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In his book "Surely you're joking Mr. Feynman", Richard Feynman relates the following story. As he was supervising a group of calculators for Manhattan project, he at some point gave them a lecture on what they were actually calculating. He claimed that this improved their performance - even though their job was, essentially, that of arithmetic gates in a computational scheme, so the lecture wasn't directly useful in their work.

Is there evidence for a similar effect in teaching mathematics?

For example, suppose I want to teach Gauss elimination. Would students learn better if I chose examples from some actual real-world problems? Or, when teaching determinants, many students are happy when presented with the "volume" interpretation, but does it actually enhance learning of the usual algebraic/computational aspects? When teaching integration or ODEs, is it beneficial to choose examples relevant to physics/biology/...?

Generally, is providing a context/big picture helpful or a waste of time when teaching "hard skills"? Disregarding the "added value" of the students knowing the big picture, being able to apply the skills, etc., does the learning of skills per se improve?

I'm mostly interested in rigorous research answering this, but anecdotal evidence is also welcome.

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    $\begingroup$ I'm not sure I understood what kind of "Feynman effect" you are referring to. Is it lecturing "the big picture"? $\endgroup$ – Mark Fantini May 15 at 11:04
  • $\begingroup$ As far as anecdotes go, I googled "Feynman effect" and found this: "When I was a physics undergrad, we also referred to a "Feynman Effect", slightly different from the above (but maybe a consequence of it): certain lecturers presented material (especially QM) to us in an exciting, dynamic way, a way that rekindled our passion for the subject, and a way which seemed utterly clear at the time; but when it came to review our notes of the lectures, a way which made absolutely no sense." $\endgroup$ – Mark Fantini May 15 at 11:08
  • $\begingroup$ I invented the term myself. But the what I'm asking is described in detail in the post: yes, essentially whether adding motivation/big picture helps develop calculational skills. $\endgroup$ – Kostya_I May 15 at 12:20
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    $\begingroup$ It really seems like the biggest effect the lecture had was that it was a morale and team building exercise. He took a bunch of people who were normally forgotten and taken for granted and instead of acting like the superior better-than-them scientist he told them how important and meaningful their work was. Unsurprisingly, people work a lot harder when they feel their work is meaningful and that they are respected and needed. $\endgroup$ – eps May 15 at 22:27
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    $\begingroup$ one more anecdotal evidence: I taught ODEs to freshmen, they told they were fed up with the stuff and asked why they should learn this. then I sketched up a seemingly trivial mechanical example that looked like high-school physics at first sight but turned out to be solvable only using DE. they got excited when they understood why they need this, and they were more motivated afterwards. $\endgroup$ – törzsmókus May 15 at 22:43
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I think it's useful but can be overdone and/or can make things more difficult.

For example, I actually think it is easier to learn ODEs first, in the mechanical way (e.g. solving the linear second order equation). Initial practice should be on more algebraic problems, not applied. This is because word problems are harder. After mastering the process with equations, you can do some applications, to see what the process is good for, and for some physical insights (especially the forcing function) and really just for more practice volume. Even here, some examples seem to click more than others. I find the kinematic examples more intuitive than the electrical ones.

One of the reasons I find most PDE texts hard is that they start immediately with a physical problem and teach PDE manipulation in the context of solving that problem. Rather than the ODE approach which is more how we learned to manipulate algebra to "solve for x" and integrals to "solve for y", before moving on to applied problems.

Note also, that there can be a big difference between saying nothing about usefulness--for some math teachers, not even knowing applications of the equations, sometimes even proud of the ignorance--and doing detailed application practice. An intermediate approach is to say something like "this is useful in XYZ field or course". Just assert it. You don't need to prove it. If done truthfully and especially by a teacher with some familiarity with other STEM classes, this just serves as motivation. If challenged, you should know enough to back yourself up, but don't need to hone the whole lecture around use of technique A in field B.

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    $\begingroup$ FYI, I've had a one-semester introduction of PDE's course three times (a consequence of my checkered and lengthy academic past), and in each case the most difficult part of the course was the first month or so when several of the main applied types are derived (various versions for vibrating strings/membranes and heat flow) and when we covered generalities such as characteristics and reflection issues for the wave equation and variable transformations that allow a general linear 2nd order PDE to be put into the three canonical forms (hyperbolic, parabolic, elliptic). (continued) $\endgroup$ – Dave L Renfro May 15 at 17:26
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    $\begingroup$ Once we got to separation of variables, Fourier series and generalized Fourier series (i.e. expressing the solution as an infinite sum of eigenfunctions, which arise from solving the separated variables ODEs), things we vastly simpler and more understandable, at least for me. If anyone's interested, the texts were The Analysis and Solution of Partial Differential Equations by Robert L. Street (Fall 1976; all of chapters 1-4, some of chapters 5 & 6, brief aspects of chapters 7 & 8), (continued) $\endgroup$ – Dave L Renfro May 15 at 17:37
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    $\begingroup$ Partial Differential Equations of Mathematical Physics (2nd edition) by Tyn Myint-U (Fall1984; nearly all of chapters 1-6, maybe a third of chapter 7, might have touched on a topic or two in chapters 8 & 9), and A First Course in Partial Differential Equations (a Dover reprint now exists) by Hans F. Weinberger (Fall 1989; almost all of chapters I-VII, possibly touched on part of chapters X & XI, one week on Chapter XII). $\endgroup$ – Dave L Renfro May 15 at 17:44
  • $\begingroup$ Thanks, Dave. Always a privilege to get your insights. $\endgroup$ – guest May 16 at 18:07
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Generally, is providing a context/big picture helpful or a waste of time when teaching "hard skills"? Disregarding the "added value" of the students knowing the big picture, being able to apply the skills, etc., does the learning of skills per se improve?

From an emotional perspective, it can be helpful. It shows why what you're trying to learn is important. It provides motivation and a breath of fresh air when you are deep down in theorems, lemmas, propositions, and definitions.

From learning and cognitive perspectives, that depends on how you present that big picture and how you keep it up over the course. From the point of view of Cognitive Load Theory (CLT) the aim of learning is to acquire schemas in the long-term memory, which in turn improves the ability to process new information, and that in return improve problem solving.

A schema is generally domain-specific and consists of all knowledge related to that domain in long-term memory. That knowledge includes, but is not limited to, the following:

  • Definitions;
  • Important theorems and propositions;
  • General problem-solving strategies related to that domain, such as the tricks of the trade for proving certain statements, what theorems get generally used for certain results, etc;
  • Important examples and counterexamples.

If that big picture does not contribute to the schema acquisition it may become one of those "nice to know" things but that won't directly contribute to anything but motivation.

Sources:

  1. Jan Plass, Roxana Moreno, Roland Brünken (Editors) - Cognitive Load Theory
  2. John Sweller, Jeroen J. G. van Merriënboer & Fred Paas - Cognitive Architecture and Instructional Design: 20 Years Later.
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