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RME is rooted in ‘mathematics as a human activity,’ and the underlying principles are guided
reinvention, didactical phenomenology, and emergent models. These principles are based on
Freudenthal’s philosophy which emphasizes reinvention through progressive mathematization
(Fredenthal, 1973, 1991). In RME, context problems are the basis for progressive mathematization,
and through mathematizing, the students develop informal context-specific solution strategies
from experientially realistic situations (Gravemeijer & Doorman, 1999). Thus, it is necessary for
the researchers who adapt the instructional design perspective of RME to utilize contextual
problems that allow for a wide variety of solution procedures, preferably those which considered
together already indicate a possible learning route through a process of progressive
mathematization.
Three guiding heuristics for RME instructional design should be considered (Gravemeijer, Cobb,
Bowers, & Whitenack, 2000). The first of these heuristics is reinvention through progressive
mathematization. According to the reinvention principle, the students should be given the
opportunity to experience a process similar to the process by which the mathematics was invented.
The reinvention principle suggests that instructional activities should provide students with
experientially realistic situations, and by facilitating informal solution strategies, students should
have an opportunity to invent more formal mathematical practices (Freudenthal, 1973). Thus, the
developer can look at the history of mathematics as a source of inspiration and at informal solution
strategies of students who are solving experientially real problems for which they do not know the
standard solution procedures yet (Streefland, 1991; Gravemeijer, 1994) as starting points. Then the
developer formulates a tentative learning sequence by a process of progressive mathematization.
The second heuristic is didactical phenomenology. Freudenthal (1973) defines didactical
phenomenology as the study of the relation between the phenomena that the mathematical concept
represents and the concept itself. In this phenomenology, the focus is on how mathematical
interpretations make phenomena accessible for reasoning and calculation. The didactical
phenomenology can be viewed as a design heuristic because it suggests ways of identifying
possible instructional activities that might support individual activity and whole-class discussions
in which the students engage in progressive mathematization (Gravemeijer, 1994). Thus the goal
of the phenomenological investigation is to create settings in which students can collectively
renegotiate increasingly sophisticated solutions to experientially real problems by individual
activity and whole-class discussions (Gravemeijer, Cobb, Bowers & Whitenack, 2000). RME’s
third heuristic for instructional design focuses on the role which emergent models play in bridging
the gap between informal knowledge and formal mathematics. The term model is understood in a
dynamic, holistic sense. As a consequence, the symbolizations that are embedded in the process of
modeling and that constitute the model can change over time. Thus, students first develop a
model-of a situated activity, and this model later becomes a model-for more sophisticated
mathematical reasoning (Gravemeijer & Doorman, 1999).
RME’s heuristcs of reinvention, didactical phenomenology, and emergent models can serve to
guide the development of hypothetical learning trajectories that can be investigated and revised
while experimenting in the classroom. A fundamental issue that differentiates RME from an
exploratory approach is the manner in which it takes account both of the collective mathematical
development of the classroom community and of the mathematical learning of the individual
students who participate in it. Thus, RME is aligned with recent theoretical developments in
mathematics education that emphasize the socially and culturally situated nature of mathematical
activity.
Traditional and Reform-Oriented Approaches in Differential Equations
Traditionally, students who take differential equations in collegiate mathematics are dependent
on memorized procedures to solve problems, follow a similar pattern of learning in precalculus
mathematics, and follow model procedures given in the textbook or by a teacher. Also, the search
for analytic formulas of solution functions in first order differential equations is the typical starting
point for developing the concepts and methods of differential equations. This traditional approach
emphasizes finding exact solutions to differential equations in closed form, i.e., the dependent
variable can be expressed explicitly or implicitly in terms of the independent variable. However, in
reality, when modeling a physical or realistic problem with a differential equation, solutions are
usually inexpressible in closed form. Therefore, as Hubbard (1994) pointed out, there is a
dismaying discrepancy between the view of differential equations as the link between mathematics
and science and the standard course on differential equations.
The teaching of differential equations has undergone a vast change over the last ten years
because of the tremendous advances in computer technology and the “Reform Calculus”
movement. One of the first textbook promoting this reform effort was published by Artigue and
Gautheron (1983). More recently, a number of textbooks reflecting on this movement have been
written (e.g., Blanchard, Devaney, & Hall, 1998; Borelli & Coleman, 1998; Kostelich &
Armbruster, 1997; Hubbard & West, 1997). Primary features of these reform-oriented textbooks
are content-driven changes made feasible with advances in computer technology. Thus, these
textbooks have decreased emphasis on specialized techniques for finding exact solutions to
differential equations and have increased the use of computer technology to incorporate graphical
and numerical methods for approximating solutions to differential equations (West, 1994).
