Saw several different people pointing out a math-realted idea from a NYT story about The Interview. One example out of the many is a blog article here:
The New York Times Doesn’t Know any Math Teachers
with follow up on twitter here:
It all leaves me a bit uncomfortable. Here’s are some reasons why:
(1) What is the message being conveyed?
If the message is something other than “ha ha, the reporter couldn’t do some fairly basic math” the message is lost on me.
I say “fairly basic” because analogous problems appear as early as elementary school – and certainly before algebra. After seeing some of the posts about the NYT article yesterday, I looked through some of the MOEMS books and found two similar problems in the first few pages of exams. For example:
(A) At a special sale, all pens are sold at one price and all pencils at another price. If 3 pens and 2 pencils are sold for 47 cents, while 2 pens and 3 pencils are sold for 38 cents, what is the cost of a set of one pen and one pencil, in cents?
(B) Admission to a local movie theater is $3 for each child and $7 for each adult. A group of 12 people pay $64 admission. How many children are in this group?
These questions are from the set of exams for kids in 3rd through 5th grade. Aside from having smaller numbers, they more or less exactly the same as the question answered in the blog article above.
Since kids can understand the problem as well as the criticism, I worry about what they will think about the criticism if they themselves can’t solve the problem.
To me the it has the familiar ring of what Jordan Ellenberg calls “algebraic intimidation” in How Not to be Wrong. In the section “Evanescent Increments and Unnecessary Perplexities” there is this passage (page 43 of my copy). The topic in discussion is a common proof of why 0.99999…. = 1:
“These arguments are often enough to win people over. But let’s be honest: they lack something. They didn’t really address the anxious uncertainty induced by the claim 0.9999… = 1; instead, they represent a kind of algebraic intimidation. “You believe that 1/3 is 0.3 repeating – don’t you? Don’t you?”
I see a pretty similar thing going on here: you believe these two equations – don’t you? Don’t you?
Which is an easy transition to the next point
(2) What about the uncertainty
Oh, there’s lots of uncertainty in this problem and the equations in the blog article ignore all of it. Put yourself in the position of a NYT editor (or fact checker). Which would you prefer:
(a) The claims in the article: “Sony did not say how much of that total represented $6 digital rentals versus $15 sales. The studio said there were about two million transactions over all.”
(b) The math from the blog: “According to our analysis of the information provided by Sony, there were 1,666,666 $6 rentals and 333,333 $15 sales.”
[post publication edit. Dan Meyer contacted me on twitter and expressed concern about part (b) above. He thought that the quotation marks made it appear that part (b) is a direct quote from somewhere else. I did not notice the potential for that confusion when I wrote this post and certainly didn’t intend for there to be any confusion on that point. To be clear, part (b) is a hypothetical submission to the editor / fact checker designed to report the solution to the math problem that led to the various posts.]
Some commentators have pointed out the error of ignoring the uncertainty in the numbers. Here’s a good graph, for example:
Jamie Cleveland’s graph in the comments to Dan Meyer’s blog article
But there are other problems too. For example – How do you know Sony’s numbers are correct? How do you know that all of the sales and rentals took place at $15 and $6 and not some other price(s)?
Blindly relying on the math to get to answers can and has lead to all sorts of problems in the business world. A great starting point for understanding some of the problems in finance is Roger Lowenstein’s book about the rise and fall of the hedge fund Long Term Capital Management, When Genius Failed. Bethany McLean’s book about the 2008 financial crisis – All the Devils are Here – is another good one. Or, if you think these books won’t interest you, just follow Nassim Taleb on twitter to learn daily that blindly relying on equations to solve problems that arise in business can lead you to answers that turn out to be drastically wrong.
Finally . . . .
(3) Is this a good example of why you should learn algebra?
For me – no. As I said above, pretty similar problems on elementary school math contests. No algebra is necessary to solve them.
I also do not think that many people would find the question of trying to back into Sony’s rental and sales numbers to be all that interesting. I’d be curious if anyone has ever tried to do a similar calculation for any other movie ever.
