How-to: Patience, Growth Mindset, and Mathematics

By Stacy Palen.

I have a TON of math-phobic students in my classes. I teach at an open-enrollment university, where the majority of students test into Developmental Math. Many of these students have such poor math skills that they are enrolled in Math 0950, which begins with counting and the number line and culminates with percentages.

We have no structure here to make sure that these students pass their quantitative literacy classes before they take astronomy.

I feel quite strongly that everyone can do basic math. More importantly, everyone should. If their numeracy does not improve, these students will be taken advantage of by banks and credit card companies and salespeople and loan officers with every major (or not so major) purchase, all the rest of their lives.

I can make an argument that is compelling to myself that the financial crash of 2008 was caused in large part by people who did not understand how to calculate mortgage payments. So the lack of numeracy in the population has larger implications than just whether they score well on an astronomy exam.

Because I don’t want to send the students out the door like lambs to the slaughter, but I simultaneously don’t want them to hate me, I’m always on the lookout for tips and tricks about learning things that are hard.

Ages ago, I learned about the “growth mindset.” That’s the idea that success comes from working hard at things, rather than innate talent.

Focusing on growth mindset turns out to be particularly useful for underrepresented groups, who for better or worse don’t see the talent route as available to them. People in underrepresented groups often internalize this. They think: if people “like them” were “naturally good” at x, more people like them would do x.

When you want to encourage students, it’s hard to think of a quick motto that encapsulates all this. And it’s not always obvious how to make use of the idea that maybe students just need more practice to feel proficient.

This is why I recently took note of this article in the New York Times, about learning patience.

The author of the article makes the point that “patience, the ability to keep calm in the face of disappointment, distress or suffering, is worth cultivating.”

I could instantly see how a more patient person would do better with mathematics than a less patient person, especially if they had learned to fear math. There’s a lot in the article about how to interrupt the function of the amygdala, which is the part of the brain that stimulates that frantic, impatient reaction to everyday frustrations like slow-moving cashiers or slow-loading web pages. Or calculator malfunction. Or algebra. It’s worth a read.

But the thing that caught my eye was this motto: “Train, don’t try.”

Mathematics is not a matter of sheer willpower: just trying harder will not make you numerate! Instead, students need to systematically practice problems of gradually increasing difficulty -- repeating as necessary -- until their ability grows and develops, just like a muscle would.

This is why I insist that they do math in my class, and it’s why I start them doing it during lab time when I (and their peers) can give them pointers on their technique and their methods.

So I’m going to try the experiment of explicitly pointing out the connection between developing patience and developing math skill. And I will encourage my students to “Train, don’t try.”

I bet I will have to try it more than once to get it right.


How-to: The Last Choice, A Questioning Strategy for Your Astro 101 Lecture

By Stacy Palen.

Often, in-class questions are presented as a binary choice: “Does the star grow, or does it shrink?”

I always couch these as, “How many of you think the star grows?" Wait for hands. "How many of you think it shrinks?" Wait for hands. "How many people think that 9:30 in the morning is an unfair time to ask that question?" (Wait for hands.)

The last choice, though it may seem frivolous, is really important.

I try to make these last choices light-hearted and a little bit funny. For example:

  • “How many of you were asleep just now?”
  • “How many of you were thinking about lunch?”
  • “How many of you were thinking about puppies instead?”
  • “How many of you wish we would just get to black holes already, and stop talking about nuclear fusion?”

You get the idea. The light humor helps them stay focused and makes it clear that I expect them to put up their hand for every question at some point, even if it’s the silly last choice. I expect them all to participate, every time.

The last choices -- and the way students react, by laughing or groaning, for example -- help me figure out where they are in their heads.  Do I have their attention? Are they feeling confident to take a risk and make a guess? Are they actually listening to me at all? Were they really thinking about puppies?

I often don’t interpret that response in the moment. After class, while I’m walking back to my office and putting my notes away, I’ll think about what was happening in the classroom right at that time when I asked the last question.

