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Tag Archives: Math Instruction
A friend of a friend posted on facebook:
What am I missing here? The article says: “According to the CDC, approximately 17 percent of American children are obese, which means their BMI lies in the ninety-fifth percentile or higher for their age and height.” Wouldn’t that mean that 5% of American children would be obese?
(The article in question was commentary on a Vogue article about putting a 7-year old girl on a diet, to which I won’t link because of profanity.)
I happen to share the commenter’s confusion; if the numbers are that far apart, shouldn’t the percentile table be recalculated?
The CDC might not think it’s necessary since the general public doesn’t seem to understand what percentiles are anyway. (If you’re a math phobe reading this, check out the first paragraph of the Wikipedia entry for a brief and clear definition.)
Percentiles are by definition norm-referenced and not criterion-referenced. In other words, the exact BMI or weight (or any other measure) required to be considered obese will change with the times as the population grows heavier, slimmer, or more or less healthy.
It’s just like grading on a curve. If everyone were obese, we wouldn’t be able to use this definition to identify most obese people because they would be less obese than the ones who are labeled obese.
My sense is that most people reading the statement wouldn’t think twice. The innumeracy!
In case you are interested in the recursiveness of this post, the answer is yes: this is commentary on a comment on an article commenting on another article.
So the facts are in: a study by behavioral psychiatrists at Stanford has shown the links between elementary aged student self-reported math anxiety and brain functioning.
I’m not sure that this study proves much of anything, except that there is a biological component to math anxiety. The amygdala, a part of the brain that is responsible for regulating fear and other heightened emotions, is overly active when solving math problems in children who report higher math anxiety. Self-reported anxiety is also associated with lower performance in math, according to the study.
The study is described in an article on the Health Imaging website.
Which leaves one big question: Two trains 100 miles apart are heading toward each other at 50 miles per hour…
I’ve been thinking a lot about math instruction for girls, so I was delighted to have the chance to chat with an extraordinary individual who’s a real authority on these subjects.
Meena Boppana is a senior at Hunter College High School who has been defying the odds her whole life by excelling in mathematics. Her enthusiasm for the subject is infectious, as you can see from her talk at TEDxHCCS last September. I’d like all of our Harlem Link scholars to have the opportunities that Meena has had and to attain the mathematical skill that she has.
Hi Meena! First of all, thanks for taking the time out to talk to me. I know you’ll be a fantastic role model for our students.
Thanks for interviewing me!
You love math so much. How old were you when you first realized that was the case?
Well, since I was three or four years old, so as long as I can remember anything. Whenever I was bored, my dad would start talking to me about something, like the prime numbers.
Lucky for you at age three, three was a prime number.
Ha—yes, three is a prime number .
OK, I’m getting right into the politics: Why do you think there are so few women who are famous mathematicians, even nowadays?
Well, I think what it comes down to is a perpetual cycle. There are so few women in math, so women don’t want to be the first to break that pattern. There are also cultural factors, especially in the U.S. In Eastern Europe there are more women in math, I believe.
What about at Hunter? Do you feel that math power, if you will, is egalitarian across gender lines among the student body?
In general, yes. Girls feel equally comfortable speaking out in math class as do boys. Also our math team is relatively balanced. When you look at the very top of the spectrum though, there are far fewer girls. Of the 32 students on the New York City math team at the Harvard-MIT math competition this weekend, there were only four girls. Last year I was the only girl.
Wow! What was that like?
That was fun because I got my own hotel room, but also potentially lonely! In the end it was not very lonely because I’m friends with all of those guys.
I take it then that you don’t feel any lack of respect from them or feeling that they look down upon you.
I do feel that I have to assert myself sometimes, but I also feel that I have their respect. You definitely have to be assertive in order to get a word in when you’re working with a team of all boys.
Boys will be boys! We find that the boys at our school often (not always) need positive attention more than the average student in order to be successful. So I’m sure you help them with your patience.
I’m curious about how you got involved coaching students in math at Girls Prep Middle School, a charter school on the Lower East Side. I’m especially interested because part of the school’s mission is to elevate girls and remove the societal barriers that send negative messages about achievement to girls.
I attended a leadership conference for high school girls called Take the Lead! last year and came up with an action project to start up a middle school math club for underserved girls. That was all hypothetical, and then I heard about Girls Prep and the school perfectly fit my vision. I reached out to the principal, and the school embraced the idea. Then I recruited several volunteers from the Hunter math team.
