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Wednesday, October 26, 2011
Wednesday, October 19, 2011
See What They Can Do
My good friend and colleague, G. Edgar Parker, Professor of Mathematics, James Madison University, has written a wonderful story for The IBL Blog. Thank you, Ed!
See What They Can Do
The bedrock for what became the basis for my teaching philosophy was forged when I was an undergraduate at Guilford College. I was in college for all of the wrong reasons; I didn’t like school, but I knew my parents expected me to go to college and I was in no hurry to find gainful employment unless I could make a living playing baseball. I was at Guilford for all the wrong reasons; I didn’t want to follow my older brother Elwood for another four years (the expectations associated with coming behind a straight-A student who was also a two-sport varsity athlete, all-conference in one sport and team captain in the other, even if they were imaginary, seemed real to me), but I went anyway because I had seen the baseball teams that my two older brothers played on there and knew that I could pitch better than anybody I had seen pitch in those six years (and besides, it had snowed on my campus visit to the school I visited that I liked the most and it was the week before the regular baseball season’s schedule back home was to start).
During the advising session before my first semester, fall of 1965, I jettisoned any ideas of following the subject that intrigued me most, Biology, because I found out that, at Guilford, you went to labs as a priority over practice and made up labs if you missed them for games. My advisor, J. R. Boyd, placed me in two mathematics courses, Calculus I and Mr. Boyd’s Linear Point-Set Theory. Interestingly enough, it was calculus that scared me. I had actually heard of that and didn’t think I was prepared. I should have guessed that Mr. Boyd was cut from different cloth; when I protested that I had only had four weeks of trigonometry in high school, he had “reassured” me by telling me, “Don’t worry, if you need to know more, you’ll learn it.”.
I was so naïve that I just assumed that the way point-set was being taught might be something that people did in college, so I just tried to solve the problems and hung on. Since Mr. Boyd never said anything negative, for all I knew, I was doing okay. I was getting some problems, or at least Mr. Boyd seemed satisfied with what he wrote on the board when I described my arguments. Since I didn’t know the difference between a hard problem and an easy problem, I was trying them in the sequence they came in the notebook. And then I ran into the problem that, looking back, could have been stated as “the continuum is not denumerable”, but was stated in a way that tempted a student to find a map from the natural numbers onto the numbers. I thought I had an argument, but Hal Phillips, whom I considered to be the best student in the class, chose the problem when his turn came. As Hal presented, I saw his ideas (which I had discovered myself) crash one by one and finally doom the argument on which I had worked so long and hard. I was, at that time, still pretty meek in public and very reticent to call attention to myself, but the ardor of the moment overwhelmed me and I burst out in class in frustration (Mr. Boyd had made no suggestion as to how to remedy the now-evident flaws in the argument), “Mr. Boyd, why don’t you just show us how to do it?”. Without missing a beat, he turned to me and said, “Mr. Parker, why should I limit you to what I know?”. It was an opportunity to learn a lesson about teaching, but at the time, I was just relieved at the way Mr.
Boyd handled the situation and didn’t add to the embarrassment I had created by calling attention to myself. Besides, I still wanted to pitch in the Big Leagues and education was just a diversion to enable me to continue to play the game I loved. Education was not my intended vocation.
Fast forward to 1977. I had finished my PhD in mathematics at Emory University, had courses given by Moore method from David Ford, Bill Mahavier, John Neuberger, Phil Tonne, and Mary Frances Neff, and chosen academe as my vocational home. I recognized by now that Moore method is what allowed me to blossom creatively and provided me with the tools to learn other persons’ mathematics as well. Still possessing, at that time, some sense of humility, I concluded that Moore method should provide the same growth potential for others it gave to me and decided to use it as the core for teaching the mathematics courses in the major at Pan American University that I gave. Over the next seven years, for each such course, I painstakingly constructed a problem sequence with the self-assurance that, anybody who could do the problems in sequence would have to be able to see how to do the next one in the sequence because the connections were so obvious to me. And each time, as often as not, the students leaped right past my lemmas to the important problems and found ways I hadn’t thought of to do the problems or ways that I considered “less natural”, or patiently proved my lemmas, but then showed me what the lemmas should have been by proving the theorem without using the lemmas.
