Math Education Research

Education is a complex and messy endeavor. We all have our own ideas about how children should be raised, about how learning happens, and about what is important for children to learn. The schools have to deal with parents on all sides of every political spectrum demanding what they think is needed for their children.

Educational research that doesn't acknowledge this messiness, that tries to buy 'scientific' cachet with control and treatment groups but frames the questions too narrowly, is more likely to reinforce the values of one group than to deepen our understanding of the learning and teaching enterprise. (Of course, we're each more likely to see flaws like this in research that doesn't resonate with our own values.) In this post, I want to dissect a flawed study, together with my friend Ben, and then give links to some studies I've enjoyed reading. I would actually appreciate it if any of you would like to point out flaws in some of those. (I'll start a new post for each one, so we don't get all tangled up.)

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“The Advantage of Abstract Examples in Learning Math”, a study done by Jennifer A. Kaminski, Vladimir M. Sloutsky, and Andrew F. Heckler, was published in Science magazine in April 2008 (you need a paid subscription to see the article) and highlighted in the New York Times soon after. I had responded to this study before I started blogging, in email to colleagues. It was brought to my attention yesterday by Ben Blum-Smith's new blog, Research in Practice, where he writes a great critique of it. I agree with all he says (here and here) and want to add a bit more.

The people who did the research, along with the unquestioning NYT author, say that the research shows that students learn math better with abstract examples than with concrete examples. Ben and I are saying that their research design is severely flawed, and that they've shown nothing useful.

Ben gives a great description of what the research folks were supposedly trying to teach, which was the properties of a "commutative mathematical group of order three". That's a fancy name for something not much more complicated than clock arithmetic. Imagine a clock that only has three hours on it, and instead of a 3 at the top, it has a 0. So 1+1 is still 2, but 1+2=0 and 2+2=1. This is the most common example used when people are first learning about these groups. The examples used in the research seem very contrived in comparison.

Both the research folks and Ben described a tennis ball factory, where you're keeping track of how many balls you have in hand, after you've put as many as possible into those 3-ball cans, but Ben's description makes a lot more sense than the one used in the research. (When I originally read this study over a year ago, I never saw the tennis ball example. In the one 'concrete' example I was able to find details for, they used a full cup for the 0, or identity element, which I would find confusing.)

People who actually study groups like this sometimes do it without numbers. The 'elements' of the group might be labeled a, b, c instead of 0, 1, 2. There are properties that can be studied, like identity elements and inverses. (0 is the identity because adding it to other elements doesn't change them. 1 and 2 are inverses because 1+2=0, the identity. These properties can make sense even when the elements aren't numbers.) So the researchers 'taught' this using 'abstract' examples for some subjects and 'concrete' examples for others. They quizzed all of the subjects using a group consisting of a vase, a ladybug, and a ring. Although concrete, these strange elements fit much better with the 'abstract' example than with the 'concrete' example. It's not surprising that the subjects whose example was more similar did better when quizzed.

There is lots of narrowly focused research like this out there. It may be useful in physics to narrow a question down to one detail, when the interactions between the small parts is clear. But in social arenas, all the parts interact, in very complex ways. Research like this cannot tell us much of value, even when its design is less flawed.

Kaminski et al want to say that their research tells us children learn math better without concrete examples. Their claim is very political. To promote it, they have done a number of studies with minor variations. (Googling their names, I see work done in 2003, 2006, and 2008.) I'd rather see education research that addresses the big, messy picture. Here's the MAA president-elect's take on this, and here is an article interviewing one of the 3 authors of the study.


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Research that I've found more interesting is much broader. It doesn't attempt a double-blind statistical power, which can only come through narrowing the questions until they become too artificial to be of use.

I've been reading Jo Boaler's work on the benefits to students at all skill levels of working together with one another. On first glance, it may seem that tracking would allow the best students to go farther, and allow slower students to get a more solid grasp on what they're studying, but tracking actually harmful to students at both ends and those in between. Boaler shows why, and shows how to make heterogeneous grouping work. One article is here, or you can read her book, What's Math Got to Do with It?