I want to see more girls on the New York City math team in the future!
Do you find that social issues, including cultural expectations, serve as a barrier with the girls?
The girls aren’t afraid of learning math, and I think any cultural barriers have been removed. Girls Prep does an amazing job with that.
We took them to the competition MATHCOUNTS last weekend. They were really inspired by seeing other kids from around the city equally excited about math.
So, not really!
That’s fantastic! When I was at Hunter, my calculus teacher, Ms. Strauss, stressed over and over again that girls benefit in math instruction from single-gender classes. Maybe she was wistful for the old days when Hunter was all-female. But you seem to have evidence that it’s true with the Girls Prep experience.
What about your own elementary school, experience? Were the boys bullying the girls in any way, preventing them from excelling in math?
I guess I don’t see it as the traditional boys bullying girls and preventing them from doing anything. But I think that sometimes the boys who are strong at math tend to shout out the answer quickly and be a little bit intimidating, so girls have to assert themselves more. And there’s the fact that at my old school, Dalton, everyone was very concerned about popularity in sixth grade. I have a friend who, after seeing the movie Mean Girls, decided not to join her school’s math team.
I’m thinking about risk taking. I’ve always believed that a strong mathematician has to be unafraid to take risks. What do you think? And as a mathematician, are you fearless?
I’m definitely afraid of a lot of things! But I do talk a lot, and so I will ask questions and that helps me assert myself in math classes. In terms of taking risks, I’m not sure how math involves taking more risks than anything else in life. Math presents a challenge, and I like rising to challenges. If someone tells me something is too hard, I’m thinking, “Bring it on!”
That’s an impressive and, I think, rare attitude.
Any career goals yet?
Not really. For a while I thought about going into academia and becoming a math professor, since I love math so much, but I also want to do something that affects society more directly. I’ve been admitted to Stanford and will probably be studying there in the fall. And I can see myself going to graduate school in math or a related field like computer science, economics, or physics. (And maybe I still will go into academia.)
So I have no idea, but math is useful in so many things so I don’t really need to know.
I’m curious about your perspective on Very Large Numbers. I mean, we are talking possibly 200 billion stars in our galaxy alone—and the latest research says the universe might after all be infinite! Do you ever get overwhelmed by the vastness of the universe, or the number of atoms in a given object?
I have always been fascinated by big numbers! One of my favorite museum exhibits is the Planetarium at the American Museum of Natural History which compares the earth to the size of the sun, sun to the solar system, etc., and keeps zooming out.
I think that the size of the universe is more awe-inspiring than scary, and I guess I don’t think about it as much as astrophysicists do.
The scope of the universe is fascinating.
I used to worry that I’d be sucked up by a black hole, and then my dad told me that if we were to be sucked up by one the whole Earth would be as well, which was somehow consoling.
Sounds like your dad has been a big influence. He must be a proud dad too.
For sure—my dad is a really great teacher and always taught me math by making it fun, not forcing me to do it. My dad started a math club at Dalton when I was in second grade, and so I had a peer group of friends doing math since then. Also he started the Math Prize for Girls competition held at MIT each year, which brings together the top high school girls from around the country.
Awesome. It must have been hard for him to let you go when you went to Hunter.
Last question: given the theme of our conversation, is there anything else you’d like to share about your passion for math?
I take math at Columbia University, and I have to say that math is definitely considered cool there. In college and beyond, having math skills is definitely an asset. And it’s only a matter of time before people figure out that what makes Mark Zuckerberg and other computer scientists successful is their math skills. (In fact Mark Zuckerberg competed in the Harvard-MIT math competition in high school )
One more thing. I believe that middle school math is just the foundation. Math gets way more interesting after that. So you have to get the basics down before you can be really wowed.
That brought a smile to my face. I want to plaster it on my fifth graders’ foreheads.
Thanks so much for taking the time. I’ve really enjoyed it and I am excited for your future.
Thank you! I’m really honored that you thought of me.
“At the end of 2011, less than 20% of the lower 48 was covered with snow, compared with more than 50% at the end of 2010″ (attribution later).
What does it mean to you?
