Mathematics created its own standards. In the mid-1990's there was a response to the NCTM standards and some people called for math to be taught in a more rigorous way and that results would be clearly measured by using standardized tests. This gave rise to the Math Wars over the NCTM Standards, and the polarization was further fueled by its media coverage.

The dispute of how to teach math has been taken over by political parties, the right versus the left facing each other on many fronts. To this day there is no end in sight for the animosity between both parties and there seems to be no desire from either part to come to an agreement.

Reflection

Having discussions about math is all together right, but when it gets personal, I think it loses any value. One has to keep focused on the important things, i.e. improving education, and not on the destruction of the enemy.

I personally consider myself a "progressivist" because I think I am more attracted towards understanding concepts and being creative when doing math. Math looks more attractive when one really appreciate the concepts, values the ideas brought up by mathematical exercises, and appreciate the beauty and elegance ingrained in mathematical thinking and its presentation.

I think that the "progressivist" is the more "ideal" approach towards math in the sense that it seems harder to follow and to measure student achievement. Therefore, in an age of massive production with a "need" for efficiency and easy-to-read measurements, rather than comprehension and quality of education, the progressivist approach is received with less than enthusiasm. Standardized tests are a logical solution for people who view school as an industrial machine so therefore want to readily compare achievement among students, schools, districts, states, and ultimately, nations to find out who is "superior." And their way to remedy educational issues usually involves changing the standards. This is a shame to me.

Sunday, September 27, 2009

What I've learned

From my Presentation



I learned that some students may say they don't like math, but what they really mean is that not always do they like math, i.e. sometimes they actually find math interesting, and some other times boring.



The teachers want to be effective and cannot stand when other teachers who are not competent stay in their teaching positions without much possibility to remove those people from the teaching profession.



I learned that some students are actually aware of the importance of math; that even though they might never have to apply the exact same concepts learned during class, math plays a role in maintaining your mind active.

From the Other Presentations

Most teachers suggest using humour as a tool to both gain students attention and dealing with difficult students. All students need humour.

Explaining too much can be counterproductive, as in the example of the student who said that "the teacher may explain and explain until I understand, but sometimes the teacher can explain and explain until I don't understand."

Students, contrary to popular belief, want to be assigned homework. They don't need a whole lot of homework, so finding the appropriate amount is a careful decision that a teacher must take.

Teachers who have taught for a long time do not need to lesson plan as much as novice teachers. Some senior teachers do not even lesson plan.

Different things work for different students. Teacher have to have a full arsenal of tools so that students do not get bored with the same techniques. The best technique cannot all the time every time.


One of the ideas that caught my attention on one of the last presentations is to ask the mathematics students to keep a math journal.

You have to use technology wisely. One of the teachers being interviewed explained how he made a mistake by simply taking the students to the computer lab and then the students, even though they had a blast at the computer lab, they ultimately did not learn anything. The students would then ask for the class to take place in the computer lab repeatedly.

Monday, September 21, 2009

Comments on "Using Research to Analyze, Inform, and Assess Changes in Instruction"

I like the way she describes her ideal classroom as "one where the students actively participate in learning process by being engaged in enriching and meaningful learning activities that help make mathematics relevant and realistic -- rather than an abstract 'thing'. . . where students appreciate each other's ideas and are not afraid to be wrong in order to accomplish learning."

I think she gives too much importance to research. In research, I feel that, after researchers have done their research, they are now trying to point to the "right" direction where all teachers must follow, if they want to be benefited from the research being done. I think that every teacher is also a human being and therefore have different ways to approach his or her students without either type of approach being the"best" one. So teacher A may in fact lecture through most of the class, while teacher B's class consists of mostly group activities, and both could equally be reagarded by the community as "great" teachers. In short, as far as teaching goes, one technique does not fit all.

On the other hand, if a teacher feels like they need to change because the way they are teaching is not a way in which he or she would feel comfortable or because they are not getting the desired results, then by all means, research should be consulted and applied. I am not saying that research is useless, but I think that it is not the answer to every teaching issue.

I like the way her class turned around. I think that she would not have noticed the shortcomings of her class had there not been a standardized test in place. I think that is sad because the only way that she could find that out was by the high number of students failing the standardized test.

Two of my Most Memorable Math Teachers

My most memorable math teacher was Alejandro "Baldor" Mendoza. Thanks to him I learned algebra. The reason I liked him was because he would talk to us and would not read from the textbook. He explained things in way that we could understand. He was very patient. Also, he could talk about things other than math, if it was appropriate.

The other teacher I remember was a high school math teacher. She would not explain much. Instead, she would have us all copying definitions and procedures from the blackboard. I did not understand and I also missed one class and instead of being helped out, I was sent to the office and got in trouble.

These two characters pretty much frame the two sides of what is wrong and what is right when teaching math. That is why I chose to write about them. A wrong thing to do would be to separate math (subject) from people (teacher and students). I think that you have to find a way to approach your students and make it personal. If a student gets behind, he should be reinforced to get back on track, not sent to the office and get in trouble.

Sunday, September 20, 2009

Homework Assignment 1

Summary of my Peers Evaluation



My fellow classmates wrote many positive comments and a few suggestions for my microteaching lesson. They all agreed that the learning objective - how to play Texas Hold'em - was clear. Most agreed that there was a "hook" in my lesson, however, there is a comment suggesting that I should have spent more time introducing the game. For the rest of the checklist, my fellow students agree that there was a pretest, a participatory activity, a check-in on learning and a summary/conclusion.

The most prevalent strength of my lesson was that it was "fun" and that "everyone participated."

On the areas where I need further work and development, my classmates have suggested that I spend more time explaining the rules of the game and that the chips that I brought for the game were not enough.




My Own Evaluation

I thought that the things that went well in my lesson were that I was calm and that the students learned how to play the game and had fun while doing so.

If I were to teach this lesson again, I would be better prepared with more chips. I would also be more organized when explaining the ranking of the hands in Texas Hold'em, and I would also spend more time explaining the system of placing bets.

I also think that presenting this topic might require more time than ten minutes --I could not get them to play by themselves without my guidance.

The things I relfected on based on my peers' feedback were that I went too fast through the introduction of the game. I think I forgot to mention basic things about the game, such as the fact that it is very similar to poker, that you need a deck of cards to play it, and that you can use chips to bet. I think I could have paced my lesson more so that instead of playing four times, we could have actually all played less times, but with everyone knowing how to play.

Thursday, September 17, 2009

Lesson Plan for Microteaching - Texas Hold'em

Materials: Playing Cards, Table of Hand Ranking for Poker, Coins


1. Bridge: Hey would you like to make some easy money? Why don't we learn to play Texas Hold'em?

2. Teaching Objectives: To learn how to engage students in an activity that will allow them to learn the rules and play a new game.

3. Learning Objectives: Students will be able to (SWBAT) play Texas Hold’em on their own, i.e. without the directions of the teacher.


4. Pretesting: Who here has played poker? - And in case that nobody has played poker- Who here has played with cards?


5. Participatory Activity: We all play a round of Texas Hold'em, with all the cards open for everyone to see, as a practice round and to answer any questions that may arise. We may have a second practice round for reinforcement.

6. Post-Test: We will have a round of Texas Hold’em with the teacher's participation, but wihtout the teacher’s directions and with the participants' cards undisclosed.


7. Summary: Today we have learned how to play Texas Hold'em, tomorrow we will learn how to bluff.