Mathematics in Fiction Class Visit

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Today I attended a colleague‘s “Mathematics in Fiction” course. This course is designed as a First-Year Seminar course, not necessarily for math majors, and has a large writing component. I was invited to attend the class as a “guest participant” so I could be part of a dialogue on the broader issues about gender & mathematics, and how women are portrayed as mathematicians in works of fiction.

Overall, I really enjoyed the discussion we had. I’m hoping the students continue to ponder the issues and questions that were raised. In our conversation, I realized I wanted to make two distinctions that the students perhaps didn’t see.

Mathematician, Math Professor, and Math Teacher
Several students said they were unsure that there are still problems about gender in mathematics, citing that they had mostly female math teachers in high school. There seems to be a cultural conflation of mathematician, math professor, and math teacher. When I tell people I have had very few female math professors, a common response is, “Well all of my high school math teachers were female.” In my mind, these three titles have different connotations. I don’t consider high school math teachers to be “mathematicians” necessarily. To me, a mathematician is someone with advanced training and who has engaged in mathematical research (and, in most cases, who is continuing to do so). The research component separates math professor from math teacher.

As far as the distinction between “mathematician” and “math professor,” I used to think the overlap between these groups was so large that we might as well call these terms synonyms. But “math professor” is an academic job title — one cannot be a math professor if one isn’t employed. Meanwhile, “mathematician” has something more to do with educational background, training, and hobbies and isn’t job related.

One Question Becomes Two
One student brought up that perhaps the gender imbalance in mathematics has more to do with interest than anything else: Could it be that girls are just less interested in math, and that’s why there are fewer female mathematicians? (I don’t believe this to be true.) Our conversation made me want to point out the following distinction, which I think is important: There is a question of whether women like math less than men like math, and then there is a question of whether women like mathematical careers less than men like mathematical careers. In my mind, these are two very different questions.

My experiences & my gut instinct make me think that the bigger issue is that women are less interested in becoming math professors, not that women are less interested in mathematics. Indeed, there has been a lot of discussion about the so-called “leaky pipeline”: While more and more women are finishing both undergraduate and graduate degrees in mathematics, there seems to be a slow-down when it comes to who is being hired into academic mathematics.

Digital Grading Follow-Up

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THE BACKGROUND
Back in March, I wrote a post called “Want Some Free Red Pens?” on my dream for digital exam grading. In my ideal world, I’d remove all the paper from my office entirely. Having only digital copies of exams would be splendid since I could get a lovely potted plant to put in place of my institutional-looking filing cabinet. Last semester, I did accomplish my goal of grading an entire set of exams without using any non-digital ink. Now I finally have the time to tell you how it went.

The exam was for our “Introductory Calculus” (MATH 120) course. It was the third exam of the semester and I had about 30 students enrolled. I gave the same exam I would have otherwise — it wasn’t an online test. If you’re really interested, you can find a copy of the test here. I photocopied it like usual, and my students took it like usual. I did choose 1-sided copies over my usual preference for double-sided to help with the scanning task.

THE PROCESS

  1. Write, photocopy, proctor, collect exam. Alphabetize exams by student lastname and remove staple.
  2. Scan exams to PDF files using department’s Xerox machine; export as e-mail attachment to myself.
  3. Use husband’s perl script to “pull apart” multi-exam PDF file into 7-page segments. Rename files “lastname-exam3.pdf”. Transfer each file to iPad and open in GoodNotes.
  4. Correct each exam, save graded copy as “lastname-exam3-done.pdf”, compile exam grades, and upload grades onto our LMS.
  5. Use LaTeX’s “pdfpages” package to combine each annotated exam with a very thorough “Solution Key” (with comments, hints, suggestions, etc) at the end. Send each student an e-mail containing their exam’s feedback with the Solution Key & notification that official exam grade is available on LMS. [This was done to avoid FERPA issues about sending graded assignments, or grades themselves, over e-mail.]
  6. Save un-graded exams in my filing cabinet in case any student wants to pick theirs up. (As it turned out, no one did.)

