After receiving these letters, I now know how much impact I will have on the lives of others. On one hand, I am excited that I can help guide and positively influence my students, but at the same time, I am conscious that my words and actions will be taken to heart with my students. I hope that as a teacher I will constantly try to be a good role model for them, but understanding that there will be days where I will say something that will perhaps effect a student in a negative way. I hope this doesn’t happen.
Realizing this now, I would be super careful of what I say, do and behave inside and outside of the classroom. Although this may seem like a big challenge, I am looking forward to it.
Wednesday, September 29, 2010
Letters from two of my students 10 years from now
Dear Mr. H,
10 years have passed since I was first taught by you in my final year in high school. I thought I share with you some of my thoughts and experiences thus far since that time.
I completed my undergrad about 6 years ago with a major in fine arts and a minor in mathematics. After that I decided to do a masters degree in architecture. I’ve finished that about 2 years ago and have been working for the past two years. Luckily, I met my significant other while I was doing my masters program. I’ve been married now for a year now.
Anyway, I wanted to write to you letting you know that you did a lot for me in that short year in order for me to reach where I am right now in my life. I felt it was necessary for me to verbalize my thanks to you. I enjoyed the way you taught math and made it interesting for someone like myself who struggled with it. More importantly, I wanted to thank you for having such high standards for all of us in that class not just academically, but also as human beings. Letting us all know that we had something important to achieve and offer to the world. I am witnessing this reality as it is unfolding in front of me, especially now that I am in the work force. I feel that I have accomplished a lot and if it weren’t for you I probably wouldn’t have tried as hard. Being able to use the skills I learned in school and broadening my vision of the world has helped me to become a successful person.
Now that I have been working for this firm in Montreal for some time now, I think I will go abroad and work somewhere where I can really put my expertise into practice and help make a change in the world. Perhaps make more eco friendly buildings now or help impoverished areas with developing homes. “The possibilities are endless” that’s what I remember you told us. Thanks for opening up the world for me.
Sincerely,
Your favorite student,
Fernado.
----------
Hi Mr. H,
I had to write to you, since 10 years have passed and I felt that I just had to let you know by now. I really disliked your classes; they were very boring and not useful. I could have learned so much more with my time if I didn’t come to your class. I felt the high standards that you had for us made me feel like I could not accomplish anything. I didn’t enjoy any math I learned with you. It was difficult and meaningless. I feel like you should have made it more interesting.
Now that I’m in university, I have great professors who are teaching me well. I’m understanding everything now. I thought I should let you know about this. I figured maybe you should change your career or something. (Just a thought) Anyway, I just had to let you know about this, I feel a weight off my shoulder. That’s all.
Bye,
Your worst student,
Jim
10 years have passed since I was first taught by you in my final year in high school. I thought I share with you some of my thoughts and experiences thus far since that time.
I completed my undergrad about 6 years ago with a major in fine arts and a minor in mathematics. After that I decided to do a masters degree in architecture. I’ve finished that about 2 years ago and have been working for the past two years. Luckily, I met my significant other while I was doing my masters program. I’ve been married now for a year now.
Anyway, I wanted to write to you letting you know that you did a lot for me in that short year in order for me to reach where I am right now in my life. I felt it was necessary for me to verbalize my thanks to you. I enjoyed the way you taught math and made it interesting for someone like myself who struggled with it. More importantly, I wanted to thank you for having such high standards for all of us in that class not just academically, but also as human beings. Letting us all know that we had something important to achieve and offer to the world. I am witnessing this reality as it is unfolding in front of me, especially now that I am in the work force. I feel that I have accomplished a lot and if it weren’t for you I probably wouldn’t have tried as hard. Being able to use the skills I learned in school and broadening my vision of the world has helped me to become a successful person.
Now that I have been working for this firm in Montreal for some time now, I think I will go abroad and work somewhere where I can really put my expertise into practice and help make a change in the world. Perhaps make more eco friendly buildings now or help impoverished areas with developing homes. “The possibilities are endless” that’s what I remember you told us. Thanks for opening up the world for me.
