Constructivism and Math Education
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Jacob West
Nov 25, 2007, 2:49:30 PM11/25/07
to Graduate Mathematics Foundations
A Constructivist Case for "Recap Courses"
Imagine that you just sat down in a University classroom minutes
before the beginning of class. The room is somewhat large, and other
students are quickly filling in the remaining seats and preparing pads
of paper for taking notes. You do the same as the Professor enters
the room and begins erasing the chalk board. As the Professor begins
to speak, you're reminded of how much you like this class: the
material is fascinating and the lectures are dynamic. Of course, one
consequence is that the Professor tends to write a bit fast, and often
talks even faster. Indeed, there's a lot of material to cover in less
than two short hours. So you focus hard on her words as you
frantically copy down whatever the Professor writes on the board.
It's sometimes difficult to process everything she says, but at least
you'll be able to look back over your notes if you miss anything, that
is, so long as she remembers to write down all the important material.
By the end of the lecture, you have four solid pages of notes and your
hand aches a little, but at least you got it all. Unfortunately,
about half way through the Professor said something that didn't quite
make sense to you, and thinking about it was enough to distract you
from whatever she said next, so that by the time you decided it was
best to figure it out later she was already more than half a
chalkboard ahead of your notes! So the remainder of the lecture you
spent catching up, not really able to listen in favor of writing fast
enough to stay ahead of the eraser. No matter though, you have your
notes.
This might be a typical lecture in most classrooms throughout the
majority of American Universities today, conducted in the traditional
format. The instructor presents some predetermined body of
information as an extended oration, while the students participate
only passively through note taking and the occasional question when
time permits. As Susan Hanely put it [1], "The current American
classroom, whether grade school or college level, tends to resemble a
one-person show with a captive but often comatose audience. Classes
are usually driven by "teacher-talk" and depend heavily on textbooks
for the structure of the course. There is the idea that there is a
fixed world of knowledge that the student must come to know.
Information is divided into parts and built into a whole concept.
Teachers serve as pipelines and seek to transfer their thoughts and
meanings to the passive student. There is little room for
student-initiated questions, independent thought or interaction
between students." Despite having a long tradition in the American
educational system, this format is not without problems. In a study
on undergraduates in a large lecture hall setting conducted at
Berkeley in 1991 [2] and reported on in [1], "it was found that only
20% of the students retained what the instructor discussed after the
lecture. They were too busy taking notes to internalize the
information. Also, after a lecture has passed eight minutes, only 15%
of the students are paying attention."
While there may be evidence and argument for the necessity of reform
in University classrooms, taking up that argument is not my intent
here. Rather, I recognize the tremendous momentum behind an
educational system that "works" more-or-less and the inherent
difficulties in implementing any educational methodology which make
radical reform not only unrealistic, but practically impossible.
Instead, I would like to propose a hopefully more innocuous change to
the current system that may provide the necessary seeds for more
widespread reform; and if not, then would at least prove to be an
interesting and low cost educational experiment. The idea, namely, is
to introduce "Recapitulations" or "Recap Courses" to augment the
traditional lecture format. Before I describe these courses in any
detail, I should first say that the underlying educational philosophy
motivating their creation is a fairly conservative form of
constructivism, as already alluded to by the title of this article.
To be clear, the 'mild' constructivist educational philosophy we
employ here originated in the work of Piaget [3] and purports that
knowledge must be actively constructed by the learner, rather than
passively received from an educator. That is, while the traditional
lecture format provides students with an opportunity to hear what the
Professor thinks is the most important material in the subject under
study, the Recap course aims to provide students with a forum for
actively discussing this material and integrating it into their
understanding of the subject as a whole. It would provide an avenue
for questions and discussions including _how_ and _why_ one should
think about the material in addition to the usual _what_ one should
think about the material. In the constructivist spirit, the Recap
course would emphasize a student driven reconstruction of the material
covered in the traditional lecture. This would provide the students
with the impetus and opportunity to review their lecture notes and a
means to assimilate the material into their current understanding of
the subject through a variety of directed activities.