THE GOOD THINGS
Here are the things I did like:

  • No crayon marks! No spilled orange juice! No paper shuffling! No page flipping! No running out of ink! Grading at home with a toddler is a tedious process, but being able to get in eight minutes of grading while also providing parental supervision was fantastic.
  • Forced Solutions. By giving every student a full Solution Key, I was able to write things like “See Remark on page 5” instead of re-writing the same paragraph of comments over and over again. Also, I didn’t have to feel guilty about printing thirty copies of said Solution Key, and I knew each and every student had been given the chance to see the solutions. (Usually, I upload the Solution Key to our LMS, but not every student bothers reading it, which is weird.)
  • Grading was Fast! During the “active grading” phase, I think it went faster than grading on paper. I didn’t have to spend time turning pages. I could Copy-and-Paste similar remarks from one test onto a different test. Because I didn’t need as much physical desk space to spread out, I was able to get in five minutes of grading here, four minutes of grading there, and so forth, so I think I was able to return the exams sooner than I would have otherwise.

THINGS NEEDING IMPROVEMENT

  • Hello, Copy Room. With about thirty students and a 7-page exam, the scanning task involved around 200 pages. It turns out that our Xerox machine does not like it when you ask it to scan anywhere near this many pages at once. After trying to scan 8 exams at once (56 pages), the Xerox’s “brain” would get hung up mid-process and a machine reboot was necessary. After this happened twice, I realized that I could only really scan 28-pages at once. So I set up four exams, pressed “SCAN”, and waited three minutes; lather, rinse, repeat. Four exams taking three scanning minutes meant about half an hour in the Copy Room I would have liked to spend elsewhere. (Thankfully, this wasn’t a total time loss since I could work on other tasks while the copy machine whirred.)

    A colleague let me know that elsewhere on campus, there exists a better copy machine that could handle this type of task more easily. But, accounting for walking to-and-from time, I am not sure this would have taken any less than thirty minutes anyhow.

  • Returning Exams. It had been my plan to use the LMS’s “Dropbox” functionality to return the exams. Unfortunately, I lost over an hour of my life trying to get this to work — without any success whatsoever. We use a Desire2Learn product, and after consulting back-and-forth with my Instructional Technologist, we concluded that you cannot return graded work unless a student has submitted ungraded work first.

    In other words, there is no way for me to return a PDF file to a student unless and until they have uploaded a (potentially blank) PDF file to me. So, basically, there is a way to “reply” to an uploaded student document, but there is no way for me to “send” a student an uploaded document first.

  • Big File Sizes. One has to be careful about writing too many GoodNotes comments. GoodNotes didn’t do a great job of compressing the PDF file size, and our LMS refused to allow me to send any file over 2MB in size as an e-mail attachment. Some of the exams were over this limit (too many comments) and others weren’t. To be fair, I am not sure if this is more annoying because of GoodNotes or more annoying because of our LMS. I also don’t know if GoodNotes has gotten better at saving from a GoodNotes document to an annotated PDF and keeping the file size smaller.

CONCLUSION

In the end, I don’t know if I’ll try this process again anytime soon. The biggest time drainers were the Xerox scanning & learning what didn’t work. If I were to do this again, I might investigate a better scanning technology. I would certainly ask my students to submit a blank PDF file to the LMS Dropbox, so I could “grade it” and instead return to them their graded test papers. My students really liked having a digital copy of their tests — it meant that when final exam week rolled around, they didn’t have to dig through their course materials to find their test. So, maybe I will revisit this idea sometime in the future? I’ll let you know if I do.

The Pregnant Mathematician Continues Teaching, Probably

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Recent Events
During the month of June, I’m teaching our “Introductory Calculus (Math 120)” course. We meet five days a week for 145 minutes, with a total of twenty class sessions running June 5th through July 2nd. Our final exam will be held on Wednesday, July 3rd. A few days ago, my very compassionate students expressed interest in knowing the probability their final exam would be cancelled due to an unexpectedly early birth of my daughter. I’m sure their inquiry was based solely on concerns for our health and not at all from them hoping to escape the intellectual challenge of a cumulative final exam!

Also recently, my husband and I went on an “Expectant Parents Tour” of the hospital where I will be delivering. Our son was born in November 2010, but at a different hospital; this time, I will be delivering at a hospital that was still under construction then! Since they have recently opened their doors, their NICU [Neonatal Intensive Care Unit] is still a “Level 1” facility. This means that they are certified to care for healthy infants born at 36 weeks or later (measured from LMP) or 34 weeks gestational age (i.e., since conception). Infants who are born with complications, or who are born before 36 weeks, are usually transferred to another local hospital. It turns out that reaching “Week 36” of my pregnancy and having lots of calculus final exams to grade will coincide perfectly.