To a global warming enthusiast, last year saw us enter January with 150% more snow on the ground than this year. To a climate change denier, this year’s figure is a mere 60% less than last year’s. Wait, it’s actually only 40% of the 2010 number—or 30 percentage points less. And what’s 30%? Not much, right?
Of course, all those statements are correct. And so I have another opinion to share on math and the populace: I think many folks are disengaged from politics partly because they have a hard time knowing what to believe when politicians and members of the media throw numbers around. They get lost among the facts, confusion and outright distortion.
Please don’t oversimplify my statement and engage in the fallacy of the single cause. This fallacy is one of the mathematical problems that results from an impatient and anti-intellectual zeitgeist. Notice the qualifiers I put in my statement above: many, partly; I know there are a slew of reasons why people are disengaged from politics. I am only saying that I believe that a lack of power over mathematical statements is a strong one.
Lies, Damn Lies, and Statistics
I want to distinguish between the proper and improper use of statistics in political statements. By improper use, I mean anything from outright lies to falsehoods. In one of the New Hampshire primary debates, Mitt Romney was caught saying that repealing the Obama health care law would save the country $95 billion. (Looking it up again, I see that he actually said $95 billion a year.) Fact checkers pointed out that his statement was false; the Congressional Budget Office (CBO) document from which he drew his claim showed that the health care law would cost the government that amount in spending in 2016, but would actually offset that spending with a greater amount in savings. The CBO estimated that the Obama health care law would actually reduce the deficit. At worst, Romney’s claim is an outright lie. At best, it’s a misleading half-truth that is technically true but completely out of context. Referee says: Improper use of a statistic!
What I think is harder for most Americans to parse is a true statement that has been interpreted properly, but is confusing for the average reader who lacks math power. These situations are ripe for down-the-garden-path framing by the speaker or writer.
In the January 23, 2012 issue of Time magazine, Bryan Walsh makes the following statement: “In December, at least half the U.S. had temperatures at least 5° F above normal” (p. 18). You know, it has been unusually warm. Good Lord—global warming is here! Perhaps the title of the article, “The End of Winter,” correctly heralds the coming Globally Warm Age.
Now hold on a minute. He only mentions (at least) half of the U.S. Did the other (at most) half of the U.S. have an average temperature of 5° lower than normal? We don’t know. In fact, it’s possible that that other half, or so, of the U.S. had average temperatures of ten or even twenty degrees below normal. We just don’t know because we, the readers, don’t have enough information.
And how abnormal is 5 degrees over the course of a month, anyway? I can tell you that I don’t know, and I’m going to guess that most of Walsh’s readers don’t know either. I remember Al Gore saying in An Inconvenient Truth something like a projected 4 degree increase in average global temperature over the next century will raise ocean levels by over a foot and flood coastal cities. (I could be wrong about those numbers; what do I know?)
Four degrees in a century, five degrees in a month. What does it all mean?
I would expect that a person who is sufficiently fluent in math to be politically engaged without feeling handicapped would walk away from a sentence like the one in Walsh’s article with the same type of concerns. I would expect him or her to think things like:
- “Give me a comparison set.”
- “Tell me the expectation.”
- “Tell me whether this occurrence is noteworthy because it is unusual.”
Luckily, Walsh does provide some comparison data in his next sentence, the one that began this post:
“At the end of 2011, less than 20% of the lower 48 was covered with snow, compared with more than 50% at the end of 2010.”
There you have it: empirical evidence of less snow on the ground from one year to the next. But aside from the problems of interpretation I noted above, this statement still suffers from a small sample size. Does comparison of one year to the next really tell us anything about climate change?
I remember the unexpected blizzard of January 1996. Parts of the Northeast were dumped with four feet of snow. It was especially noteworthy for me because I remember my father digging out a trail to the family car on the way to my sister’s wedding. Guess what? A year later, January 20, 1997, there was no snow on the ground in New York City, says the National Weather Service.
So what does it mean? Was 1997 the aberration, or was it 1996? Was 1997 the year snowfall in New York ended?
None of the above. Climatologists tell us that global warming is imperceptible on a day to day, and even year to year, basis. They don’t look at a couple of data points to draw their conclusions; they go back thousands of years to study the cycles of air temperature change, and collect thousands of other contemporary data points to describe and define today’s weather patterns.