Some of the math that comes into play behind the scenes in When Genius Failed and All the Devils are Here may actually be a good example. An excellent starting point is Zvi Bodie’s paper:
On the Risk of Stocks in the Long Run
If you think the math behind $15 million in movie sales would interest kids, just think about the math behind trillions and trillions of dollars in derivatives!! (to be clear: snark)
So, the whole shaming of the NYT for this bit of math leaves me uncomfortable. I think the implicit message it sends about the “we know math” crowd vs the “you don’t know math” crowd is terrible. I also think that the calculation is much more subtle than is being let on and ignoring those subtleties is something that’s caused lots of problems in the business world over time. Thinking that these sorts of problems can be solved by simply plugging into formulas is thinking that is mistaken. Finally, even ignoring the first two points, I do not see the problem itself as that compelling, and especially not compelling as a reason to learn algebra.
Mike's Math Page 
I re-read the NYT piece, and the author had no interest in the “proportion of sales by quantity” problem, so he didn’t need the help of an algebra “expert”. As you say, it is not even an algebra problem. One problem with the teaching of algebra these days is due to the desire to make everything “mathematical”. The idea that mathematics is about the formal processing of symbolic statements is so wrong that it is laughable.
Here is a stupidly simple example:
Conversion of temperature from Fahrenheit to Celsius (and back).
Method 1: subtract 32 and multiply the result by 5/9
Method 2: let F be the temp in Fahrenheit and C be the temp in Celsius.
Then C = (F – 32)*5/9
If we are lucky it is pointed out that the graph of this equation is a straight line, but oh dear, it is not in slope-intercept from and will be difficult to plot.
Now substitute your value of F into the equation and evaluate the expression….
and so on, all purely formal, having forgotten what the original problem was.
I am not saying that algebra is a waste of time, but if the study of algebra sheds no light on the way of the world then it is pointless.
Asked to convert C to F the method 1 person thinks “” do this in reverse”, the “algebra” person has to find the inverse of the function……
You raise some worthy points here, Mike. I agree that those of us who publicize mathematics need to be mindful of those who are less comfortable with the subject. And I also agree that we should be wary of the other extreme–applying equations and algorithms routinely and mindlessly.
That being said, you’ve got to cut bloggers and tweeters a lot of slack. I doubt Dan Meyer crafted that post and tweet thinking “This will be my message to the world about the value of Algebra”. I’m guessing it was meant more as a math-teacher in-joke (and I laughed, heartily and repeatedly). Who could have predicted it would be retweeted 5,000+ times? I don’t share your discomfort about its tone or message, but what resonates with people isn’t under our control.
And as far as I’m concerned, there is a legitimate criticism here. The journalist made a point to assert that the company did not provide a piece of information. The assertion, itself, suggests that the missing piece of information is valuable. His lack of understanding of basic algebra prevented him from properly doing his job as a journalist. A newspaper that sees fit to publish a piece like “Is Algebra Necessary?” maybe needs to think a bit more about the role mathematical literacy plays in reporting.
To be fair, I know I have very strong feelings about this topic. I found Hacker’s Op-Ed laughable (he suggested replacing algebra with algebra) and I wrote a response for the New York Times Learning Network (“N Ways to Use Algebra with the New York Times“).
“So, the whole shaming of the NYT for this bit of math leaves me uncomfortable. I think the implicit message it sends about the ‘we know math’ crowd vs the ‘you don’t know math’ crowd is terrible.”
I’m not criticizing an introductory algebra student here. This is the United States’ paper of record. And I’m not criticizing someone for attempting a mathematical analysis and failing either. (Though, again, since this is the NYT, I wouldn’t hesitate.) I’m criticizing the NYT for failing to even pick up the algebraic toolbox while simultaneously publishing an argument that the toolbox is useless.
“But there are other problems too. For example – How do you know Sony’s numbers are correct? How do you know that all of the sales and rentals took place at $15 and $6 and not some other price(s)?”