Some Probability
While on our tour, I started wondering about the chance that I would go into labor early enough that it would affect my summer class. Not only would this be an unfortunate inconvenience for my students, it would also mean that I would probably have to deliver at a different hospital since I wouldn’t be at 36 weeks yet. What’s the chance this happens?

First, I should point out that I’ve had a relatively uneventful pregnancy — thankfully, both my unborn daughter and I have been in excellent health (if not a little grumpy from having to share the same circulatory system and oxygen supply). Second, my son was born within a few days of his due date, a little on the early side. Third, I’m not carrying multiples, nor am I expecting any major complications as my due date approaches. So we will just assume that this is an average pregnancy as far as the medical issues are concerned.

One of the things I talked about in my original “Pregnant Mathematician” post was how due dates are calculated using Naegele’s rule. Also, there was a rather large (n=427,582) study done in Norway [See Duration of human singleton pregnancy—a population-based study, Bergsjφ P, Denman DW, Hoffman HJ, Meirik O.] that found the mean gestational length for singleton pregnancies was 281 days, with a standard deviation of 13 days.

Let’s assume a mean of 281 days and a standard deviation of 13 days. What’s the chance a woman goes into labor 251 days or earlier (corresponding to 36 weeks)? Notice that 251 is about 281+(-2.31)*13, so giving birth prior to 36 weeks means you’re about 2.31 standard deviations away from the mean. By the Empirical Rule, I know this would be quite rare: There’s less than a 2.5% chance!

We can use Wolfram|Alpha to compute the exact probability. Our input is the command “CDF[NormalDistribution[mean, stdev], X]”; in this case, we are taking mean=281, stdev=13, and X=251. Wolfram|Alpha returns a result of 0.0105081, meaning there is about a 1.1% probability that I will give birth early enough to impact my current students.

It’s Gonna Be How Hot?!!

According to tomorrow’s weather forecast, the heat index here will reach 110 degrees tomorrow. This is miserably hot for everyone, but especially if you’re pregnant. Although my actual due “date” is in the first few days of August, I’m sincerely hoping that the birth occurs sometime in July. August heat in South Carolina is no joke! Wolfram|Alpha comforted me with its computation that there’s about a 41% chance I will give birth before I have the opportunity to be pregnant in August.

I’m also going to assume my chances of a July delivery are even higher than this, since human gestation doesn’t exactly follow a normal distribution. While a measurable percentage of moms give birth two or more weeks early, nearly none give birth two or more weeks late. By that point, OBs usually induce labor because of declining amounts of amniotic fluid and concerns for the health of the newborn. I’m going to take this and a “fingers crossed” approach and assume that a July birthday is at least 50% possible.

[There’s a name for such a distribution; I thought it was a truncated normal distribution, but that doesn’t seem to be quite right. The statistician who told me the term isn’t in his office at present! Anyone know what it’s called?]

Postscript: Dear Students,
Regardless of when I deliver, I can assure you that your calculus final examination will occur as scheduled. I know lots and lots and lots and lots of people who really enjoy torturing unsuspecting college students with tough calculus exams, and it would be easy for me to cajole one of those people into proctoring your test! So, don’t fear: You will certainly have the opportunity to demonstrate all the calculus you have learned this summer & feel proud of your scholastic achievement upon completing our course — including its final exam. 🙂

The Pregnant Mathematician Drinks Coffee

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In our math department’s faculty lounge, one can often find a liquid-y substance some people refer to as “coffee.” This designation seems questionable to me, so instead I have opted for a Starbucks prepaid card. I don’t drink a lot of coffee, but I do have one cup in the morning.

According to Starbucks, a tall (12 oz) cup of Pike Place Roast contains about 260mg of caffeine. Personally, I prefer the Blonde Roast, but I haven’t found its nutritional data. One of my students inquired at Starbucks and the barista told him that the Blonde Roast contains more caffeine per ounce than the others; apparently, since it is less roasted, less caffeine is lost during the roasting process, leading to more in the final product. I wonder if this is true.

How long does caffeine hang out in your body?

Wikipedia reports that the biological half-life of caffeine in an adult human is around 5 hours. The half-life of a substance is the amount of time required for half of the material present to metabolize. In other words, if the half-life of caffeine in your system is 5 hours and you consume 260mg of caffeine at 8am, then five hours later (at 1pm) we would expect 130mg of caffeine to remain in your system — provided you haven’t consumed any more caffeine since your morning coffee.

