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For Parents: Math Facts and Weird New Ways of Solving Problems

At my favorite coffee shop this week the barista asked me if I had any advice for helping her daughter with her math homework, and helping her daughter with her math facts. I sort of talked her ear off until she had to help the next person. Today it occurred to me that other people might have the same question, and also I had a lot more to say about that.

Math Facts

So many students come to me in 7th or 8th grade struggling because they don’t know their math facts well. If they don’t know their math facts (ie 7×7=49) then when they’re trying to do things like long division or multiplication or even algebra things like foiling, every time they run into something they need to multiply they have to stop and figure it out by counting on their fingers or adding up a whole list of 6’s, and it completely distracts them from the problem they’re working on. Lots of times it looks like they’re struggling with what they’re learning, but really it’s just that they haven’t memorized their math facts well enough.

However, I also meet students in high school who hate math (and therefore science and most of the really profitable careers) because someone tried to force them to learn their math facts in a way that made them feel really ashamed and frustrated and bitter.

So, it’s a tight rope, and it needs to be done, but it needs to be done carefully. Ideally, learning math facts can be fun.

A great place to start is with computer games like In Search of Spot or Number Munchers.

Flash cards can be great, if you can make it fun. (i.e. positive. Praise them for getting facts correct, don’t say anything if they get it incorrect. Say things like ‘whoa that was fast!’ when they answer a fact more quickly than usual.) 5-10 minutes each day is ideal, don’t make it too long or it stops being fun.

I like these flash cards.

You can also get infinitely many free math drills here.

Weird New Ways of Solving Problems

A lot of time students come in with a familiar type of problem, but their teacher is having them solve it in a way I’ve never seen before. Long division methods change all the time. So do factoring methods. This is really hard, because you want to help and explain things, and the way you’ve always done things is so familiar. But, learning multiple methods is really confusing for kids. It’s almost always better to try to teach them the method they’re learning in class. Because their teacher will build off that later.

This, of course, means that if you’re going to help them with their homework you have to first try to understand this new method yourself. The first thing you need to know is that it’s probably more similar than it looks at first, and if you think about it for a while and look at slides/notes from the teacher you can usually figure it out. There are also lots of online resources that will explain these new methods. My two favorites are Khan Academy and Purple Math.

Also, if you run into something you’re stuck on, post a comment here or email me and I can try to help.

In general, especially for younger students, the absolute most important thing is that it’s fun for them, and that they start to feel like math is something they can be good at. That idea will get them through many future math classes and will help them be successful eventually, no matter how hard it is.

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Magnetism for the Non-Mathematically Inclined

A few weeks ago I came across Magnetism: A Very Short Introduction while looking at other books in the same category as mine. It’s written by a physics professor at Oxford, and it’s completely fascinating and entertaining. Not what I expected at all. I was expecting something pretty dry, and focused on the physics. Instead, Stephen Blundell, the author, gives a whole historical background complete with descriptions of how people used to think magnets could heal them of psychic illnesses.

He also tells the stories of the physicists who discovered these phenomena. I loved seeing them as human beings, with personalities, rivalries, getting passed up for jobs/promotions, and being embarrassed that they weren’t very good at math.

It doesn’t have any equations or math in, it’s all about the concepts and the history.

What I wish someone had told me when I was just starting out as a physics major

(And a girl. Not that I was starting out as a girl. I had been a girl my whole life, but I was just starting out being a girl physics major.)

It’s not supposed to be easy

I had this idea that if you were smart things were easy. You either knew something or you didn’t. You either got it or you didn’t. My first few physics classes reinforced this. I had already taken high school physics, so the intro classes weren’t so bad. Then I got to my first real physics class, Mathematical Physics, taught by an old Russian professor. I assumed it would be easy. Then the first midterm came. Three problems. I had no idea what to do for any of them. I had no idea how they even related to anything we’d done in the class so far.

