Tanya Khovanova's Math Blog

Alice and Bob are good friends. Bob caught a cold and called Alice for help. He wanted Alice to go to a pharmacy and bring him some cold medicine. Alice did that and I would like to assign a number to this act of giving. How can we quantify this favor?

First, we need to choose a scale. Usually favors cost us in time, money and emotions. Alice spent half an hour driving around, plus $5 on the medicine (we’ll skip the cost of gas for simplicity). It also cost her emotionally, especially because the traffic was really bad.

Measuring everything on three scales is complicated. I would like to convert everything to one scale, because in the future I intend to compare this act of giving to other favors Alice does. For example, Alice knows for sure that this favor for Bob was a less costly favor than her phone call yesterday to her ex-mother-in-law, even though the phone call took only five minutes and didn’t cost any money.

We probably can convert everything to dollars, but I am trying to resist this money-driven society that measures everything in dollars. So, I prefer to use points. Each dollar translates to one point, but time and emotion are more subjective.

Alice makes two calculations in her head: what she really spent and what she is owed.

Here’s what she spent: Alice counts 5 points for the medicine. She also views her time as money. She charges $100 an hour for consulting and values all her time at this rate. Hence, she adds 50 points for time spent. Traffic was bad, but not so bad. She thinks that her traffic stress cost her 15 points. Since she also had to cancel her date with her boyfriend, she estimates her annoyance with this at 100 points. On the other hand, she got this warm feeling from helping Bob and she was happy to see him. So she thinks that she got back 30 points. Adding all this up, we get a total of 140 points. This is how much Alice thinks she spent for this particular favor.

Does it mean that Alice thinks that Bob owes her 140 points? Usually not. The calculation of how much Alice thinks Bob actually owes her is completely different. She thinks that he owes her 5 points for the cost of medicine. Also, she knows that Bob earns much less than she does and values time differently, so she think that he owes her 30 points for her time. Since Bob is not responsible for traffic, she doesn’t add traffic points. Also, she never told Bob that she had to sacrifice her date for him, so she doesn’t think it’s fair to want Bob to be thankful for the sacrifice he doesn’t know about. At the same time she hopes that one day Bob will sacrifice something for her. She can’t ignore this sacrifice completely, so she adds 10 points for that. Altogether she thinks that Bob owes her 45 points.

Do you think Bob feels as if he owes Alice 45 points? Like Alice, he also has two numbers in his mind. One number is the amount of points he received as a result of this favor and the other number is how many points he officially owes Alice.

He actually was planning to ask his neighbor to buy the medicine, but for some reason he called Alice first and she offered help. Alice was delayed at her work and arrived at Bob’s place much later than he expected. She also brought the worst flavor of the syrup. Bob doesn’t value time as much as Alice, so he thinks that Alice spent 10 points driving and 5 points on the medicine. Bob felt ill throughout Alice’s visit and did not enjoy seeing her. Combining that with her late arrival with the wrong syrup, he thinks that he was annoyed for about 15 points. So he thinks that he got zero points from this transaction.

At the same time he wants to be fair. Bob knows that Alice did her best to help him; besides he never specified the flavor he likes. As a result, he doesn’t count his annoyance in how much he owes Alice. So he thinks that he owes Alice 15 points. What Bob really did to thank Alice, I will discuss in a later blog entry.

In conclusion, let me remind you of my system. I measure all favors in points. And for each favor I assign four numbers:

• the giver’s official favor value (in our example 45 points)
• the giver’s hidden favor value (in our example 140 points)
• the receiver’s official favor value (in our example 15 points)
• the receiver’s hidden favor value (in our example 0 points)

I had a dream that sometime in the future I am babysitting my two-year old granddaughter-to-be Inna.

Me: Here is one apple; here is another apple. How many apples will you get when we put them together?
Inna: Two.
Me: So, 1+1 is… ?
Inna: Two.
Me: Good girl.
Inna: How much is 2+2?
Me: Shhh. We can’t talk about that.
Inna: Why? Will a big bad wolf come and eat us?
Me: Sort of. It is copyrighted and I do not have enough money for the private use license.
Inna: Did you spend all your money on a 1+1 license?
Me: No, honey. Google owns the rights and they released it for public use.
Inna: What about 3+3?
Me: We might be able to talk about it in a couple of years. The government is discussing the purchase of the rights, though it would be half of their annual education budget.
Inna: What about 4+4?
Me: 4+4 is approximately 8.
Inna: Don’t you know if it is 8? Do you think it could be 7?
Me: No, I know exactly how much it is. But the copyright has a loophole. You can’t say the exact sentence, but it doesn’t forbid variations. Have you heard that Stephen Colbert is being sued for saying how much 4+4 is on his show? Colbert argues that his grimace constitutes a complete reverse in meaning.
Inna: What about 5+5?
Me: Your father’s brother’s nephew’s cousin’s former roommate is a lawyer and he says that the 5+5 license doesn’t permit the answer to be in the same sentence as the statement. So, to be on the safe side, you should always go like this: “5+5 is a number. I like ice cream. My favorite flavor is chocolate. The number I’ve mentioned several sentences ago is 10.”
Inna: Can this lawyer find a loophole in the 2+2 license?
Me: The copyright agreement itself is copyrighted and too expensive for him.

Later my son, Alexey, comes to pick up his daughter. I continue the conversation with him.

Me: Your daughter is gifted in math. Is there any chance that her public school can teach her how to add 10+10?
Alexey: I know that our public school bought a limited license. They can discuss additions only in a designated room on Mondays from 11:00am to noon and only after 8th grade.
Me: Why the heck wait until 8th grade?
Alexey: They are required to study copyright laws first and pass the state exams.
Me: Have you considered private schools? Inna is so gifted — she might even get a scholarship.
Alexey: Our private school was able to copyright only three questions for their scholarship evaluations. And everyone knows that the answers are A, D and D.
Me: I have an idea. I am subscribed to Russian TV. They have a channel that broadcasts an educational math show in English.
Alexey: How could that be? The U.S. blocks all foreign non-copyrighted broadcasts in English.
Me: Their English is so bad, everyone thinks it is French.
Alexey: Ah, I was wondering where my boss’s son got his new horrifying accent.

At Inna’s next visit, Inna came up with some ideas.

Inna (in a low voice): I think I know how much 2+2 is.
Me: You can’t tell me that. But maybe you have a new favorite number. You can tell me your favorite number.
Inna: My favorite number is 4.
Me: Do you know how much 3+3 is?
Inna: I changed my mind. My new favorite number is 6 now.
Me: Good girl.
Inna: How come we are talking about addition and you never told me what the number after 10 is?
Me: Shhh. We can’t talk about that… —

Everyone knows that math education in public schools in this country is pathetic. If you looked at this problem from an economics point of view, the first question would be, “Qui prodest?.”

Who profits from bad math education? I know one place — the lottery. People who understand how the lottery works rarely buy tickets. They might buy an occasional ticket as entertainment, but never as an investment. No wonder they say that the lottery is a tax on people bad at math.

Huge money from lotteries goes to states and towns, and a big portion of that goes to education. That means towns, schools and math teachers have direct financial incentive not to provide good math education. This conflict of interest creates a situation in which, in the long run, it is profitable for schools to hire very poor math teachers or cut their math programs.

The situation is unethical. I think that lottery organizers should at least pretend that they are resolving this conflict and spend part of the lottery money to educate people not to play the lottery.