Alan Schoenfeld wrote the book Mathematical Problem Solving, in which he describes his very detailed research into the process followed by math students versus mathematicians while attempting solution of a hard problem. He taught a course in problem-solving strategies, using Polya's framework, and gave a pre-test and post-test to those students. Compared to students taking a more typical math course, these students' problem-solving skills improved significantly more. Here is an article of his on a different topic, how mathematical conversation in the classroom promotes learning.

Both Boaler and Schoenfeld compare groups with and without the 'treatment' they believe is effective, and show evidence for their belief, as Kaminski et al did. There are other sorts of research, which attempt to understand children's learning processes, but which don't have 'control groups'. One such project I'm interested in following is Measure Up, which introduces math through measurement and algebraic reasoning. Here's an article from that project.

Our biggest problem may not be understanding better how children learn, but implementing the good ideas that come from this research. When most elementary teachers are uncomfortable with mathematics, what we need to focus on is how to help them. Liping Ma's book, Knowing and Teaching Elementary Mathematics, compares elementary teachers in the U.S. and China. The Chinese teachers understand the math much more deeply. A good summary of the book is here. And an article by Ma is here. And here's a piece by another researcher, Hung Hsi Wu, on the depth of understanding needed by elementary teachers.

What math education research have you found valuable?

What math is used for...

When I first got out of college, in 1979, I wanted to do something with my math degree besides teaching. I knew I wanted to eventually become a teacher, but I figured I'd be a better teacher if I had a deeper understanding of what math is used for.

I think I had an interview at that time at Bechtel. (It was someplace with more security than I'm accustomed to.) I know I thought a lot about how most of the employers who might want my talents were doing things I didn't approve of. Code-breaking sounded fascinating, but back then code-breaking meant working for the government (or so I thought), and I was no happier then than I am now about my government's warring tendencies.

I ended up doing some computer programming for a very small company. It was business reports - not very exciting, but nothing terrible, either. After a bit more than a year, the company folded, and I headed toward teaching.

I haven't had to think about this issue much in the 30 years since then. I've seen much broader uses of math, and coding-breaking has come to be identified more with information security than with espionage. But of course the war-makers are still big employers of mathematicians, and that was made clear to me this morning by a post that looked really fun at first.

Liz, at STEM-ology, posted It's a Math World, After All, about a cool new 'ride' at Disney World called Sum of All Thrills, that lets kids (and adults?) design their own ride, and then "experience it on a giant robotic arm simulator." (It reminded me of the turtle geometry I was just reading about in Mindstorms.) I loved it!

But then I followed her link to the original Yahoo article, and saw this:

"Sum of All Thrills" sponsor Raytheon has nothing to offer the average consumer. But the high-tech defense and homeland security contractor does have jobs for those passionate about engineering...

I wasn't planning on going to Disney World any time soon, but Disneyland was the high point of a big trip my family took when I was 12, and I might be willing to take my son some day. This is a reminder to me of how much corporate propaganda is built into places like that. My comment on Liz's blog ended with "That's a show-stopper for me. War-mongers get way too much access to our children, and I have a problem with that..."

Have any of you struggled with math's less wholesome uses?

Math Teachers at Play

Dan has posted Math Teachers at Play 17 at his mathrecreation blog. (There were two issues of MTAP #15, so this is really the 18th issue.) I've been negligent about linking to each issue of MTAP, so here's an archive of links to all the past issues (click on 'past posts').

Two of the posts I especially enjoyed:

John Cook has posted on the math behind musical scales in his posts Circle of fifths and number theory and Circle of fifths and roots of two at his blog The Endeavor.

Alison Blank has put together an inspired and inspiring Prezi presentation, Math is Not Linear, and posted about it on her blog Axioms to Teach By.