As an example, suppose we take as our ambient "educational
environment" that of an undergraduate course in mathematics at a
typical American University. In these classrooms, the most common
method of teaching is the traditional lecture format, with material
usually drawn from some standard textbook. In addition to the
regularly scheduled lectures, most such courses also allot 1-2
"discussion sections" per week where students meet with teaching
assistants. The actual content of these discussion sections can vary
widely, but the usual intent is to reinforce the lecture material.
Some typical approaches to this include working through example
problems relevant to the week's assigned homework, addressing and
working through specific homework questions, or even administering
short quizzes. However, in most cases, these are typically organized
again in a traditional lecture format.
In contrast, a Recap course would maintain the goal of reinforcing the
material at hand, but would take a more "student-centered" approach.
What follows is meant to outline the general flavor of how a Recap
course might be run, rather than provide a specific or detailed
guideline. Roughly, a weekly Recap course might develop as follows:
before the beginning of class, the instructor writes 3 problems in one
corner of the board. The first should be 'easier' than the others,
perhaps drawn from an example already worked out in the text or
presented during a lecture earlier that week. The second should be a
homework problem that the instructor anticipates the students
struggling with more so than the other homework problems. The final
problem should involve material not yet covered, but that the students
will be expected to know how to solve by the following week. These
problems will serve as a general guideline for the Recap discussion.
The Recap would begin with student summaries of the material covered
in lecture that week. What are we doing? This can be done
effectively in several ways, depending on class size, with possible
options being open discussion, short written summaries of the key
points followed by some sharing and directed discussion of these
points, or even short group presentations. These openings should then
develop into a discussion of why this material is relevant to the
development of the course, the subject under study, and possibly other
areas of study outside of this particular course. Why do we care?
Once the key ideas have been discussed and some motivation has been
provided, the Recap should shift focus to the problems on the board.
What can we now solve? In small groups, the students should discuss
the first problem, write up a solution, and give a short explanation
of where the key ideas entered into solving the problem. All
together, the class might then discuss the second problem in a
brainstorming type format, with the instructor making a list of
possible ideas, and perhaps carrying out some of those ideas under the
guidance of the class. Finally, the third problem should be discussed
in the context of how our current ideas might be extended to solve
this new (possibly nonsensical in the current context!) problem.
Again, this discussion could be left entirely open, or guided by first
taking a short time for individual written responses. What have we
learned? The Recap should always end with students giving written
recap of the discussion and what was learned, as well as any feedback
or remaining unanswered questions. These responses would then be
incorporated into the following week's Recap.
Clearly, Recap courses would be a marked departure from the
traditional lecture format, but would serve to augment rather than
displace the traditional lectures by providing an alternative format
and teaching methodology that may greatly benefit those students upon
whom traditional lectures fail. Moreover, most Universities already
have a mechanism in place for implementing Recap courses during the
usual discussion sections, so that no additional administrative
overhead would be required and any given Professor's preferred method
of teaching would not be disrupted. Therefore, Recap courses
represent a simple and relatively easy to implement adjustment to the
standard University curriculum that nonetheless may vastly benefit the
student learning process.
References:
[1] "On Constructivism", Susan Hanely. Maryland Collaborative for
Teacher Preparation, 1994. Web address:
http://www.inform.umd.edu/UMS+State/UMD-Projects/MCTP/Essays/Constructivism.txt
[2] T. A. Angelo (Ed.), Classroom Research: Early Lessons From
Success. New Directions for Teaching and Learning, no. 46. San
Francisco: Jossey-Bass, 1991.
[3] "Constructivism and Education: A Shopper's Guide",
M. A. Boudourides. International Conference on the Teaching of
Mathematics, 1998. Web address:
http://thalis.math.upatras.gr/%7Emboudour/articles/constr.html
Imagine that you just sat down in a University classroom minutes
before the beginning of class. The room is somewhat large, and other
students are quickly filling in the remaining seats and preparing pads
of paper for taking notes. You do the same as the Professor enters
the room and begins erasing the chalk board. As the Professor begins
to speak, you're reminded of how much you like this class: the
material is fascinating and the lectures are dynamic. Of course, one
consequence is that the Professor tends to write a bit fast, and often
talks even faster. Indeed, there's a lot of material to cover in less
than two short hours. So you focus hard on her words as you
frantically copy down whatever the Professor writes on the board.