Was there a better way to explain before I asked the question? Had I been talking too many minutes in a row? And so on. I might make a note in my lecture notes about a sticky concept, or an analogy that worked particularly well. This reflection afterwards helps me improve for next time.

Most importantly, the last choice gives students an “out.” It is an acknowledgement on my part that they might not know the answer, and that’s OK. I expect them to go ahead and guess sometimes! Giving the last choice makes it clear the question is not a referendum on how smart they are. I am genuinely asking the question because I am trying to figure out what they have understood so that I can help them.

Really, that’s what the last choice question is all about: it’s a less intimidating way for them to say, “I don’t know.”

The last choice helps students to stay focused because they know there will be a moment when they can answer honestly, and often it will come with a laugh.

A closing note on classroom technology: Sometimes I use “clickers." Sometimes I use a piece of paper divided in 4, with A, B, C, D written on each square; students fold the paper to show me the letter of their answer. Sometimes I just have them put up their hands, because the question is an extemporaneous one that just happened naturally in the course of my lecture. In this post, I talk about the questioning strategy as though it applies to extemporaneous questions. But of course, you could use this strategy for a planned questions, too.


Current Events: We've Landed on Mars! Again!

By Stacy Palen.

As of this writing, InSight has just landed successfully on Mars!  This mission is a little bit different from other recent missions: InSight (short for Interior Exploration using Seismic Investigations) is a lander, not a rover. Because it’s in the news, this is a great opportunity for a brief in-class discussion!

 

Insight-landing
This illustration shows a simulated view of NASA's InSight lander descending on its parachute toward the surface of Mars. Credits: NASA/JPL-Caltech

InSight is designed to investigate the interior of Mars: the crust, mantle, and core.   

A seismometer will measure surface vibrations, which will be used to determine the size of the core, the thickness and structure of the crust, and therefore the thickness of the mantle as well. These same measurements will be used to measure how frequent and how powerful the tectonics are on Mars, as well as the frequency of meteorite impacts.

Heat flow measurements will be taken using a probe that is hammered 5 meters (16 feet) down into the surface. These measurements will be used to determine the temperature of the interior. All of these measurements will lend insight (ha!) into the formation and evolution of Mars.

Radar soundings back and forth to an orbiting spacecraft will be used to measure the wobble of Mars on its axis, which in turn is affected by the structure and composition of the core.

The InSight lander is the result of many technical advances. It’s the first lander to pick up and place instrumentation from the top of the lander onto the surface of the object being studied.

The plan is for InSight to place its seismometer on the surface of Mars, then place a heat and wind shield directly on top of the seismometer. The lander needs to make these placements autonomously, something that has never been done before.

In another major technological advance, InSight arrived on the Martian surface through a complex series of steps involving parachutes and retrorockets. The successful operation of these kinds of tools is a pre-requisite for future human exploration. Just InSight's successful landing represents a major step forward in the study of Mars.

 

Insight-on-mars
An artist’s rendition of the InSight lander operating on the surface of Mars. Image Credit: NASA/JPL-Caltech

Discussion questions for students may be wide-ranging:

  • How does this mission connect to earlier discussions about the formation and structure of planets? (Hooray for final exam review!)
  • Why do we care about the structure of Mars? What implications might a study of the geologic properties of Mars have for future Mars exploration?
  • Do you expect the meteorite impacts to be frequent (more than one per day) or rare (less than one per sol)? Why?

And here are some quick “Google It and Think” questions for small groups:

  • How long did it take for this mission to get to Mars? What factors determined this “flight” time?  What challenges would humans face due to this travel time on a trip to Mars?
  • How many spacecraft, rovers, and landers are currently functioning on Mars? Why is Mars a unique target for missions like these?
  • There are three separate stages for the entry, descent, and landing sequence. What are they?  What problems can you think of that might have occurred in each stage? What steps did the engineering team take to make these problems less likely?

Reading Astronomy News: Astronomers Spot One of the Oldest Stars in the Entire Universe

By Stacy Palen.