I added quite a few blogs from this issue to my already long list at Google Reader. I think I'm really going to like the quirkiness of this blog:

Vlad Alexeev shows us an impossibly small book of impossible figures in the post Mini Books of Anatoly Konenko at his blog Mathematical Paintings and Sculptures.

Next issue is in two weeks, right here. After that, it looks like we're switching to a once a month schedule, on the 3rd Friday of each month (with Carnival of Mathematics taking 1st Fridays).

Blog Action Day - Climate Change

Today is Blog Action Day 09 - Climate Change. (Thanks for the pointer from squareCircleZ, who wrote a great post on it.)

What's that got to do with math? Well, we won't have time for the cerebral pleasures of math if we're dealing with floods and droughts, famine and war.

Here's a site that give some numbers to help you think about it. I try to live simply, but if everyone lived like I do, we'd need over 2 planet Earths.

I wanted to write more, but this is already coming out late in the day...

Links on Thursday

I've been saving these up, I guess ...

I keep hearing about thought experiments in physics lately. Here's a good post on that idea.

This post is mostly about John Conway, the game of life, and how that relates to segregation (watch the video).

Here's a post on the n-queens problem. (For n=8, put 8 queens on an 8x8 chessboard so none are attacking any others.) On Sunday evening I played around, trying to find a solution, and couldn't. On Monday morning I showed the problem to Artemis (the boy I tutor), who said, "That has 92 solutions, unless you don't count rotations and reflections. Then it has 12 solutions." I wanted to get past his memory and work with thinking about it. He put 4 queens on a graphpaper board, and said "now go from there". I told him I'd done that much and gotten stuck. He and I crossed out places queens couldn't go, and I suddenly saw a solution. I told him he was a good teacher. [He also showed my his attempt to multiply out (A+B)^5. He isn't yet grounded in why you do the steps you do; he did all the right steps plus a bunch more. We'll get there... :^) It sure doesn't feel like work, tutoring him.]

This one is not really math. NASA is planning to crash something into the moon tomorrow, and it should be visible with garden-variety telescopes. Lesson plans available, and I might do something with the kids I teach, if I can pull it together in time.

Tanya Khovanova doesn't like IQ tests, and has made a funny question up that she's pretty sure won't appear on anyone's IQ test.

This woman knows how to take learning into her own hands! She had a concussion, and decided to create a multi-player role-playing game called SuperBetter to help herself recover. Wow! (Thanks, Dan, for pointing me there.)

Sweeney Math has a nice systems of equations project for an algebra II class, or a stat class. "Students find data online that they are interested in comparing. (Sales of video games v sales of movies, Wins of their favorite sports team v wins of their friend's favorite sports team, Women's race times v Men's race times, Success of movie with many sequels v another, Sales of Abercrombie v sales of American Eagle, etc) They graph and find best fit lines for each set of data, then answer some thought provoking questions about the results." One of the questions is the point of intersection and what it means. I like it.

At God Plays Dice, there's an interesting (to me) book review post titled Counterexamples in X, where X is a field in mathematics.

I don't have an interactive whiteboard available where I teach, but I do have a 'smart classroom' (computer hookup and internet projection). Most of the examples in this Interactive White Boards interview can be used in my classroom, I think.

I can't remember now why I did this, but I'm still intrigued... I went to Wolfram Alpha and typed: factor 1782^12+1841^12. It's just a bunch of big numbers. Why do I like it?

Oops! There was one more. I remember from when I was young, a letter to Ann Landers, claiming that you get wetter in the rain when you run than when you walk. Here's a good physics analysis debunking that.

Mindstorms: Children, Computers, and Powerful Ideas

I have a new hero. Seymour Papert writes so brilliantly about math, learning, and how it all fits together, I think I'll have to read his book a few times to absorb it all. He wrote Mindstorms: Children, Computers, and Powerful Ideas back in 1980. (Why I never read it until now is a mystery to me. I've taught programming and math since the early 80's and could have used these ideas.) I expected a book about computers from the 1980's to be pretty severely dated, but the ways in which it's dated are surprisingly trivial. Papert's notions about why programming a turtle is valuable are still true, powerful, and not widely applied. But the book goes way beyond programming turtles.