The half-life of caffeine in your system is related to lots of factors: Your age, your weight, what medications you’re taking, how well your liver is functioning, and whether or not you’re pregnant.

Maternal Caffeine Consumption & Half-Life
It turns out that if you’re pregnant, the half-life of caffeine increases quite a bit. In other words, it takes your body longer to metabolize caffeine. Today’s quick search yielded these few medical studies that agree about this:

According to Golding’s study, by the 35th week of pregnancy, the half-life of caffeine increase to a high of 18 hours. (For comparison, the Knutti et al. study cites a half-life of 10.5 hours during the last four weeks of pregnancy.) Since I am not yet in my 35th week of pregnancy, let’s assume the half-life of caffeine in my body is 12 hours. How is this different from my (assumed) non-pregnant state, when its half-life is only 5 hours?

Suppose I consume a Pike Roast Tall coffee at 8am that contains 260mg of caffeine. What time will it be when only 50mg of caffeine remain in my system? When not pregnant, it would take my body about 11.9 hours, so by 8pm less than 50mg of caffeine would be found in my system. Meanwhile, a half-life of 12 hours means it would take my body 28.5 hours — that’s over a full day!

We discussed this calculation as part of my PreCalculus course a few semesters ago. One student, who was usually rather quiet and didn’t ask many questions, raised his hand. He asked, “So, Dr. Owens, what you’re telling me is that to save money on Starbucks coffee, I should get pregnant?” Laughter ensued, and I assured my students that getting pregnant as a cost-savings measure was really not an optimal strategy.

As far as the risk to maternal and neonate health, the American Congress of Obstetricians and Gynecologists concluded “Moderate caffeine consumption (less than 200 mg per day) does not appear to be a major contributing factor in miscarriage or preterm birth” in 2010 [1] [2].

My Conclusion: Okay, it’s probably best if I limit my caffeine intake while pregnant. But I also have to think about my overall happiness and my enjoyment of life — as far as caffeine goes, today’s science seems to imply that my occasional Starbucks habit is a net positive (happiness minus risk), even when taking into account its expense (increased work productivity minus $2 per cup).

“The Pregnant Mathematician” Drinks Glucola

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Glucose Challenge Screen for Gestational Diabetes
As I posted about a few days ago, this week I had a one-hour glucose challenge test to screen for Gestational Diabetes (GDM). Today I received a phone call from my OB’s office informing me that my results were back and they were within the “normal” levels. Getting a negative result is comforting, but then I went back to hunting for statistical data on what this result really means.

According to an article I found in Obstetrics & Gynaecology, a 1994 study (“Poor sensitivity of the fifty-gram one-hour glucose screening test for hyperglycemia“) by van Turnhout HE, Lotgering FK, Wallenburg HC reported the sensitivity and specificity of the 1-hour glucose challenge test were 27% and 89%, respectively, with a prevalence rate of 5%.

In statistics, sensitivity and specificity are markers of how good of a test you’re considering. The sensitivity of a test tells you, “Out of all the people who have the condition, what percent of them will test positive?” Similarly, the specificity of a test tells you, “Out of all the people who don’t have the condition, what percent of them will test negative?”

If a test were perfect, we would expect both of these to be 100%. This would mean that 100% of people who have the condition really test positive, and 100% of the people who don’t have the condition really test negative. Of course, in the real world, this never really happens.

What Can I Conclude?
Another way we can gauge the performance of a test is to find its positive predictive value and its negative predictive value. I’m going to assume the sensitivity and specificity in the study cited above are correct. The same study above also gives a positive predictive value of 11% and a negative predictive value of 96%, but what do these numbers mean?

Let’s assume we give the same 1-hour glucose challenge test to 10,000 pregnant women. With a prevalence rate of 5%, we would expect 500 women to have GDM and 9500 not to have GDM. Of the 500 with GDM, since the sensitivity is 27%, we know 27% of 500 would screen positive, for a total of 135 women. These are women who have GDM and whose screening will come back positive. Meanwhile, of the 9500 women without GDM, since the specificity is 89%, we would expect 89% of 9500 or 8455 women to have a true negative result. The status of all of our 10,000 participants is displayed in the table below:

Women with GDM Women without GDM
Women who test positive 135 1045
Women who test negative 365 8455
Total 500 9500

According to this table, a total of 135+1045=1180 women would test positive. Of the women who get a positive result, only 135 of them really have GDM; this is the positive predictive value and, in this case, it’s 135/1180 = 11.44%.