Fast forward to the summer of 1988. I had been at James Madison University since 1984 and I now understood how to follow the students’ ideas in the major courses, but still thought, for some reason, that non-majors had to be led to the fountain before they could drink. That summer I was assigned yet another section of Mathematics 103: The Nature of Mathematics, the lowest numbered mathematics course in the catalogue and a course that I had taught every semester since it was introduced in 1985 (I had developed such a course for a less well-prepared clientele at Pan American, so, naturally, even though I had voted against our department offering such a course for General Studies at James Madison since I thought a school of our pretension should make calculus the core requirement, I was picked to give one of the initial offerings and was blessed with it on a continuing basis.) . On a lark, I had the following conversation with myself: “The students are in this course because all they want is a mathematics credit. So no one will fail the course. But I will not tell them that and I will teach them as if they were majors.” So I designed a course in which I “taught” them rigor by having them justify, on the basis of the field axioms applied to the words number, +, and $\times$, some of the fundamental techniques from high school algebra, and then put them to work on proving, as theorems, the field axioms stated for the ordered pairs of numbers with addition and multiplication defined so as to make the structure the complex numbers (without having i a part of the notation, I hoped nobody would recognize that they may have studied this algebra already and I was correct in this guess). The only modification of Moore method I made was that the class was split into eleven groups, each with two or three students, and each group was responsible for proving a single theorem (chosen by lot), with all students responsible for certifying the correctness of the arguments and for reproducing and using what they certified to be correct. The class got them all! It was not that the students populating this class could not do the mathematics, it was my not believing they could do it that never gave them a chance to do it. The summer of 1988 may have been an act of God. I have used this problem set many times since, and no other class has gotten them all. But every class has gotten some of them, and most classes have gotten the existence of reciprocals, the problem I consider the most difficult in the problem set.
Dr. Parker, why should you limit your students to the way you think about things, and why should you impose limitations on them?
The longer I teach, the more I wonder why I didn’t ask myself this question earlier.
G. Edgar Parker, James Madison University
See What They Can Do
The bedrock for what became the basis for my teaching philosophy was forged when I was an undergraduate at Guilford College. I was in college for all of the wrong reasons; I didn’t like school, but I knew my parents expected me to go to college and I was in no hurry to find gainful employment unless I could make a living playing baseball. I was at Guilford for all the wrong reasons; I didn’t want to follow my older brother Elwood for another four years (the expectations associated with coming behind a straight-A student who was also a two-sport varsity athlete, all-conference in one sport and team captain in the other, even if they were imaginary, seemed real to me), but I went anyway because I had seen the baseball teams that my two older brothers played on there and knew that I could pitch better than anybody I had seen pitch in those six years (and besides, it had snowed on my campus visit to the school I visited that I liked the most and it was the week before the regular baseball season’s schedule back home was to start).
During the advising session before my first semester, fall of 1965, I jettisoned any ideas of following the subject that intrigued me most, Biology, because I found out that, at Guilford, you went to labs as a priority over practice and made up labs if you missed them for games. My advisor, J. R. Boyd, placed me in two mathematics courses, Calculus I and Mr. Boyd’s Linear Point-Set Theory. Interestingly enough, it was calculus that scared me. I had actually heard of that and didn’t think I was prepared. I should have guessed that Mr. Boyd was cut from different cloth; when I protested that I had only had four weeks of trigonometry in high school, he had “reassured” me by telling me, “Don’t worry, if you need to know more, you’ll learn it.”.
I was so naïve that I just assumed that the way point-set was being taught might be something that people did in college, so I just tried to solve the problems and hung on. Since Mr. Boyd never said anything negative, for all I knew, I was doing okay. I was getting some problems, or at least Mr. Boyd seemed satisfied with what he wrote on the board when I described my arguments. Since I didn’t know the difference between a hard problem and an easy problem, I was trying them in the sequence they came in the notebook. And then I ran into the problem that, looking back, could have been stated as “the continuum is not denumerable”, but was stated in a way that tempted a student to find a map from the natural numbers onto the numbers. I thought I had an argument, but Hal Phillips, whom I considered to be the best student in the class, chose the problem when his turn came. As Hal presented, I saw his ideas (which I had discovered myself) crash one by one and finally doom the argument on which I had worked so long and hard. I was, at that time, still pretty meek in public and very reticent to call attention to myself, but the ardor of the moment overwhelmed me and I burst out in class in frustration (Mr. Boyd had made no suggestion as to how to remedy the now-evident flaws in the argument), “Mr. Boyd, why don’t you just show us how to do it?”. Without missing a beat, he turned to me and said, “Mr. Parker, why should I limit you to what I know?”. It was an opportunity to learn a lesson about teaching, but at the time, I was just relieved at the way Mr.
Boyd handled the situation and didn’t add to the embarrassment I had created by calling attention to myself. Besides, I still wanted to pitch in the Big Leagues and education was just a diversion to enable me to continue to play the game I loved. Education was not my intended vocation.
