Then the developer formulates a ten

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.