What about the women who, like me, get a negative result? There are 8820 of us, and 8455 of us don’t have GDM. This gives a negative predictive value of 8455/8820 = 95.86%.

Was This Worth $40?
My results were negative, so I am one of the women with a negative result. The values above tell me that since I got a negative result on my glucose screening, I can assume there’s about a 96% chance I don’t have GDM. I’m waiting to be billed for this screening, but I’ll go with my initial $40 estimate. Even after this analysis, I’m still wondering if the knowledge I gained was worth the $40 I paid for it. (“Everything is worth what its purchaser will pay for it,” so I suppose this must be too.)

I feel unqualified to answer the “Worth it?” question because I don’t know a way to quantify the importance of this test. It seems clear that if a condition is really, really awful, then finding out you’ve got it is probably worth $40, and so is finding out you’re home free.

Is GDM really, really awful? It certainly has the potential to affect both my health and the health of my unborn child, so it seems better to know about it than not. But there are lots and lots of things that could affect our health that I don’t know about, and won’t be screened for, and probably won’t ever hear about.

One thing I wish I did have, for this screening and all of the others I have been (or will be) offered, is data ahead of time. I want to know the false positive and false negative rates. I want to know the sensitivity and the specificity and the predictive values. And I want to know how much money it’s going to cost me, and how much of a hassle it’s going to be. Lastly, I would like to know more about the medical significance of the condition, and since I’m not a medical doctor, I need it in some kind of quantifiable metric for when I do these kinds of calculations.

“The Pregnant Mathematician”

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I’d like to start by pointing out I know nearly nothing about medicine or obstetrics. I wouldn’t call myself an expert in statistics. I am a mathematician: In my life, what this means is that I was trained to approach problems from a very particular viewpoint. When confronted with choices in my own life, I can’t help but think of them as a math professor would. I would love to find someone trained in obstetrics or medicine to collaborate with on the issues I’ve written about below! Do you know of anyone who would be interested?

“The Pregnant Mathematician”
There isn’t a lot written about what it’s like being a pregnant mathematician. While I would guess that the overlap of these two population groups is small, a better reason might be that it is exhausting to be both a mathematician and pregnant at the same time, and the idea of adding “blogger” to that list seems insane.

Nevertheless, it’s difficult for me to think about having either description without the other. I Tweeted yesterday one of my “daydream goals”:

What would I write about? Well, here’s how my pregnancy has become mathematized over the last couple of months.

Glucola for Breakfast
This morning I was screened for Gestational Diabetes (GDM). This required fasting overnight (nothing but water), arriving at the lab, drinking a very sugary drink, waiting an hour, and having my blood drawn. I’m not exactly sure how much I’ll be billed for this test, but my guess is that it will cost about $40 out-of-pocket.

One of the things I’m always interested in is what the false positive rate for these types of screenings is. In other words, if my doctor phones me in a few days and tells me that my screening test came back positive, what is the chance I really have the underlying condition?

During my hour wait in my doctor’s office, I spent some time trying to find out this answer. According to this New Zealand-based maternity site,

“Approximately 15 – 20% of pregnant women test positive on the [glucose screening] test although only 2 – 5% will have any form of diabetes.”

In other words, if we give the same test I took this morning to 100 pregnant women, we should expect 15-20 of them to have a positive result. However, it will turn out (after further screening) that only 2-5 of them will actually have any form of diabetes. So if my OB tells me that today’s screening has come back positive, then this means I am one of the 20 women with positive results; but only 2-5 of us actually have gestational diabetes. Supposing that 5 of us have the condition, that means 5 out of the 20 positives are true positives.

If my screening this morning comes back positive, there’s around a 5/20 = 25% chance I have any form of diabetes. Looking at this another way, without taking the screening test, my “best guess” of my chance of having GDM is between 3% and 10%, based on the incidence rate for the overall population. So I can feel comfortable that there’s between a 90% and 97% chance that I don’t have GDM. But now I have taken the test; if it comes back positive, this means there’s still a 75% chance that I don’t have GDM.

Basically, I think I just paid $40 to find out if my risk is 10%, or if it’s really higher and is 25%.

