Common Sense Mathematics

I’m at the console in front of the class where I can monitor what each student is doing on his or her mac. It’s intermittently sad, though perhaps not a sad as at other times.

One student has Excel open to create a column of figures that grow at a 5% rate. Here’s his algorithm:

• type the starting value in cell A1
• when cell An has been filled
• – go to an online calculator
• – enter the value from cell An (all dozen or so digits)
• – multiply by 0.05
• – add the value from cell An
• – type the answer (not even copy paste) into cell A(n+1)

I don’t even want to think about this.

This same student tried first to solve this problem by looking in the text (he actually found the problem), then in the instructor’s manual (where he hoped to find a solution). I hoped that he might actually be lurking on this blog during the exam, would see my comments, know I was talking about him, and mend his ways. But no. He left the exam early. I hope he passes the course. He’s one of the cleverest students in the class and has refused to learn anything new all semester… that’s sad.

Another student was asking Yahoo how to solve exponential equations.

It’s as if they trust the internet (or go to it) rather than trusting their own intelligence – or even trying to remember what was taught in the course. Of course that might be the best solution in the ten year time frame I want to think about …

I’ve said here that it’s not as sad as at other times because a much smaller than usual percentage of the class is trying to answer the questions this way.

Last class.

Complex semester. In my 50 years teaching mathematics I’ve generally tried to teach a course no more than three times in a row. After that it gets stale. I hear myself saying things I’ve said before and wonder why the students don’t remember – when of course it was a different group of students. I’ve now taught QR N times in a row, N >> 3, and will continue to, until the book is out (and I retire?). I need to find a revolutionary strategy for the fall.

Today we’ll make up the final, maybe do last year’s. I feel more than usually inclined to prep them for it directly.

I won’t get to update what actually happened until tomorrow afternoon. I hope I remember to do that – and remember what happened.

Next to last class. Thinking ahead ten years, there’s no point trying to squeeze in any more “material”.

This is what I suggested in email to the class:

One good way to review for the final: find one or two problems (maybe from homework) that you are on the edge of understanding (you can almost get right). Ask about them in class.

Note “on the edge” – it’s too late to try to learn something you’re absolutely clueless about, and a waste of study/class time to think about stuff you know well.

Let’s see what they come up with.

I should have graded/returned papers today, but forgot and did the xword/sudoku on the T instead. Sorry.

Worked the regression problem on US energy consumption. The class went well or badly depending on your feelings about watching the teacher make mistakes and recover from them. The recovery is an interesting process that should be made explicit, but boy is it confusing for the students.

In this case we prefaced constructing the regression line with a discussion of why energy usage increased in the last half century – population growth and more gadgets. To understand the former we computed energy use in Twh/person. The first attempt gave something like 100  kwh per person per year – which was (to me) obviously ridiculously small. It’s just about enough to keep a 100 watt bulb on for 1000 hours (3 hours per day). So I went looking for three more zeroes. First had to check the Tera was 10^12, not 10^15. Then found the zeroes in misreading a `’` for a `.` in the data. Then the correct calculation gave an answer that was several orders of magnitude too big – the answer is probably in the meaning of “domestic” energy consumpion. That might be “household” but more likely is “United States” and counting all forms of energy, converted to Twh.

Thinking these things through is hard – and really only possible if you have a kind of confidence that the students don’t – for perfectly good reasons.

Finished with useful Excel work drawing the trendline. Boring by comparison.

Plan: Term paper consultations (particularly spreadsheet help). Questions. False positives. Course evaluations.

I showed several students (individually) how they might arrange their data in Excel to support conclusions in the text of their papers – rather than doing the arithmetic on the fly in the text.

We discussed false positives in several instances: terrorist screening, pregnancy tests, plagiarism detection. When asked about a perfect test all said it had to find all the baddies. It was harder to lead them to see that accusing innocents was a less than perfect thing to do. We did some made up numerical examples.

As has been happening more and more often lately, I think more learning happens with these kinds of qualitative arguments than with particular examples, even with real data.

Plan: The Massachusetts lottery. The homework due today. Course evaluations.

Small exciting class. We did the Mass lottery house advantage (1%) – a good short review of percentages. Most of the class was about pari-mutuel betting at the races. ending with a discussion of Intrade and crowdsourcing. It wasn’t very quantitative but it was informative.

Starting probability. I’m going to do the checkerboard H/T exercise for Charlie as a guest lecture right before my class. Perhaps I’ll do it again in my class, even before introducing any of the other ideas.