According to Boyce (1995), the primary benefit of incorporating computer technology in
differential equations is the visualization of complex relationships that students frequently find too
complicated to understand. For example, a typical differential equation, u’’+0.2u’+u=coswt,
u(0)=1, u’(0)=0, can be easily executed with technology, and students can understand the behavior
of the system by using technology to draw a three-dimensional plot as a function of both w and t.
The main reasons to use computers in a differential equations course are that geometric
interpretations of solutions through the use of computer software help students to understand basic
concepts such as initial value problems, integral curves, direction fields and flows for dynamical
systems (Lu, 1995). In addition, many concepts including phase portrait, stability, stable and
unstable manifold, bifurcation and chaos can better be understood by introducing a computer
program for teaching and learning. However, the current reform movement in differential
equations emphasizes a combination of analytic, graphical, and numerical approaches from the
start. Although different from traditional approaches to differential equations, this movement is
quite similar to traditional approaches in the way in which conventional graphical and numerical
methods are used as the starting point for students’ learning, as Rasmussen (1997, 1999)
documented. Thus, as is the case with the traditional approach, students typically do not participate
in the reinvention or creation of these mathematical ideas associated with graphical and numerical
methods, the representation that conventionally accompany these ideas, and the methods
themselves. The learning that occurred was characteristic of mindless graphical and numerical
manipulation in the reform-oriented approach. In these respects, the learning demonstrates little
improvement over traditional approaches where mindless symbolic manipulation was the prevalent
mode of operation.
The current curriculum-oriented reform movement in differential equations has some contentbased
advantages. The approach being developed here seeks to build on and complement these
positive aspects by adapting principled perspectives and approaches that have informed the rethinking
of mathematics learning and teaching at the elementary and secondary level to the rethinking
of mathematics learning and teaching of differential equations.
reinvention, didactical phenomenology, and emergent models. These principles are based on
Freudenthal’s philosophy which emphasizes reinvention through progressive mathematization
(Fredenthal, 1973, 1991). In RME, context problems are the basis for progressive mathematization,
and through mathematizing, the students develop informal context-specific solution strategies
from experientially realistic situations (Gravemeijer & Doorman, 1999). Thus, it is necessary for
the researchers who adapt the instructional design perspective of RME to utilize contextual
problems that allow for a wide variety of solution procedures, preferably those which considered
together already indicate a possible learning route through a process of progressive
mathematization.
Three guiding heuristics for RME instructional design should be considered (Gravemeijer, Cobb,
Bowers, & Whitenack, 2000). The first of these heuristics is reinvention through progressive
mathematization. According to the reinvention principle, the students should be given the
opportunity to experience a process similar to the process by which the mathematics was invented.
The reinvention principle suggests that instructional activities should provide students with
experientially realistic situations, and by facilitating informal solution strategies, students should
have an opportunity to invent more formal mathematical practices (Freudenthal, 1973). Thus, the
developer can look at the history of mathematics as a source of inspiration and at informal solution
strategies of students who are solving experientially real problems for which they do not know the
standard solution procedures yet (Streefland, 1991; Gravemeijer, 1994) as starting points. Then the
developer formulates a tentative learning sequence by a process of progressive mathematization.
The second heuristic is didactical phenomenology. Freudenthal (1973) defines didactical
phenomenology as the study of the relation between the phenomena that the mathematical concept
represents and the concept itself. In this phenomenology, the focus is on how mathematical
interpretations make phenomena accessible for reasoning and calculation. The didactical
phenomenology can be viewed as a design heuristic because it suggests ways of identifying
possible instructional activities that might support individual activity and whole-class discussions
in which the students engage in progressive mathematization (Gravemeijer, 1994). Thus the goal
of the phenomenological investigation is to create settings in which students can collectively
renegotiate increasingly sophisticated solutions to experientially real problems by individual
activity and whole-class discussions (Gravemeijer, Cobb, Bowers & Whitenack, 2000). RME’s
third heuristic for instructional design focuses on the role which emergent models play in bridging
the gap between informal knowledge and formal mathematics. The term model is understood in a
dynamic, holistic sense. As a consequence, the symbolizations that are embedded in the process of
modeling and that constitute the model can change over time. Thus, students first develop a
model-of a situated activity, and this model later becomes a model-for more sophisticated
mathematical reasoning (Gravemeijer & Doorman, 1999).