How valuable is this knowledge? Is it worth fasting overnight? Is it worth taking time off from work to sit in the waiting room for an hour? Is it worth the $40 I’ll be billed? I don’t know. I never know the answers to these questions, but it seems I always choose to follow my OB’s advice anyway — her practice suggests screening of 100% of their maternity patients, so I just trusted their expertise.

Due Date Calculation
One of the things people always ask when they find out you’re pregnant is, “When are you due?” My obstetrician has some date written down on my medical records, labeled EDD (“Expected Date of Delivery”), that will happen this summer. The basic way the EDD is calculated is using Naegele’s Rule: Take the day of your last menstrual period (LMP), add one year, subtract three months, and add seven days. Example: If LMP date was in January this year (say, January 18th, 2013), adding one year gives 1/18/2014, subtracting 3-months gives 10/18/2013, and adding seven days gives a final EDD of October 25, 2013. There are lots of online calculators that will perform this calculation for you, like this one. This calculation gives 280 days post LMP for an estimated delivery date. But how accurate is that?

Suppose we take forty weeks (280 days) as the mean length of human pregnancies, measured from LMP to delivery date. It seems reasonable to expect that even if your actual delivery date isn’t your EDD, at least it’ll probably be in the same 7-day window. Unfortunately, this isn’t true either. Not only is it not very likely for you to deliver on your EDD, but it isn’t very likely you’ll deliver that week.

A study done in Norway (Duration of human singleton pregnancy—a population-based study, Bergsjφ P, Denman DW, Hoffman HJ, Meirik O.) involving 427,582 singleton pregnancies found a mean gestation length of 281 days, with a standard deviation of 13 days. By the empirical rule, this means more than 30% of women won’t give birth in the 26-days surrounding their EDD!

Example: Your EDD is June 15th. According to the study cited above, there’s more than a 30% chance that your real delivery day will be either before June 2nd or after June 28th. There’s a very real possibility you won’t even give birth in the month of June. (After talking about this with a few colleagues, one of them found a Statistics book that cited a standard deviation of 16 days!)

Here’s a great article about this same topic: http://spacefem.com/pregnant/charts/duedate0.php

Nuchal Translucency Screening
One non-invasive genetic screening test offered to all pregnant women in the United States is the nuchal translucency screening (“NT screening”). This screening is done in the first trimester (between weeks 11 and 14) and involves an ultrasound and a blood test. It screens for Down Syndrome (trisomy 21) and other chromosomal abnormalities caused by extra copies of chromosomes. The ultrasound image is read by measuring the width of the nuchal fold, found at the base of the neck. If the measurement is outside of the normal range, further testing is indicated. The cost of the test varies widely, but is somewhere in the neighborhood of $200.

The risk of carrying a baby with chromosomal abnormalities increases with maternal age. This page has a large table of the risk of Down Syndrome (and other trisomy abnormalities) as a function of maternal age. I’ll use a maternal age of 29, which carries with it a risk factor of 1 in 1000 for Down Syndrome. This means that out of 1000 moms (all age 29), one will have an affected baby.

One of the problems with the NT screening is the false positive risk. This would happen if your test comes back positive, when in fact you do not have the condition. In other words, this is the “false alarm” risk. According to my OB, the false positive rate for the NT scan is 5%.

Let’s go back and talk about our 1000 women (all pregnant and age 29, with no other risk factors). If we screen all of these women, then a 5% false positive rate means that 5% of them will have a positive test but whose babies do not have Down Syndrome. So that’s 50 women who will get a “false alarm” from this test. Also, one woman will have a true positive: She’ll get a positive test and her baby will have the condition. Altogether, out of the 1000 women, 51 of them will get a positive test. And out of these 51 women, only 1 will have an affected baby. The other 50 women will undergo lots of unnecessary worry.

I was offered the NT screening during the early portion of my pregnancy. Given my age, my child’s risk of Down Syndrome was around 1%. I was given the option to pay $200 to have the screening. If it came back negative, then I’d be worry free. If it came back positive, this would mean my child’s risk is about 1/50 or 2%.

In other words, without taking the screening, there was a 99% chance my child does not have Down Syndrome. Or, if I wanted, I could pay $200 for a test that, if positive, still means there’s a 98% chance my child does not have Down Syndrome. Is this test worth $200 to me?

As with the GDM screening, I followed my OB’s advice and had the nuchal translucency test a few months ago. The results were normal, which was comforting. The bill was around $250.

It’s tough having to juggle my emotions as a mother, my knowledge of statistics as a mathematician, and my interest in minimizing unnecessary financial expenses.

