The ten day week, now resolved!

This post is by James

Last week, I published a simple question:  if the week were ten days long, and we got three days off, instead of two, would be get more or less days off than we currently do, with the seven days week with two days off?

Of course, there can only be one right answer, but it turns out there are lots of different ways of getting there!  Here is my personal solution:

Let us consider the smallest number which is a multiple of both ten and seven.  In our case, that is 70.  Now this means that 70 days contains either 7 ten-day weeks, or 10 seven-day weeks.  Since we get two days off per week in the seven-day week, this means that the seven-day week gets 20 days off in seventy days.  The ten-day week has three days off per week, but only seven weeks, so we get 21 days off with the ten-day week.  Thus we conclude that the ten-day week gives us slightly more time off.

There are other ways of doing this.  Ten-day week solutionMy father, for instance, when he was working it out, phrased the problem differently.  His logic was that in a seven-day week, we get one day off for every 3 and a half days of the week.  Thus to get three days off, at the same rate, the week would have to be 10 and a half days long:  seven days for the first two days off, and three-and-a-half days for the third day off.

Because 10 and a half is slightly bigger than ten, the ten-day week actually gives us more time off.

If this all seems a bit complicated, hopefully the picture will help.

Both of these are perfectly good ways to solve the problem!  They both give the right answer, and we can extend both of them to solve other problems (like how much more time you would get off each year with the ten-day week.  It’s about five days.)

This is relevant for math education because the original problem contains the exact same math as the dry-as-dust problem:

\text{Circle the bigger fraction: }

\frac{3}{10}\;\;\;\;\;\; \frac{2}{7}

Yes, our original perfectly sensible and interesting problem can be turned into a seemingly pointless problem which makes kids wonder why anybody ever bothers with the stuff!  We bother, of course, because in the hands of somebody who understands the math, the dry-as-dust description also happens to be reliable and fast.  But without understanding the problems behind the math, is it any wonder students think that math is all about “circle the bigger fraction,” rather than exploring fun and exciting what-ifs?

The other reason we use the dry math is because it works!  It always works, and it always works the same way.  For instance, if somebody is cooking and comes across a bizarre recipe which calls for 2/7 of a cup of sugar and the user only has 3/10 of a cup, do they have enough sugar?  Mathematically, actually solving the problem uses the exact same math as this!  It’s not obvious, but it is true.  And once you understand both problems, you can see it.  But without motivation, who actually gets that far?  Very few.  And so education needs the context before people engage.

Now I’m curious.  Can anyone else thinks of anyone contexts which result in this problem?  The stranger, the better!

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Just for fun: A ten-day week?

This post is by James

The title says it all.  What if the week had ten days?

This all started when I was talking with Jason (my brother and co-blogger) the other day and somehow we started talking about the metric system (centimeters, kilometers, grams, etc.)  The strangest thing about the metric system is that it changed how we measure everything—except time.  Unlike everything else metric, in the metric system, we have 60 seconds in a minutes, 60 minutes in an hour, and 24 hours in a single day (give or take a very little bit.)  This is all very strange. Everything else in the metric system is related by powers of 10.  1 kilometer is 1000 meters and 1 milligram is 0.001 grams.  Why did we stick with these strange numbers for time (24 and 60)?

We didn’t come up with an answer (we aren’t historians of, well, anything, so we didn’t really try), but we did remember that during the French Revolution, they tried to create decimal time.  They had ten hours in a day, one hundred minutes in an hour, and 100 seconds in a minute.  (I’m guessing they finally gave up on that because it made an hour far, far too long.)

I dare say that most of us could live pretty well with a different way of measuring what hour it is.  Most of us only really care about “It’s-far-too-early-in-the-morning,” “time-to-wake-up,” “lunch-time,” better-not-nap-time,” and “evening.”  The really interesting bit is when they tried to fiddle with the length of the week.  Yes, they tried to implement a 10 day week!  I don’t know what effect the ten-day week really had in France in 1800, but it would certainly have some odd effects today.  What would we call the extra days?  Would we still have “hump day”?  Would TGI Friday’s have to change their name?

Most important of all, if we tried to switch to such a system today, you would run into an interesting problem [1], namely that people would really object to getting fewer days of the week off if we just kept the weekend two days long.  Getting two days off out of ten is clearly more work than getting two day off out of seven.  It seemed to us that if a government didn’t want to have instant riots when they made the switch, they would have to offer some kind of compensation to workers.  The obvious choices would be to give the workers more than two days off during the week; it doesn’t much matter when.  Then the workers would have three days off out of ten, instead of just two.  Surely this is fair?  How can we tell?

And this is where we finally get to the math.  How do we compare the two systems?  Which gives you more time off, 3 days out of 10, or 2 days out of 7?  Can we do this kind of comparison in general?