Sincerely,
Your favorite student,
Fernado.
----------
Hi Mr. H,
I had to write to you, since 10 years have passed and I felt that I just had to let you know by now. I really disliked your classes; they were very boring and not useful. I could have learned so much more with my time if I didn’t come to your class. I felt the high standards that you had for us made me feel like I could not accomplish anything. I didn’t enjoy any math I learned with you. It was difficult and meaningless. I feel like you should have made it more interesting.
Now that I’m in university, I have great professors who are teaching me well. I’m understanding everything now. I thought I should let you know about this. I figured maybe you should change your career or something. (Just a thought) Anyway, I just had to let you know about this, I feel a weight off my shoulder. That’s all.
Bye,
Your worst student,
Jim
Tuesday, September 28, 2010
Response and Summary to Battleground Schools: Mathematics Education
This article was read at the perfect timing, just when I was wondering about the history of mathematics in North America. It proved to be very insightful in a concise manner. The article goes on to outline the 3 major movements of mathematics in the 1900:
1. Progressive Reform (1900-1940)
2. The New Math (1960)
3. “Math Wars” NCTM Standard reforms (1900-now)
I wanted to know about the history of mathematics to better understand the reason for why math has been difficult to learn and not-well received by most. It seems almost too puzzling to understand why this is the case. Nonetheless, I’m glad the article outlines a few reasons:
1. Mathematics is hard.
2. Only necessary for a small elite.
3. People who like Math tend to be socially challenged people, and not normal.
4. No shame in not knowing Math.
I would have to agree with all these reasons. There are perhaps much more. Like the article mentioned, generally people are afraid or math phobic in a way, and not just a few feel this way but quite an outstanding number of people do.
In addition to outlining the above, the article mentions that there are two polarizing views of mathematics, progressive and conservative; the former being more flexible in the approach of teaching and the latter taking the traditional views of how math should be taught using strict rules and methods. I personally was taught in a conservative manner; however, I believe that math should strive to become more balanced with the progressive attitude, or at least teachers should understand when to exercise both methods of teaching in different situations. Certain students may learn best in either way. The concern is not trying to find which one to focus on but rather see the merit in both ways. Too much of either would not give the whole picture of mathematics to our future students. My hope is that students will become active agents in the process of learning math.
1. Progressive Reform (1900-1940)
2. The New Math (1960)
3. “Math Wars” NCTM Standard reforms (1900-now)
I wanted to know about the history of mathematics to better understand the reason for why math has been difficult to learn and not-well received by most. It seems almost too puzzling to understand why this is the case. Nonetheless, I’m glad the article outlines a few reasons:
1. Mathematics is hard.
2. Only necessary for a small elite.
3. People who like Math tend to be socially challenged people, and not normal.
4. No shame in not knowing Math.
I would have to agree with all these reasons. There are perhaps much more. Like the article mentioned, generally people are afraid or math phobic in a way, and not just a few feel this way but quite an outstanding number of people do.
In addition to outlining the above, the article mentions that there are two polarizing views of mathematics, progressive and conservative; the former being more flexible in the approach of teaching and the latter taking the traditional views of how math should be taught using strict rules and methods. I personally was taught in a conservative manner; however, I believe that math should strive to become more balanced with the progressive attitude, or at least teachers should understand when to exercise both methods of teaching in different situations. Certain students may learn best in either way. The concern is not trying to find which one to focus on but rather see the merit in both ways. Too much of either would not give the whole picture of mathematics to our future students. My hope is that students will become active agents in the process of learning math.
Friday, September 24, 2010
Responses to the 5 burning questions from Teachers and Students
Teacher Responses:
I had the opportunity to ask two teachers these few questions; they will be referred to as teacher A and teacher B. Teacher A has been teaching high school math for about 10 years now and has taught mostly in international schools in Asia, namely Macau, China and Korea. She graduated from Australia. On the other hand, teacher B has taught in Canada for a total of 9 years, 2 years in Ontario and 7 years in British Columbia. He is teaching in a large public school in the Vancouver region. He graduated from Ontario.
