mathematics

Exit 245. Intentionally written to be confusing.

Every few months, a meme circulates asking people to solve this math problem (or one like it):

6 ÷ 2(1 + 2)

People begin arguing over what the answer actually is, with most people saying (for this problem) either 1 or 9. They insult people with the other answer, because apparently that’s what people do on the Internet. And several of my friends tag me every time this goes around, either to resolve an argument or just to read what people are saying.

The correct answer is 9, but I’ll come back to that later. The issue is that this problem is intentionally written to be confusing.

I find the whole thing annoying. For one thing, this is something that we were supposed to learn somewhere around sixth grade. Apparently, we as teachers and educators aren’t doing our job if adults’ mathematics skills have atrophied to the point that no one can do a simple yet fundamental problem from sixth grade. So, naturally, everyone tags their friend with the college degree in mathematics (me) to resolve a sixth grade problem. I wonder, do these same people tag their friends with degrees in English when they forget to spell a word? Do they tag their friends with degrees in music theory when they don’t remember the name of a song? I’m writing this so that when this meme goes around again, I won’t have to type a long response; I can just share this link.

Anyway, Order of Operations says this (paraphrased, my own words): First, resolve expressions in parentheses and other grouping symbols. Then resolve exponents. Then evaluate multiplication and division from left to right. Then evaluate addition and subtraction from left to right. This was somewhat of an arbitrary distinction, but mathematicians and others in related fields have done it this way for centuries, and an organized set of rules needs to exist in order for most problems to have a well-defined solution (for example, whether you follow these rules or not determines whether the answer to the above problem is 9 or 1 or something else entirely). Math books and teachers often abbreviate this rule with the acronym “PEMDAS,” which stands for Parentheses, Exponents, Multiply and Divide, Add and Subtract.

So, the correct answer as written is 9, and this is why:

6 ÷ 2(1 + 2)
Do what is in parentheses first: 6 ÷ 2(3)
Multiply and divide from left to right. As you read from left to right, 6 divided by 2 comes first, and that equals 3, so now the problem is 3(3).
This equals 9.

There are several common incorrect interpretations of these rules which lead to the arguments. Before I continue, it should be pointed out that the disagreement has nothing to do with “Common Core” or “new math” or anything like that. The Common Core State Standards say nothing about changing the order of operations from what it has been for centuries.

Misinterpretation #1:

6 ÷ 2(1 + 2)
Do what is in parentheses first: 6 ÷ 2(3)
2(3) still has parentheses in it, so do it before the division: 6 ÷ 6
This equals 1.
The rule, and the fundamental concept of what parentheses are, is that what is INSIDE the parentheses is done first. Parentheses go AROUND the operation that needs to be done first (or out of order). The parentheses are not AROUND the multiplying by 2, so the fact that 2 is next to the parentheses has nothing to do with changing the usual order.

Misinterpretation #2: “I typed it into my calculator and it said 1.” Your calculator was programmed by a human being who apparently interpreted the rules differently from someone who got 9 as the answer. I have actually seen photographic evidence that some calculators give 1 as the answer and some give 9. Calculators will do whatever their users and programmers tell them to.

Misinterpretation #3: In my observations, this one is the most common. The issue here is a lack of understanding of the acronym PEMDAS:

6 ÷ 2(1 + 2)
P: Parentheses first: 6 ÷ 2(3)
E: there are no exponents.
M: Multiplication comes next, so do 2 times 3. 6 ÷ 6
D: Division comes next, and 6 divided by 6 is 1.
A, S: there is no addition or subtraction, so the problem is done, and the answer is 1.

As I explained above, multiplication and division are evaluated on the same step; it is not correct to evaluate all of the multiplication and then all of the division. Addition and subtraction work the same way. Evaluating all of the multiplication, then all of the division, then all of the addition, and finally all of the subtraction introduces other problems as well. Multiplication and division have to be evaluated on the same step, because every division can be changed to a multiplication using fractions (e.g., 6 divided by 2 is the same as 6 times ½). Addition and subtraction have to be evaluated on the same step, because every subtraction can be changed to an addition using negative numbers (e.g., 6 minus 4 is the same as 6 plus -4).