It's sometimes difficult to process everything she says, but at least
you'll be able to look back over your notes if you miss anything, that
is, so long as she remembers to write down all the important material.
By the end of the lecture, you have four solid pages of notes and your
hand aches a little, but at least you got it all. Unfortunately,
about half way through the Professor said something that didn't quite
make sense to you, and thinking about it was enough to distract you
from whatever she said next, so that by the time you decided it was
best to figure it out later she was already more than half a
chalkboard ahead of your notes! So the remainder of the lecture you
spent catching up, not really able to listen in favor of writing fast
enough to stay ahead of the eraser. No matter though, you have your
notes.
This might be a typical lecture in most classrooms throughout the
majority of American Universities today, conducted in the traditional
format. The instructor presents some predetermined body of
information as an extended oration, while the students participate
only passively through note taking and the occasional question when
time permits. As Susan Hanely put it [1], "The current American
classroom, whether grade school or college level, tends to resemble a
one-person show with a captive but often comatose audience. Classes
are usually driven by "teacher-talk" and depend heavily on textbooks
for the structure of the course. There is the idea that there is a
fixed world of knowledge that the student must come to know.
Information is divided into parts and built into a whole concept.
Teachers serve as pipelines and seek to transfer their thoughts and
meanings to the passive student. There is little room for
student-initiated questions, independent thought or interaction
between students." Despite having a long tradition in the American
educational system, this format is not without problems. In a study
on undergraduates in a large lecture hall setting conducted at
Berkeley in 1991 [2] and reported on in [1], "it was found that only
20% of the students retained what the instructor discussed after the
lecture. They were too busy taking notes to internalize the
information. Also, after a lecture has passed eight minutes, only 15%
of the students are paying attention."
While there may be evidence and argument for the necessity of reform
in University classrooms, taking up that argument is not my intent
here. Rather, I recognize the tremendous momentum behind an
educational system that "works" more-or-less and the inherent
difficulties in implementing any educational methodology which make
radical reform not only unrealistic, but practically impossible.
Instead, I would like to propose a hopefully more innocuous change to
the current system that may provide the necessary seeds for more
widespread reform; and if not, then would at least prove to be an
interesting and low cost educational experiment. The idea, namely, is
to introduce "Recapitulations" or "Recap Courses" to augment the
traditional lecture format. Before I describe these courses in any
detail, I should first say that the underlying educational philosophy
motivating their creation is a fairly conservative form of
constructivism, as already alluded to by the title of this article.
To be clear, the 'mild' constructivist educational philosophy we
employ here originated in the work of Piaget [3] and purports that
knowledge must be actively constructed by the learner, rather than
passively received from an educator. That is, while the traditional
lecture format provides students with an opportunity to hear what the
Professor thinks is the most important material in the subject under
study, the Recap course aims to provide students with a forum for
actively discussing this material and integrating it into their
understanding of the subject as a whole. It would provide an avenue
for questions and discussions including _how_ and _why_ one should
think about the material in addition to the usual _what_ one should
think about the material. In the constructivist spirit, the Recap
course would emphasize a student driven reconstruction of the material
covered in the traditional lecture. This would provide the students
with the impetus and opportunity to review their lecture notes and a
means to assimilate the material into their current understanding of
the subject through a variety of directed activities.
As an example, suppose we take as our ambient "educational
environment" that of an undergraduate course in mathematics at a
typical American University. In these classrooms, the most common
method of teaching is the traditional lecture format, with material
usually drawn from some standard textbook. In addition to the
regularly scheduled lectures, most such courses also allot 1-2
"discussion sections" per week where students meet with teaching
assistants. The actual content of these discussion sections can vary
widely, but the usual intent is to reinforce the lecture material.
Some typical approaches to this include working through example
problems relevant to the week's assigned homework, addressing and
working through specific homework questions, or even administering
short quizzes. However, in most cases, these are typically organized
again in a traditional lecture format.
In contrast, a Recap course would maintain the goal of reinforcing the
material at hand, but would take a more "student-centered" approach.
