Wednesday, April 10, 2013

Why all the problems with Math?

Nine out of every seven adults have trouble with fractions. It's a sad, but true, statistic.

Earlier this week, I was sent a link to an info graphic about America's Math Problem from my good friend Allison of, Well, I haven't actually met her yet, but I am sure she would be a good friend if we ever did meet. She sure seemed nice in the email. Here is one graphic from the webpage she sent me to:
Yet another depressing statistic

Anyway... the infographic shows a bunch of the usual depressing statistics about math education and the state of math abilities here in the U.S. For example, eighth graders in the U.S. rank 25th in the world in math skills. Only one in three are considered proficient.

One in five adults are innumerate. Why does this happen?

John the Math Teacher Guy

Years ago, I had a hobby job teaching remedial math at a local university. This zero-credit class was basically a review of high school algebra. Passing this class or testing out of it was a requirement for getting a diploma at this university.

It was not hard for me to get this job. Why? They were desperate for teachers. Every semester, 7% of the students at this university of 20-some thousand students were taking this class. (If my math is correct, that's 1,400 students.) The need for teachers was so great that basically anyone within 50 miles of the university who could conjugate a quadratic equation was asked to teach.

Over the course of eight semesters, I developed some opinions on why so many people have trouble with math, and of course, I have some opinions on how to fix it.

"It is obvious"
My eighth grade math teacher

There is an old joke, told by old math guys. A professor was going along in his lecture and said "It is obvious, then, that the cosine of this angle is..." A student toward the front timidly raised a hand. "I'm sorry, I don't understand. Why is that obvious?"

The professor started to speak, and then checked himself. He took out a piece of paper and started alternating between scribbling, pacing, and tugging at his beard [1]. After a few minutes, he ran out of the room to his office. Ten minutes later, he came racing back into the room, waving a piece of paper, saying "YES!!  It is obvious!!"

Here is my opinion... There is a tendency for professors and teachers to present a lesson so as to make the exposition seamless. When the teacher is at the board, the class does not get to hear the wrong turns that were taken the first time he or she saw this problem. "Hmmm... maybe I can make a perfect square if I add seven to both sides? Wait, that doesn't work. Oh fudge. Wasn't there some fudge left in the fridge?" I always gain weight when I do math anywhere near a fridge.

The student is left with the impression that the teacher just automatically knew that factoring the denominator would lead to the solution. This inevitably leads the student to the next thought... "I didn't immediately see that, so I guess I don't have a math brain." 
Normal brain (on left) compared with the math brain (on right)

As much as I would like to claim superiority over mere mortals for my stupendous math brain, the whole "math brain" thing is a myth [2]. Twice this past week, I found myself helping folks with Excel. Both of these folks were giving me variations on the "I can't do math" theme. And the thing is, both of these people actually did understand what they were asking me about. All I did, in both cases, was say "yeah, you're doing it right."

Don't get me wrong here -- I think most math teachers are good. A typical student has 13 teachers who learn them math by the time they get a high school diploma. If only 10% of math teachers manage to intimidate, then a typical student has a 130% chance of running into a teacher who derails them [3]. Once the student learns that they can't do math, it is very hard to get them back in the saddle.

Problems with stories
Glen had five story problems and Gail had six migraines. If the hypotenuse went on sale for 30% off, then how long would it take Glen and Gail to go crazy together? Story problems are the source of much anxiety. It is a little known fact that story problems led to the Crimean War. Absolutely true. And story problems also caused the bubonic plague.

Edward MacNeal argues that at least part of the problem is that the linguistic aspect of math is often separated from the math side of math. At least that's what I think he is saying. I dunno. I don't do words so good. 

The non-statistical human brain
I have blogged before about how people just don't make good statisticians. Surprisingly, I am not the first to bemoan this problem. John Allen Paulos did a whole bunch of bemoaning this in his book "Innumeracy". Here he talks about the traps that many of us fall into:

"[This] book is largely concerned with various inadequacies - a lack of numerical perspective, an exaggerated appreciation for meaningless coincidence, a credulous acceptance of pseudosciences, an inability to recognize social trade-offs, and so on."

"In an increasingly complex world full of senseless coincidence, what's required in many situations is not more facts - we're inundated already - but a better command of known facts, and for this, a course in probability is invaluable."

