*This is guest post by Liam McDaid, Astronomy Coordinator and Professor of Astronomy at Sacramento City College. He is also a Senior Scientist at Skeptic Magazine. This guest post is the first to result from my effort to get more education (e.g., Astro101) and public outreach content on AstroBetter and it focuses on the application of estimation in the Astro101 classroom. I also think there are lessons to learn here that are applicable to mentoring research students. How often do we ask our students to estimate the results before doing the calculation/data reduction? Or maybe ask them to draw what they expect the plot to look like before generating it. What are the order-of-magnitude estimates that we should expect our budding Astro PhDs to be able to perform? It’s something that most of us are aware that we need to know how to do, but are we actually teaching the skill to our undergraduate and graduate students? — Kelle
*

One of the things we hear about students (particularly in the US) is their terror of math. The fear of math is oddly selective. The same students who hate/fear math are the ones who balance their checkbook each week and know their grade in class to four significant figures. Clearly, they can do arithmetic when it’s important to them. Over the summer, I read an interesting book called How to Measure Anything, by Douglas Hubbard (it does indeed show how to measure anything). I realized that a big key to the problem of students and math lay within. For whatever reasons, most people don’t know the value of information they have an interest in. For example, if a piece of info would save your company $5 million, how much should you pay for it? How do you quantify the value of a piece of information? Clearly this is an important task for a company, yet few know how to do it.

The solution is to get people to start estimating. I don’t mean solving problems, I mean getting them to create rough estimates of something. The endgame is to make people into what Hubbard calls “calibrated estimators”. For example, if you ask your students when the Roman Emperor Domitian died, it is unlikely that they will know. But they can estimate, can’t they? They may know that the Roman Empire was on top from about 150 ce to 300 ce (roughly). The fact Domitian was an emperor means he likely lived and died in that range. In astronomy, a couple of examples could involve:

- How long is the Moon’s sidereal period? (This only works before students learn the value). It’s interesting to see what the upper limits for this one are—–-longer than a year is not unheard of.
- What is the photospheric temperature of the Sun? This works as a good order-of-magnitude exercise to see if students think of it in terms of thousands or millions of degrees. Since dealing with the Sun involves both temperature scales, confusion is common. To make things a bit more concrete, you can frame it around controlled nuclear fusion. We don’t have it, so which temperatures are more likely involved?
- How far away are the stars we see in the sky? There is no single answer but one of the most luminous stars easily visible to the eye (Deneb) is only about 1500 ly away. So the range would have an upper bound of a few thousand ly, or 1 kpc if you have been habituating your students with those units.
- Fermi’s paradox. Aside from being ultra-sexy content, it lets students hammer out estimates for how long we should wait for the aliens to contact us. Or whether we should expect it to happen at all, as one of the scariest implications of Fermi’s paradox is that it’s entirely possible that we are the most advanced civilization in the Galaxy. That we are the cavalry for sentience. Think about that for a moment. A
*rough*estimate can be found here.

Quantification is always a good thing and it helps students (and us) wrap our brains around something. In astronomy we almost have to give up the hope of emotionally understanding the vast scales we casually throw around every day. Understanding them intellectually is vitally necessary. It’s also one of the guidelines for critical thinking as laid out by Carl Sagan’s Baloney Detection Kit, which, along with supporting material, is available for $5 from the Skeptic’s Society. (Pardon the commercial.) Having a class that can estimate properly goes a long way toward attacking both math anxiety and improving critical thinking.

The next step is to get them to give you an estimate with a 90% confidence level. Students will likely at first have trouble with this so make it clear in our example of the death of Domitian that they can make the range of dates as broad as they want, so long as they are 90% sure that the death year of Domitian lies within. Some students will simply say between today and 3000 years ago, so get them to tighten it up. Once they have a range, ask them if it is, in fact, their 90% confidence estimate. When they say yes, make it interesting. Say that if they were to get $1000 for getting it right, would they stick with their estimate or would they rather spin a wheel where 90% of the wheel is marked “win”. Many times, students will go with the wheel; explain to them that their estimate is clearly not at 90% confidence if they choose the wheel.

Hubbard claims that he can train anyone to be a calibrated estimator in a one-day workshop. Someone who can estimate things has taken a large step toward being like Fermi and their fear of math may recede. Making students do this in class (lecture/seminar/lab/observation session) can make some students uncomfortable, but it does force them outside of the box. It also gives them a tool at least as sharp as that legendary razor that Occam left us.

What are some other examples of where we can get our students estimating? Share in the comments.

{ 4 comments… read them below or add one }

This highlights the second point Liam makes: not only must we teach our students how to estimate the values of parameters, but how to estimate the

error bars. In this case, Liam is off by several hundred years (the Roman Empire technically began ~27 BCE, and lasted well into the 5th century CE – Domitian died in 96 CE, 246 years outside of the range Liam indicates).I agree, though, that estimation is indeed an important skill that needs practice.

I should add, this also highlights the usefulness of bouncing ideas off of people, and collaborating in the solution of “Fermi problems”. I freely admit that when I read this article I had never even heard of Domitian, and had to look up the date of his death. But I

didknow that the Roman Empire started after the death of Julius Caesar in ~50 BCE (yes, 44 BCE, but I didn’t remember that exact date either). (I also realize it’s possible Liam mistakenly used “BCE” when he meant “CE”, in which case the approximation he makes isn’t so bad. The historical equivalent of a minus sign error 🙂 )Sanjoy Mahajan‘s order of magnitude physics might be relevant to this post.

How about the date of Justinian’s ascension to emperor? 🙂