More than once, I have pointed out this error to people, and the response is something like “then my teacher/textbook/etc. taught it wrong, because I’m doing what I learned.” Now that is possible; I have seen teachers inadvertently teach incorrect subject matter. A more likely explanation, however, is that the person making this statement either never learned it properly or remembered it wrong. One who only remembers the acronym PEMDAS and what the letters stand for, with no context, would be very likely to make this error, because they follow the order of the letters. The rules for order of operations written as complete English sentences say something very different.

For this reason, I have often encouraged students to write “P-E-MD-AS” to emphasize that some of the operations have equal predence. I have also started to hear some students in middle school say that their elementary teachers told them not to just memorize “PEMDAS,” probably for this reason. And that is what good teachers should be doing, teaching students to understand rather than just memorize out of context.

Misinterpretation #4: “Multiplication without a times sign, just writing two numbers or letters next to each other, creates an ‘implied grouping,'” which would then make the work identical to misinterpretation #1 above. In a simpler example of this thinking, these people would say that “10 ÷ xy” would mean to divide 10 by whatever you get when you multiply x by y, instead of to follow PEMDAS strictly and divide 10 by x first, then multiply this answer by y.

I find this hardest to argue against. Although the order of operations rules say nothing of this implied grouping, it doesn’t always look right. I read “10 ÷ xy,” and my first thought would be to find out what xy is, and then divide 10 by this answer. This violates PEMDAS, but maybe it looks right because x and y are written more closely together. There is mathematical precedent for this implied grouping as well; to find the cosine of 3x, one usually writes “cos 3x,” and this is almost universally interpreted as the cosine of 3x, rather than whatever the cosine of 3 is, multiplied by x.

Ultimately, there is a simple solution to this issue of implied grouping: be like every mathematician ever, and every high school and college textbook, and don’t use the ÷ symbol in the first place. Many people don’t remember math beyond elementary school and buttons on a calculator, so they may have forgotten that in more advanced math, division is usually written as a fraction. So now, if division is written as a fraction, it can be made extremely clear which order of operations the author of the problem intended, and the so-called implied grouping becomes an expression within the numerator or denominator, which is grouped explicitly by the fraction bar.

And if you are typing instead of writing by hand, use an extra set of parentheses to make sure that your intended order is clear. In other words, even though the answer to the original problem is 9, it is written in a way as to be intentionally ambiguous. So stop arguing about it and do something more productive with your time.

Exit 227. Taking my own advice.

Two unrelated things happened this week that, when juxtaposed, say something interesting about me.  

The first was a conversation I had on Tuesday with a former student who is now in high school.  I’ll call her “Lambda-2 Fornacis.” Lambda was in my class three years ago, the same class as Protractor Girl, The Boy I Have No Memory Of, and The Kid Who Sat Behind Me At A Basketball Game Once.  She was the kind of student that most teachers love to have in their class. She did her homework, it was neatly written, and she always was one of the top students in my class.  I think she had straight As all through middle school. I normally tell students that they can add me on social media after they finish middle school and go on to high school, but somehow (probably because these kids have older friends who talk) she found my Instagram (which doesn’t have my real name anywhere on it) and started following me the year after she had my class, when she was still in middle school.  I didn’t do anything about it, though, because I figured she wasn’t the type to cause trouble, although I didn’t follow her back until the day after she finished middle school.

Anyway, Lambda asked me something about a recent post on Instagram, I replied, and then I asked her how she was doing.  She mentioned that she had dropped precalculus. This year has been the first time she had ever struggled in math, she didn’t like the teacher she had this year, and she had been rethinking her career plans.  I have to admit, that was a little disappointing to hear at first, because she was such a great student for me, and I’m always disappointed to hear when people don’t love math as much as me. However, I completely understand where she is coming from, and I told her so.  I told her about hitting the same proverbial wall with physics my freshman year at UC Davis, how I struggled so much with that class at first, and while I still did well, it just didn’t feel as natural for me as math did. It was during that first physics class when when I decided for sure to major in mathematics and not physics, and I didn’t take any more physics after I was done with the minimum that would be required for the math major.  I told her that there’s nothing wrong with changing your mind about your future plans, especially since she’s only 15. I told her that as late as age 19, I was telling people that there was no way I would ever be a teacher. And I told her that I took all the most challenging classes in high school, to the point that I had some very long days senior year, but I wasn’t doing it because I had a career plan. For me, it was because I felt like school was the one thing I was good at, and I would be a failure if I didn’t.  This is not a mentally healthy outlook. I know that Lambda is going to be successful no matter what direction she takes her education.

