What are landmarks and landscapes for rough quantitative reasoning? And why do they matter? Here's a blog-ish, conversational answer.
Teaching estimation is very popular. K-graduate. It's a standard part of 1st grade curriculums. (By "estimation", I mean creating rough answers to quantitative questions about the physical world. "How many beans are in the bag?" Other kinds of estimation, of linear measurement, and as part of computation, are even more common.)
But if you talk with undergraduates who have taken an introductory physics course emphasizing estimation, you hear things like the following. 'In the beginning, the estimation problems were about length, and time, and pizzas, and I had a feel for what reasonable values were. But second semester... magnetic field density?!? Who has a feel for that? And the estimation problems stopped having anything to do with the course material, becoming pointless estimation drills.' Asked if any attempt was made to develop such a feel, as by systematically teaching landmarks (aka referents), the answer is invariably no. It's not even part of the implicit curriculum.
And so for more than half a century, there have been letters to the editor, in physics, and across the physical sciences, from professors bemoaning their doctoral candidates lack of a quantitative feel for the field they've been studying for years. And suggesting greater use of rough quantitative approximate reasoning (aka Fermi problems).
So what's missing? Well, a couple of things. But one of them, is a failure to explicity teach landmarks and landscape.
What does that mean? It is common to have a lecture slide with a few values. Or a textbook sidebar. Or appendix.
Some are good landmarks, others not so much.
It's rare to support students in chosing and learning landmarks. Though occasionally some value and mnemonic gets mentioned in passing.
Earth radius - "Six ee Six meters" (6 × 106 m, aka 6E6, -6%, ±<1% )
|106 × 6.38 | 6.36 m||±<1%||equitorial | polar radius [Cow]|
|106 × 6.37 m||±<1%|
|106 × 6.4 m||+0.3% | +1%, ±<1%|
|106 × 6 m||-6%, ±<1%||"six ee six is down by six percent"|
|107 m||+58%, ±<1%|
|106 m||-84%, ±<1%|
And while exponential scales have been around for years, only a few units tend to appear in education. Length is common. And wavelength. But few others.
And even when present, they are rarely used. Scale, as a powerful organizing concept, is rarely used to powerfully organize content.
We do have diagrams, like the one on the right, in engineering. Which lay out an overview, a landscape of values.
Students learn to plug-n-chug torque equations. They don't learn 1 N⋅m is another name for "screwdriver".
What if such diagrams were instead used pervasively? And were interactive. And were linked to lots of fun stories?
What we do now can be surprisingly ineffective. A high-profile intro physics text, repeatedly reviewed and revised, has long had an ideal gas law problem... with PVT values for solid Argon. Now that would be wonderful, if it was intentional. But we don't include problems to test awareness of an approximation's applicability, or awareness of context. We don't even teach it. So year after year, both professors and students, plug-n-chug.
But is teaching landmarks & landscapes a right way to address this? I'm sympathetic to a counter argument of "Here in the future, why memorize what can be googled in a few seconds?" But... nice interactive landscape diagrams would need to exist, before they could be googled. And currently don't. So we need to get at least that far. And then, if you use them, you find yourself remembering landmarks. But creating nice landarks is non-trivial, benefiting from creativity, insight, and analysis. So it's nice to have someone lend a hand with that.
And while there's lots of intent, in various fields, to make introductory courses more quantitative, that usually means "more like physics". With its half century of failure.
Every science and engineering field has its gap between what skills professionals report needing, and what is taught. Back of the envelope calculation is a common one. And every field is realizing that teaching itself has been going much less well than was thought.
Would making content roughly quantitative, pervasively, be an improvement? That's an open question. And since improvements are often not independent, it's also unclear what other improvements might be needed to enable it.
If your students came to you, living and breathing estimation, steeped in years of experience, how would you then change content to maximally benefit from that opportunity?
Perhaps, if the benefits appears big enough, we can start with the content, and motivate creation of the skills.
Hasn't gelled yet. The final section doesn't flow. Are the topic questions actually answered? What exactly is the target audience and objective? Permission for graphics. What else?