As an example of our over-willingness to attach meaning to coincidences, here is a quote from the brilliant scientist and mathematician, Michele Bachmann: "I find it interesting that it was back in the 1970s that the swine flu broke out under another, then under another Democrat president, Jimmy Carter. I'm not blaming this on President Obama, I just think it's an interesting coincidence."  Gerald Ford was president at that time, but there are just all kinds of flaws in her logic.

Here is another quote, from author H.G. Wells: "Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write."

How do we solve this problem? I think this concept must be taught not once, but throughout our education. Intuition is a good thing, and can be helpful in leading us to solving problems. But, intuition is not knowledge. I am just kinda thinking out loud here, but maybe math teachers need to reward students when they guess, rather than discouraging them from guessing.

Math Anxiety
Each semester that I taught, my first assignment was a touchy-feely one. I asked the students to write about one of their experiences of being taught math. The next class day, I asked the students to share their stories (if they were comfortable sharing). The most heart-wrenching story was one woman who told of being forced to do long multiplication at the board. She was so anxious that she wet her pants.
OMG!  How do I add these two fractions?!?!!?

In my opinion, math anxiety is the single most pervasive cause of innumeracy. The anxious brain can't be expected to function well. Below is a quote from a pair of authors who agree with me. Their book is something of a self-help book on math anxiety.

"Math anxiety, which is widespread and poses a serious problem in our society, often stems from childhood experiences, including intimidation and humiliation. It is reinforced by cultural and family messages and transmitted by teachers who suffer from it themselves. More frequent in women than in men, it is manifest in self-sabotaging behavior that leads to unsatisfactory work lives."
"Where Do I Put the Decimal Point?", by Elisabeth Ruedy and Sue Nirenberg

How do we deal with it?  As a teacher, I had a lot of ways to get students beyond the anxiety. First off, humor. It is not possible for a person to be anxious while they are laughing. I know it may be hard for readers of my blog to believe this, but I am capable of being funny once in a while.

Another useful technique is just plain being honest about math anxiety. That was the whole point at the start of my class--to let everyone know that math anxiety is not rare. We all have it. I might get my membership in the Society for Pompous Math Guys revoked for saying this, but even I get a little math anxiety once in a while.

When I was doing a math problem for the class, I would often put a magnifying glass on my own anxiety, going into public panic attack mode when I started to get confused. The first instructive element: we all get moments of self doubt and confusion. The second instructive element was for me to demonstrate techniques to overcome that anxiety.

The authors of the infographic that inspired this blog post point out that math anxiety can be cured by inspiring confidence. Here is another snip, from the end of their page:
The first quiz of every class I taught, there would be at least one student, usually in the back, who I saw crying. Literally shedding tears. I didn't have to look at the actual quiz to see that they did not do well. 

When I saw this, I would ask the student to work the quiz with me, one-on-one, during my office hours. I would ask them to share their brain with me for the moment, and tell me everything that they were thinking as they approached the problem. That's when the second guessing and negative self-talk came rushing out. All they needed was a little assurance that they were thinking things through "correctly". 

I am happy to say that all these students passed the class. The lowest grade any of them got was an A-.

Popular girls don't do math
Big hair, or big brains?

I have a favorite book on overcoming math anxiety. It's called "Overcoming Math Anxiety", by Sheila Tobias. Her main thesis of the book is that math anxiety is the single most pervasive cause of innumeracy[5],  and that math anxiety can be cured.

All that aside, the main thesis of the book is that women have more of an issue with math anxiety than men do. She gives all sorts of facts and figures, and explains that it is not an issue of capability. There is no math brain, right?  The issue, she explains, is cultural. When young women hit puberty, there is a barbaric onslaught of peer pressure that says that being a math genius is uncool. "Genius" in the case, applies to anyone who gets a passing grade in either algebra or geometry.

Not everyone agrees with the assessment that there is a difference in math performance among females. Here are two articles that cite contrary evidence, one from Huffington Post, and the other from Time magazine. Perhaps Tobias' research was flawed, or perhaps times are changing. At any rate, I am encouraged. It is still disappointing that there are so many people tormented by math, but at least there are signs that it isn't so much a gender issue anymore.

[1] All math professors have beards. John the Math Guy has a beard. Therefore John the Math Guy is a math professor.

[2] While the mythical math brain has been debunked as an urban legend, the raw sexual magnetism of applied mathematicians has been demonstrated again and again in both the lab and in field studies.

[3] I just thought I would throw in that little bit of innumeracy. It's not really a 130% chance. That would be silly. Hopefully, this is a key to a person who thinks they are not good at math to reconsider.

