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 238. The relationship goes both ways.

I saw a coworker the other day. Her children, two of whom had been students of mine in the past, were with her. The oldest one is now in high school. She was in my class three years ago, in that same memorable class as the girl who dropped precalculus, the guy I had no memory of, Protractor Girl, and the friendly guy I saw at the basketball game.

I waved. She waved back.

“Good,” she said. Or something like that; I don’t remember the small talk part word for word. “How are you?”

All I could think of to say was, “I’m really stressed right now.” It’s true. I am really stressed right now. I have a lot of things at home that need fixing. My house is a mess. I have a lot of school responsibilities I’m trying to juggle.

“It’s okay,” she said. “You’ll get through it.”

There are plenty of stories out there about teachers inspiring students. Most people have a favorite teacher who inspired them in a particular way, whether or not this teacher even taught the subject matter that the student in question enjoyed. But, after almost two decades of working in education, I would venture a guess that there are just as many stories of students inspiring teachers. Students and teachers are a significant part of each others’ lives for a time, and the relationship goes both ways. My former student is correct here. I will get through this.

And so will all of you. Happy Easter/Resurrection Day, friends.

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.

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 220. No memory of this kid.

Every year, on the first day of school, I give my students an assignment where they answer some questions about themselves.  It gives me a chance to do some necessary paperwork while they are writing, but it also gives me a chance to learn a little bit about who is in my class that year.

One of the questions I ask is who lives at your house.  That way, I can see if a student has a large family, or if they live with both parents, or if a relative other than Mom or Dad is raising them.  I added to that question two years ago: “If anyone in your house has had me as a teacher, circle their name.”  I looked at my class list that year and saw a few familiar last names, most likely younger siblings of students I had had before, and by that point I had been at the school long enough that I was probably going to be getting younger siblings of students I knew every year until I retired (unless I end up at a different school for whatever reason).  So I added this, just in case there were any students whose siblings I knew but I didn’t notice that they were related.

Students aren’t good at following directions, of course.  I’ve had a few students just see the words “circle their name” underlined, and they circle the names of everyone in their family.  And the reverse happens too; I had one this year name her older sister on that paper and not circle her name even though her sister was in fact in my class.  Whether this was due to not circling the name or just not knowing whether her sister had been in my class and being too lazy to ask, I don’t know.  With this student, it could have gone either way.

Sometimes I can tell right away when a student has an older sibling whom I know.  This year, there is one boy in my class who very much resembles a girl from three years earlier with the same last name, except that he’s a dude and not built like a gymnast.  On the first day of school, I told him a funny story involving his sister and a protractor, which he said he remembered hearing about back when it happened.

One girl this year circled her older brother’s name.  I just assumed it was a mistake.  The name didn’t seem familiar, and it’s a fairly distinct last name that I would have remembered.  I never asked her whether it was a mistake or not.  But about a week ago, a student sitting next to this girl mentioned that she had heard stories about me from an older friend who was in my class last year.  I just kind of chuckled.  The girl who had circled her brother’s name then said, “My brother told me he liked having you as a teacher.”

I made some noncommittal remark, something like “That’s good, I’m glad.”  But that really got me thinking.  Apparently this girl did in fact have a brother who was in my class, and I had no memory of this kid.  I thought maybe he never actually had my class.  Maybe his friends were in my class, so he knew who I was.  Maybe he was a student who liked to hang out in my room after school and do homework, because I’ve had students do that sometimes some years.  Or maybe he came to the club that I sponsor once a week after school.  But surely I would have remembered him if he had actually been in my class.

I got curious a few days ago.  I clicked on the archives of previous years of the student information system and started checking class lists, going back to the first year I was at that school.  And eventually I found him.  The girl was right, and I was wrong.  He was in my class, in 2015-16, my second year at this school.

And I had no memory of this kid.

That was a pretty memorable class, too.  Some of the students I remember the best were in the same class as him, the same period in the same year.  Like Protractor Girl.  And the student who sat a few rows behind me at a Kings game once.  And one of the handful of students who have been consistently in touch with me since they left the school.  And the daughter of a coworker who had a hilarious quote about one lesson that I’ve shared with every class since.  But I don’t remember this kid at all.

I feel bad when I realize that there are former students who I don’t remember.  A few years ago, I wrote (warning: there are a few of you with whom I’ve discussed some of my fiction writing other than what has appeared on this site, and clicking the following link may contain spoilers about the events that inspired that writing) about a particularly memorable experience about forgetting a former student.  But in that case, eleven years had passed in the time since I had had that student, and I had moved.  This time, it was not nearly as long, and I’m still at the same school, with his sister in my class right now.

I just got out the yearbook from his year to see what this kid looked like.  And he wasn’t there.  That made this whole thing look even more creepy at first… but probably not, he was probably just absent on picture day.  I found his picture in the yearbook for a different year, though.  And he still doesn’t really look familiar, except for the fact that I can see the resemblance to his sister who is in my class right now.  While I was looking through the yearbook, though, I saw so many other names and faces whom I hadn’t thought about in years.

I’m sure I’m not the only teacher who goes through this.  I’m sure it’s perfectly normal, after having 140-150 students every year, that I’m not going to remember every single one.  It just makes me feel bad.