And here is where I fiendishly leave you hanging.  Can you figure it out by yourself?  Do you know?  Which would you pick if time off was the only thing that mattered?  Let me know!


[1] There are other reasons why people would object to this than the ones we list here.  People doubtless objected in 1800 France for the same reasons as well.  We are doing this only as a what-if; we don’t actually want the week length to change.  Because some of the reasons are sensitive, we insist on not going beyond the simple “what if” as given.  Remember that it’s all in fun.

The wrong way to teach math?

This post is by James.

I recently ran into a New York Times article by Andrew Hacker, provocatively titled “The wrong way to teach math?”  You can read it at the link here.  Looking at his other published works, which includes “Higher Education?: How Colleges Are Wasting Our Money and Failing Our Kids – and What We Can Do About It” and an article called “Is Algebra Necessary?” published in the New York Times, it’s clear he’s made a bit of a career challenging common assumptions in math education.  This article is more of the same.

I doubt I agree with all of his conclusions, but I found this article to be a very good read, and I agree with many of its important points.  If I had to summarize it in a single sentence, it would be:

“We need to teach people how to solve practical word problems rather than advanced mathematics.”

If you’ve read the rest of this site, you can tell that somewhat agree with the spirit of this statement.  My opinion is that if you can’t solve word problems, you can’t really do math.  Nobody is fluent in a language if they can’t speak it with somebody they don’t know in a situation they’ve never seen before.

My biggest qualm with the article is that I think it underplays the advanced mathematics more than it should and overplays the success we have in K-12.  For instance, near the end of the article, he says:

We teach arithmetic quite well in early grades, so that most people can do addition through division.  We then send students straight to geometry and algebra, on a sequence ending in calculus. Some thrive throughout this progression, but many are left behind.

I would argue that ideally, the arithmetic curriculum should be more similar to that of geometry so that when students hit geometry, it isn’t a big change but a continuation of what they’ve been doing the whole time.  Real-world arithmetic is very much as complex as geometry and we need to stop hiding that fact.

On the other hand, if we had to have either the current math education system or Andrew Hacker’s system, I would probably go with his as being more practical.  I don’t think those are the only two choices, though.

Despite this qualm, he makes really good points about the kind of math people actually run into in real life.  He teaches a class at Queens College which he calls “Numeracy 101,” which focuses on making students fluent in real-world math.  This includes vital things like being able to read a graph of real-world data without being scammed by clever tricks often used by big companies and governments.  It also includes the ability to estimate how much something would cost.  If you want more details, I suggest that you read the article.

So what do you think?  I’m pretty sure he’s overstated his case, but I can’t argue that the kind of things he’s talking about teaching are pretty useful.

Edited for clarity.

Memorization?

This post is by James.

About two months ago, I signed up for Quora.  Its been interesting following the questions and answers there, whether I agree with the most popular answers or not.  I feel that I get more information the questions the answers.  What people are curious about, and willing to ask anonymously on the Internet, shows you things about people that I’m not likely to run into in my everyday life.

The most useful feed for me is Math Education.  I range from “expert” to “generally competent” in the  other topics I follow, but because I am not personally a teacher (that’s Jason, my brother), the Math Education forum on Quora is a great way to discover common ideas about what math is, what it is used for, how to learn, and so on.  In particular, you manage to read stories which really bring home how much “math as many people learn it” and “math as STEM professionals use it” are different.

Which brings me to today’s topic.  Recently I saw two different posts on Quora which brought up a common theme.  The first was a question:

Do good math students do anything besides purely memorizing formulas and theorems when it comes to tests?

The second post was an answer to a different question.  I don’t remember exactly what the question was, but it involved memorization.  The answer was from a mathematician in India who told a truly terrible story about how she grew up being told by her family she was bad at math, and would never amount to anything in the world, because she had trouble memorizing the multiplication table up to 20 by 20. (The ending was less depressing; she found her own way at college and discovered that nobody else important cared at all.  She became a mathematician and is more than happy to put that behind her.)

Both of these touch on the question:  what is the relationship between mathematics and memorization?  Both of the questions start with the assumption that memorization is an important part of mathematics, and continue from there.  Large parts of the answers are teachers/other students/college professors saying that it really isn’t.  What’s going on here?

Before we get deeper into this problem, it will be good to define a couple of things.  First and most importantly, what is memorization?  Like many words, it’s hard to pin down.  For instance, the first definition that pops up when I Google “definition memorize” is “commit to memory; learn by heart.”  This isn’t really what I’m talking about today.  By this definition, if you’ve learned something well enough not have to look it up,  then you’ve memorized it.  Some of the synonyms I find online include “learn; understand; recall.”  I doubt the people answering the Quora questions above would have answered the same way if the questioners had replaced “memorize” with “learn” or “recall.”

