Your Past is not Your Future
Difficulty with mathematics can be a rough experience. Sometimes frustration goes back a long way. When I was teaching at the local community college I frequently worked with students who had not taken math in twenty years – and were still afraid of it. Whether you are still in school and having trouble, or you have a long and difficult history with mathematics, the thought of “going through math again” is difficult for many people.
There is a condition related to difficulty with mathematics known as “math anxiety”. Math anxiety is an attack of sweaty palms, a racing heart, and a complete conviction in your own mathematical ineptitude. Its effects can be debilitating. In 1978 Sheila Tobias wrote a groundbreaking book about this affliction called, Overcoming Math Anxiety. I bought a copy when I was a sophomore in college because I was having trouble in yet another math course. Tobias provided some helpful tips on working your way to a more normal relationship with math. I used what I could and passed my math class. I also vowed never to take another math class. I had suffered long enough. Math was dead to me.
Except that the registrar at the small college I attended automatically signed me up for the next math class because I had completed the first one. When I got my schedule I laughed out loud because I was registered for “Math/Physics II”, a course that combined second semester calculus with physics. I planned to drop that course at the first opportunity. I wasn’t even planning to attend the first class.
Except that I was also a new young father who was constantly and desperately trying to appease his often-colicky daughter. I happened to be wandering the college lobby with my sobbing baby when the first Math/Physics II class was about to start. My friend Erin came right up to me and said, “Don’t you have class now? Let me take her while you’re in class. She’ll be fine.” Erin was sweet, capable, and most importantly, willing. I didn’t see how I could respond, “Yeah that’s great but would you take her if I don’t to go to math class?” I was grateful for the break, even if it meant sitting through a math class.
Except that this class turned out to be different. For starters, there were only four of us. The course was taught by an oceanographer from MIT named Carl Ketchum. Those people and that course changed my view of mathematics and, in doing so, changed my life. (Yep, it can really happen)
Carl believed, way back in the ancient 1970’s, that math was not well learned as a spectator sport. We learned by solving problems and having discussions. Discussions in a math class! Can you imagine that? What is there to discuss? You learn the rules and then you practice them, right?
Wrong. Carl showed us that math was like a game, a tantalizing puzzle that might take days or hours to unravel. He hooked us, addicted us, to the feeling you get when you finally figure a problem out. Once we got a taste of that we kept coming back for more.
One of my classmates, Annette, came in one day with a problem. She told us that she had been watching a drip in her bathtub. Initially she put a tuna can under it to see how long it would take the drips to fill it up. (She was already showing signs of math addiction.) But as she watched the drips, she noticed that the drips didn’t bounce out of the can. She began wondering how high up the drip would have to fall for some of the water to bounce out. Distance the drop would have to fall… velocity of the water… inertia of the can… We spent 45 minutes working that problem out.
Daisy came in with a problem about the amount of water in Jordan Pond. We spent almost an entire two-hour class working on that. The solution was elegant – simple and accurate. We ended up treating the pond as if it were an egg and then slicing it in half.
My own epiphany came over a problem of an orange falling from the ceiling of an elevator while the elevator was going up. I struggled with this for two days (I was hooked). On the third day as I was swinging my legs up into bed – I knew! The answer came to me completely formed. I got up and wrote it down.
My experience with mathematics changed dramatically for several reasons. I had a good teacher, a small class, and interesting curriculum. I was lucky for all of that. Yet I think many of us can have these things and still struggle with math. Why?
The first obstacle that many of us face is our history. We don’t succeed because our minds are sure that we can’t. Because we’re sure that we can’t, we don’t engage with the work. Because we don’t engage, we don’t persist. And because we don’t persist, we don’t succeed.
Are you sure you can’t succeed at mathematics? Many people are. Their learning experiences provide them with clear evidence that they are genetically incapable and should avoid anything technical or “mathy”. There is no math gene. You aren’t missing it. If you can graduate high school and pass your other courses there is no reason you can’t learn algebra. If you can graduate college there is no reason you can’t learn calculus. There is nothing special about these subjects except for the abysmal ways they are sometimes taught and the history of struggle that students bring to them.