Even More Weather Data

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I’ve had fun over the last few days chatting with colleagues, friends, and family about the March-related weather in Charleston. See my previous blog posts to find out the background of this information. Here I’ll outline two new developments that came up today. Again, all of this pertains only to the month of March and only in Charleston SC.

(1) When do we ever use calculus, anyway?

In an e-mail yesterday, Dan Jarratt remarked that he was surprised by the result that the today-to-tomorrow temperature change had an average (mean) of +0.037 degrees Fahrenheit. In other words, given today’s high temperature, on average we expect it’ll be about 0.04 degrees warmer tomorrow. This isn’t a very big difference, as Dan remarked; it was lower than he thought it would be. This made me wonder if I too thought this was a small temperature change.

I would guess that the hottest month in Charleston is August. (That is, August is the month that has the highest average temperature.) Also, I would guess that the coldest month is six months from August, so that would mean February. Assuming that the temperature shifts in a sinusoidal fashion, we’d get a nice sine function with a period of 12 months; a local maximum in August; and a local minimum in February. This led to the following question, which I asked my Calculus students today:

When is the temperature in Charleston increasing most rapidly?

First, we had some discussion on how we could rephrase this question into one about calculus. If the temperature is increasing most rapidly, that would mean that the slope of the tangent line is its largest; this would occur halfway between February and August. We agreed that this would be the month of May. Let T(x) be the temperature at time x. Graphically, if the temperature is increasing the most rapidly, then this is where T'(x) has a local maximum, so T”(x) changes sign from positive to negative. In calculus, we call this a point of inflection: a place on the graph where the tangent line increases most rapidly (should such a place exist). Alternatively, it is where the graph changes concavity — in this case, from being concave up to concave down.

(2) What would a numerical simulation tell us?
Both my College of Charleston colleague Jason Howell and Dartmouth professor François G. Dorais suggested my predictive model wasn’t great, and

Thankfully, Jason was willing to help write some code toward this goal. (His code is given below, written for MATLAB, in case you’re interested!) He gathered historical averages for high temperatures for each date in Charleston, restricted to the month of March, from weather.com. The averages were computed using data from 1893 through 2013. Given that the average temperature change was +0.037F and the standard deviation of this temperature change was 8.39, we can run a number of trials to answer the question

How many days in March can we expect to be at or below their historical average temperature?

Jason’s code simulated one million different March months, given a starting temperature on March 1st of 68 degrees F. Here’s a histogram of results:

resultsOf course, March 2013 hasn’t finished quite yet. But this histogram does tell us that if we end up with 23 or 24 days with an “at or below average” temperature, this isn’t exceedingly rare — or it isn’t as uncommon as I had thought it would be.

Here’s a graph of the daily average temperatures (based on the same historical data):

historical

Jason’s Code:

function y=weatherexp(start_temp, num_trials)

%set historical averages, from weather.com
hist_avgs = [62 62 62 63 63 63 63 63 64 64 64 64 64 …
65 65 65 65 65 66 66 66 66 67 67 67 67 67 68 …
68 68 68];
%initialization
num_days=length(hist_avgs);
temp_diff = zeros(size(hist_avgs));
num_days_below = zeros(1,num_trials);
%parameters for normally distributed daily temperature changes
%from months of March from 1893 to 2012
stdev = 8.39;
avg = 0.037;
%loop over trials
for j=1:num_trials
%set temperature for start of a simulated month/week/etc.
curr_temp = start_temp;
for i=1:num_days
%get temperature change
temp_change = avg+stdev*randn(1,1);
%new temp
curr_temp = curr_temp + temp_change;
%how far from average?
temp_diff(i) = curr_temp – hist_avgs(i);
end
%count number of days below average
num_days_below(j)=sum(temp_diff<0);
%temp_diff
end
%histogram of the num_days_below data
figure
hist(num_days_below,[0:31])
y = num_days_below;

More Weather Data

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Dan Jarratt just e-mailed me a graph displaying the temperature difference from today until tomorrow, where we focus our attention only on Charleston SC in the month of March over years 1893-2013. Here are the summary statistics:

  • n = 3250 day-to-day changes, measured only when dates are in 3/1 to 3/27 over years 1893 to 2013
  • Mean = +0.037F, median = +1F
  • Standard deviation = 8.39F

daily-temp-changes

 

What is this graph telling us?