Suddenly I felt like I was out of my depth. I went to go talk to the professor. I had skipped class a couple of times and I think he knew that because he wasn’t that keen on helping me. What he said was “Maybe you’re just not going to get it.” After that I panicked. I felt deep down that I just wasn’t smart enough to study physics, even though it was something I had loved since I was a kid. I thought maybe I had just hit the limit of my natural intelligence.

I hadn’t. I had simply hit the limit of the place where I had learned things before. In my experience, 90% of the things that look like talent are actually practice. When you see someone with a perfect golf swing or a genius math brain, you’re only seeing the result, you’re not seeing the great coach they had, the hours they spent practicing, or all the times when they were a little girl sitting in the passenger seat of their dad’s truck talking about the differences between even and odd numbers.

I didn’t know this. I thought I had just hit the ceiling of what I was capable of. I wish I could go back and tell myself that there is no ceiling. There is no capable, not capable, or highly capable.

I work with a lot of students, and I can see some of them falling into this trap. They encounter a difficult problem and they start thinking they’re dumb. They’re wrong.

It’s hard because it is made of ambiguous problems. A lot like life. You learn some tools, some equations, some concepts, and then they set you loose on problems. At first, you will have no idea what to do. This is not bad. This is a feature of real problems. (See my post on problem solving for ways of tackling these things.)

In order to be successful in physics, you have to make the mind shift from “I’m not smart enough” to “this is hard.”

It’s okay. It’s okay if it’s hard. It’s okay if you look at a problem and have no idea what to do. It’s not supposed to be easy.

Unlike in most of your previous schooling, you’re not just learning one thing at a time and being asked to do something repetitively. You’re being asked to do something crazy hard, which is, stumble around until the pieces fit together. Or maybe they won’t, and you’ll ask for help. But not from someone who will tell you that maybe you won’t ever understand. Because if you keep trying you will.

Some people pretend they know more than they do

Another thing that left me feeling stupid was how much my classmates seemed to know. For some reason, the reputation that physics has for being difficult brings out some pretty intense egos. Be careful not to assume, just because someone is talking about something fancy-sounding that you’ve never heard of, that they know more than you do.

In this same first physics class, I met several people. We studied together. They talked a lot about things in ways that made me feel like they knew what they were talking about. Again, I felt like an idiot who didn’t belong.

Then we got our grades back. I did the worst I had ever done in a class, but it was still a 3.1. Those same people who I thought knew everything got 2.5’s or worse.

Never think someone knows more than you do just because they’re speaking authoritatively. They might just be covering insecurity with arrogance.

Ask for help

The next quarter I started my next physics class. I was completely panicking. Sure I would fail. Sure I wasn’t good enough. I had to take mathematical physics II.

Then, randomly, I did the best two things I’ve ever done for myself in the context of my physics degree.

1. I introduced myself to some other girls in that class.

2. I went to the professor after class, told him I was worried I was unprepared and that I wouldn’t do well.

The first thing immediately diminished my anxiety. Suddenly, I had people to commiserate with, to study with, to hang out with and laugh with. It was amazing and great and so much fun and the opposite of isolating. They told me later that they were so glad I had introduced myself, because they were feeling terrible and scared, too. It’s hard to do that, because sometimes other people look arrogant when they’re scared. Or they look closed and unfriendly when they’re scared. But, if you just introduce yourself it can be amazing. Having allies is probably the most important thing when you’re trying to do something hard.

The second thing was really good for my grade. Luckily, this professor was an extremely, extremely, nice guy. He laughed a little when he heard what my previous grade was. He told me it would be ok. He gave me a book. This book was basically a more detailed version of the textbook we’d used the previous quarter. It was waaaay better. (Incidentally, I do not recommend you use any textbook by Mary Boas. She leaves out waaay too many steps when she solves math equations.)

In that class I got a 3.9, and I made some awesome friends.

At some point in your life you will (hopefully) reach a place that will feel like the limit of your ability. It is not. It is the end of the mapped territory. It is the end of the things you already know how to do. It is a place where you will need to make a large step up in order to continue. Maybe physics will be this for you, maybe it won’t. Maybe it will be easy for you. If so, that’s great, but I hope you eventually find yourself in that place where you think you might just not be good enough.