I don’t spend a whole lot of time reading those papers about my students.  Maybe I should get them back out every few months as I get to know the students.  And in the meantime, I’m glad that this student thought I was a good teacher, even though I don’t feel like one since I don’t remember him.

Exit 162. Not the new guy anymore.

I told someone recently that the upcoming school year will be my 18th year teaching (not including 2005-06, when I was traveling for half the year and substituting the other half).  How is that possible?  The students who recently graduated from high school and are starting college this year were newborn babies when I started teaching.  Where did all the time go?

And more importantly, why do I still feel like a new and inexperienced teacher?

Part of the reason is because I haven’t been teaching in the same place for very long.  I haven’t been in any one public school or school district for more than four years.  Every time I have started over, I have felt new again, since students and their parents don’t know me, and I am unfamiliar with the school culture and the curriculum.  I spent seven years at a tiny private school, and that’s kind of a different world, not to mention that there were only nine teachers and many of them had been there for a long time, so I still felt new in some ways after a while.

But I think I’m finally starting to feel like I’m not the new guy anymore.  My school has had a lot of turnover since I was hired in June 2014, with several retirements, several others taking other positions elsewhere in the district, a few moving away for family or financial reasons, and one death.  Even though I’m only going into my fourth year at this school, I think I’ve been there longer than about half the staff, and among the six math teachers, I have been there the second longest, and I am tied for second in terms of how long I have been a full time teacher in the district.

I have started preparing for the upcoming school year, and I have gotten to meet some of my new coworkers.  And the idea of not being new anymore is finally starting to sink in.  I am able to help some of my new coworkers find their way around the school, get the computers to work, and, in the case of math teachers, learn how the curriculum works.  And this really seems to be helping my confidence.  I’m not quite as shy or reticent among my other coworkers as I used to be.  I feel more like I belong, and less like I’m always rubbing people the wrong way.

I have written before that my principal has told me that she could see me being a leader among the teachers.  Maybe she’s right after all.

(By the way, I missed another week on this blog.  Sorry.)

Exit 123. You’re tough.

Since I teach math, I have had many students over the years tell me that I was one of their favorite teachers, despite the fact that they hate math, or they are bad at math (they think), or both.  I know that feeling well, although as a student, math was never the class I hated.

I recently saw a post, on the Facebook group for alumni of the high school I went to, saying that a former physical education teacher and coach had passed away.  I’ll call him Mr. F.  I saw him much the way that the students in my classes whom I described above see me: I hated PE.  I was never very good at running or lifting or any physical activity.  But I loved Mr. F as a teacher, mostly because he was really funny.  Sometimes he would say things completely unexpected out of nowhere.  One time, I told him, quietly, nervously that my stomach hurt and asked if I could use the bathroom before we started running or doing whatever we were doing that day.  He pointed toward the bathroom and said, loudly enough for everyone to hear, “Yeah!  Go take a big sh**!”  I have not stayed in touch with Mr. F, I haven’t seen him since I finished high school, and I don’t know anything about his passing other than someone on this post mentioned cancer.

But when I saw that he passed away, this was not the story I shared on that post.

In the summer of 1991, right after the year I had Mr. F’s PE class, I worked out in the weight room with the football team.  A lot of my friends told me I should play football, mostly because of how I was built.  But I was not an athlete.  I liked to eat too much, and I did not like to run.  But football players were the cool kids, you know how high school stereotypes are, so I worked out with the football team nevertheless.

There was another problem, though: I didn’t really understand football.  I understood the basics, touchdowns, field goals, first downs, and such.  So in addition to working out all summer, I solved this other problem the only way I knew how: I did my research.  I did a lot of reading that summer about football.  I learned about football rules, the roles of the different positions on the field, different types of plays, strategies, and the history of American football.  And when the first day of double practices came, just after my 15th birthday, I was ready.

No, I wasn’t.  Who am I kidding…

I was in the locker room getting ready that morning, and I saw Mr. F.  I had not seen him all summer, and I wasn’t sure if he knew that I was going to try out for football.  He seemed happy to see me, and he asked how I was doing.  I said that I was nervous, and that it looked like practice today was going to be tough.  “But you know what?” he replied.  “You’re tough.”  It really meant something to me that he believed in me, despite the fact that I could never run very fast or do a pull-up in his class the year before.

My football career lasted one day.  I lasted that morning and that afternoon, and I didn’t come back.  I was in way over my head.  I was badly out of shape.  But something positive did come out of that experience in the end.  It took a few months for me to get over the disappointment of not being good enough to play football, of letting down Mr. F and all my friends who encouraged me to play.  But by the time the following football season started, in the fall of 1992, I enjoyed watching football much more than I ever had in the past.  The time I spent learning more about the game helped me enjoy watching it much more, and this has stayed with me to this day.

It’s okay that I couldn’t handle football, and that I wasn’t very fast or strong in Mr. F’s PE class.  Not everyone is an athlete.  But I still found inspiration from Mr. F.

And it’s okay that some of the students in my class did not understand everything I attempted to teach them.  Not everyone is a mathematician.  But my students can still find inspiration from my class.