He starts the book with a story from his childhood, about how he was in love with cars, and at two knew about "the parts of the transmission system, the gearbox, and ... the differential" (more than I know even now). He adds:

I became adept at turning wheels in my head and at making chains of cause and effect: "This one turns this way so that must turn that way so..."
Gears, serving as models, carried many otherwise abstract ideas into my head... I saw multiplication tables as gears, and my first brush with equations in two variables (e.g., 3x+4y = 10) immediately evoked the differential. By the time I had made a mental gear model of the relation between x and y, figuring how many teeth each gear needed, the equation had become a comfortable friend. (page vi)

But of course not all children will fall in love with gears the way he did, hence his "attempts ... to turn computers into instruments flexible enough so that many children can create for themselves something like what the gears were" for him. He points out many ways in which the gears encouraged his understanding of mathematics, including affect (he loved them), body knowledge (he could turn his hand or body the way the gear turned while he was thinking about it), and flexibility as a model for mathematical structures. His creation on the computer, the LOGO language, included a turtle on the screen (or a robotic turtle) that could be moved around.

He worked with Piaget for years, and has a similar clarity about the deep learning that must happen for children to understand things that seem very basic to us adults. He has differences with Piaget, though, and the most salient here is his conviction that the cultural environment makes a difference in when kids will learn things. To learn formal systems like mathematics, it helps for kids to have a fun "world" to play in that uses formal systems, like LOGO. So Piaget saw the 'formal reasoning' stage of development happening around 12, and Papert thinks much younger children can do formal reasoning if given the right environment.

He has a lot to say about the damage wrought by the culture associated with schooling:

Our children grow up in a culture permeated with the idea that there are "smart people" and "dumb people". The social construction of the individual is as a bundle of aptitudes. There are people who are "good at math" and people who "can't do math". Everything is set up for children to attribute their first unsuccessful or unpleasant learning experiences to their own disabilities. ... Within this framework children will define themselves in terms of their limitations, and this definition will be consolidated throughout their lives. Only rarely does some exceptional event lead people to reorganize their intellectual self-image in such a way as to open up new perspectives on what is learnable. (page 43)

Of course kids in school hate making mistakes, and want to throw the mistakes away, or run away themselves. But if they're doing programming on a project they care about, the mistakes become bugs that need fixing, not testaments to their inadequacy, and they become willing to debug. The more they get into that habit, the more willing they'll be to deal with future 'mistakes' that way.

Most of us learned Euclidean geometry in high school, with its axioms, straightedge and compass, and our first taste of proofs. (There are alternates to this, non-Euclidean geometry and Origami geometry, that still use a system of axioms and step-by-step deductive proofs.) Analytic geometry uses the x and y coordinate system to connect algebra and geometry. Papert mentions those two and then talks about how turtle geometry is both easier for kids to connect with (tell the turtle how to move in a circle, by figuring out how you'd do it) and more sophisticated (it has a deep connection with calculus). Once a child has really played with turtle geometry, they're likely to feel more at home as they learn about other geometries. Papert goes into how using turtles to think about physics is likely to lead into some deep science learning, too.

Reading Mindstorms motivated me to find and download Scratch, a modern descendant of LOGO, and start learning it. Scratch has 'sprites' instead of the turtle. You can create as many sprites as you want, and give each one a script. This week I've brought my computer in to Wildcat, where I teach kids in a very free-form environment, so they can play with Scratch. They are loving it. I'll probably post soon about that.

While I was online, searching for more information about Papert's recent work, I discovered that he'd been in a tragic accident. While in Hanoi in December 2006 for a conference, he was hit by a motorbike and suffered a severe brain injury. There is hope he will eventually recover, but he hadn't yet as of July of 2008. Here's the news article from then. I've searched and haven't found anything more recent. I'm wishing him well.