That was Tuesday.  On Wednesday night, I got very little sleep.  I discovered another important thing here at the house that needed to be fixed.  I started to panic under the pressure of everything that needed to be done. I was behind on grading papers.  I had errands and chores that were piling up, and the kitchen sink was full of dirty dishes. I had now four important home repairs that needed to be dealt with as soon as possible, one of which was already making life more inconvenient in very tangible ways, and another of which had the potential to do so if left unchecked.  I couldn’t sleep, and I wasn’t sure if it was related to stress, recent changes in medication, other health problems I didn’t know about, lack of exercise, or what. It’s very hard for me to get these home repairs and chores done sometimes, because I’m rarely home during business hours and my schedule isn’t very flexible. I don’t get a lot of exercise this time of year, because I’m only home when it’s cold and dark.  And I couldn’t call in sick and take a day to recover from the lack of sleep and deal with these problems, because the classroom is such a mess that a substitute wouldn’t be able to find what they needed, and the kids would get behind anyway because my curriculum doesn’t work well for people who haven’t been trained and aren’t well-prepared.

I went to work on one hour of sleep (and I had gotten three hours the previous night).  I made an important decision while I was tossing and turning: long story, but basically I sent an email to the administrators saying that I needed to back out of one of my weekly commitments.  This would give me one more day of the week that I could get home a little earlier when needed, if I needed to deal with something before it got dark and places closed. Thankfully, they were very understanding.  But, I told the principal, I still feel like I do so much less than so many other teachers. Some of them are working on graduate degrees. Many of them attend more professional development workshops than I do. Some of them are department chairs, or serve on committees.  And many of them have young children of their own. I feel like there is something wrong with me, that I have such a hard time handling my own job, let alone all that extra stuff.

And then it hit me.

Why do I have such a hard time taking my own advice?

Just a day and a half earlier, I was messaging Lambda telling her that it was okay not to burden herself with hard classes that she didn’t need.  So why can’t I tell myself that it is okay not to burden myself with stressful commitments that I don’t need?

Everyone’s brain works differently.  I get more easily stressed and overwhelmed, and I’m fighting demons from the past that many of my coworkers don’t have.  If I really believe what I told Lambda, then it’s hypocritical to insist upon myself that I take on extra commitments that I don’t get anything out of.

It’s now Saturday, and I feel so much better.  Getting out of that extra commitment allowed me to leave earlier than usual on Thursday, which gave me time to make some phone calls to start the process of dealing with the two most pressing home repairs.  I didn’t get completely caught up on grading, but it’s now a three-day weekend, so I’ll have time to catch up.

I’m going to be fine.  :)&[4].

Exit 32. Welcome to Warp Zone.

I recently returned from attending the California Mathematics Council North conference at Asilomar Conference Grounds in Pacific Grove.  This really is an amazing place: a conference grounds built right next to the beach, in a forested area where deer occasionally are spotted running around, in a town full of big trees and old houses that is one of the best places I know of to take a walk.  But as much as I love talking about different parts of northern and central California, that’s not where I’m going with this today.

A few days before the conference, I asked the principal and the other math teachers I work with if there was anything specific they wanted me to look for.  No one said anything in particular, although one coworker asked if Dan Meyer was speaking, since he is one of her favorite speakers at events like this.  (I have not had time to properly vet Mr. Meyer’s blog, so if you click the link, be aware that the content belongs entirely to Mr. Meyer, and Highway Pi does necessarily endorse any of the opinions or writings shared my Mr. Meyer.)  I was planning out my weekend, and I saw that the title of Mr. Meyer’s talk was “Video Games and Making Math More Like Things Students Like.”  With a title like that, how could I not attend this talk?

Dan Meyer is an alumnus of UC Davis, my alma mater, and he is currently working on his Ph.D. at a major private university in Northern California which (no disrespect to Mr. Meyer) I dislike so strongly that it shall not be named in this blog.  He is a young guy, probably younger than me, considering that he received his B.S. from UC Davis a full five years after I did.  He does a lot of these talks, apparently.  I went into the talk willing to put aside my bias against his affiliation with Voldemort University (I did not know at the time that he also had a UC Davis connection).  I was expecting something that looked a bit like what I already do in the classroom, where I’ll make up word problems on quizzes about Mario and Link to get the students more engaged in what they are doing.