This means that in this post, we need to be more careful.  Language is tricky and the above definitions just aren’t good enough to really answer the questions people were asking on Quora.  This is because if you used the most general definitions of memorization I found online, then there are definitely things most people need to memorize in order to be fluent in math.  Being able to quickly recall the addition and times tables up to about 10×10 or 12×12 is very helpful, even in advanced mathematics.  How you become fast at arithmetic is in theory not important, but in practice, most people need to memorize something.

So what are math educators talking about when they condemn memorization?  Well, some of them are actually talking about knowing the basic math facts.  This is surprising to me, but it turns out this issue is a bit contentious.  From the context of the answers, though, I think that far more of them are talking about a different meaning of memorization.  I’m going to call this kind of memorization “foreign song memorization,” named after one of the most common places where this kind of memorization happens.

This name comes from personal experience.  Despite my families constant disbelief, I have actually been in a couple of choirs which didn’t immediately kick me out after listening to me sing (to be fair, these are the kind of choirs which don’t kick anybody out.).  In a few of these choirs, we’ve actually been kind of ambitious and sung things in other languages, like Italian or Latin.  When we did this, we generally had to learn how to pronounce the words on the page.  This means that we had to do a lot of memorizing in order to know how the song was supposed to be sung.  We’d often spend a lot of time carefully practicing the sounds and making sure we had the whole sequence down and in the right order.  The music helped, of course, because we could use the written words as triggers to remember what we were supposed to sing.

One thing we practically never learned was what the words meant.  Even if the choir director knew what the words meant, they seldom taught us; we just didn’t have enough time.  We did sometimes spend quite a while learning how to read the language phonetically, because that was very important in getting the sound right. When we were finished, though, none of us would ever claim that we had actually learned Latin.  Unless we did extra work on our own time, none of us would claim that we had learned even a little bit of Latin.  We had the sounds right (maybe), but we had no understanding.  If you had dropped us somewhere where the original language was spoken, we would be helpless.

This is the kind of memorization that drives math teachers bonkers.  When students treat learning math facts the way we treated our Latin texts, it can be very impressive—as long as you stick to the script.  But as I’ve said elsewhere on this site, math is a language.  Drop somebody who has learned math like I have “learned” Latin into a place where real ideas are being discussed, where there are real problems to solve, and they will instantly be in over their head.  I suspect that many former students, faced with this, decide that they are simply bad at math, when the truth is that they have simply learned the wrong things.

Exactly why this type of memorization is bad is a topic for a different post.  Today I’ll just point out the biggest problem:  when you make a mistake in repeating something memorized this way, there is no way for you to tell. If you don’t know the language, then you could easily say “he” where you are supposed to say “she,” or say “tater-roll” where you were supposed to say “table.”  You can easily spout total nonsense with no way of knowing.

So, what to do?  Well, as I noted above, students still need to learn their math facts.  Fluency requires it.  It would be a very odd foreign language class which didn’t include vocabulary and grammar, and math facts are the vocabulary and grammar of arithmetic.

So lets extend the analogy.  When a student learns a foreign language, they do need to spend some real time learning the vocabulary.  Good teachers do not stop there, though.  They also give the students lots of time to practice their vocabulary in “real” situations, either with themselves or other students.  They force the students to use the words they have learned in situations which aren’t in the book and weren’t in the lesson.  They expect the students to understand.  And when they test, they test for understanding.  One method I remember from my German classes was watching a video with two people talking in German.  After watching it, we were forced to answer detailed questions about what the people were talking about and doing.  These situations used words from the class, but the actual situation wasn’t in the book or in the class.  We were expected to understand it anyway.

I think that this model is how we must understand math, and especially the changes in math.  Many of the attempts to reform math are simply ways to make the questions we ask the students more like a foreign language class and less like singing Latin in a choir.  We throw new situations at students, confident that if they are learning the language, they will be able to handle it.  We ask questions that they have never seen before, and we provide more context for them to do it in.  In practice, this means more word problems and more careful explanation than is required if we just want them to repeat the correct words.

As for memorizing the math facts—we have to do it.  The good news is that we probably don’t have to do it all by simply reciting the facts.  Practicing speaking a language with other students helps the student remember the vocabulary.  Every time the student has to pull a word or a fact from deep in their mind, it is made stronger and easier to remember.  Time that we lose in raw memorization is recovered during actual use.

As for actually doing it—well, that’s not really my specialty.  It’s what many teachers and educators are trying to do, and it’s certainly what actually worked with me.  All I’m saying with this blog post is:  help them out.  When it looks like math is being made more complex than it really is, ask yourself:  will this question make my child more fluent in math?  Is there an idea that this teacher is trying to get across which perhaps my teacher didn’t teach me?  Even better, ask the teacher what they are trying to do, and if the goal is fluency, help them out.  It is hard work.  We need help.