If you’re going to get better at math you have to do something about your math baggage. You have to believe that you can succeed – given enough time and using the proper tools. And you have to cultivate complete disinterest in the internal voice that comes into your head whenever you get stuck:
“I don’t get this. I’ll never get this. This is too hard for me. I’m not this smart. I’m going to be rich enough to pay someone to do this stuff for me. But if I can’t do this I’ll probably end up selling hamburgers…”
You know how this dialog goes. Let it be. Don’t engage with these thoughts because they won’t help you. Negative thoughts about mathematics come from a belief in your own ineptitude. What’s a belief anyway? Here’s an important idea:
Beliefs are ideas you’ve repeated until they’re automatic.
You develop your beliefs about mathematics based on your experiences. If you have a bad experience in math class, you might think, “I’m no good at math.” If you think this often enough, you’ll live a self-fulfilling prophecy – you’ll be “no good at math.” The good news (the very good news) is that you can create a new belief the same way you created the old one: repeat a different idea. If you can affirm your ability, if you can talk to your unconscious mind, if you can conjure up a different attitude, you can change your belief about you and math. Changing that belief will have a huge impact on your success.
One way to start on developing a new idea is to hear from an expert (that’s me!) that math is not as hard as you’ve made it out to be. In thirty years of teaching mathematics here’s an observation I would swear by:
The major difference between mathematicians and struggling math learners is that when mathematicians come to something they don’t know, they don’t panic.
Panic, or anxiety, has an interesting effect on your thinking. Most of your complicated thinking is done in your working memory. Working memory is like the conductor of an orchestra. It’s got a lot to keep track of. Your working memory is paying attention to the math you’re working on, hearing the sirens on the street outside and noticing the spinach stuck in the teacher’s front teeth. Working memory also probes your older memories for clues that might help solve the problem you’re working on while simultaneously trying to make sense of it. When you are stressed or anxious your working memory resources shrink.
Shrinking working memory resources are like a shrinking conductor: Eventually only the first violinist can see him. There are lots of sounds going on around him but he isn’t controlling any of them. When doing math, a shrinking working memory means that you’re not doing all kinds of things that would be helpful to your understanding. You’re not making connections to things you’ve learned before. You’re not trying out approaches to evaluate their potential usefulness. At the point where you panic there’s a good chance you’ve forgotten what the problem is actually about.
You don’t have a problem with mathematics. You have a problem with thinking about mathematics. Your mental conductor is waving his hand to a single violinist who is playing, “I can’t do this,” over and over. In several clinical interviews I’ve conducted with struggling learners I have watched a student’s internal resources shrink before my eyes as his anxiety increased. I have watched students – when stressed – actually lose the ability to recall number facts that I have proof they know. Maybe you’ve felt like this when you’ve sat in front of a difficult math problem.
What can you do? I recommend eating ice cream.
There are varied opinions about why I would recommend ice cream, particularly. It could be that, being a Vermonter, I have an unusual appreciation for Ben and Jerry. It could be that I think ice cream is one of the nicest things a person can give himself or herself. Actually, ice cream is just a metaphor. When you’re stuck on some mathematics, stop before you get into panic mode. Do something nice for yourself. Eat ice cream. Or organic green tea if you’d rather have anti-oxidant goodness instead of sugar and fat. (I don’t have any trouble making that choice.)
The point is that learning mathematics is cyclical and almost always involves coming to a point where you have to struggle to understand something. YOU ARE GOING TO GET STUCK. You need to not take the getting stuck part so personally and so seriously. Getting stuck is part of the game and, actually, part of what makes it so much fun. What kind of game has no challenge and is still fun? Why do you think they put the chutes in Chutes and Ladders?