I want to include so many quotes, but I think I'll just write more posts on this later. If you want to think deeply about how children (and adults) learn, read this book. If you want a fresh perspective on how computers might be used with children, read this book. If you want more reasons to shake your head over the current testing craze in the public schools, read this book.

Why Math? Why School?

Part I. Why Math?

Deborah Meier (author of The Power of Their Ideas) co-writes the Bridging Differences blog with Diane Ravitch. Meier's topic today is 'Why School?' She is discussing what she hopes students will learn in school. Deborah Meier has done amazing work, and I usually like what I read at her blog. But her conception of math seems terribly shallow to me:

Sufficient mathematics to make sense of what they find in the media—statistics, probabilities, forms of graphing, percentages, et al to a high degree of sophistication by the time they are 16. Basic arithmetic computation by 13.

I agree that one reason to learn some basic math is to be able to have intelligent opinions about national issues: Is it more costly to have single payer health care or what we have now? Is social security doomed because more and more of our population is the elderly? I would never have expected to be interested in a blog on the tax code, but Mary O'Keefe (who runs the Albany Area Math Circle) writes great posts about issues like this on her tax blog. How can you understand national budget questions if you get nervous about numbers?

So yeah, these are reasonable goals, but dreadfully insufficient.

Here's what I wrote in response:

Deborah, I have a deep respect for you and the work you've done. So I was distressed to see your opinion of what math students should know - mostly arithmetic and statistics. Well, that's a fine start, but it is not enough.

Shouldn't they know enough math to understand science? Shouldn't they see the beauty of math? Two books, accessible to anyone, that I'd highly recommend, are The Cat in Numberland, about infinity, and Powers of Ten. (I've blogged about a number of fun math books at my blog, Math Mama Writes.)

What Diana said above about literature and history applies to mathematics as well: We will teach mathematics because it is important and beautiful. We will teach it not because it will save our society, not because we "must" know particular techniques, but because we simply do not have it in our hearts to do otherwise.

I don't feel like I was particularly eloquent. If you think math is important for more than these basic uses, please go on over there and say your piece.

And I'd love to hear from you here. Why is math important? And what math is vital for schoolchildren to learn? I love math, but I don't feel clear on why people who don't love it should learn it (beyond the basics discussed above).


Part II. Why School?

Meier labeled her post 'Why School?' in response to Mike Rose's new book with the same title. I've enjoyed his previous books, Lives on the Boundary and Possible Lives, so I expect to like this one too. Whether it will answer some of the questions I find most difficult remains to be seen.

My personal vision of the ideal school is more like a kids' community center, where the children decide how to spend their time, and are surrounded by resources and adults who want to share in the learning adventure. Deborah and her respondents talk about what should be 'required'. I don't think it's possible to require students to learn anything. The best we can do in a system based on requirements is to show the students (who are usually still eager to learn, if it appeals to their own values and priorities) why our subject is vital. That's why I love the work Dan Meyer and Kate Nowak are doing making high school math topics relevant for their students.

These two questions go deep for me: Why Math? Why school?

The Joy of Tutoring

Artemis* is 8. He arrived for his first tutoring session last week ready to learn more about trigonometry. He doesn't yet do algebra, and a few days ago said "I can't subtract", but trigonometry is what he's enamored of right now. (He can subtract, he just doesn't know how to use the standard algorithm, yet.) He's full of extremes like this. He was reading before he was 2, but had very little control of his body until recently.

For the past year he's been coming to the math salon I host, along with his parents and twin sister. The first time he came, he was so excited he just had to twirl around and get his whole body moving. (He reminds me of myself. When I'm really excited, I just have to wiggle and wag my tail.) He's still excited, but now he can be part of a group of people working together on a math problem. Part of his excitement during our tutoring session was that he got to have me all to himself. He snuggled up next to me on my sofa, and we dove in.