 

Basic Orbital Dynamics

This explanation was written up by Caleb Pool, who very kindly gave me a bunch of awesome feedback on my Kinematics book, and then wrote this cool explanation of Orbital Dynamics that I thought you might find interesting/useful/entertaining.

Basics of Orbital Dynamics

When we look outside, we see ground that is pretty flat.  Trees go straight up, buildings stand up, and the ground is perpendicular to those.

orbits1

 

Even buildings look like they stand up straight.

 orbits2

Water looks very flat when we stand next to it.

 orbits3

If one side of a pond is higher than the other, the water flows until it is level again.

orbits4

Gravity pulls the water equally until it looks flat from a close perspective.  But what about when we start zooming out?

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The water still looks flat to us.  We zoom out some more.

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And some more:

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And some more:

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And a lot more:

orbits9

It seems the earth is round.  But we knew that.  How can all the water stay on without falling off?  Gravity pulls objects (and water) toward the center of the earth.

orbits10

Although, from our perspective, things fall down and water looks flat, the bigger story is that things fall toward the earth’s center and water curves around the earth.

 

The earth spins about an axis passing through the poles. This rotation causes a force that is largest around the equator, and it makes the earth’s water and land bulge slightly:

orbits11

This bulge makes the earth’s radius through its equator about 3,963 miles (6,378 km), when its radius through the poles is 3,950 miles (6,356 km).  A bulge that causes a 13-mile difference in radius between the poles and equator seems like a lot, and it is, but that bulge is not very big compared to the earth’s overall size.   That bulge plays an important role in many orbits.  The basics of an orbit work even if we consider the earth to just be a sphere for now.

 

Suppose you find an extremely high mountain that reaches above the earth’s atmosphere and you hike to the top, carrying an extremely powerful cannon.

orbits12

You fire the cannon.

orbits13

But that’s not far enough.  You increase the power and fire again and again:

orbits14

You keep increasing the power on this magic cannon and firing away from your mountaintop.  The cannonballs fly so far that, as they fall toward the earth, the earth starts curving away underneath them.

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You aren’t satisfied with just shooting cannonballs so far away they land on the opposite side of the earth, so you continue to increase the power:

orbits16

And look what just happened!  The cannonball went all the way around the world!  If you step out of the way, you get to watch it zoom by really fast (remember, you had to fire it fast enough to go that far!).  What happens now?  The cannonball doesn’t just stop dead in its tracks after it goes around once; it keeps going around.  And around.

It turns out that you have just made a satellite and launched it into orbit.  Until it gets slowed down by the thin atmospheric gas at the edge of space and burns up, your cannonball will continue to fly through space, following the same path as it circles the world.  Your special cannonball now joins the moon (earth’s only natural satellite) and a host of about 40,000 man-made objects as it endlessly circles in orbit.

 

As gravity pulls the cannonball toward the earth, it falls, but the earth’s surface curves away just as fast, so it never gets any closer to the earth as it goes around.

 

Now that you are satisfied with your powerful cannon, you look at other orbiting objects to see how they behave.

orbits17

Objects in orbit that are farther away from the earth go around it more slowly.  This is primarily because earth’s gravity is weaker the farther away from the earth you go.

 

Example:  Which is farther away from the earth?  The moon, which goes around once a month, or a GPS satellite, which goes around twice each day?

 

Answer:  the moon is farther away.

 

Fun Practical Applications of Physics

Here’s a fun video of some physics-based tricks. (Called 10 Amazing Bets you will Always Win, by Quirkology)

A couple of them (moving liquid from a bottle to a glass and separating two glasses) use air pressure creatively.

Physics can also be used to make some super-cool art.  Have you seen the Cloud Gate in Chicago?  On the outside it’s an amazing panorama of the city skyline, on the inside it’s a crazy multidimensional universe. That’s the magic of curved mirrors.  

Vsauce on YouTube has a really interesting video about curved mirrors.