Unfortunately for members of the media and politicians, the unsexy answer to most questions involving statistics that describe complex phenomena such as human decision making and the global climate is, “It’s complicated” or “There isn’t enough information to say definitively one way or the other.” And in the political sphere, no one wants to hear that. A more mathematically informed populace would know better.
How Many Homeless Kids Are There in Florida?
A final anecdote to illustrate my point begins a couple of months ago, at the start of this warm and unsnowy winter. I was in the diner on West 187th Street and I couldn’t help but overhear a decidedly one-way conversation about the previous night’s news report. Maybe it was because the protagonist was yelling at his listener across the breakfast bar, and not giving him much of a chance to chime in.
I hardly ever watch the TV news, but in this case I had recognized a report from the night before. It was a now famous and terribly compelling 60 Minutes story about homeless families.
It sounded even more compelling when the gentleman was yelling at his friend. “Did you know,” he screamed, “that one in four children living in Florida are homeless? Can you believe it? I knew things were bad, but that’s really bad!”
There are just over 4 million children living in Florida. That puts the number of Floridian homeless children at around 1 million. That’s a lot of homeless children.
But the gentleman in the diner remembered the story wrong. The 60 Minutes website quotes the story as saying, “The number of kids in poverty in America is pushing toward 25 percent. One out of four.” I too remembered something about 1 in 4 children in Florida, but I wasn’t confident enough to shout it across a public place, and I certainly would have taken greater notice if I thought 1 in 4 children in Florida might be homeless.
Indeed, the Coalition for the Homeless of Central Florida states that approximately 84,000 homeless children live in Florida, or about 5% of the national total. That’s 1 in 50, not quite the same as 1 in 4.
I’m not writing to understate the problem of homelessness. We certainly have our share of families in transitional housing at Harlem Link and I know the impact is has on children. But I believe we’ll never get anywhere with our public policy discussions if we don’t have a vibrant and informed populace working over the actual and accurate facts.
A commenter on my previous post noted somewhat derisively about Gödel, Escher, Bach: “I wouldn’t say it has much to offer as far as understanding math.”
I think the commenter misinterpreted my reason for mentioning this book in my post. It’s not because GEB helps a reader understand math, but it inspires a love of all things mathematical and is a model for the value of wonder that drives so many mathematicians.
But the more I think about it, the more I realize that I do think that this inspiration belongs in the category of “helping to understand math.”
It’s true: if you don’t have a fundamental understanding of the structure of our number system that we call number sense, the patterns that recur throughout this system and a good grasp of logic then you’ll quickly get lost in GEB. But as an educator, I have come to believe that the love of math is one powerful precursor to understanding math.
It provides the confidence to take risks and make mistakes–the coin of the realm for professional mathematicians.
It flushes the face with blood and the brain with endorphins when a lush, generative problem arrives–the rush of trying and experimenting, the gambler’s craze.
When you’re seven years old and you don’t know what to do but you think you might have the answer, you’re trying different things and ready to stake your life (or your reputation among your peers) on your next best guess, you can’t hold back, you have to share your ideas with your peers and teachers, you have to try adding this to that and subtracting the other thing…when you are still thinking about it when you get on the school bus to go home and you tell your family about it, maybe even your dog…you are practicing for a life of building math power.
Love of math is not automaticity. It’s not conceptual understanding. But it’s the building block for those. If understanding spreads like a bacterium, love of math and fearlessness for trying are the agar in the Petri dish.
I believe math power snowballs, one way or the other. When you’ve got it, that lust for more leads to an increasing sense of mastery over the number system. When you don’t, each successive stage from algebra to trigonometry to calculus seems only increasingly foreign and inaccessible.
Finally, when I read that comment I could not help but think of gender bias and elementary math instruction. I don’t have any research to cite here (I’m hoping my new readers can help crowd-source or channel Carol Dweck), but I know from my experience, my mentors and Larry Summers’ public comments that girls are typically not treated as mathematically capable.
And the snow gathers as time goes by and children grow into adults. Not a single text inspiring the love of math from my last post and comments was written by a woman (they were written by John, Peter, Douglas, David, Roger, Leonard, Tobias, George, Rafael, Scott, Martin and another John, if you were wondering).
Imagine what would happen if every little girl entering kindergarten were welcomed with, “You’re just going to continue to grow to love math and the mysteries that will unfold before you!”
Now tell me–can the love of math lead to math understanding?