New math that isn’t new

Kimberly at Number 2 Pencil scoffs at a breathless story about a seventh grader “discovering” a new way to solve math problems using negative numbers. It ain’t new to people who know anything about math, which apparently excludes the girl’s teacher and the reporter.

Brenda of Isomorphisms mocks a similar story about a 16-year-old girl who’s allegedly revolutionized math with her new “Lizzie Method” for solving quadratic equations. The story says the girl “has the math world spinning.” Or yawning.

Responding to an e-mail inquiry, the (University of Michigan math) professor writes, “This method was the standard method taught when I was in school, and I suspect that in many parts of the USA it is still the standard method.”

The Lizzie Method works well for equations with small numbers, which are common in basic high school math. Brenda, who teaches math, is frustrated:

Imagine if a thirteen year old science student tries to dissolve salt in oil, and fails. Or if a ninth grader writes a short story that makes use of flashbacks, rather than exposing the events chronologically. Neither of these would make it into any newspaper. Neither would be portrayed as earth-shattering events. Neither’s teacher would be lauded for not giving their pupil a failing grade. People are familiar enough with science and literature that they tell the difference between inquisitive, creative kids, and world-changing inventions. Yet innumeracy is so deeply ingrained that a teenager’s slight departure from the a small set of methods for doing mathematics is beyond the grasp of her school’s teachers and her city’s journalists.

I can’t speak for teachers, but most journalists are profoundly innumerate. At the Mercury News, our best number-crunching reporter would circulate an occasional reminder for colleagues about the meaning of 100 percent or the dangers of looking at an increase without mentioning the base number; sometimes he’d teach a little lesson on fractions. And people were grateful for the help. If you look at the story on Lizzie, it has loads of color. But it’s clearly written by someone who thinks solving quadratic equations is a job for Albert Einstein.

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  1. I agree that these aren’t revolutionary. It’s nice that the kids are thinking. Maybe the big deal is being made because they’re girls.

  2. Sean Kinsell says:

    Lizzie seems to me to be FOILing backwards, unless my non-specialist mind missed something. And as for Killie, good for her, but what is a seventh grader who needs an intervening step to figure out that 3 – 1/5 = 2 4/5 doing getting a math achievement award? But yes, it’s nice that even in this day of scientific calculators, some children still like to mess around with numbers.

    BTW, Laura. I’ve almost never been involved in a math class (as either student or teacher) in which it wasn’t the case that most of the best students were boys but the very, very best student was a girl.

  3. “Maybe the big deal is being made because they’re girls.”
    Or maybe the big deal is being made because the teachers, administrators and journalists are morons.

  4. Richard Cook says:

    Why do you think that?

  5. Ken Summers says:

    “I can’t speak for teachers, but most journalists are profoundly innumerate”

    Sadly, I think it is also true for teachers, at least outside the sciences. Sadder still, I know it is true for some math teachers.

    The really weird thing about innumeracy is that educated people who are innumerate sometimes seem almost proud of it.

  6. I posted this over at Brenda’s site, but I’ll post here too.

    The key to Lizzie’s “method” lies more in the fact that teachers tend to give homework assignments and quizzes with whole-number answers than anything else.

    I used the same method nearly 30 years ago.

  7. Richard: Do you not remember the uproar a few years back when talking Barbie said, “Math class is hard!” All we heard about was how girls were already behind boys in math and this was going to discourage them even further.

  8. Ken Summers says:

    My daughters would take issue with that “girls are behind in math” nonsense.

  9. Richard Brandshaft says:

    I’d like to respond to Ms. Jacobs on innumerate journalists, which has come up before.

    There are 2 problems:

    1) An ignorance of the very basics. My favorite is things like a “300% decrease” in something where negative numbers make no sense. That kind of thing is good for a laugh, but it isn’t the most important case.

    2) An ignorance of how numbers are used, and what they mean. You can understand a lot of pitfalls WITHOUT knowing how to do the calculations. For example, you don’t have to know how to do statistical inference to understand why hormone replacement therapy studies initially reached the wrong conclusion. Most (“most” does not mean “all”) computational errors effecting social policy are the kind of thing one can understand without knowing much about math. Journalist’s ignorance is a massive cultural failure of journalism schools and newspaper editors.

    Combining 1) and 2) — very elementary but effecting social policy — is the unwarranted importance placed on averages. Averages are too sensitive to extreme cases. To steal Paul Krugman’s example, if Bill Gates were to walk into a bar containing a few ordinary people, the average worth of a person in the bar would be over a billion dollars. If you must use a one number summary — which is seldom a good idea — the median is preferable.