Response to question 1:
Both of the teachers expressed that they try to give students questions that they can handle, in order for them to feel successful and to gain some confidence, this is where they would gain their interest in math. For example, “working with these students one on one and making sure they know how pleased I am when they do well on tests, quizzes or assignments.” was teacher B’s response.
Response to question 2:
Both teachers use computers and certain softwares to teach their lessons. Teacher A works in a school where all students have a computer and are networked with the teacher’s computer, so she can do things live on her computer in order to teach her students, this is very engaging for students living in a age where they are always around technology. Teacher B does not have the same luxuries as teacher A, nonetheless he also uses a computer and a projector for his class.
Response to question 3:
Teacher A said that AP calculus is the goal for most students at her school, and she feels that it prepares them quite well. She finds no issues of the curriculum that she is teaching right now. Teacher B on the other hand hasn’t found any concerns so far in the new grade 10 curriculum, since it is the beginning of the year. However, he did mention that there is less material to teach so there is more room to explore with new activities that he has thought of so far to help with the students learning. He also mentioned, “I think students learn differently than before internet became available and technology became so advanced. Now, students are accustomed to retrieving information instantaneously and networking in larger groups. There almost seems like an impatience to learn when the Socratic method of teaching is used.” I found this to be the most useful knowledge to know about how students like to learn these days.
Response to question 4:
Teacher A said that she has felt boredom but it was because she was teaching the same things in the same way due to not having much access to resources. However, in order to get over her boredom she started doing more research online for new ways to use technology and read up on tweets about all these things other teachers were doing in other schools to improve her teaching style. Teacher B on the other hand, mentioned that he has never been bored since he has been always creating new lesson plans each year he teaches.
Response to question 5:
Both teachers expressed that they are not interested in administration; however that might change in due time.
Student Responses:
Student who enjoys mathematics (Based on responses given):
Math gives us a basic knowledge to be able do everyday things, such as handling a budget and ones finances and is an important skill to know how to apply math and to show some understanding of simple math in many careers.
For many students math seems to be a rather daunting task, this can be attributed to the “wordiness” of many math problems which makes simply understanding the problem an issue, never mind actually solving it. Furthermore, it is one thing to see a teacher do an example on the board and have it look nice and easy but, for a student to do the math on their own is more difficult, especially when the question is not worded very clearly.
When trying to help students better understand mathematics it often helps to capture their interests by presenting real life examples and applications as opposed to just showing an example that works.
From the perspective of this student, some important qualities that a math teacher should possess are to be well organized, knowledgeable of the subject area and has clear goals for their teaching and learning objectives, and is available and willing to help students as often as possible.
Group projects can be seen as a good way boost marks but more importantly it gives students a chance to learn about math in different ways. A project is a nice change of pace from doing math problems day in and day out. This could be something good to do at the end of a unit, term, or the end of the year. However, it would be a good idea to keep in mind that, especially in the higher grades, most students already have a lot of other projects and larger assignments to do and that we should be aware of this and not overburden them with something that they really wouldn’t appreciate doing at the end of the year. If you are going to give a group project to students in either math 11 or math 12 it may be received better nearer the beginning of the year instead of nearer the end of the year.
Student who doesn’t enjoy mathematics
I was to interview the student who disliked mathematics. Based on his response to our 5 burning questions, it was obvious to note that this student does not favor mathematics. However, even though he disliked mathematics, he was able to admit that mathematics did have useful benefits in reality. An example that he gave was related to purchasing items to renovate a house. The student also mentioned that he was not interested in the applications of mathematics in reality; he would simply be content to know the steps to get to the right answers.
Like many other students who struggle with mathematics, the common issue among them is the need for more time to get their work done and constant practice to help them remember the steps more efficiently. It is true that mathematics is one of the hardest subjects to learn in high school, from this student’s perspective, he seemed to prefer instrumental mathematics over relational mathematics due to the fact that mathematics is not really an interest of his.
