Can we teach differential equations in a modeling first approach?
At SIMIODE - Systemic Initiative for Modeling Investigations and Opportunities with Differential Equations we BELIEVE students can learn differential equations using mathematical modeling to motivate, introduce, and develop understanding and skills. But is that true?
Our informed answer is a resounding, "YES!" From colleagues who have tried this approach we have found that sometimes they are met with initial resistance from students who believe they are in a "technique and tricks" course - they heard about it from their classmates who had taken the course. Students say, "What is all this stuff about modeling, data, reality? Just give us the equations and tell us the techniques and we will practice them." However, somewhere along the course using modeling the students' eyes open up, they see the added value of this approach and the lack of necessity or value in solving equations by hand methods and tricks that machines can do instantly and they actually relish the chance to build differential equations from real situations and then work on solution strategies, be they simple by-hand techniques or using technology. They get excited to see and experience the wide range of applications of mathematics. They gain confidence in seeing how mathematics is introduced, how it comes alive, how it is used, why it is important, and how to translate its use into other fields because that is how it was just introduced to them, namely, from other fields outside of mathematics.
Have you found or used successful instances of introducing an application setting or modeling scenario BEFORE you introduced the mathematics, any area of mathematics? Share these with our community here or if these instances involve differential equations consider contributing your ideas and materials to our growing repository of materials at www.simiode.org.
In our career of teaching for over 50 years we have found that students learn best when the mathematics is in context, when it relates to their reality, and when it solves an interesting problem. This has been a guiding notion for many faculty, indeed, for a number of movements, including the Calculus Reform. But it is now more than all that, for with technology and data we can present scenarios in which serious mathematics of change (which is what differential equations is all about) is offered and students can discover that mathematics for themselves and certainly see a reason for studying the mathematics.
In SIMIODE we are attempting to build community (of teachers and students) based on these ideas and we invite you to join us. Come see our illustrations of this learning in action and bring your own experiences to the community. Join our merry band! We welcome you.