RME’s heuristcs of reinvention, didactical phenomenology, and emergent models can serve to
guide the development of hypothetical learning trajectories that can be investigated and revised
while experimenting in the classroom. A fundamental issue that differentiates RME from an
exploratory approach is the manner in which it takes account both of the collective mathematical
development of the classroom community and of the mathematical learning of the individual
students who participate in it. Thus, RME is aligned with recent theoretical developments in
mathematics education that emphasize the socially and culturally situated nature of mathematical
activity.
Traditional and Reform-Oriented Approaches in Differential Equations
Traditionally, students who take differential equations in collegiate mathematics are dependent
on memorized procedures to solve problems, follow a similar pattern of learning in precalculus
mathematics, and follow model procedures given in the textbook or by a teacher. Also, the search
for analytic formulas of solution functions in first order differential equations is the typical starting
point for developing the concepts and methods of differential equations. This traditional approach
emphasizes finding exact solutions to differential equations in closed form, i.e., the dependent
variable can be expressed explicitly or implicitly in terms of the independent variable. However, in
reality, when modeling a physical or realistic problem with a differential equation, solutions are
usually inexpressible in closed form. Therefore, as Hubbard (1994) pointed out, there is a
dismaying discrepancy between the view of differential equations as the link between mathematics
and science and the standard course on differential equations.
The teaching of differential equations has undergone a vast change over the last ten years
because of the tremendous advances in computer technology and the “Reform Calculus”
movement. One of the first textbook promoting this reform effort was published by Artigue and
Gautheron (1983). More recently, a number of textbooks reflecting on this movement have been
written (e.g., Blanchard, Devaney, & Hall, 1998; Borelli & Coleman, 1998; Kostelich &
Armbruster, 1997; Hubbard & West, 1997). Primary features of these reform-oriented textbooks
are content-driven changes made feasible with advances in computer technology. Thus, these
textbooks have decreased emphasis on specialized techniques for finding exact solutions to
differential equations and have increased the use of computer technology to incorporate graphical
and numerical methods for approximating solutions to differential equations (West, 1994).
According to Boyce (1995), the primary benefit of incorporating computer technology in
differential equations is the visualization of complex relationships that students frequently find too
complicated to understand. For example, a typical differential equation, u’’+0.2u’+u=coswt,
u(0)=1, u’(0)=0, can be easily executed with technology, and students can understand the behavior
of the system by using technology to draw a three-dimensional plot as a function of both w and t.
The main reasons to use computers in a differential equations course are that geometric
interpretations of solutions through the use of computer software help students to understand basic
concepts such as initial value problems, integral curves, direction fields and flows for dynamical
systems (Lu, 1995). In addition, many concepts including phase portrait, stability, stable and
unstable manifold, bifurcation and chaos can better be understood by introducing a computer
program for teaching and learning. However, the current reform movement in differential
equations emphasizes a combination of analytic, graphical, and numerical approaches from the
start. Although different from traditional approaches to differential equations, this movement is
quite similar to traditional approaches in the way in which conventional graphical and numerical
methods are used as the starting point for students’ learning, as Rasmussen (1997, 1999)
documented. Thus, as is the case with the traditional approach, students typically do not participate
in the reinvention or creation of these mathematical ideas associated with graphical and numerical
methods, the representation that conventionally accompany these ideas, and the methods
themselves. The learning that occurred was characteristic of mindless graphical and numerical
manipulation in the reform-oriented approach. In these respects, the learning demonstrates little
improvement over traditional approaches where mindless symbolic manipulation was the prevalent
mode of operation.
The current curriculum-oriented reform movement in differential equations has some contentbased
advantages. The approach being developed here seeks to build on and complement these
positive aspects by adapting principled perspectives and approaches that have informed the rethinking
of mathematics learning and teaching at the elementary and secondary level to the rethinking
of mathematics learning and teaching of differential equations.
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ASDAR SYAM
Label:
Mathematics,
Realistic Mathematics Education
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