I started with the Pythagorean Theorem. He knew there were hundreds of proofs, but I don't think he'd really walked through one before. The proof I'm most familiar with involves a bit of algebra, and for him that was the complicated part. (Maybe he's ready for lots of heady stuff but not yet algebra? We'll see.) Just now I looked up proofs to try to find the one I used. Didn't get a good link for that one, but here are two I'll show him next Monday, both completely visual: One with the triangles hinged, the other with them sliding.

[In a previous post I mentioned mathematical holes that can cause students grief for years and years, like not learning your times tables in 3rd grade because you were out sick. I was unsure whether I wanted to say that because Artemis and others like him were in the back of my mind somewhere. When a student isn't expected to know things in a particular order, it's not too hard to work around them, and get to them later.]

I showed Artemis a few more basic geometry proofs, like the angles in a triangle adding to 180 degrees. In the middle of our one-hour lesson, he got so excited by it all he just had to move, so he took a 10-minute break on the trampoline. At the end of our lesson, I lent him Geometer's Sketchpad, Who Is Fourier?, and Mathematics: A Human Endeavor. He's been reading the Fourier book since then, and came in this week excited about one of the formulas he saw in it.

[Ooh, this is my first time doing that. I like it! I used codecogs.]

It seemed to me that he was intrigued by the fact that sine isn't additive. So I played the mystery box (or Guess My Rule) game, where I have a function in mind, and he figures it out by giving me inputs to see what outputs I give him. It gave me a fun way to talk about functions, input, output, domain, range, etc. With each of the functions I used, I then drew a graph, and we looked at whether it would be additive. I talked about it as linearity.

He wanted to think about , so I pulled out my TI calculator. I tried to keep chatting with him, but I found myself saying "Look!" a few times, and belatedly realized he was too entranced by the calculator to do anything else. So we looked at things like , which I knew went with what he'd been reading in the Fourier book. I let him borrow the calculator, and later that day his mom went out and bought him one.

Next week he wants to take a walk and find math all around us. Sounds fun to me. (It took me a moment to let go of the notion that we had to do something more industrious.) ;^)

I am like a kid in a candy shop myself, getting to work with someone who loves math so much. It feels like jazz improv, taking his lead and doing a riff on it. Wow! I'll be taking on a few more students in the coming months. I wonder if any of the others will lead me as well as he does, so I can learn more about how to teach by following.


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*He decided to use the pseudonym Artemis for my blog posts because he likes the Artemis Fowl books.

I Love Nerds!

First off, you gotta know that I've been trying to reclaim the word nerd as something positive for years. One of my favorite bands in the 80s was The Roches. The refrain in their song Nurds has the line "Nurds, I'm so glad I am one!" And googling got me this charming link for a radio show about nerds.

Maria Andersen just posted this Calculus Rhapsody video on her blog, and I'm still giggling! (I've linked to YouTube. If that's blocked where you are, check out Maria's embed.)

So are there any fun, complimentary words for us nerds? (None found in the online thesaurus I just checked. In fact, nerd is connected to stupid person and fool there more often than anything else.) Our culture clearly has issues with the power of intelligence...

Math Relax: A Guided Visualization for Overcoming Test Anxiety in Math

If you are nervous during tests, try listening to this every night for a few weeks. (Although I don't feel the quality is perfect, and eventually hope to redo it, many students have found it very helpful.)

If you like it, please let me know.

Click Here for Soundtrack

This recording combines relaxation techniques with a guided meditation focused on enjoying math (and tests) more. It may seem absurd now, but if you repeatedly imagine yourself actually looking forward to a test, then you’ll eventually find your outlook at test time to be at least a little bit more positive. The techniques I use in this recording are taken from the work of Margo Adair in her book on applied meditation, Working Inside Out.