  10. What I don’t understand about the quadratic equation is why she’s getting praised for a method that’s so difficult. I’d doubt if I could remember that for a quiz or test. Far easier is to simply remember that “x equals negative b, plus or minus square root, b-squared minus four a c, all over two a” can be sung to the tune of “Pop Goes the Weasel”. I learned it five years ago, haven’t used it in four years (graphing calculators, now! Just use Program QuadForm!), and can rattle it off any time.

  11. “The Lizzie Method” is just factoring polynomials – a fairly important operation in college-level math and in the mathematical sciences, but it only works well when there is a whole-number answer. That is, with random coefficients there’s a very tiny chance that there is an answer that this method can find. Or it is possible to derive the quadratic formula by trying to apply this method to arbitrary coefficients, through several pages of algebra.

    As for Kellie’s use of negative numbers – I was using fancier tricks than that to do calculations in my head in the 4th grade. It’s obvious when you understand numbers.

    But evidently these were not obvious to their teachers. Kudos to the kids for understanding things that their teachers didn’t even know. These ideas aren’t new, but they were new to these kids. It’s evidence that quite a lot of mathematics is hard-wired into the brain – if you are smart enough and don’t let semi-innumerate teachers cause you to block your natural abilities.

  12. Fred Jacobsen says:

    Journalists’ ‘innumeracy’ (love that word)was particularly pernicious when the SF Chronicle breathlessly announced in an above-the-fold front page headline that ‘Teen AIDS Cases Rise 50%’ (I paraphrase). The headline writer and reporter had ignored the baseline. Reading the CDC report, upon which the article was based, it turned out that teen cases had increased from something like six to nine (in a nation of 240 million).

    By the way, Ms. Jacobs, I was wondering when someone would come up with a Samarra/O’Hara pun. Kudos.

  13. markm has it right. Most of my fellow math majors in college could summon the “Lizzie” method in their heads–when it provided a simpler or quicker way–but I doubt we’d bother to name the method.

    It’s something you just know.

  14. Richard Cook says:


    I remember that one but also remember it got a spasm of play then sank from sight. I think the articles involved were rather breathless about relatively non-unique occurances. To be frank the articles stunk and tried to puff something up that shouldn’t have been. As far as making a big of it because they are girls…any thinking journalist would realize that girls have been outpacing boys for awhile now (its lead to several books being written on the subject) for whatever reason. If the reason for the puffery was because they are girls that’s a pretty damn poor reason.

  15. Richard, the doll was actually yanked from the market.

    I think there are a lot of things out there that everybody “knows”, contrary to evidence, and one of them is that girls are bad at math. Hence the to-do in the press about poor Barbie.

    Thinking journalists? I think I’ve run across some …

  16. I dont even that was a real Barbie. It was Malibue Stacy, the “Barbie” clone of the Simpsons. And as far as I can remember Lisa Simpson was the only one to be outraged

  17. Rod Wiltshire says:

    A few years ago there was a Dilbert cartoon that made fun of women (specifically women) who don’t realize that if you decrease a number by 25% and then increase the result by 25% you don’t end up where you started. (I forget the specifics, but this was the arithmetic the joke turned on.) I cringed, awaiting an uproar, but there was no uproar. Apparently no one who is much upset by such things, man or woman, followed the joke.

  18. Rob – you are wrong. The Simpsons did a parody with Malibu Stacy saying something similar.

    “But it’s clearly written by someone who thinks solving quadratic equations is a job for Albert Einstein.” Have to disagree, JJ, I think this was clearly written by someone who thinks she knows all she needs to know about math. Why would she not, I don’t know, call a local math proff and ask for some background and an expert quotes? One phone call and the story is dead. Why wasn’t that phone call made?

  19. Andrew Allers says:

    When I saw this, I wondered how she (Lizzie) came up with it. The story says:

    “I just started playing around with the numbers,” Lizzie remembers, “and trying to see what I could do with them until I ended up getting the answers that were [on the answer key].” When she found a way that worked, Lizzie figured the lesson had been learned.

    Trial-and-error is a very non-mathematical way to solve problems. The fact that she seems to have just started playing around with numbers in the equation until something worked against the answer key is not a very good reason to reward her (and refer to her as a ‘math whiz’).

    It seems that all she did was stumble across another recipe to use instead of the traditional recipe. If she’d derived this new recipe at least that would be a little interesting.

  20. Yeesh, I find it frightening the level of innumeracy among those critiquing the article (see my blog for specifics).

    The technique that Lizzie came up with is not one that has never been seen before, but it’s also one that for whatever reason has languished in obscurity. I’ve taught factoring many times to many different levels of students, and having read the article, I’m getting better results with my students than I’ve ever seen before.

    I’d be curious to know why this algorithim has fallen into obscurity, but I doubt any of the jackdaws here would be likely to do that sort of research (or even be capable of it, given that many seem to think that factoring is only useful for solving contrived quadratic equations).


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