Since mathematics is not interesting to this student, he would want to get his work done and over with as soon as possible. Thus he preferred learning and working on mathematics in a private tutoring environment, where it is only him and either a teacher or tutor to teach him the mechanics of solving the math questions.
When I asked him what he thought about the concept of group projects, he replied by saying that he would not be able to concentrate on both talking with group mates and working on mathematics. He preferred sticking to the curriculum and keeping it simple, so that he can finish his mathematics education quickly.
(Niyaz, Howard, Matt)
I had the opportunity to ask two teachers these few questions; they will be referred to as teacher A and teacher B. Teacher A has been teaching high school math for about 10 years now and has taught mostly in international schools in Asia, namely Macau, China and Korea. She graduated from Australia. On the other hand, teacher B has taught in Canada for a total of 9 years, 2 years in Ontario and 7 years in British Columbia. He is teaching in a large public school in the Vancouver region. He graduated from Ontario.
Response to question 1:
Both of the teachers expressed that they try to give students questions that they can handle, in order for them to feel successful and to gain some confidence, this is where they would gain their interest in math. For example, “working with these students one on one and making sure they know how pleased I am when they do well on tests, quizzes or assignments.” was teacher B’s response.
Response to question 2:
Both teachers use computers and certain softwares to teach their lessons. Teacher A works in a school where all students have a computer and are networked with the teacher’s computer, so she can do things live on her computer in order to teach her students, this is very engaging for students living in a age where they are always around technology. Teacher B does not have the same luxuries as teacher A, nonetheless he also uses a computer and a projector for his class.
Response to question 3:
Teacher A said that AP calculus is the goal for most students at her school, and she feels that it prepares them quite well. She finds no issues of the curriculum that she is teaching right now. Teacher B on the other hand hasn’t found any concerns so far in the new grade 10 curriculum, since it is the beginning of the year. However, he did mention that there is less material to teach so there is more room to explore with new activities that he has thought of so far to help with the students learning. He also mentioned, “I think students learn differently than before internet became available and technology became so advanced. Now, students are accustomed to retrieving information instantaneously and networking in larger groups. There almost seems like an impatience to learn when the Socratic method of teaching is used.” I found this to be the most useful knowledge to know about how students like to learn these days.
Response to question 4:
Teacher A said that she has felt boredom but it was because she was teaching the same things in the same way due to not having much access to resources. However, in order to get over her boredom she started doing more research online for new ways to use technology and read up on tweets about all these things other teachers were doing in other schools to improve her teaching style. Teacher B on the other hand, mentioned that he has never been bored since he has been always creating new lesson plans each year he teaches.
Response to question 5:
Both teachers expressed that they are not interested in administration; however that might change in due time.
Student Responses:
Student who enjoys mathematics (Based on responses given):
Math gives us a basic knowledge to be able do everyday things, such as handling a budget and ones finances and is an important skill to know how to apply math and to show some understanding of simple math in many careers.
For many students math seems to be a rather daunting task, this can be attributed to the “wordiness” of many math problems which makes simply understanding the problem an issue, never mind actually solving it. Furthermore, it is one thing to see a teacher do an example on the board and have it look nice and easy but, for a student to do the math on their own is more difficult, especially when the question is not worded very clearly.
When trying to help students better understand mathematics it often helps to capture their interests by presenting real life examples and applications as opposed to just showing an example that works.
From the perspective of this student, some important qualities that a math teacher should possess are to be well organized, knowledgeable of the subject area and has clear goals for their teaching and learning objectives, and is available and willing to help students as often as possible.
Group projects can be seen as a good way boost marks but more importantly it gives students a chance to learn about math in different ways. A project is a nice change of pace from doing math problems day in and day out. This could be something good to do at the end of a unit, term, or the end of the year. However, it would be a good idea to keep in mind that, especially in the higher grades, most students already have a lot of other projects and larger assignments to do and that we should be aware of this and not overburden them with something that they really wouldn’t appreciate doing at the end of the year. If you are going to give a group project to students in either math 11 or math 12 it may be received better nearer the beginning of the year instead of nearer the end of the year.