When you are stuck with your math you need to learn to walk away and then come back to it. You may have to do this several times. But if you know that this will be the case, and if you know that every mathematician does this (anyone you’d care to name), that should make it easier. Remember, don’t panic.
When Andrew Wiles solved Format’s Last Theorem (never mind what it was, it was VERY hard) he worked in secret on the problem for seven years. Every day he came back to the same problem. Every night he went to bed without having solved the problem. If he had panicked (“Oh my god why am I not smart enough to solve this problem?!) he would have given up. Because he had learned to live with not knowing (I don’t know if he was into ice cream or not…) he was able to persist and solve a problem that had confounded mathematicians for 300 years.
If you panic, you will not persist. So when you come to something you don’t know, or something that seems confusing, practice smiling and saying to yourself, “So this is where it gets interesting.” Then go eat ice cream.
But come back. Always come back. Math isn’t magic (OK, maybe it is a little bit). You have to think about where you’re stuck before your mind can help you understand. You need to be upbeat and persistent.
Persistence is critical for success in mathematics. By persistent I don’t mean that you should practice the same formulas over and over, though some practice is helpful. Persistence isn’t doing 50 more arithmetic problems after finishing 100 of them. Working equation after equation to rehearse the same rules is not what mathematicians do to improve at mathematics. Despite what you’ve heard, repetition is not the silver bullet. (“If only he had done 1,000 more quadratic equations he could have gone to MIT… no.”)
Persistence is when you are willing to let a problem sit and then come back to it over and over until you have solved it. It’s what mathematicians and successful math students do.
TIMSS is an international study on math performance. When you read in the paper that American math education is going to hell in a handbasket, you were probably reading about TIMSS. In the TIMSS study, countries are ranked by how well their students do. The tests are given at fourth, eighth, and eleventh grade. The U.S. did pretty well at the fourth grade level but dropped much lower in the standings in the upper grades.
TIMSS data came from the answers students from over 30 countries gave to questions on a math test. After working on the test, students were asked to fill out answers to questions on a questionnaire. The TIMSS questionnaire asked questions like, “What language is spoken in your home?” and “About how many books are there in your home?”
One researcher thought it would be interesting to investigate the questionnaire that accompanied the TIMSS test. He wasn’t so interested in specific answers to the questions. He thought there might be a relationship between finishing the questionnaire and being persistent. So, by country, he calculated the average number of questionnaire questions that students finished. Then he ranked countries by the results. Ordering countries by the average number of questions completed on the questionnaire had virtually the same result as ordering them by math score. There was a very strong correlation between the two. If you knew the average number of questionnaire answers for a country you could predict what their math success was likely to be. You could say (though there are many empirical reasons to be cautious about this) that persistence predicted success.
Five Steps to Make Your Future with Math Different from Your Past
1. Change what you believe about yourself and math. Repeat these sentences out loud (maybe in your backyard or bathroom so no one wonders about your well-being):
Everyone can do math.
I am capable of being successful with math.
I intend to change my relationship with math.
My past is not my future.
Having a new intention is the first step towards changing old patterns.
2. Take some time to make a list of how your life would change if you were good at math. Would your life be different in significant ways? Would you change your career path? Would you feel more confident? Are there activities you would try if you were good at math that you don’t try now?
3. Practice being patient with yourself. You can do this by taking on a difficult task and practicing patience when you don’t complete it immediately. A good place to start is with the crossword in the newspaper. Do as much as you can, tell yourself that you’ve done a good job so far and let it go. The goal isn’t to finish but to practice being supportive of your own efforts.
4. Practice and affirm your own persistence. If you’re serious about understanding mathematics you need to spend some time every day thinking/working on it. Try some problems or puzzles. Have a shot at Sudoku or logic problems if you’ve never tried them. Start with a little at first, until you begin to enjoy yourself. Doing math is like any other worthwhile pursuit. Some days will be productive. Some days you’ll be spinning your wheels. Just be patient and keep coming back.
5. Feel good about your effort. Applaud yourself for learning to do something you never thought you could do.