Whether or not the toolbox is actually useful in this particular instance is a matter of interesting debate, a lot of which I saw on Twitter. There is actually much less variation in the $15 and $6 price points than you seem to think. (The article doesn’t concern theater tickets, which vary widely, rather online sales, which Sony could carefully control.)
Much was made on Twitter about the word “about,” which introduces enough variation to make precise figures impossible to calculate. I’ll still criticize the NYT for not making a qualified statement, even one about the relative quantities of rentals vs. sales.
“Finally, even ignoring the first two points, I do not see the problem itself as that compelling, and especially not compelling as a reason to learn algebra.”
Whether or not this problem is compelling is a matter of taste, of course. 4,000 retweets would seem to argue in my favor, however.
“Whether or not this problem is compelling is a matter of taste, of course. 4,000 retweets would seem to argue in my favor, however.”
Oh gosh, not the “1 billion Chinese can’t be wrong” fallacy again.
Besides, you’re dealing with aggregated data here which doesn’t provide any evidence for your argument. How many retweets found this a compelling argument to learn algebra, how many retweeted for the humor, and how many retweeted to show other people what kind of “bullies” the “we know math” crowd “really” is?
If my kids were older, this is where I would hope to take a discussion about the actual math in that article:
Someone else’s nice desmos graph
No math teachers at the NYT? You mean like the ones who *don’t know* if 120 and 30/5 is equal to 126 (http://marilynburnsmathblog.com/wordpress/we-ask-we-listen-we-learn/)?
Maybe the whole newspaper is *run* by math teachers.
There is quite a bit of missing context in your entire argument, even though Mr. Meyer made the connection to the Hacker op-ed piece thoroughly unambiguous.
Those of us snickering at the NYT are not at all math-shaming Cieply, the latest reporter — it was an honest miss and indeed factually accurate in re information explicitly disclosed — we are just having fun with the confluence of the two articles. If anyone is being teased it is Hacker, but with you I doubt many of us consider the involved calculation to be a justification for mandatory Algebra instruction.
It’s just funny, is all. What goes around comes around and all that.
“[W]e are just having fun with the confluence of the two articles.”
I have no problem with this. I did find it funny and a good jibe at the ridiculous 2012 Hacker op-ed piece. However:
“it was an honest miss and indeed factually accurate in re information explicitly disclosed”
The first half of this sentence is inaccurate, because it’s clear that it wasn’t an “honest miss” – it wasn’t even a “miss” to begin with. The phrase “Sony did not say” is pretty standard language for flagging a couple of things:
1. Sony did not reveal precise figures for the two categories of sales, i.e. rentals and purchases.
2. The reporter wants to point this out so as to raise the question, in the reader’s mind, of why Sony did not reveal this.
What Mr Meyer then went on to argue, in his above comment as well as in the Twitter conversation that ensued from his tweet, is that it behooves the reporter to engage in blatant speculation by using algebra to come up with some numbers based on figures that were only estimates.
This is where I part company with Mr Meyer, as I believe the reporter did not have to, and in fact MUST NOT, do what Mr Meyer has suggested. To do so would be to mislead the reader into thinking the reporter had more accurate information than what he actually had access to, and appears to me to be behavior that would be considered to be rather unethical.
Math is important, but so is the professional integrity of reporters, which can be sorely lacking these days. In this case, I think Mr Cieply did exactly what he had to do, and it is Mr Meyer who has shown, in his commentary, his ignorance about what reporters can and cannot say.
I finally had a chance to read the Zvi Bodie paper. I was curious about his claims in part 4: mean reversion doesn’t matter. Have you read the Lo and Wang paper he references? I’m not convinced that the shortfall insurance does increase in this case, but I haven’t been able to sort out fully the relationship between the different time values in their formula (tau, T, t). The suggestive point is that the tau-period variance achieves a maximum value, rather than increasing linearly with time. That means, as the holding time increases, the shortfall threshold becomes farther out of the money.
I’m not sure whether that is sufficient to make the options actually cheaper as holding periods increase.
Happy to discuss this privately if you are interested. You’ve got my email, I believe.