Student who doesn’t enjoy mathematics
I was to interview the student who disliked mathematics. Based on his response to our 5 burning questions, it was obvious to note that this student does not favor mathematics. However, even though he disliked mathematics, he was able to admit that mathematics did have useful benefits in reality. An example that he gave was related to purchasing items to renovate a house. The student also mentioned that he was not interested in the applications of mathematics in reality; he would simply be content to know the steps to get to the right answers.
Like many other students who struggle with mathematics, the common issue among them is the need for more time to get their work done and constant practice to help them remember the steps more efficiently. It is true that mathematics is one of the hardest subjects to learn in high school, from this student’s perspective, he seemed to prefer instrumental mathematics over relational mathematics due to the fact that mathematics is not really an interest of his.
Since mathematics is not interesting to this student, he would want to get his work done and over with as soon as possible. Thus he preferred learning and working on mathematics in a private tutoring environment, where it is only him and either a teacher or tutor to teach him the mechanics of solving the math questions.
When I asked him what he thought about the concept of group projects, he replied by saying that he would not be able to concentrate on both talking with group mates and working on mathematics. He preferred sticking to the curriculum and keeping it simple, so that he can finish his mathematics education quickly.
(Niyaz, Howard, Matt)
Thursday, September 23, 2010
My reflection on teaching how to juggle a soccer ball
Feedback from peers:
Strength:
1. Explained learning goals clearly.
2. Well explained steps on how to juggle a ball.
3. Was encouraging and patient.
4. Got everyone involved.
Areas of Development:
1. Time management.
My own reflections:
I felt like the overall lesson went well, and my reflections on my strengths were very similar to what my peers observed and interestingly enough, the area of improvement was also the same, time management. I should have thought how long each part of the lesson would have taken prior to teaching it. I also felt that my explanations could have been better as well.
Strength:
1. Explained learning goals clearly.
2. Well explained steps on how to juggle a ball.
3. Was encouraging and patient.
4. Got everyone involved.
Areas of Development:
1. Time management.
My own reflections:
I felt like the overall lesson went well, and my reflections on my strengths were very similar to what my peers observed and interestingly enough, the area of improvement was also the same, time management. I should have thought how long each part of the lesson would have taken prior to teaching it. I also felt that my explanations could have been better as well.
Wednesday, September 22, 2010
Monday, September 20, 2010
Response to Video: Dave Hewitt
I found the video to be useful for us. Teaching math in an auditory method is rather different according to my experience. However, I think there are some advantages and disadvantages.
To think of the positive aspects of it, I think it makes students think of math mentally and it enables them to use their ears in terms of understanding mathematical expressions to a certain extent. I’ve notice that if I’m trying to explain a mathematical problem (a simple one) to my friend over the phone, it is very hard for them to picture it, just because they haven’t had the practice of listening to math. Most of people’s mathematical education was done on paper, and sort of ignores the other senses such as the ears and mouth.
To think of the down fall of teaching math in an auditory way, it may just not work as perfectly as the video showed it to be, especially for students who can’t learn well with their ears. Hearing mathematical expressions and picturing them in one’s own mine may not be so easy for them, or the expression they think of may not be exactly what you said. Although, this may be a problem, it still can be used as a learning opportunity. For example, the places where he added the brackets and which expressions were to be divided by 3 and so on. These may be places where the teacher can correct the students’ thinking, by of course writing the expression out.
Overall, I found this method of teaching math quite interesting and will think of ways of how it can be implemented while teaching certain topics in mathematics.
To think of the positive aspects of it, I think it makes students think of math mentally and it enables them to use their ears in terms of understanding mathematical expressions to a certain extent. I’ve notice that if I’m trying to explain a mathematical problem (a simple one) to my friend over the phone, it is very hard for them to picture it, just because they haven’t had the practice of listening to math. Most of people’s mathematical education was done on paper, and sort of ignores the other senses such as the ears and mouth.
