Tuesday, April 26, 2011

Reflections on mathematical modeling (II)

Brain data dump...

levels of formalization:
  • what do previous data tell us - deterministic models (e.g., average, linear regression)
  • what do previous data tell us - stochastic models (e.g., range, standard dev., standard error)
  • increased sophistication (e.g., non-normal forms of stochasticity: null models, interesting parametric distributions).
  • meta-analysis - combining previous empirical studies
  • models with and without feedback or loops
Learning a language, learning modeling concepts.

Modeling data, modeling dynamics.

Learning by,
  1. copying,
  2. applying,
  3. combining,
  4. creating.

Friday, April 22, 2011

Discussing scientific papers in classes - what do we DO?

Should we demonstrate understanding during class time, or should we just jump ahead? I think we need to demonstrate understanding in class, if only to make sure people actually work at reading the assigned papers. However, we could even read the paper out loud, but that would not guarantee understanding. So, it seems to me that in each class we should address at least the following questions:
1. Is the question addressed in the paper interesting?
2. Do the data address the hypotheses?
3. Do the results support the conclusions?
4. What are the implications of the conclusions (or of the results)?

In class, we might start with #2, then #3, #4, and then maybe return to #1.

Pedagogical and scientific goals

I posit that understanding is the core value of mathematical modeling. There are (at least) two levels of understanding, the understanding of our own questions. The first aspect of understanding enhanced by modeling is making our spoken language precise with the aid of mathematics. The second aspect of understanding is providing an unambiguous structure to our ideas that the scientific community can use, that is, the development of useful theory.

I like to think of the scientific process of knowledge creation as a 3D spring, coil, or spiral, where a single loop represents a complete cycle of the scientific process (question, hypothesis, test, interpretation), and progress occurs as we repeat the process through multiple cycles, traveling down length of the coils. Mathematical modeling can help us at different phases of a single coil.

I think that making ourselves formalize our conceptual models helps us see and understand our ideas to a greater degree. Formalization helps us become ever more specific and thereby operationalize our hypotheses and thereby generate more testable predictions. Going through the formalization process helps us understand what a mathematical model is and and how mathematical models provide structure to theory. The process helps show us and convince us of how models are used in Science.

Thursday, April 21, 2011

Reflection on teaching modeling, or why should non-modelers try to model?

At the moment, I believe that non-modelers (students or faculty) benefit from attempting to model simple systems. I believe that it helps them become better scientists.

I am near the end of a semester in which we tried to incorporate a little bit of modeling into an otherwise basic graduate level ecosystems course. I think I would like to reflect a bit.

For years I have helped teach a population/community grad course, where we included basic population and food web models, and a smidgen of other stuff. In that course, we started everyone out making the same assumption of ignorance for all, and we taught just enough for students to implement simple models in R. I am not sure how satisfactory it is. I think I want to teach more basic R so that students learn about R in a modeling context, not just their stats classes. I think by learning R they will learn about models even more effectively.

This semester (Winter/Spring 2011), in the ecosystems course, we started students thinking along two tracks, one of conceptual models of ecosystems and the other learning the R language. Our thought was that by the time they had learned enough about ecosystems, to create conceptual models, they would have learned enough R to begin formalizing their conceptual models. However, that has not been the case, for at least two reasons.

The first cause of sub-optimal pedagogy may have been that students new to a language (e.g., R) need to work with it at least three days/week (preferably 4-6), but I did not structure the assignments that way. They need both carrots and sticks, and assignments that require daily turn-around (e.g., automated release and deadlines, or email with 24 hours to upload answers). I would not even have to grade every one of them - just mark them turned in or not, perform spot checks, and provide detailed answers. Why didn't I do this? Several not-very-good reasons:
  • I felt sorry for them,
  • I wasn't 100% convinced that I should push programming and math that hard,
  • it would have been more work for me,
  • not everyone needed that kind of practice,
  • those that needed that kind of practice COULD have done self-study.
The second reason for suboptimal pedagogy was that I tried to be more flexible with the modeling assignments than I was easily capable of -- I could create stuff, but some of it took longer than was convenient. In brief, we asked students to come up with a scientific question, explain what is known and unknown regarding that question and their study system, and design a conceptual model that captures the essence of their question and/or system. Students were then asked to formalize their conceptual model using mathematics or computer code or both. The students conceptual models were not all ecosystem models with merely pools and fluxes of all the same units and element(s). Rather, most were a hodge-podge of different sorts of variables that related typically in a mechanistic fashion, but were not comprised of, for instance, pools and fluxes of carbon. Therefore, the relatively low programming ability of the students (see first reason, above) and my desire to be flexible with regard to acceptable topics meant that I had to invent lots of unique code for each different student. And that, Virignia, is the second reason why my pedagogy was sub-optimal.

However, I think that forcing students to formalize their conceptual models has helped them see and understand their own conceptual models to a greater degree. Formalization helps them become ever more specific with regard to their conceptual model and this helps them generate more testable predictions. Formalization helps them understand what a mathematical model is and and how mathematical models provide structure to theory. The process helps show them how models are used in Science, and last, it helps them see indirect connections more clearly and accurately. Well, ... I hope it does all that.