Summary: A red dwarf star in the Milky Way barely contains any heavy elements at all. Its age is estimated at 13.5 billion years.

Article: http://www.astronomy.com/news/2018/11/red-dwarf-is-one-of-the-oldest-in-the-universe.

Questions for Students:

1. Why does the lack of heavy elements imply that the star formed very soon after the Big Bang?

Answer: Because since the Big Bang, stars have been making heavy elements and returning them to the interstellar medium. Young stars have more heavy elements than older stars.

2. Why do astronomers think there must have been at least “one ancestor” before this star formed?

Answer: Because it has some heavy elements in it.

3. How is the birth of this small star connected to the first generation of stars, which were probably ALL very massive?

Answer: Supernova explosions from those first stars could trigger the formation of smaller stars.

4. Where would this star lie on an H-R Diagram?

Answer: This star, because it is a very small red dwarf, would lie at the lower right on a H-R diagram.

5. This star is one-seventh (about 0.15 times) the mass of the Sun. Which of the following is a reasonable main sequence lifetime for a star with that mass?
a. 10 million years
b. 100 million years
c. 1 billion years
d. 10 billion years
e. 1 trillion years

Answer: e.

6. Astronomers can confidently state that all stars like this one (with similar mass) are still around, and none have died yet. Why can they state this so confidently?

Answer: Because 1 trillion years is a lot longer than the age of the universe.


Classroom Stories: Calendars, Leap Years, and Graphs

By Stacy Palen.

Discussing the calendar can bring a “science and society” learning objective into the astronomy classroom.

Lunar Calendars

Islam, for example, uses a lunar calendar. The resulting gradual drift of holidays and festivals such as Ramadan through the seasons opens discussion not only to the use of calendars, but also to the earliest observed crescent phase that marks the beginning and end of the fasting month.

Asking students to imagine, and then explain, how Ramadan differs when it is celebrated in different seasons, can give students a better appreciation for why seasons and calendars matter. When we discuss the lunar calendar, Muslim students often raise their hands to add stories of Ramadan, such as waiting for the first observation of the crescent moon that signifies the end of the fasting period.

A few years ago, a Muslim student, Kimi, would come to intro astronomy class wearing traditional attire and veil. Kimi was quiet until we reached this discussion of the calendar. She stepped in to talk about the meaning and practice of Ramadan. This talk, in turn, transformed the views of other students, and they welcomed hearing about her experience.

Later in the year, Kimi had some problems in her neighborhood. Her car was broken into, and her door was vandalized. Kimi stayed after class to explain that this was why she was late that day. Three other students waited to walk with her across campus and offer moral support. We do teach more than astronomy in the astronomy classroom.

The Gregorian Calendar

Students find it surprising that the date of a long-ago event does not tell you precisely how many days ago it happened. This is due to the number of days in a year (365.25) not being an integer. A random error occurs because early calendars did not take the fraction of a day into account, so they needed adjusting once in a while. Students are often surprised to hear this.

For example, in 1582, when the calendar changed from Julian to Gregorian, 10 days were deleted: Thursday, October 4 was followed by Friday, October 15! The Gregorian calendar was invented to correct the systematic errors, so that the random adjustments were no longer necessary.

Explaining the Gregorian calendar is cumbersome: every fourth year is a leap year, except if it's a centennial. The only exception to this is if the centennial is divisible by 400. The effect can be difficult to visualize.

800px-Gregoriancalendarleap_solstice.svg

Image created by Wikipedia user BasZoetekouw using Astrolabe data and used under Creative Commons Attribution 3.0 Unported license.

This graphical representation shows how the date of the summer solstice changes over the course of 400 years. This graph shows that the summer solstice moves 1/4 day later each year, until the fourth year, when it resets to, almost, the original date. As time passes, those almost errors accumulate, until they add up to just about one day after 100 years.

That “just about” error accumulates until it adds up to a full day after 400 years. And then the leap day is skipped.

Asking students to visualize how far the blue line would rise without the leap-year reset helps them understand that the date of the summer solstice would change significantly, by nearly a month, over the course of just one human lifetime. A visual representation of the effect helps students grasp a difficult concept with ease.

