We are holding NSF-funded workshops on order-of-magnitude estimation for teachers of all grades (K-community college), in order to provide expert training in solving order-of-magnitude problems and assist in developing lessons to implement these problem-solving techniques in the classroom. This is a joint effort between Dartmouth College (NSF Award 1515364) and the University of Wyoming (NSF Award 1515404). The first workshop was held at Dartmouth on November 12, 2015, and a mirror workshop took place at the University of Wyoming on June 30th, 2016. We are keeping in contact with participants to get feedback on how the implementation of the techniques learned is impacting their classrooms, and to solicit problems to build a database to share with the community.

Thank you to everyone who has participated so far!

**Our Goals**

Learning science is not about memorizing facts, equations, and numbers, but developing conceptual understanding and problem solving intuition. These are the most translatable skills we can help our students develop, and they are useful in wide array of fields, not just the sciences. A well-known tool for honing these abilities is order-of-magnitude estimation (also known as “Fermi problems”). When solving order-of-magnitude problems, students use existing knowledge to make educated guesses about the approximate values of parameters and simple arithmetic to estimate a quantity. The “right answer” is just within a factor of ten (or so) — we aren’t concerned with exact calculations, but gaining some general insight into the question at hand.

These techniques are commonly taught to and used by advanced students in the sciences, but younger students would benefit from practicing them early in their education. Astronomy is particularly well suited to OoM estimation, as the numbers (and their associated uncertainties) are often quite large. However, estimating astronomical values can be a daunting task, until you realize that the same techniques you can use in every day life apply universally!

Our goal is to introduce you to order-of-magnitude techniques with examples from here on Earth, guide you through extending these approaches to astronomy, and finally to help you develop your own problems suited to your subject area and students’ needs. You can then implement these in your classroom, using us and your colleagues as a resource to help you along.

Beyond the introductory workshop, we will be following your progress and employing various methods to test the efficacy of the program. This will include requesting regular updates and lesson plans from you, pre- and post-testing, as well as working with a control classroom that is not utilizing order-of-magnitude problems.

Next year, we will reconvene for a second workshop to discuss your perceptions of the program, improve our methods, and gather a database of problems for dissemination to the larger community. Stay tuned, and thanks for participating!

**Some Examples**

Our approach in developing OoM questions for astronomy (or any other science subject) is to begin with order-of-magnitude thinking about everyday situations that are familiar to students, and then explore analogous questions in an astronomical context. Here are some sample questions, with general approaches to solving them, to get an idea of what we mean (there are more thoroughly worked examples on the “Problems” page).

– How long does it take to walk around the planet, if you didn’t stop, and could walk on water?

– How long does it take for the Sun to go around the galaxy?

– If a tree’s leaves all fell off, what thickness (in terms of leaves, like, two leaves, or five leaves, or whatever) would the pile be under the tree?

– If two galaxies crash into each other, how many stars crash into each other?

– Which is flatter, a pancake, or Kansas?

– What is the minimum size of the universe, based on the measured curvature?

**Approaches to solving these example questions**

*(*Thanks to* Dr. Kevin Hainline* for input developing and testing these in the context of astronomy courses at Dartmouth)

**– How long would it take to walk around the planet’s circumference?**

The general way to approach this questions is to figure out average walking speed, and then how big the planet is, remembering things like America is 3000ish miles, and it comprises four out of the full 24 time zones.

**– How long does the sun take to go around the galaxy?**

This is much harder, because it requires a knowledge of something like Kepler’s law of planetary motion, but you can figure out the mass of the galaxy if you know the size roughly and the fact that stars are separated by parsecs.

**– If a tree’s leaves all fell off, what thickness (in terms of leaves, like, two leaves, or five leaves, or whatever) would the pile be under the tree?**

You can use a rough unit-based approach, where you have to figure out how many leaves / the density of leaves for a tree, and the size of each leaf, and then you can multiply and figure out that generally, there’s only a bed of about 1-2 leaves thickness under a given tree, which makes sense since if it were a larger number, that would imply more leaves than the tree could support. The problem is the same as asking how many leaves does a random light ray strike as it moves through a tree, which should be one as otherwise there’d be leaves that did not receive light)

**– If two galaxies crash into each other, how many stars crash into each other?**

Similar problem set up, but the end result is that it’s very unlikely, since stars are very small as compared to their separation.

**– Which is flatter, a pancake, or Kansas?**

Here, all you need to know is an estimate for the height of a mountain in Kansas, the highest point is actually 4000 feet above sea level, so if you use that fact, and then take pancake and scale it up, you find that the general pancake, with mm to half-mm size fluctuations, is in fact more hilly than Kansas.

**What is the minimum size of the universe, based on the measured curvature?**

Based on the limits of the curvature, and the size of the observable Universe, one can estimate the radius of curvature of the *entire* Universe, which is enormous, with the exact value depending on what limits for the curvature you adopt.

I would like to attend this workshop.