To think of the down fall of teaching math in an auditory way, it may just not work as perfectly as the video showed it to be, especially for students who can’t learn well with their ears. Hearing mathematical expressions and picturing them in one’s own mine may not be so easy for them, or the expression they think of may not be exactly what you said. Although, this may be a problem, it still can be used as a learning opportunity. For example, the places where he added the brackets and which expressions were to be divided by 3 and so on. These may be places where the teacher can correct the students’ thinking, by of course writing the expression out.
Overall, I found this method of teaching math quite interesting and will think of ways of how it can be implemented while teaching certain topics in mathematics.
Friday, September 17, 2010
Memorable Math Teachers
I always had a passion for math. However, there were a few math teachers of mine who really increased this passion of mine. One of whom was named Mrs. Z. She taught me throughout grade 3 to 5. She was a professor at university who decided to start teaching the younger grades. She had high expectations of us, so that was rather intimating, nonetheless I was always up for a good challenge.
What made her a great teacher was that she taught us the content of math, and then gave us the little tricks on how to deal with certain problems, and little short cuts here and there. It not only made the subject easier to understand, it also gave us an opportunity to look for our own short cuts and sure enough we developed our own. She knew how to explain things very carefully and told us how to look for certain words in problem solving questions that would help us figure out how to set up the problem.
In addition, she cared a lot for her students and making sure that we did well. She knew how much capacity each student had and set her expectations according to our capacities.
Looking back on the things that my various math teachers did that I disliked, I can’t recall that many. However, there were times when I saw how my teachers dealt with a certain student who were not so good at math. For example, asking a student to do a problem in front of the class when they were not paying attention to begin with. I felt sorry for those students, but I guess that was the way the teacher wanted to teach them a lesson about paying attention in class. Not the best way in my opinion.
That’s all I can think of right now, I would write more but there is not enough time to look back at all of my math teachers.
What made her a great teacher was that she taught us the content of math, and then gave us the little tricks on how to deal with certain problems, and little short cuts here and there. It not only made the subject easier to understand, it also gave us an opportunity to look for our own short cuts and sure enough we developed our own. She knew how to explain things very carefully and told us how to look for certain words in problem solving questions that would help us figure out how to set up the problem.
In addition, she cared a lot for her students and making sure that we did well. She knew how much capacity each student had and set her expectations according to our capacities.
Looking back on the things that my various math teachers did that I disliked, I can’t recall that many. However, there were times when I saw how my teachers dealt with a certain student who were not so good at math. For example, asking a student to do a problem in front of the class when they were not paying attention to begin with. I felt sorry for those students, but I guess that was the way the teacher wanted to teach them a lesson about paying attention in class. Not the best way in my opinion.
That’s all I can think of right now, I would write more but there is not enough time to look back at all of my math teachers.
Five burning questions for teachers and students of math
Teacher
1. How have you motivated students who are not naturally excited about math?
2. Do you make use of different technologies in your class to encourage different learning styles, if so what are they and how do you implement them?
3. How do you feel about the current math curriculum, what is good what isn’t? Do you feel that there is a growing disconnect between math 12 and what is expected/necessary for first year university math?
4. From the time that you started teaching in a high school until now, have you ever been bored? If so, how did you deal with the issue of boredom?
5. When do you plan to retire? Do you plan do go into administration?
Student
1. Do you think mathematics is useful in life? why/why not?
2. Why do you think students dislike or like math?
3. When learning math does it help to see real life applicable examples and applications or is it enough to know that something works and to see an example of it whether or not it has an obvious application?
4. What would you change about the way mathematics is being taught if any?
5. Would doing group research projects be helpful and is it something that would make your math class more enjoyable for you?
Here are our group's five questions.
Group: Mathew, Howard, and myself.
1. How have you motivated students who are not naturally excited about math?
2. Do you make use of different technologies in your class to encourage different learning styles, if so what are they and how do you implement them?
3. How do you feel about the current math curriculum, what is good what isn’t? Do you feel that there is a growing disconnect between math 12 and what is expected/necessary for first year university math?