Image retrieved from: https://commons.wikimedia.org/wiki/File:Gregoriancalendarleap_solstice.svg


Classroom Stories: Establishing a Common Vocabulary

by Stacy Palen.

Sometimes, you do a thing in class, and you think, “Why did I never do that before?!” That happened to me this semester, when for the first time, I gave my introductory astronomy students a first assignment in vocabulary building.

This was basic, absolutely basic. I gave them a list of astronomical objects. Then, I told them to go to the library, look at an introductory astronomy book, and find a description of each object. The objects included: planet, meteor, comet, star, nebula, galaxy, and so on. 

I asked my students to rewrite each definition in one or two sentences, and then to hand in their definitions in during the second class. I did not give my students anything “tricky.” No words like dwarf planet or energy.

The assignment was a simple census of the Universe: what’s out there, and how can we talk about it?

The assignment was a bit of a desperation move. Often, my students are not prepared for class on day one, but I hate to waste the opportunity to establish a good homework habit by waiting until the second week. This seemed like a nice compromise that would get them engaged with the material, even if they didn’t have their book(s) yet.

Most students did not go to the library (of course). But two of them did, which sort of shocked me. And all of them completed the assignment in its entirety. It took me only a few minutes to grade because every student gave reasonable working definitions of the objects. I was not looking for detail.

But since then, the real magic has happened in classroom discussion.

For many students, it’s been a long time since they thought about space, and they've forgotten a lot of what they knew. For another group, they never learned about these objects to begin with. And for others (the most difficult group), they think they know what these objects are, but they are mistaken.

In Utah, astronomy appears in the 3rd and 6th grade core. Therefore, unless their high school made a special effort to offer an astronomy class, my students may not have talked about space at all since 6th grade. That’s a long time to ask anyone to hold onto unused knowledge.

This semester, I’ve noticed the advantage of our common vocabulary when talking about physical laws. For example, when I talked about orbital motion, I was able to say, “These laws govern the behavior of all kinds of orbits, from planets to comets to stars orbiting the center of the galaxy to extrasolar planets. We will use these laws over and over again.”

And students had an idea, from the census that they took in their vocabulary assignment, how broadly I was applying those laws. The feedback from students -- in the completely non-scientific form of nods -- was more positive than past feedback had been.

Before, I felt that my students were not sufficiently amazed by the universality of physical law. Now that we have a common vocabulary in place, I sense that they better understand my own amazement, and that in turn helps them develop a deeper appreciation for our Universe.


Reading Astronomy News: Jocelyn Bell Burnell and the $3 Million Breakthrough Prize

by Stacy Palen.

In September of 2018, Jocelyn Bell Burnell won a $3 million prize in recognition of her outstanding discovery of pulsars. This article presents an opportunity to link science and society while recalling and applying information about radio telescopes, the motion of the sky, and pulsars.

Article: https://www.npr.org/2018/09/06/645257118/in-1974-they-gave-the-nobel-to-her-supervisor-now-shes-won-a-3-million-prize

Questions for Students:

  1. It may be difficult to visualize the data Bell Burnell was taking from the radio telescope. The chart recorder used to record the data is very similar to a seismometer, a machine that records earthquakes. The radio telescope chart recorder scrolled through 96 feet of paper every day. How much paper did Bell Burnell use for the month of observations between when the blip vanished and when it returned?

    Answer: 30 days * 96 feet per day = 2,880 ft

  2. Why did Hewish think the signal must be man-made?

    Answer: He thought it must be a man-made radio interference because the signal disappeared and then reappeared.

  3. Bell Burnell figured out the signals were coming from space. What observation about the pulses led her to that conclusion?

    Answer: Bell Burnell observed that the source moved at the same speed as the stars.

  4. Prior to Bell Burnell’s discovery, astronomers thought that neutron stars might not be observable. Why might neutron stars be difficult to observe?