What follows is meant to outline the general flavor of how a Recap
course might be run, rather than provide a specific or detailed
guideline. Roughly, a weekly Recap course might develop as follows:
before the beginning of class, the instructor writes 3 problems in one
corner of the board. The first should be 'easier' than the others,
perhaps drawn from an example already worked out in the text or
presented during a lecture earlier that week. The second should be a
homework problem that the instructor anticipates the students
struggling with more so than the other homework problems. The final
problem should involve material not yet covered, but that the students
will be expected to know how to solve by the following week. These
problems will serve as a general guideline for the Recap discussion.
The Recap would begin with student summaries of the material covered
in lecture that week. What are we doing? This can be done
effectively in several ways, depending on class size, with possible
options being open discussion, short written summaries of the key
points followed by some sharing and directed discussion of these
points, or even short group presentations. These openings should then
develop into a discussion of why this material is relevant to the
development of the course, the subject under study, and possibly other
areas of study outside of this particular course. Why do we care?
Once the key ideas have been discussed and some motivation has been
provided, the Recap should shift focus to the problems on the board.
What can we now solve? In small groups, the students should discuss
the first problem, write up a solution, and give a short explanation
of where the key ideas entered into solving the problem. All
together, the class might then discuss the second problem in a
brainstorming type format, with the instructor making a list of
possible ideas, and perhaps carrying out some of those ideas under the
guidance of the class. Finally, the third problem should be discussed
in the context of how our current ideas might be extended to solve
this new (possibly nonsensical in the current context!) problem.
Again, this discussion could be left entirely open, or guided by first
taking a short time for individual written responses. What have we
learned? The Recap should always end with students giving written
recap of the discussion and what was learned, as well as any feedback
or remaining unanswered questions. These responses would then be
incorporated into the following week's Recap.
Clearly, Recap courses would be a marked departure from the
traditional lecture format, but would serve to augment rather than
displace the traditional lectures by providing an alternative format
and teaching methodology that may greatly benefit those students upon
whom traditional lectures fail. Moreover, most Universities already
have a mechanism in place for implementing Recap courses during the
usual discussion sections, so that no additional administrative
overhead would be required and any given Professor's preferred method
of teaching would not be disrupted. Therefore, Recap courses
represent a simple and relatively easy to implement adjustment to the
standard University curriculum that nonetheless may vastly benefit the
student learning process.
References:
[1] "On Constructivism", Susan Hanely. Maryland Collaborative for
Teacher Preparation, 1994. Web address:
http://www.inform.umd.edu/UMS+State/UMD-Projects/MCTP/Essays/Constructivism.txt
[2] T. A. Angelo (Ed.), Classroom Research: Early Lessons From
Success. New Directions for Teaching and Learning, no. 46. San
Francisco: Jossey-Bass, 1991.
[3] "Constructivism and Education: A Shopper's Guide",
M. A. Boudourides. International Conference on the Teaching of
Mathematics, 1998. Web address:
http://thalis.math.upatras.gr/%7Emboudour/articles/constr.html
Jacob West
Nov 25, 2007, 2:59:27 PM11/25/07
to Graduate Mathematics Foundations
I realize that this is a little long, so I also put up a copy that
might be easier to print here:
http://math.ucsc.edu/~jwest/recap.pdf
> Teacher Preparation, 1994. Web address:http://www.inform.umd.edu/UMS+State/UMD-Projects/MCTP/Essays/Construc...
might be easier to print here:
http://math.ucsc.edu/~jwest/recap.pdf
megan
Nov 26, 2007, 2:25:23 AM11/26/07
to Graduate Mathematics Foundations
I think that Jake's idea of a "recap course" is a great idea. It
seems to me to be a social constructivism idea, which I tend to agree
with most. I don't think that students' education should be entirely
left up to the student, because as I said before I think many would
opt to do nothing. I do believe that they should be promoted to
interpret material individually, or in small groups. Many of my
teachers in math have promoted group work, but there was no time for
it in class. I have observed in my own experience, that the students
whom are most likely to start an outside study group are ones who
excel in their individual work. Students, who are less likely to ask
questions to their peers or the teachers, tend to feel inadequate
about their abilities in that subject and are embarrassed if they ask
a "stupid question". I think that if small group work or individual
work was implemented in classes early on, this could be somewhat
avoided.