Credits
Voice: Sue VanHattum
Flute music: Wayne Organ
Applied Meditation Concepts: Margo Adair
Script: Sue VanHattum
Recording Studio: Contra Costa College
Preliminary Release: Muskegon Community College
Tested by: students at Muskegon Community College and Contra Costa College


If you find this helpful, please let me know. If it has made a big difference for you, please consider making a donation in Margo Adair's name to one of the following organizations. Margo Adair died of cancer on September 2, 2010. Please make sure the organization is still active before donating.

• Community Alliance for Global Justice 
• Detroit Summer
• Seattle Young People's Project

How To Use This Recording

Before listening to this recording for the first time, read “That’s How Math Is” (below), which talks about math learning, and includes a summary of problem-solving steps, so you can approach math from a good perspective.

The more often you listen to this recording, the more effect it will have. I recommend listening at bedtime every night. (Don’t worry if you fall asleep during the recording, your subconscious will still hear it.) If possible, start at least 2 weeks before your next test. Whenever you start, keep using this recording through at least two months and two tests. Whether it’s helpful or not, I’d like to hear from you.

Contact me if you'd like a script of the recording. (Useful if you want to make a recording in a different voice, or with changes to the words, perhaps to use for a different subject.)

Jean Harvey, a student who used this while taking the Beginning Algebra course at Contra Costa College, says it took about 3 weeks of listening to it every night for it to make a difference. She didn’t expect to pass 118 and ended up earning a B in the course. She went from a 69% on the first test to a 96% on the second test.


“That’s How Math Is…”

Some things to know about learning math:
• In an ideal world, everyone would have lots of hands-on experiences to help them internalize ideas related to number, would always learn at their own pace, on their own schedule, and would have access to tutors and mentors who love math and love helping people find a good path to follow in order to learn it.

This is not that world - many students learned math from teachers who were themselves uncomfortable with it. (I’d guess about 4 out of 5 people are uncomfortable with math, and elementary teachers probably aren’t any better than the general population in this.) Those teachers were not able to explain math concepts in a way that made them make sense, and were often tense and would do ineffective things like requiring students to follow the book’s method. So the cycle of discomfort continues.

• Math concepts build on the ones before in a way that’s not seen in any other subject area. Even with good teachers, if you miss a few months in third grade (for example), that hole may cause you grief forever. If you recognize that there are holes in your past learning, it will be especially helpful to work with a good tutor or mentor to fill them in.







• Math is not about memorization; it’s about understanding. Ask why every step of the way, and you’ll learn math in a deeper, more satisfying way.

• Once you understand something well in math, it suddenly seems so easy that it’s hard to understand why it took so long to ‘get it’. This is true with any new concept, but it’s particularly noticeable with math: Imagine… You’re in class, struggling with a problem that seems impossible, and the person next to you blurts out “That’s easy!” You feel like a fool. It’s happened to just about everyone, including that person who thought it was easy. This happens partly because new synapses (connections between neurons/brain cells) are made as you learn – once they’re made the thing that seemed impossible now seems easy.

Even mathematicians are likely to feel dumb at first when looking at a new problem. That sensation of having no clue how to get started can be overwhelming. But the good mathematician has had enough successful experiences in their past that they find it easier to tell themselves they can do it. (When faced with a problem that’s hard for me, I often have this argument going on in my head: I can’t do this! Yes you can. No I can’t…)

Good mathematicians also have some techniques for problem-solving that help them break things down. Here are the 4 steps that George Polya proposed (but there's much more to it than this). More on this here.
1. Understand the problem.
2. Make a plan for how you might solve it.
3. Carry out your plan.
4. Look back. (Check your work, see how it might apply to other problems, etc.)

Solving math problems can be a real struggle, but the satisfaction once you do solve your problem can be quite powerful. Think of math problems as puzzles to solve, think of yourself as a detective, and have fun!



Added on 11/21/11: Test anxiety can be addressed in many ways. This guided visualization is one way. Googling 'test anxiety' will help you find many other ways. One method you might find helpful is described here, along with the research supporting it.