One path to the correct answer is to look at the chance that a student has 13 teachers who are not intimidating. This chance (assuming there is no correlation from one grade to the next) is 90% raised to the 13th power. This is about 25%. The other 75% of students can be expected to have at least one math teacher who teaches them to be a mathaphobe.

Stepping back from the math problem for just a moment... I realize that I just did what I said teachers shouldn't do. I did the "It's obvious" step. It's obvious that the way to solve the problem is to turn it around the other way. It was obvious to me only because I have seen variants on this problem umpty-googol times.

Another way of looking at the problem is to take it one step at a time. What's the chances involved in two years of school? There are four possibilities: good teacher for first grade and good teacher for second grade, and so on. Multiply the probabilities together, and now you know the likelihood of all combinations.

That doesn't solve the question yet, so take it to three years. There are 8 possible outcomes... you can compute the probabilities, and away you go. Eventually, in going through this approach, the student sees the pattern.

In my opinion... that would be a good way to teach the solution to this problem.

[4] Contrary to popular belief, Paulos did not coin the word innumeracy. He wrote a book about it and made the word famous. It was first coined by Douglas Hofstadter.

[5] Sounds like something I just got done saying. Guess who got me thinking along those lines?


  1. John I believe math problem is huge in India also. ...its very common, I think problem is teacher can't clear concepts of basic mathematics they just tell students to accept it as they also don't now especially in 7-10 grade. When I asked for explanation they told me to accept as formula latter on I learned basic concepts and found how those formulas are derived then I understood so many basic concepts....

  2. Good evening!
    I have to say that math problem is just the same all over the world.
    It' s a global problem, unfortunately.

  3. I just got the smack-down on an observation report (as an adjunct for a remedial college math class) saying that I should have been better prepared. I responded to this basically with what you wrote in your post: "I intentionally under-prepare because I don't want students to think that there's no work and no struggle in this!"

    Thanks so much for writing this!

  4. I appreciate the comments Awadhoot and Μ+Λ, that this is not just a US problem. At the gut level, I don't find it surprising.

    Bon, I love your comment. It is sad that you were smacked-down for being a better teacher rather than conforming to a bad way to teach!

  5. One thing I do with my Algebra 1 class is occassionaly have them cite the variety of knowledge they need to solve a particular type of math problem.
    I recently did this with solving a system of equations and we stopped when we got up to 30 math concepts. I wanted them to understand how many of the 30 things they already knew (add, subtract, etc) and that only a few had been taught in the current unit. I stressed anxiety and lack of patience is a bigger problem than lack of knowledge.
    - Tom

  6. Hmm Interesting blog and great topic.. Well I think the math problems is same it is discussed.
    Let's talk about the solution.
    In my opinion we must realized what to teach and how to clear the concept of the students.
    Second we must tell them how to use these mathematics in real life... Many of my students ask this questions....

    1. Hello Ahtisham Shah.
      I would like to ask you how old are your students.
      And if your recommented solution is effective to all of them.
      Thank you.

  7. Math is indeed a challenging subject for many people. I struggled with math for years, and unfortunately I passed the math struggles down to my daughter. Although she attends a very decent public school in California, she still experienced issues with the subject matter especially during test time. But I vowed to not let my daughter continue to fail at math like I did, so I got her a math tutor in San Jose. The tutor helped her tremendously!

  8. John,

    I'm not sure if you'll see this, since it's such an old post. I'm wondering about your technique of letting yourself be flustered in front of the class. I think it sounds like a great idea, but how did you go about it?

    Did you add in some of that anxiety-erasing humor, and make it obvious that it was at least partially a show? If so, did this insult the students at all, like you were making fun of the confusion?

    Or did you really show your true confusion, making them feel related to? If so, did this make them feel like you were unqualified?


  9. I don't think any were offended, since I had established rapport with the class.... and they were used to me being funny..

    The little acting job was a caricature - a deliberate exaggeration of the feelings that even real mathematicians have when they encounter an unfamiliar problem. I would read the problem, then put on the "deer in the headlights" look and start moaning and crying... then verbalizing some of the self-defeating self-talk that convinces us that we are dumb. Then I would make an overshow of me trying to regain composure. It's alll about acting and channeling my own insecurities!

  10. This is great. I met an obviously highly intelligent Russian programmer a few years ago, whose professor-parents spoke math to him from early childhood. They also made him write formulas down everyday, though he had no idea what they meant. He told me that in 4th grade, the lights came on and he became a math wiz. Fascinating concept.