One thing that I worry about in this kind of learning environment is,
will students choose groups that will benefit them in the long run?
The best group work I have done, has been one in which everyone has
different talents but contributes equally. I think if groups are
chosen by the students themselves, they will pick people they are
friends with and not necessarily those who will add to their learning
experience. Also, outspoken gifted students tend to run group
activities leaving little for anyone else to contribute. Perhaps a
way to get around this is to rotate the groups throughout the
quarter.
To add to Jake's idea, I think it would be important to either have
all the TA's for the course, or a smaller classroom size for this type
of instruction. This simply due to the fact that when students work
with their peers, different opinions on an approach to a problem can
lead to a halt in progress. When this happens they tend to turn to
the teacher for an answer, and if many small groups were at work it
would help to have more teachers on hand. Furthermore, different
teachers have different approaches to problems, which could promote
this idea of self-learning. When I taught Pre-College programs I had
one of the best experiences of my teaching career, in which I team
taught 3 classes of Algebra with 3 teachers. We would each lecture one
class a day while the others moved around the classroom answering
individual questions. This schedule would change from day to day, so
that we were not teaching at the same time everyday. This was great,
because each of the teachers learned from the others, resulting in the
last lecture being the most comprehensible. Students, especially the
3rd class, seemed to benefit greatly from this experience. So not
only do I think students could benefit from group work, but also
teachers.
seems to me to be a social constructivism idea, which I tend to agree
with most. I don't think that students' education should be entirely
left up to the student, because as I said before I think many would
opt to do nothing. I do believe that they should be promoted to
interpret material individually, or in small groups. Many of my
teachers in math have promoted group work, but there was no time for
it in class. I have observed in my own experience, that the students
whom are most likely to start an outside study group are ones who
excel in their individual work. Students, who are less likely to ask
questions to their peers or the teachers, tend to feel inadequate
about their abilities in that subject and are embarrassed if they ask
a "stupid question". I think that if small group work or individual
work was implemented in classes early on, this could be somewhat
avoided.
One thing that I worry about in this kind of learning environment is,
will students choose groups that will benefit them in the long run?
The best group work I have done, has been one in which everyone has
different talents but contributes equally. I think if groups are
chosen by the students themselves, they will pick people they are
friends with and not necessarily those who will add to their learning
experience. Also, outspoken gifted students tend to run group
activities leaving little for anyone else to contribute. Perhaps a
way to get around this is to rotate the groups throughout the
quarter.
To add to Jake's idea, I think it would be important to either have
all the TA's for the course, or a smaller classroom size for this type
of instruction. This simply due to the fact that when students work
with their peers, different opinions on an approach to a problem can
lead to a halt in progress. When this happens they tend to turn to
the teacher for an answer, and if many small groups were at work it
would help to have more teachers on hand. Furthermore, different
teachers have different approaches to problems, which could promote
this idea of self-learning. When I taught Pre-College programs I had
one of the best experiences of my teaching career, in which I team
taught 3 classes of Algebra with 3 teachers. We would each lecture one
class a day while the others moved around the classroom answering
individual questions. This schedule would change from day to day, so
that we were not teaching at the same time everyday. This was great,
because each of the teachers learned from the others, resulting in the
last lecture being the most comprehensible. Students, especially the
3rd class, seemed to benefit greatly from this experience. So not
only do I think students could benefit from group work, but also
teachers.
megan
Nov 26, 2007, 2:27:42 AM11/26/07
to Graduate Mathematics Foundations
I just wanted to point out an interesting article from TIME Magazine
http://www.time.com/time/magazine/article/0,9171,1653653,00.html
http://www.time.com/time/magazine/article/0,9171,1653653,00.html
Jacob West
Nov 26, 2007, 10:22:51 PM11/26/07
to Graduate Mathematics Foundations
Also, these were the teaching resources I mentioned in class today:
http://teaching.berkeley.edu/bgd/teaching.html
http://teaching.berkeley.edu/bgd/teaching.html
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