Trust, Modeling, Curiosity, and Learning
In the current issue of The New Yorker (4 Mar 2019 on page 57) there is a cartoon of a man and a woman looking down at their baby who is sound asleep in a crib. They say to each other "Should we wake her before she wakes us?" They have trust that the child will awake. Indeed, they are beyond trust as they worry about timing and the inevitable, which is that the baby's awakening will wake them up on the baby’s terms, not theirs.
The Oxford English Dictionary defines trust as "confidence in or reliance on some quality or attribute of a person or thing, or the trust of a statement." While Rushworth M. Kidder writing in an essay "Trust: A Primer on Current Thinking" (An Institute for Global Ethics Research Report) - see here for more (www.globalethics.org), says, "Trust is commonly used today, involves two subsidiary concepts: trustfulness, whereby an individual expresses a sense of confidence in others, and trustworthiness, wherein an individual acts so as to engender trust and be worthy of the confidence of others."
Trust is something our students place in us. They trust our judgements to bring to the classroom or laboratory appropriate ideas and concepts (the college catalog says such and such, so class ought to have comparable stuff). These young people trust our judgement in culling the vast literature in our fields to bring something of value. After all they quite often are going into deep debt to benefit from what we offer. Further, they trust our ability to give them feedback on their thinking and productivity through comments, discussion, and grades.
We put a great deal of trust into our students; some would say we pour our souls into the enterprise of teaching. So when a student does not stop by to pick up an essay we have labored over and took our precious time to communicate information and improvements we are justifiably hurt. They have violated a trust between student and teacher.
In David Brooks’ excellent book, The Social Animal, published by Random House in 2011 (and by now sitting on too many shelves collecting too much dust, as all “the rage” books suffer) he says about Step One of Learning, “Benjamin Bloom has found that teaching doesn’t have to be brilliant right away: `The effect of this first phase of learning seemed to be to get the learner involved, captivated, hooked, and to get the learner to need and want more information and expertise.’”
There you have it! Students trust us, but we have to “hook” them. Indeed, we CAN use that trust to open their imaginations to what our area of interest/expertise/passion offers. We have to engage them and “captivate” them. We have to bring them to the point where they “need and want” from us, from our field, from our view of the world. How do we do this?
We at SIMIODE contend that we do not engender trust or create need by grinding out technique after technique for solving differential equations, endlessly and without motivation except to say, “You will need this in - fill in the blank for the cognate subject that required them to take our DE course.” Indeed, there are many who would say, “Let a machine grind out the solutions!” with SAGE, Maple, Mathematica, etc. in mind. In order to keep our students’ trust we need to engage them, offer them something which piques their curiosity.
We try to support a modeling-first approach to learning differential equations in all we do in SIMIODE. And if not modeling-first then concurrent or soon thereafter the mathematics at hand is introduced.
Modeling, or applying the mathematics, can be a great hook and create in students a need for what we have to offer in terms of mathematical facts, skills, and relations. Most importantly modeling creates a curiosity and meaningful engagement (sometimes struggles!) with mathematical AND contextual notions in the area of applications. So the students want to know more about how you do a certain thing; indeed, IF you can do a certain thing and whether or not it would be useful in “solving” what is in front of them at that moment, in the current context which has engaged or hooked them.
The modeling-first philosophy closely parallels the underlying principles found in problem-based learning, inductive learning, and inquiry-based learning, which all suggest that students learn best by doing and retain best when they construct their own paradigms. Additionally, by putting the mathematics into the context of real world problems - modeling, it makes the subject meaningful, applicable, interesting, and powerful in the eyes of the students, which can aid with student attitudes about mathematics, resulting in increased curiosity, persistence, and perceived usefulness. Silvia in a very nice article (Silvia, P. J. 2008. Interest - the curious emotion. Current Directions in Psychological Science. 17(1): 57-60.) points out how curiosity can draw our students into learning, into the situation at hand, while driving them on; all because they trust us, by the way!
So consider mixing it up a little in your coursework – trust, modeling, curiosity, and learning. You will like the results and your students will enjoy the process.