4. From the time that you started teaching in a high school until now, have you ever been bored? If so, how did you deal with the issue of boredom?
5. When do you plan to retire? Do you plan do go into administration?
Student
1. Do you think mathematics is useful in life? why/why not?
2. Why do you think students dislike or like math?
3. When learning math does it help to see real life applicable examples and applications or is it enough to know that something works and to see an example of it whether or not it has an obvious application?
4. What would you change about the way mathematics is being taught if any?
5. Would doing group research projects be helpful and is it something that would make your math class more enjoyable for you?
Here are our group's five questions.
Group: Mathew, Howard, and myself.
Monday, September 13, 2010
Response to Relational and Instrumental Understanding
To my surprise, I was shocked and excited to read this article, only to find that I was thinking about this debate of relational and instrumental understanding not too long ago in my own head, something I began formulating once completing my undergraduate degree in mathematics. I realized that during my university studies, I felt that the math that I was being taught in university was rather different from the math I was being taught in high school, and the difference I now realize is exactly the difference between relational and instrumental mathematics.
It was confusing at first what the author meant by these two understandings, but with a few examples and some of my own experiences I soon realized the distinction between the two. I too, like the author, feel slightly inclined towards agreeing with relational understanding, simply because unfortunately, I was taught mostly by instrumental understanding. This harmed me in university. Since most of the higher level courses in mathematics at university require you to understand the subject relationally; or else you would not survive learning mathematics at that level. From a stand point of becoming a teacher of mathematics, I think it is important for me to understand everything in a relational manner, just so that I will be able to explain things to students and answer the “why” questions that my students will have. Which I think I would be able to do for high school mathematics. The way I made sense of these two understandings in my own head was:
1. Instrumental understanding equaled the mechanics of solving a problem, without questioning why. The emphasis is on how. (e.g. how do I get the answer)
2. Relational understanding equaled understanding the principles of why certain processes work while solving a problem. The emphasis is on why. (e.g. why do I do…)
I would have to completely agree with the author that instrumental understanding is sufficient for a short time, but then breaks down further down the road, as it was in my own experience going through university. I realized that one can actually do well in mathematics just by memorizing a combination of rules and using them in practice. This tool did NOT last and in fact is not at all sufficient at a higher level of mathematics.
This brings me to my other thought that, are we teaching math to students so that they can apply it to their daily situations involving doing some math or are we training PhD math students? I understand that my lectures in university were doing exactly the latter, while the former was being implemented at the high school level so that I would know how much change I should get when I buy something. The answer to this questions is still not yet clear to me.
It is also true that to teach relationally, it will take a longer period of time to explain and for students to understand, however, it would be useful as the author explained it (e.g. point 4, 6, 7). I would have to disagree with point 5, since it’s a little contradicting with how he mentioned that instrumental is easier to understand, which I have to agree. It is much easier to learn the process of doing something rather than learning why to do such a step by step process.
My thought on why most teachers teach instrumentally or why students know math instrumentally is: (to add a little from what the author already mentioned)
1. It’s the easy way of teaching it and it makes the students seem like they are doing well and getting good grades.
2. Most students are not interested in “why” the process works.
3. The majority of students were taught math instrumentally from the very beginning.
The reason for this (number 2) I think is because they are not only learning math in school, they are learning a variety of other subjects, so to learn things efficiently they want to know the easy way of solving the problem, that’s it. The only students who are concern with the “why” are those students who have a natural passion for mathematics and they will be the ones who go on to do further in the field of math. For the passionate students, I think it is essential for them to know the subject relationally and it wouldn’t hurt for the rest of the students to know why. The problem is that students are forgetting to ask the “why” questions. They are so caught up in the process of learning a subject that they lose their natural curiosity for subjects. Nonetheless, I think it is essential for a math educator to understand when it is necessary for the student to know the subject relationally and implement that style of teaching whenever possible. This invariably will help the student become motivated for learning math.