    Answer: Neutron stars might be difficult to observe because they are incredibly small. Even if they are very hot, they will not be very bright.

  5. What is it about Bell Burnell’s discovery that earned her the Breakthrough Prize?

    Answer: No one had ever dreamed that an object could act in this way.

  6.  Some people in the scientific community see this award as righting a long-standing wrong. Does Bell Burnell see it that way?

    Answer: No, actually. She seems to be perfectly fine with it. But then, she’s giving all the prize money to promote diversity and fight unconscious bias. So maybe she’s just being graceful.

  7. According to Bell Burnell, why did she not receive the Nobel Prize in 1974?

    Answer: Bell Burnell says that at that time, the committee was not awarding early career scientists.

  8. Do you think that was a fair decision of the Nobel committee?

    Answers will vary.

Share your own questions in the comments!


Reading Astronomy News: A Third Neutrino Source Is Found!

by Stacy Palen.

Until the result discussed in the article linked below, only two distinct neutrino sources were known: the Sun and Supernova 1987a. Now there is a third: a distant blazar.

This article complements material about active galactic nuclei, neutrinos, scientific instrumentation, and the process of science. Following are some questions that I thought of as I read the article. Share your own (with answers!) in the comments.

Article: https://www.eso.org/public/blog/pinpointing-the-source/

  1. What is a blazar?

    Answer: A blazar is a particular kind of active galaxy in which the jet points at Earth.

  2. A blazar is a little bit similar to a pulsar, but not exactly the same. Compare and contrast the two objects.

    Answer: A blazar is detected by the emission coming from its jet. In this way, it is something like a pulsar, which is observed when its jets point toward Earth. The pulsar, however, is much smaller and spins rapidly so that the jet points toward Earth only some of the time.

  3. You have learned that there are many, many neutrinos passing through a human body in one second: 100 trillion, just from the Sun. How many neutrinos were detected from this blazar?

    Answer: Only one neutrino was detected! It is somewhat surprising that one neutrino out of so many can be important.

  4. How many different regions of the electromagnetic spectrum were observed in this project? What are they?

    Answer: Five regions of the electromagnetic spectrum were observed: gamma ray, radio, infrared, optical, and X-ray.

  5. This discovery is an example of what astronomers sometimes call "multi-messenger astronomy." What do they mean by that? If the neutrino had not been detected, would the discovery still be "multi-messenger?"

    Answer: Multi-messenger astronomy means that astronomers are getting information about an object from light (electromagnetic radiation) AND another source, like neutrinos or gravitational waves. If the neutrino had not been detected, this would not have been multi-messenger because all the other detections were made from observations of light.

  6. Why has a blazar like this never been discovered before? Do you expect to see more discoveries in the future? Why or why not?

    Answer: Astronomers did not have equipment capable of discovering these neutrinos until IceCube became operational just a few years ago.

  7. This discovery took many people working together, at many different facilities. The end of the article focuses on some of the difficulties and advantages of this approach. Describe one difficulty and one advantage of involving many scientists, particularly different kinds of scientists, in a scientific project.

    Answers will vary.

Reading Astronomy News: Neutron Stars and General Relativity

by Stacy Palen.

Here is a nice little article from NRAO that corresponds to material in Chapter 13 of Understanding Our Universe and Chapter 18 in 21st Century Astronomy: https://public.nrao.edu/news/neutron-stars-fall.

Questions for Students:

  1. Make a sketch of this triple-star system to show how the three objects move in their orbits as time passes.

    Answer: A sketch with a pair of stars in a small orbit around each other and the combined system making a much larger orbit around a third body.
  2. Anne Archibald says that they can “account for” every pulse since they began their observations. What does she mean by that: does she mean they observed every pulse or they can calculate the time of every pulse?

    Answer: The astronomers can calculate the time of every pulse.

