Republic of Mathematics blog

My colleague Sigal Gottlieb this morning raised an issue with me she found very upsetting. She normally thinks of the bag of mathematical tricks she has acquired as a research mathematician and professor as adding to her understanding.

Instead she’s finding it is, in some circumstances,  obscuring her vision to think and making her lazy.

The issue stems from the following problem that she posed to her oldest son, from the Singapore Math Curriculum:

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In a large group of children there are 3 boys for every 2 girls. 30 boys leave the group and then in the smaller group that’s left there are 2 boys for every 3 girls. How many boys and girls were there in the original group?

What concerned Sigal is that she began, in her mind, to solve this problem by setting up algebraic equations, when further thought showed a much more direct way of thinking through the problem. Sigal was concerned that her (advanced) mathematical training and experience had obscured and suppressed an ability to think. She said that if she keeps on becoming expert in mathematics soon she will not be able to think at all!

Sigal wrote earlier about a similar experience with her children in which she, the expert, failed to see a simple and direct solution to a problem because her advanced training led her to formalize the problem instead of thinking about it.

To emphasize how one might think about the ratio problem, here is a simpler version:

Two children Amy and Bob, each have collections of books. Bob has twice as many books as does Amy. Amy’s mother buys her 24 new books and then Amy has twice as many books as does Bob. How many books did Amy and Bob have to begin?

I am interested in how you FIRST begin to approach these problems? What is your first thought about how to tackle these problems? And how did you find the answer? What insights do you gain by thinking differently?

Is your ability to think, dear reader, also becoming obscured by knowledge?

This is a true story that I tell my statistics students.

When they hear it they tell me to listen to my doctor and my wife, both of whom have my best interests to heart, and to follow their advice.

So, here it is.

I have had checks on my blood pressure and tests for my cholesterol levels.

My blood pressure is a little high: 140/90. My LDL cholesterol levels are high and  HDL cholesterol levels are low (I do not remember the exact results).

My family physician, a gentle, concerned doctor who does not like doing tests to excess, tells me he is worried.  The Framingham Heart Study data suggests, given my age, weight, blood pressure, and cholesterol levels, that I have  a little more than a 25% chance of a major heart incident within 10 years.

At this point, my dear wife, who is in the consultation room with me, is worried that I am about to have a heart attack. My doctor explains that a “major heart incident” is not necessarily a heart attack.

Being a mathematician I put the following to him. Since I have a 25% chance of a major heart incident within 10 years, according to the Framingham Heart Study data, the odds are with me: I have a 75% chance of NOT having a major heart incident within 10 years.

Frustrated by my impeccable quantitative reasoning, my doctor adopts a different tack. He tells me that he has a large practice, and that he sees many patients.

“You” he says, “are an outlier in terms of indicators for heart problems”.

He tells me that there are 4 people in his practice who are out on a limb, as it were, for indicators of potential major heart issues, and I am one of them.

I understand this reasoning. Outliers stand out. That’s why they are called outliers. I am really out there on a limb. Who would not be concerned?

Then he tells me the deadly clincher: “One of these four died last week from a heart attack.”

This, of course, was meant to drive home the seriousness of the situation.

But, ever resourceful, I pointed out to him that, there you are: the one in four has had the heart attack already, so I’m safe!

He had no reply. My wife was dumbstruck.

Taking pity on my physician for his inability to attack my logic, I promised him, and my wife, that I would ignore my impeccable logic and follow his advice.

Thus far, EKG’s and ongoing tests indicate I am healthy.

The moral?

Statistics is a vital part of quantitative reasoning, but just because there’s very little chance of being hit by lightning doesn’t mean you won’t be.