To conclude, I think a computer would be able to do any math work instrumentally, but it wouldn’t be able to, on its own, work the math relationally. So with that in mind, I think it’s important to know why certain processes work in math problems, and knowing what is going on behind the scene. After all, if aliens come and kill all the mathematicians in the world, I’m hoping that the rest of the people would know why we do things a certain way in math. To my understanding, I think learning something instrumentally is just touching the surface of the matter, but learning it relationally is diving deep into the subject. If you would know it relationally, you would be more capable of explaining it to someone who is learning it. The difference between instrumental and relational to me is like knowing how to use a computer vs. knowing how a computer actually works. It really depends on what you are interested in.
It was confusing at first what the author meant by these two understandings, but with a few examples and some of my own experiences I soon realized the distinction between the two. I too, like the author, feel slightly inclined towards agreeing with relational understanding, simply because unfortunately, I was taught mostly by instrumental understanding. This harmed me in university. Since most of the higher level courses in mathematics at university require you to understand the subject relationally; or else you would not survive learning mathematics at that level. From a stand point of becoming a teacher of mathematics, I think it is important for me to understand everything in a relational manner, just so that I will be able to explain things to students and answer the “why” questions that my students will have. Which I think I would be able to do for high school mathematics. The way I made sense of these two understandings in my own head was:
1. Instrumental understanding equaled the mechanics of solving a problem, without questioning why. The emphasis is on how. (e.g. how do I get the answer)
2. Relational understanding equaled understanding the principles of why certain processes work while solving a problem. The emphasis is on why. (e.g. why do I do…)
I would have to completely agree with the author that instrumental understanding is sufficient for a short time, but then breaks down further down the road, as it was in my own experience going through university. I realized that one can actually do well in mathematics just by memorizing a combination of rules and using them in practice. This tool did NOT last and in fact is not at all sufficient at a higher level of mathematics.
This brings me to my other thought that, are we teaching math to students so that they can apply it to their daily situations involving doing some math or are we training PhD math students? I understand that my lectures in university were doing exactly the latter, while the former was being implemented at the high school level so that I would know how much change I should get when I buy something. The answer to this questions is still not yet clear to me.
It is also true that to teach relationally, it will take a longer period of time to explain and for students to understand, however, it would be useful as the author explained it (e.g. point 4, 6, 7). I would have to disagree with point 5, since it’s a little contradicting with how he mentioned that instrumental is easier to understand, which I have to agree. It is much easier to learn the process of doing something rather than learning why to do such a step by step process.
My thought on why most teachers teach instrumentally or why students know math instrumentally is: (to add a little from what the author already mentioned)
1. It’s the easy way of teaching it and it makes the students seem like they are doing well and getting good grades.
2. Most students are not interested in “why” the process works.
3. The majority of students were taught math instrumentally from the very beginning.
The reason for this (number 2) I think is because they are not only learning math in school, they are learning a variety of other subjects, so to learn things efficiently they want to know the easy way of solving the problem, that’s it. The only students who are concern with the “why” are those students who have a natural passion for mathematics and they will be the ones who go on to do further in the field of math. For the passionate students, I think it is essential for them to know the subject relationally and it wouldn’t hurt for the rest of the students to know why. The problem is that students are forgetting to ask the “why” questions. They are so caught up in the process of learning a subject that they lose their natural curiosity for subjects. Nonetheless, I think it is essential for a math educator to understand when it is necessary for the student to know the subject relationally and implement that style of teaching whenever possible. This invariably will help the student become motivated for learning math.
To conclude, I think a computer would be able to do any math work instrumentally, but it wouldn’t be able to, on its own, work the math relationally. So with that in mind, I think it’s important to know why certain processes work in math problems, and knowing what is going on behind the scene. After all, if aliens come and kill all the mathematicians in the world, I’m hoping that the rest of the people would know why we do things a certain way in math. To my understanding, I think learning something instrumentally is just touching the surface of the matter, but learning it relationally is diving deep into the subject. If you would know it relationally, you would be more capable of explaining it to someone who is learning it. The difference between instrumental and relational to me is like knowing how to use a computer vs. knowing how a computer actually works. It really depends on what you are interested in.
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