The Accidental Mathematician

Too innumerate


In case you missed it, the New York Times has just published an opinion piece aptly named “Is Algebra Necessary?”. The title basically says it all: the 3 pages of the article boil down to the argument that math is hard and students don’t like it, so we should cut it out from most school and university programs, replacing it by “citizen statistics” or something like that.

Think of math as a huge boulder we make everyone pull, without assessing what all this pain achieves. So why require it, without alternatives or exceptions? Thus far I haven’t found a compelling answer.

Let me try to help. A friend has linked to this page describing the new British NHC policy on availability of testing strips to people with diabetes. I don’t have diabetes myself, but I know folks who do, so here’s a little bit of background based on what I know from talking to them and reading up on it.

When you have diabetes, your body either does not produce insulin at all or does not produce enough of it to metabolize sugar properly. To maintain blood sugar levels within an acceptable range, diabetics need to monitor strictly their food intake and inject insulin when necessary. This is a tight balancing act. Not enough insulin, and your blood sugar level skyrockets, leading to serious complications. If you inject too much, your blood sugar level drops too low, with diabetic coma and death as a possible consequence. So, the amount of insulin has to be just right, relative to your metabolism and food intake.

How do you know when your blood sugar level is within an acceptable range? Mostly, you don’t. You can only find out by testing. That’s where the test strips come in. You prick your finger, smear the blood drop on a test strip, put the strip in a meter, and read the results. Each strip can only be used once. Frequent testing is essential, not only to know whether your blood sugar level is acceptable at a given time, but also to figure out how it depends on your food and insulin intake over time, for example how quickly it increases right after a meal and decreases afterwards. With that knowledge, you can better time your meals and insulin shots.

Fortunately, I have not had the experience, but here’s what the Female Science Professor had to say about it:

I had a little booklet in which I recorded my blood sugar level, but I also started keeping track of the results in a spreadsheet and I graphed the results every day. I got interested in the shape and magnitude of some of the blood sugar highs and lows, but my initial sample spacing (in time) was too rough to get a satisfactory graph of these spikes, and there were other aspects of the data that I didn’t understand when I did the minimum number of recommended tests.

So, despite my loathing for jabbing myself in the finger with a sharp object, I started collecting more data. I tracked the blood sugar spikes so that they were defined by more than one point and I could really see their shape and I was certain of their maximum values. I collected data day and night. I dreamed of a device that could provide a continuous readout of my blood sugar and make perfect graphs. Even with my primitive data collection techniques, however, I made beautiful graphs and I did things with the graphs in terms of how I analyzed them over different time periods and how I displayed the data. I was obsessed with these graphs.

Now, back to the NHS policy. Here’s the money quote from the linked page (emphasis mine):

A number of research studies have been conducted to assess whether self-monitoring for blood glucose is cost effective for the NHS.

The studies have found self-monitoring has not been effective at improving blood glucose control in cases where patients have either not known how to interpret their results or have not been themselves committed to self testing.

The new policy, therefore, is that patients “should be assessed at least as often as annually on the appropriate frequency of their testing and whether self-monitoring is benefitting the patient.” This is followed by advice for patients whose testing supplies have been restricted by the new guidelines:

Your healthcare team may not be aware of the importance of self monitoring to you. Outline how self testing benefits you.

The following examples may help to improve your chances of being prescribed more test strips:

  • I experience hypoglycemia
  • Testing has improved my post meal blood sugar levels -cite examples
  • Since testing my HbA1c has improved by X%

“Interpreting the results” means more or less what FSP describes in the earlier quote. A non-scientist might not draw the charts and analyze them as well as FSP did, but a simplified version would still be much better than nothing. There might be calculations, inequalities, estimates, maxima and minima, possibly graphs. For example, the more rapidly your blood sugar level tends to change, the more often you need to test; some of us know this concept under the name of “derivative”. In other words, calculus with some statistics.

Now, first of all, imagine a major medical association or regulatory agency issuing guidelines to the effect that a lifesaving medication should only be prescribed to those patients who can prove that they can read and understand the instructions on the bottle, and that their ability to do so should be reassessed on an annual basis. Just… imagine that.

Second, a patient who understands graphs and inequalities will have a much better control of their health than someone who has trouble with even basic arithmetic. This is not a case of mathematics being a prerequisite for a job, the usual argument for numeracy. This is mathematics making a difference in your own health.

Third, and perhaps most importantly: it’s not only the patient who needs to understand the numbers here. It’s also his or her doctor. FSP’s doctor, luckily for her, was very appreciative of her efforts:

The first time I brought my graphs and spreadsheet to a doctor’s appointment, the doctor was stunned. He called all the other doctors and nurses over to look at it. He asked my permission to make copies and fax them to other doctors. He asked me to start sending him my graphs between appointments. He stopped talking to me like I was a slow child and started discussing with me what the data might mean. He gave me suggestions for ways to get more useful data. All of this helped get me through a difficult time.

Unfortunately, it’s not very difficult to imagine a less optimistic scenario. I’ve seen too many doctors whose default assumption is that they know better. If the patient disagrees with them, that’s because the patient doesn’t know what she’s talking about. Now imagine a patient arguing with them about the test strip allowance. Presumably, the patient will often have to make a mathematical argument as described above. But what if the doctor himself does not understand the mathematics involved? What if you are laying out the graphs and estimates in the doctor’s office, but he either interprets them incorrectly or simply has no idea what you are talking about? But the NYT piece has this to say about doctors:

Medical schools like Harvard and Johns Hopkins demand calculus of all their applicants, even if it doesn’t figure in the clinical curriculum, let alone in subsequent practice. Mathematics is used as a hoop, a badge, a totem to impress outsiders and elevate a profession’s status.

Doesn’t figure, indeed.

I was going to stop there, but actually since we talking about arguments and disputes, there’s also a fourth item. In mathematics, truth is never a matter of opinion. It is never decided based on rank or authority. There are of course certain things we know from experience, for example estimates that we expect to be true because we’ve had to prove dozens of similar estimates before and they all worked out. Still, there is always the questioning, the need for the actual argument. I may have 20+ years of experience as a professional mathematician, but if I make an error and a freshman student points it out to me, I have to listen. I don’t get to dismiss what he says just because he’s a freshman. I’ve met more than one doctor who could have benefitted from internalizing that attitude.