Fast forward to 1977. I had finished my PhD in mathematics at Emory University, had courses given by Moore method from David Ford, Bill Mahavier, John Neuberger, Phil Tonne, and Mary Frances Neff, and chosen academe as my vocational home. I recognized by now that Moore method is what allowed me to blossom creatively and provided me with the tools to learn other persons’ mathematics as well. Still possessing, at that time, some sense of humility, I concluded that Moore method should provide the same growth potential for others it gave to me and decided to use it as the core for teaching the mathematics courses in the major at Pan American University that I gave. Over the next seven years, for each such course, I painstakingly constructed a problem sequence with the self-assurance that, anybody who could do the problems in sequence would have to be able to see how to do the next one in the sequence because the connections were so obvious to me. And each time, as often as not, the students leaped right past my lemmas to the important problems and found ways I hadn’t thought of to do the problems or ways that I considered “less natural”, or patiently proved my lemmas, but then showed me what the lemmas should have been by proving the theorem without using the lemmas.
Fast forward to the summer of 1988. I had been at James Madison University since 1984 and I now understood how to follow the students’ ideas in the major courses, but still thought, for some reason, that non-majors had to be led to the fountain before they could drink. That summer I was assigned yet another section of Mathematics 103: The Nature of Mathematics, the lowest numbered mathematics course in the catalogue and a course that I had taught every semester since it was introduced in 1985 (I had developed such a course for a less well-prepared clientele at Pan American, so, naturally, even though I had voted against our department offering such a course for General Studies at James Madison since I thought a school of our pretension should make calculus the core requirement, I was picked to give one of the initial offerings and was blessed with it on a continuing basis.) . On a lark, I had the following conversation with myself: “The students are in this course because all they want is a mathematics credit. So no one will fail the course. But I will not tell them that and I will teach them as if they were majors.” So I designed a course in which I “taught” them rigor by having them justify, on the basis of the field axioms applied to the words number, +, and $\times$, some of the fundamental techniques from high school algebra, and then put them to work on proving, as theorems, the field axioms stated for the ordered pairs of numbers with addition and multiplication defined so as to make the structure the complex numbers (without having i a part of the notation, I hoped nobody would recognize that they may have studied this algebra already and I was correct in this guess). The only modification of Moore method I made was that the class was split into eleven groups, each with two or three students, and each group was responsible for proving a single theorem (chosen by lot), with all students responsible for certifying the correctness of the arguments and for reproducing and using what they certified to be correct. The class got them all! It was not that the students populating this class could not do the mathematics, it was my not believing they could do it that never gave them a chance to do it. The summer of 1988 may have been an act of God. I have used this problem set many times since, and no other class has gotten them all. But every class has gotten some of them, and most classes have gotten the existence of reciprocals, the problem I consider the most difficult in the problem set.
Dr. Parker, why should you limit your students to the way you think about things, and why should you impose limitations on them?
The longer I teach, the more I wonder why I didn’t ask myself this question earlier.
G. Edgar Parker, James Madison University
Saturday, October 15, 2011
The Teaching Axiom of Choice
I was reminded of this quote by Bertrand Russell, while watching Ken Robinson's TED talks.
One of strongest and most consistent beliefs among the IBL teachers is that creativity can be taught. This is quite evident at the Legacy of R.L. Moore conferences -- IBL practitioners have this unending supply of belief in their students, and not assuming what a student can or cannot do.
I have internalized this for myself as the Teaching Axiom of Choice, which is stated as "Every student has the capacity to learn Mathematics." Our educational system (in the U.S.) has a tendency to label students at a young age. Mathematical ability it thought of by some as a fixed, unchanging attribute that cannot be altered significantly by effort.
What this does to teachers is set expectations, which then in turn affects teaching decisions. It goes like this. If you are a teacher who believe in fixed attributes of students and are given a track of students who are "low level," then you are likely to assign "easy" tasks and not challenge students. Effort is to be minimized and the goal is to just get some of these students through the material so they can pass and get on to the next level.
It is unlikely that problem solving, deep engagement in rich mathematics, and a developing other higher-level thinking would be part of the class. So then the labels lead ultimately to a dry, barren math experience (though unintentional -- no one means to do harm).
It's a choice. We can, though not always easily, believe in our students. Give them our best, and work hard on our teaching methods to provide the best possible chance for learning. I am the first to realize that the system plays a strong role in what we are allowed to do in the classroom. With that said, the Teaching Axiom of Choice then is a precondition to providing transformative experiences for students. If you do not believe in students (or at least willing to suspend belief), then you are unlikely to give your students the kinds of tasks and structured freedom to let them compose, to dream, and to be creative.