  1. In the month of March, on average it’s going to be 0.037 degrees (F) warmer tomorrow than it is today.
  2. But half the time, the temperature increases by over 1 degree (F).
  3. The temperature jumps are roughly normally distributed.
  4. The cumulative percentile jump of 0 degrees (F) is 49.85%.
  5. Graphs are neat.

Probability and Weather

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Warning: I know very little about probability. I know even less about weather phenomena! The post below describes something I was thinking about today, because I find it interesting and I’m procrastinating when I should be grading a giant pile of calculus exams instead!

The following image appeared on my Twitter feed, courtesy of @LCWxDave:

This made me wonder, “What is the probability that at least 21 of the last 27 day’s highs have been at or below average?”

First, let’s make two (probably bad) assumptions: (1) High temperatures are normally distributed, and (2) the events “Today’s high temp” and “tomorrow’s high temp” are independent. If this is the case, then the probability requested above is

(27!/(21!6!)+27!/(22!5!)+27!/(23!4!)+27!/(24!3!)+27!/(25!2!)+27!/(26!1!)+27!/27!)*(0.5^(27)) = 0.0029623

or about three-tenths of one percent. This struck me as being exceedingly rare! Then Dan Jarratt pointed out,

It puzzled me that my computed probability was 0.3% but the actual collected data suggest happened 8% of the time! What’s going on?

First, I still have no idea about my assumption about temperatures and normal distributions. Second, I really ought to do a more careful calculation and not treat each day’s high temperature as independent from the next day’s high. Surely, if it’s very cold on Wednesday, it is probably pretty likely it’s going to be cold on Thursday, too.

So instead of treating the 27 days as 27 different events, let’s consider them as 13 two-day events. Out of these 13 two-day events, about 10 of the two-day events have been colder than average. New question: “What is the probability that at least 10 of the last 13 two-day’s highs have been at or below average?”

In this case, the computation yields

(13!/(10!3!)+13!/(11!2!)+13!/(12!1!)+13!/(13!))*(0.5)^(13) = 0.046142578125

So my computed probability is 4.6% with the data suggesting 8%. The comparison between these two seems far more reasonable. What this tells me is that today’s weather seems quite dependent on yesterday’s weather, which isn’t surprising. After discussing this with my colleague Garrett Mitchener, he pointed out that a great way to predict tomorrow’s weather is to say that it will be exactly like today. Hopefully, we are better mathematicians than weather predictors.

Digital Plan for Digital Action

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It turns out that several people had some great suggestions about my wish for digital exam grading. I’ve decided to attempt it for my next Calculus exam, scheduled for Tuesday, March 26th. Here’s an outline of the plan:

  1. Photocopy exams single-sided and unstapled. Place a copy of each exam into an empty file folder.
  2. Subject unsuspecting Calculus students to grueling exam on these topics: Related Rates; Linear Approximation; Mean Value Theorem; Derivatives and Graphs.
  3. Alphabetize exams as they are turned in according to course roster. For absent students, place blank exam where theirs should be.
  4. Use department copy machine to scan all ~350 pages to a single PDF file and send it to me via e-mail.
  5. Thank my husband profusely for writing pdftk bash script that will take the single PDF file and break it apart, at every ~9th page, and rename the files according to last name (keeping alphabetical order in place). If this works, I should end up with 36 PDF files where each student has a file called “Owens-Calculus-Exam3.pdf” or something similar.
  6. Create Dropbox folders for the ungraded exam PDFs and the graded exam PDFs. Use GoodNotes to grade the exams on my iPad. Export the finished product back to Dropbox.
  7. Disseminate graded exams and grades to students.

It’s likely my first attempt at this will take longer than nondigital grading. One of the things I will have to do as I go is come up with “Correction JPGs” for those errors that happen most frequently and store them somewhere on Dropbox. I think these should be easy to add to each exam using the “import JPG” feature of GoodNotes. Usually I estimate that grading will take no longer than 10 minutes per exam. For my 36 calculus students, this means regular grading should take me about six hours. Hopefully this digital grading effort won’t take too much longer than this.

For Step 7, I also need to find out about FERPA. Provided I have a “sign for consent” on my exam header page, is that enough for it to be okay for me to e-mail each student her graded exam? Alternatively, is there a way using our Desire2Learn-Dropbox (on our Learning Management System) to return the exams to the students in some easy way?

Wish me luck!