My calculus students are getting ready for their first midterm, and for some reason this makes me think of what it's like to take on a life of new frugality. It's a bit daunting (in the calculus realm) to face that first exam, or (in the frugality realm) to think about "giving up" things you've gotten used to.

As my students study, here is the first and most important piece of advice I give them:

And isn't that true of so many things? Watching Michael Jordan won't teach you how to do a lay-up: doing lay-ups teaches you how to do lay-ups. Reading about yard sales or bulk purchasing won't save you money; actually trying yard sales or doing a few bulk purchases will save you money. Well, it ought to save you money eventually. Although you'll probably make a few mistakes at first. But it's not until you try it that you learn how the system really works for you.

Which leads me to the other piece of advice I try to share with my students (although not this bluntly)

I hope my students won't fail their exam (although I know that some of them will). But in class and on homework, they're going to make mistakes, and the best students are willing (even eager) to do so. They make stupid guesses, see whether those guesses work, and learn something from what goes wrong. They do this so easily, they don't think of the wrong guesses as "failing". But the weak students stare at a blank piece of paper, unwilling to write anything down for fear of writing something incorrect.

And this, this is true for so many parts of life. I've had so many people tell me they can't make bread; a few say that they tried and it bombed. I could tell you my own failed-bread stories about orange juice bread (yucko), about the no-yeast failures, about the salt/sugar mix-ups in my own life. And yet, somehow, I mostly got through this. Bad bread is a learning experience: the sugar and salt are now VERY well labeled. Bt nowadays when my family finds out I'm making bread, there's a minor celebration. The mistakes were definitely worth the jumping-hugging-praising routine I get to go through now.

One of the joys of being frugal is that mistakes are often not very costly. This summer, I tried a homemade dishwasher detergent; it left a film on all our plates and glasses. For that little lesson, I'm out 45¢ worth of borax, washing soda, and salt. A few weeks ago, I bought a pair of yard-sale pants that I later discovered my son had just barely grown out of: another 25¢ down the drain. It's hard to think of those as "failure", but those are the kinds of stories that can keep a novice from even trying frugal strategies.

On paper, I'm teaching my students about the slope of the tangent line, but in reality, what I'm hoping they learn is so much bigger. I want them to bang their head against new ideas, to play with their own mistakes, to practice doing what they can't yet do. And I want them to come out the other side, experts on some simple math thing (yes! they can factor a quadratic!), ready to make new and bigger mistakes at higher and higher levels.

As my students study, here is the first and most important piece of advice I give them:

Do NOT read the book! |

Whenever I have a student fail a test and come talk to me about it, he always said he studied by reading the book. But math isn't about reading and understanding what someone else wrote, it's about actually solving problems. So don't read the book; instead, solve problems. Do the same problem over and over again, until it's easy for you. Practice

*doing*the math, not*reading*the math.
Here's another way to bomb the test. Do the homework problem; get it wrong; look up the answer in the back of the book; say, "oh, I understand now!", and then move on. The big mistake comes in that last step -- the moving on to a new problem. Instead of looking for new problems, grab a sheet of blank paper and start over. Do that darned problem again (and again) until it's easy to get it right. Make sure you can do it yourself, not just appreciate how the book or your roommate did it.

And isn't that true of so many things? Watching Michael Jordan won't teach you how to do a lay-up: doing lay-ups teaches you how to do lay-ups. Reading about yard sales or bulk purchasing won't save you money; actually trying yard sales or doing a few bulk purchases will save you money. Well, it ought to save you money eventually. Although you'll probably make a few mistakes at first. But it's not until you try it that you learn how the system really works for you.

Which leads me to the other piece of advice I try to share with my students (although not this bluntly)

Prepare to fail. |

And this, this is true for so many parts of life. I've had so many people tell me they can't make bread; a few say that they tried and it bombed. I could tell you my own failed-bread stories about orange juice bread (yucko), about the no-yeast failures, about the salt/sugar mix-ups in my own life. And yet, somehow, I mostly got through this. Bad bread is a learning experience: the sugar and salt are now VERY well labeled. Bt nowadays when my family finds out I'm making bread, there's a minor celebration. The mistakes were definitely worth the jumping-hugging-praising routine I get to go through now.

One of the joys of being frugal is that mistakes are often not very costly. This summer, I tried a homemade dishwasher detergent; it left a film on all our plates and glasses. For that little lesson, I'm out 45¢ worth of borax, washing soda, and salt. A few weeks ago, I bought a pair of yard-sale pants that I later discovered my son had just barely grown out of: another 25¢ down the drain. It's hard to think of those as "failure", but those are the kinds of stories that can keep a novice from even trying frugal strategies.