"Is a man what he seems to the astronomer, a tiny lump of impure carbon and water crawling impotently on a small and unimportant planet? Or is he what he appears to Hamlet? Is he perhaps both as once?” -- Bertrand Russell.The power of imagination is built quite strongly into the fabric of our very existence. One of the traits that separates us from other animals is that we read poetry and listen to Miles Davis. Other animals communicate and make sounds, but they don't write plays or compose music.
One of strongest and most consistent beliefs among the IBL teachers is that creativity can be taught. This is quite evident at the Legacy of R.L. Moore conferences -- IBL practitioners have this unending supply of belief in their students, and not assuming what a student can or cannot do.
I have internalized this for myself as the Teaching Axiom of Choice, which is stated as "Every student has the capacity to learn Mathematics." Our educational system (in the U.S.) has a tendency to label students at a young age. Mathematical ability it thought of by some as a fixed, unchanging attribute that cannot be altered significantly by effort.
What this does to teachers is set expectations, which then in turn affects teaching decisions. It goes like this. If you are a teacher who believe in fixed attributes of students and are given a track of students who are "low level," then you are likely to assign "easy" tasks and not challenge students. Effort is to be minimized and the goal is to just get some of these students through the material so they can pass and get on to the next level.
It is unlikely that problem solving, deep engagement in rich mathematics, and a developing other higher-level thinking would be part of the class. So then the labels lead ultimately to a dry, barren math experience (though unintentional -- no one means to do harm).
It's a choice. We can, though not always easily, believe in our students. Give them our best, and work hard on our teaching methods to provide the best possible chance for learning. I am the first to realize that the system plays a strong role in what we are allowed to do in the classroom. With that said, the Teaching Axiom of Choice then is a precondition to providing transformative experiences for students. If you do not believe in students (or at least willing to suspend belief), then you are unlikely to give your students the kinds of tasks and structured freedom to let them compose, to dream, and to be creative.
Friday, October 7, 2011
Nuts and Bolts: Small Groups
The topic of this post is group size in collaborative learning environments. There are differing opinions about this, and I will not go into any data regarding small group sizes. Rather I'll talk about what I do, and you can use this as your starting point.
Size: I like groups of size 2 or 3. Clearly 2 is the smallest group size. Once groups get to size 4 or greater, students can hide. This is something I want to avoid.
Mix 'em up -- generally it is a good idea to move students around. One can use the Random() command in excel to create a random list of numbers, and then sort students on that column. Then take bunches of 2 or 3 students to form the groups.
Random shuffles aren't always the best, however. For instance, there may be certain personalities you want to keep away from each other. In this case, one has to do some group engineering. Experience tells me that as you get closer to the boundary of a classroom (i.e. the walls), there exists a greater probability of personalities, especially if groups were allowed to form naturally. I also look for students who are quiet. Then I make a list of personalities and quiet students. Seed each group with exactly one from this list, and then fill the groups with the other students.
The number of people in the group is but one aspect of heathy groups. Students can get noisy, chatty, or a person can start dominating discussions. It is important to be an active coach and mentor. An instructor might be called upon to manage personalities and guide through gentle questions/directions like, "Can I count on you guys to work as a team today?" or "Let's stay in mathland (to the student texting)."
It can also be useful to talk about what it means to be a good group member. A good group member shows up to class ready and on time, listens carefully to their partners, is supportive, and offers appropriate levels of help at the right time (i.e. no blurting out answers and having a genuine respect for learning).
Short version:
Size: I like groups of size 2 or 3. Clearly 2 is the smallest group size. Once groups get to size 4 or greater, students can hide. This is something I want to avoid.
Mix 'em up -- generally it is a good idea to move students around. One can use the Random() command in excel to create a random list of numbers, and then sort students on that column. Then take bunches of 2 or 3 students to form the groups.
Random shuffles aren't always the best, however. For instance, there may be certain personalities you want to keep away from each other. In this case, one has to do some group engineering. Experience tells me that as you get closer to the boundary of a classroom (i.e. the walls), there exists a greater probability of personalities, especially if groups were allowed to form naturally. I also look for students who are quiet. Then I make a list of personalities and quiet students. Seed each group with exactly one from this list, and then fill the groups with the other students.
The number of people in the group is but one aspect of heathy groups. Students can get noisy, chatty, or a person can start dominating discussions. It is important to be an active coach and mentor. An instructor might be called upon to manage personalities and guide through gentle questions/directions like, "Can I count on you guys to work as a team today?" or "Let's stay in mathland (to the student texting)."
It can also be useful to talk about what it means to be a good group member. A good group member shows up to class ready and on time, listens carefully to their partners, is supportive, and offers appropriate levels of help at the right time (i.e. no blurting out answers and having a genuine respect for learning).
Short version:
- Find a good size (2 or 3, in my opinion) and mix of students.
- Coach and mentor group dynamics until they work as teams.
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