That is not what the talk was about at all.

At one point, Mr. Meyer was talking about real-world relevance of mathematics tasks.  This is a big thing with textbook writers.  The example he gave (probably from a high school pre-calculus textbook, although considering he’s from Voldemort University and the Silicon Valley, where so many people are so well educated, his classes probably do this in Algebra II) involved graphing fourth-degree polynomials.  This is normally a pretty dry topic, so the textbook he was citing from as an example had a picture of a snowboarder and made the graph be the number of Americans who participated in snowboarding.  Now, all of a sudden, according to textbook author logic, fourth-degree polynomials are cool… but that doesn’t really help students who don’t get it in the first place.

This was humbling to me, because what the textbook author did here is pretty much the same thing I do when I retype quizzes and make them about Mario and Link.  I’m still going to keep doing that, because the students seem to enjoy it, and those problems are still less dry than the ones that come with the textbook.  I still believe that these make my class more enjoyable for students.  But the point that Mr. Meyer was making was that this is not the kind of fundamental change that brings student success.  And those are not the connections with video games that he was there to talk about.

The talk was not just about connecting video games to the classroom; the point Mr. Meyer was making was about how video games engage students, and how we can engage them in math the same way.  For example, real world relevance the way textbooks do it is not what determines students’ engagement, because video games are much more engaging, and they do not take place in the real world.  Video games capture students’ attention despite the fact that none of these students have ever seen a portal gun, a Goomba, or an angry bird with a slingshot in real life, so contriving real world situations like the snowboard example above doesn’t help students.  Video games (at least most of them less than 30 years old) leave the path to the goal open-ended, so we should construct math problems that give students multiple options for how to reach understanding of the topic.  Video games deal with failure by giving you another chance, so we should give students multiple chances to demonstrate their learning rather than base their entire success or failure on one test.

This talk, as well as many other things I heard this weekend, drove home the point that I really have a lot of room for improvement in my teaching.  With the new curriculum standards, the focus is turning from students’ ability to get the right answer to students’ ability to reason and understand concepts.  I’ve always treated reasoning and understanding as an ideal goal in my teaching, but in the past, students have still been able to get by in my class by memorizing and getting the right answer.  The new standards and the new curriculum are forcing me to bring my actual teaching in line with those ideals.  It’s difficult, and I’m having to do a lot of things differently from what I’ve done before, but it’s exciting too.

One of my favorite books is Ernest Cline’s Ready Player One.  This 2011 novel is set in a dystopian 2040s, in which society is falling apart, and everyone escapes from their reality in a giant virtual-reality video game called OASIS that had grown over the decades into a social network, an operating system, and so much more.  (Imagine if World of Warcraft and Facebook had a baby.)  The OASIS’ late creator, who was born in 1972, was obsessed with the popular culture of his childhood, and he hid a series of very difficult puzzles in the OASIS, essentially a treasure hunt, that will lead to fame and fortune for whomever solves them first.  Wade, the protagonist who tells the story in first person, is trying outwit a bunch of shady corporate bigwigs, who form the villains of the story.  The puzzles make reference to early video games and 1970s and ’80s music and movies, which is what made it such a fun book to read.  I’m glossing over a lot of the back story; you’ll just have to read it yourself.

At one point, everything looks hopeless, and the corporate bigwigs appear to be winning.  It is in this hour of darkness that Wade says my favorite quote from the entire book: “Like any classic video game, the Hunt had simply reached a new, more difficult level.  A new level often required an entirely new strategy.”

Sometimes, playing video games as a kid, I would get to a level that was really hard, and I would find some way to skip it.  Super Mario Bros. 3 comes to mind, with the cloud, or the warp zone whistle, or the P-Wing that would give me the power to just fly over all the enemies.  My career is at a new level.  The last level got really hard, so I got a new job; this is analogous to using the warp whistle, to escape to a different level.  And this new level is harder.  Welcome to Warp Zone.  There are a lot of new challenges before, and what I did before may not get me through them.  But I’m working on a new strategy.  And every day, I have another chance to try new strategies for the challenges I face in life.  It’s like getting an extra life.

(P.S.  Because I know some of you are curious, here is a link to the same talk that he previously gave at a different conference: http://vimeo.com/113714091)