  3. Think back to Ole Roemer’s observations of the speed of light. Roemer observed that the moons of Jupiter passed behind the planet sooner than expected when Jupiter was closer to Earth in its orbit because light did not have as far to travel from Jupiter to Earth. In addition, he observed that the moons passed later than expected when Jupiter was farther from Earth in its orbit. That’s because light had a greater distance to cover. From this, he was able to measure the speed of light to fair accuracy.  The experiment conducted in the article used a different type of “clock”, created not by orbiting moons, but by a rotating neutron star. Explain how the experiment described in the article is related to Roemer’s experiment. Remember, we now know the speed of light quite precisely.

    Answer: This experiment solves the problem “backwards”. It used the known speed of light with the early arrival of a pulse to determine that the pulsar is closer.  A late arrival means the pulsar is farther away.

  4. “Gravitational binding energy” can be thought of as analogous to “nuclear binding energy”. Where in this course have you seen “nuclear binding energy”?

    Answer: Nuclear binding energy appears in discussions of nucleosynthesis, the proton-proton chain, the CNO cycle and the enrichment of the galaxy in heavier elements.

  5. Why is it important to test a scientific idea over and over again?

    Answer: It’s important to repeatedly test a scientific idea because there may be limits in which the idea fails.  These limits become more accessible over time as technology improves.

  6. Suppose that the result had been different. Imagine if the neutron star fell differently than the inner white dwarf. What would astronomers conclude about Einstein’s Equivalence Principle?

    Answer: Astronomers would conclude that the Equivalence Principle might be wrong for very dense objects. They would test this again in another system, if possible, as well as further test some of the alternative ideas mentioned in the article.

 

What other questions would you ask your students, based on this article? Feel free to leave suggestions in the comments!


How-to: A Day One Strategy for Improving Classroom Engagement

By Stacy Palen.

We often wish that students were more engaged in class. Sometimes we complain that students won’t ask questions, or that they won’t answer the ones we ask.

This is a training problem: we have to train students to know what our expectations are. Think of it this way: expectations are different in every classroom. In some classrooms, especially large ones, students are expected to sit passively and quietly, taking notes, watching videos, or just letting the professor “get through” the material. In some classrooms, students are expected to “discuss.” In others, they are expected to “do.”

How does a student know which kind of classroom they are in? On day one, you can make your expectations clear to students, but it may require allowing yourself to be pretty uncomfortable on that first day! 

First, tell students that you expect them to contribute during class. Then, give them an immediate opportunity to do so! Ask your students what they want to learn in the class, and then stop talking.

Let them tell you. Make a list on the board or screen, where you faithfully show that you hear their contributions. This is not the time to say, “Well, we aren’t really going to talk about constellations.” This is the time to say, “Constellations. Good.” And write that down and then say, “What else?”

This lets students know that you are not looking for a “right” answer. You are taking their input seriously, even if it’s “aliens.” It takes courage for students to speak up in front of their peers and risk being wrong. They need help with that.

Second, tell them that you expect them to answer questions. Then, give them an immediate opportunity to do so! Ask a question that has no “wrong” answer. For example, ask about a recent astronomical event. At the moment, the solar eclipse is a good one.

“How many of you heard about the solar eclipse?” Hands will go up.

“How many of you witnessed it?” Hands will go up.

Ask those students to share their experience with the group. Then ask a question that clearly has a “right” answer: what causes a solar eclipse? WAIT, for a long time if necessary.

Seriously.

For a long time.

I sing “Happy Birthday” to myself to force myself to wait long enough for students to know that they have to take responsibility for answering the question. Sometimes, I even tell them this is what I’m doing--they think it’s hilarious.

Often, in the first few days, as I wait, a student will speak up and say, “Wait. What was the question?” That's because their attention drifted away. They did not know that I was expecting them to stay focused and answer questions--because they didn’t know what kind of classroom they were in.

WAITING is particularly important in the first few days of class. It’s how you show that you actually expect your students to respond to you, unlike those other professors who ask rhetorical questions. Wait. Every single time. How will students know it’s not a rhetorical question, unless you prove it to them by waiting, even though it makes you uncomfortable?

That’s enough for you to try to keep track of on the first day! Try it out and let me know how it goes in comments below.