I did the H/T experiment twice, once for Charlie and then in my class. It worked well both times, better the second. In my class I segued to house advantage after a discussion of why doubling your roulette bet must fail. I think the switch of chapter order (multiple independent events before the economics of betting) worked. I can suggest it as possible in the teacher’s manual for CSM.

In Charlie’s class the discussion of likes/dislikes led to a pretty uniformly expressed wish for more structure in the problems – like “a real math course”. They were polite, but forceful. And the only response is “it’s not supposed to be like what you think of as a ‘real’ math course.” I said that I’d try in the future to announce that clearly at the start of the semester. People who just want to check off the requirement by learning algebra all over again should do just that – this course is in fact harder than algebra. The hope is that it’s more interesting and more useful. That turns out to be true for some but not all the students. How can I increase the fraction?

Plan: answer homework questions. Then I need to say something about compounding and the formula (1+r/n)^n. But simply calling that APR is wrong. APR/EPR is much more complex, and does not always mean the same thing. (The wikipedia article is pretty convincing on that point.) Maybe just do the payday loan example from the book. The real ten year takeaway is, as usual pay attention!

I’d like to get on to probability. It’s a nice change of pace. Perhaps I can do the easy stuff (just counting), so the decks will be cleared for gambling next week. If I decide not to get to money today then I might want to change the order of topics from the book and do repeated events, preparing the way for independence.

Interesting class. I didn’t do anything like what I’d planned. Started by working the Chinese R&D expenditure problem. I thought that starting with the exponential growth spreadsheet would be the best strategy, but in fact just working backwards fom $70 billion by multiplying repeatedly by 0.8 was better: quicker and more instructive.

I was ready to move on to probability when I realized I’d said nothing about doubling times and half lives. So we discovered the rule of 70, I told them they’d need ideas from calculus to understand why it was try. Having discussed doubling times I moved on to half lives. That called for an explanation of radioactivity – and a discussion of the chemistry and the history of nuclear bombs and nuclear power. I did the talking, the students were all ears. None of that will be on the exam but it will probably stick with them for years …

Probability next week.

Credit card interest last time. Paying off a debt today – and saving for retirement. I might do the Globe quote:

If you file your tax return on time or get an extension, but fail to pay, the IRS will charge you interest on unpaid taxes. That rate currently works out to about 3.25 percent and is compounded daily.

The IRS also will charge you a late payment penalty of one-half of 1 percent of any tax not paid by April 17. That translates to a $25 penalty if you owe $5,000. It is charged each month or part of a month the tax goes unpaid, up to 25 percent, or $1,250 on that $5,000.

That interest rate can jump to 1 percent, however, if the tax bill hasn’t been paid within 10 days after the IRS issues a notice of intent to levy. But if you work out a payment plan with the IRS, it will reduce the rate to one-quarter of 1 percent.

link (free only to subscribers)

There are three kinds of questions to ask about this quotation.

The narrowest would ask for the cost of that $5000 nonpayment, for, say, six months. That would require careful reading along with an understanding of “daily compounding”.

The second would ask which costs more – the interest or the penalty? A quick estimate shows that the penalty dwarfs the interest, even without worrying about the details.

The third question is “what’s the ten year lesson?” And the answer is easy: pay your taxes when they’re due. No need to do any of the arithmetic at all!

We opted for the third one, in class. Then we talked about mortgages, paying for retirement, and the car deal you can’t believe exercise.

Plan: exponential growth. Start by recalling the distinction between constant absolute change and constant relative change. The first leads to a straight line. We’ve seen the second in percentage change and the 1+ trick. Compare linear and exponential growth, then go directly to credit card discussion.

– unless there are unanswered homework questions that take precedence.

The good news and the bad news is that I did pretty much everything I’d planned to do. We read the linear vs exponential spreadsheet. It’s more sophisticated than ones they could write, but reading is easier, and valuable. In particular, naming the cells there means that the Excel formulas read just like the ones I write on the board.

One student was particularly intent on learning what negative slopes meant, so we digressed into depreciation. They understood – for a fleeting moment – the analogy (positive vs negative) is to addition as (greater than 1 vs less than 1) is to multiplication. And laughed when I acknowledged that although they understood now they might not in an hour or a day from now. Even in a course where the goal is to teach things that will matter for the next ten years, sometimes understanding is temporary.

We talked about how credit card companies make their money – some from merchants, who willingingly pay for the convenience, some from people who don’t pay their bills in full. 1.65% interest per month is even more than 12*1.65% per year – we worked out (1.065)^12.