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Thoughts on A Case for Domain-Specific Curiosity in Mathematics

I recently finished listening to this review on curiosity in mathematics forwarded to me by Bethany Lisi at the University of Massachusetts Center for Teaching and Learning. As I read I’ve had some thoughts: 

First: The distinction between curiosity and confusion and frustration is important:

  1. In curiosity a solution is visible even if not immediately visible one can see how to approach the solution. 
  2. In confusion, one doesn’t see exactly how to get to a solution but believes that it is possible. 
  3. In frustration one doesn’t even have enough information to begin to see how a solution is possible. 

This is very important for the instruction of physics as well because students often report feeling confusion or frustration when what I’m really going for is curiosity. A key to making to helping students with this transition could be making sure that they have sufficient information to approach the problem, and letting them know about these different distinctions themselves because of course I can’t possibly provide enough information for everyone all the time.

Another important thought from the paper is that students generally believe that any math problem math homework problem should be solvable in 2 minutes. Furthermore, the amount of time that they report being willing to spend on a problem before giving up is somewhere between 11:00 and 12 minutes this is an important thing to begin to address particularly for 131 students but also for my physics 181 lab. It is really important to know that many students have this expectation in fact it might be worth asking them what their expectation is and clarifying how much time they should spend on various problems. However, if you do that you need to make sure that your homework assignments reflect that time expectation. I think, I’m going to ask faculty graduate students and my new undergrads how much who are Majors how much time they think they should spend in designing an experiment before giving up before they think that that path of exploration is in fact ultimately fruitless. If I can get this information before class on Friday then I could fold it in to my lecture.

Why the PaperCast is Quiet

The PaperCast is quiet right now as I am listening to the audio book of Life as No One Knows It by Sara Imari Walker. This is a very interesting book which explores the concept of Assembly Theory: a new conceptual paradigm for physics in which the lineages of objects takes center stage. In effect, it folds the idea of evolution into physics. I am not yet sure what I think. However, I do want to run an Honors seminar around this book – ideally including students from both my 132 and my quantum II class. The motivation for such a seminar would be an investigation on how physics is a living discipline and that we may not even yet have the “final” paradigm. I am also interested because Prof. Walker seems a physicist who is very fluent in the cultural ways of the life sciences, which is also of interest to me.

Thoughts on “Using Framing as a Lens to Understand Context Effects on Expert Reasoning”

After listening to this paper, a few things are interesting right out of the gate:

  1. The resource model of cognition hasn’t really propagated to biology yet.
  2. Two they point out that experts have learned to reason across disciplines using cross-cutting concepts. Could it possibly be that experts have gained this ability because they were exposed to it in different contexts? I don’t think this is true but it’s worth thinking about the fact that it could possibly be.
  3. I like the idea of engineers as existing in the middle of the idealized/real-example continuum with physics on the idealized end and biology on the other.

Teaching the “Energy Story” Next Time

As described in the page on my Physics 131, I like to present content in terms of concepts first. I want students to think conceptually and then move to translating that story to mathematics. I refined this process this past semester and next time, I want energy the energy story to be presented thus:

  • What types of energy do you have at the beginning/end?
  • Are the totals the same or did some come in or out?
  • What came in or out?

In general, ask them to structure it as:

  • Initial energies
    • What went in/out and as heat/work.
      • Final Energies

Also, as we did on the forces unit: go through all the problems for the unit doing just the story first. Ask them to keep their work. Then cycle back through and solve.

Francis Bacon as Applied to Physics 131

I just finished listening to a podcast lecture about Francis Bacon and his ideas on philosophy of science in the Renaissance. His distinction on the importance of inductive reasoning as the foundation of science as opposed to the concept of deductive reasoning that was so critical to the Scholastic epistemology got me thinking about how we teach physics and what part of physics education might be best for lecture or for lab. In the lecture part of my class, we use a deductive method: students are given facts and then asked to apply those general rules like Newton’s laws to specific situations and use that deductive reasoning to gain insight of specific situations. This is someone analogous to Medieval natural philosophy: in medieval natural philosophy the axioms and conclusions of figures such as Aristotle or taken as being axiomatically true and only deductive reasoning was required to apply those fundamental Universal Concepts to particular situations.

Now, of course, the concepts that I’m teaching in my physics class, such as Newtonian mechanics and the like, are grounded in inductive approaches hypotheses and experimental verification but they’re presented to my students as being true axioms that they have to apply deductively. However, it’s clearly important for students scientific education that they also become familiar and adept at the inductive reasoning that is truly the philosophical underpinning of modern science. Where should students get this understanding, and this experience?

My thought after listening to the series of lectures, was that perhaps this is what the lab is fundamentally for in a physics class: the lab provides an excellent environment for students to engage in these inductive processes of model building making predictions testing predictions falsifying claim these types of skills. Now, we already do a lot of this in the 131 labs that Chris Ertl has been developing. However, I feel that this insight into the contrast between inductive and deductive approaches provides a new lens through which we can look at the different types of instruction that occur in a lecture and lab portions of a Physics course. Moreover, it gives us an opportunity to actually talk about these differences in logical approaches with our students and do some of that, what I consider to be critically important, instruction into the fundamental philosophy of science.

From my collaborations with colleagues in the integrated introductory life science Mutual mentoring group one thing that became quickly apparent, was that throughout introductory biology lab introductory physics lab and introductory chemistry lab no one was explicitly teaching the scientific method. Nor, were they teaching some of the fundamentals of the philosophy of science. In contrast, this framework of thinking about lecture as essentially deductive instruction and lab as inductive instruction provides an excellent basis from which to actually begin talking about these important ideas regarding the philosophy of science. It makes that transition natural. I’m curious, as I begin to develop labs for our first semester majors for fall 2024, to think about this framework. Moreover, I would love to talk about this with somebody at the upcoming American Association of physics teachers meeting in Boston this summer to get additional insights. Particularly, from those folks who have spent careers thinking about effective methods for Laboratory instruction.

In addition to this particular new perspective on what could potentially be the fundamental logical underpinnings differentiating lecture versus lab instruction, this entire experience has also added to my growing appreciation of the importance for all teachers to read widely. Only by reading across disciplines do we get these new ideas and new perspectives of ways to approach and think about are disciplinary based instruction. Moreover, I think that those of us who are teaching faculty are in a unique position to this trans-disciplinary approach. We are not subject to the “publish or perish” pressures that so many of our tenured colleagues are subject to. We often have the freedom to read and think a little more broadly within our official positions. In short, it’s another experience in intellectual humility that other disciplines, in this case the history of philosophy, can provide insights that inform science instruction.