On paper, I'm teaching my students about the slope of the tangent line, but in reality, what I'm hoping they learn is so much bigger. I want them to bang their head against new ideas, to play with their own mistakes, to practice doing what they can't yet do. And I want them to come out the other side, experts on some simple math thing (yes! they can factor a quadratic!), ready to make new and bigger mistakes at higher and higher levels.

Thanks for this post :) I keep beating myself up over not being good at all of this yet, but it's good to be reminded that practice makes perfect, not intention.

ReplyDeleteWhen I was your age, Steph, I used paper towels. I shopped at malls. I bought lots and lots of packaged food. You're enough of a math geek (and a theist) to know it's not the position vector, it's the direction of the velocity vector that matters. Courage and persistence! -MM

DeleteThis was so good! I've used a similar analogy for learning to sew. You can't expect to make a haute couture dress on your first try. You didn't learn to speak or walk without stumbling and making mistakes but you didn't give up. You kept practicing and trying until it became second nature. I really like the math/calculus analogy. It's okay to make mistakes because that means you're learning. My husband is a great teacher because when he learns something he breaks it down into small steps and figures things out then he can teach someone else without leaving out important details, and explain a lot of whys to the student. I'd love to take math from you, you sound like a great teacher too. (Math was always my worst, most feared subject.)

ReplyDeleteThanks, Rozy! Well, I haven't learned to make a haute couture dress on my Nth try (where N is a lot of years of sewing). But I can put a mean patch on a pair of pants, and I sure get a lot of practice at that! --MM

DeleteWhen I would study for tests, I wrote my own tests. That's another good way -- I practiced tripping myself up.

ReplyDeleteAnd I totally agree about reading the book and about problems. I would always have to sell myself on the correct answer. I had a prof who had lots of practice tests available. I'd take the tests, look at my wrong answers, and then write out why I got it wrong and why the correct answer was better.

Practice tripping yourself up -- that's a good one. Yes, I like the idea of pushing yourself even beyond the assigned problems. Of course (and I'm thinking of certain students here) . . . that assumes you've done the assigned problems in the first place. Sigh. --MM

DeleteAlgebra was so hard for me in hs and college. After I got my MA, I taught GED until I could get a job. I ended up loving teaching GED. My whole digestive system was so fouled up the first morning of teaching, all because of Algebra. Finally, I developed a system of mnemonics for myself so I could even remember without looking at the book. I taught in many steps.

ReplyDeleteI am a word person not a number person. I could easily teach number people and knew how to reach word people. Even though I am an English teacher, I had rather teach math to anyone. Of course, I am not certified in math. Finally, I got to the point where every day I would do algebra and geometry for fun.

When I handed students their GED consumables, I put a huge X though the directions and told them to listen to me, NOT to read directions. One girl sat for ten minutes without putting pen to paper. She just looked at the problem and the blank piece of paper.

I asked her what she was doing. She said--thinking. I told her to quit thinking and make a mark of some kind on her paper. She was shocked because she said teachers always told her to think about what you are doing. That paralyzed her. She tried to figure out how to work the problem before she even wrote it down!!! You have validated advice I gave, advice that other GED teachers criticized.

I can sew anything that can be sewn! Any mistake in sewing is usually just covered up or embellishment is added...lol.

I like what you say about having students listen to you. It's really hard to write about math, because the words can't point: `divide by this x over here; then see how this term and that term cancel? ' You can't say that in writing. But you can say it in person; thanks for being a good teacher!

DeleteI like this post a lot! I keep struggling with the courage to do the writing that I want to do, and what you have here goes in line with what my husband keeps saying: don't read about the "right" way to write (and don't worry about reading too much in the genre anyway if you want to create your own "rules" for your story--at first I was getting too tied up in what I thought was appropriate research.) AND to just do it. That the first drafts can be fixed, scrapped, etc and that I need to practice getting things on paper more than anything else.

ReplyDeleteI think the secret of good writing (and good math and good photography) is to do it SO much that you're happy to throw most of it away, because you know either you've got good stuff left over, or because you can write/solve/paint something at least that good again easily. Once you know there's a good chance you're going to scrap whatever you start, you don't have to freeze up about whether it's "right" before you put pen to paper.

DeleteYou've just summarized Carol Dweck's research. She talks about kids who have neen praised so much about being intelligent that they are afraid to make a mistake or try something hard. The other type of learner asks questions, challenges themselves, goes to tutors, and tries new things. I'd recommend her book.

ReplyDeleteAh! This is a book I've not read, but I've seen similar studies. Because of her/them, I make sure the praise I give my students is (1) specifically about the particular work they just did, and then (2) about a character trait involving persistence, not brains. For example, I'll say, "you did really well on that last homework. I can tell you've been working hard, and I admire that." I should go get her book, huh? --MM

Delete