I Hate Math!

If there’s any subject a student fears above all other subjects, it’s Mathematics. And the “I hate Math” chorus is sung until the person dies. The next few articles will broadly explore this phenomenon through a series of short stories.

The Local Bar

I headed to the local bar last night to get some reprieve from all the snow, slush, and cabin fever (what better way to get rid of cabin fever than to go from one indoor environment to another, right?). As I went to open the door, the bouncer stopped me and said, “ID.”. I glanced at him a little bewildered thinking of saying, “You do realize that I’m closer to forty-five than I am to twenty-one, right?”, but I chose to bite my tongue given the arithmetic involved, the sameness in size of his biceps to my thighs, and the fact that the other thought that went through my mind was, “Still got it!”. I pulled out my wallet.

Into the bar I went, pulled up a seat ordered nachos and a beer. The barfly in her mid-40s next to me introduced herself, “Hi, I’m Ginny.”.

“Hi, I’m Manan.”
“Monad?”
“No, no, Manan.”
“Spell it?”
“M-A-N-A-N”
“Oh! Manan! Is that an ethnic name?”
“As ethnic as Ginny is.”

The nachos arrived and my beer with them.

“Oh, you got the nachos?”, Ginny said.
“Apparently so.”
“What kind are they? Chicken or beef?”
“Neither. Just some cheese, lettuce, jalapeños, tomatoes, the usual vegetarian stuff.”
“Are you a vegetarian?”
“I’m really more a vegetarian who cheats.”
“Haha! So you’re not full of yourself. That’s nice. Are you catching the train?”
“No, I live nearby.”
“Oh, I’m sorry. Usually whenever I see someone nice in here, they’re catching a train.” Thanks?

Then Ginny’s drink came, and she was distracted. As I merrily ate my nachos, drank my beer, and watched curling (Sochi, Winter Olympics) trying to figure out what in the world makes this a sport, some not-so-nice-looking men walked in. They grabbed their seats and ordered their beers, just like everyone else. And so the night went.

As I was finishing my nachos and debating if it was a good time to make the slush-ridden, two-block trek back home, a youngish couple, each probably closer to twenty-one than to thirty, walked in. I wondered if they got carded by Tommy the Tank. From their appearance, mannerisms, and the general awkward distance they kept from each other, it looked like a first or second date. Probably a first date since Valentine’s Day was just this past Friday, so they were probably both tiptoeing around that minefield.

He ordered a “vodka and a soda” — yep, probably twenty-one — for himself and a beer for his date. As he went to grab a chair next to me, the barfly’s friend, two chairs over from me, yelled, “Seat’s taken!”. He apologized and searched for another chair. Since people tend to choose their seats leaving a one seat gap between themselves and the person who was already sitting at the bar, there were not two, empty adjacent seats. So the couple remained standing behind me.

They were definitely on a first date and he was torpedoing any chance for a second date. “I’m not really a fan of beer, it’s too heavy and it takes too many beers for me to feel the effects.”, he said. Buddy, she’s holding a beer that you just handed to her and you’re holding something that will apparently get you to “feel the effects” much sooner. After a few inane comments, the topic of school came up. Most certainly, twenty-one.

“You know, I was always a creative person. I always liked English, and History, and the Arts. I hated Math and Science.”, he started.
“I loved Math and Science. I was always really good at it.”, she replied.
This is a train wreck, there’s no second date.

He ranted on, obviously feeling the effects of the vodka and soda, “Well, for me, I remember Algebra with Ms. Doodledee [name obviously altered and made-up]. It was terrible. I didn’t understand anything that was going on. We just had to keep doing these problems over and over and over again. Then when I got to college, I had to take Algebra again because I didn’t place out because of my SAT scores. And that course sucked too.”. At this point, it was time to head out. I’d heard this story a thousand times.

Did you see the Math?

There were math concepts all throughout this story. Did you catch them?

1. How old could I be, if I am closer to forty-five than to twenty-one? I am at least thirty-four years old, if we stick to the integers.
2. “Monad” is a technical term from Category Theory. Here’s a good definition of Monad from Wolfram Alpha
3. Ginny, the barfly, uses elementary logical operations in her question, “chicken or beef?”. The “or” is not the logical “OR”, but rather exclusive-or (“XOR”) to mean “one or the other, but not both” (I know I’m using “or” in the definition of exclusive-or — more about this later.).
4. Ginny, the barfly, is informally using statistics, coupled with set theory, in her racial profiling question, “Is that an ethnic name?”. Though she probably meant no harm, she likely had in her brain an index of “ethnic names” and “non-ethnic names” — whatever that may mean. Those two, mutually exclusive (more set theory) categories, are likely subdivided into further mutually exclusive subsets (more set theory). She probably was unable to find “Manan” in her brain and (intuitively) recognizing that her lists were likely incomplete, she asked if the name was ethnic. At least her question, is less racially charged, than “What foreign country is that name from?”.
5. I’m using conditional probability (informally) to make a statistical conclusion about the couple’s age group (which later stops being statistical since I end up being 100% convinced of the couple’s age — at least the guy’s age, and then using some vague understandings of social norms and customs (another type of selection bias), I’ll deduce that the gal in the date was the guy’s age to within a year, but not less than twenty-one — boundary conditions!).
6. The seating problem of having enough seats to seat \(N\) people, yet \(N\) seats are rarely filled because of gaps that are left, is a nice combinatorial problem. These problems usually begin with “How many ways can you arrange \(N\) objects under the following constraints …”
7. Joe Not-So-Smooth’s comments about the rate at which he will feel inebriated should he have a beer vs a “vodka and soda” is really what an inequality is. It’s an ordering of objects under an appropriate metric. In this case, “beer \(\lt\) vodka and soda” under the metric of “which will get me drunk faster?” — vodka and soda is better at that.

A Set Of Seven Math-Misery-Causing Sins

Joe Not-So-Smooth’s math misery tale is a tale I’ve heard often and regularly. So often and regular that it is pretty convincing that this is a systemic problem. How is it that practically everywhere one goes and when the topic of math in school comes up, Joe’s reaction is a common one? For this, we have to take a good, long, and hard look at what happens. I’ll give broad “chapter topics” on this here.

1. Worksheets, homework assignments, and drills are fine — so long as the teacher knows how to use these education objects. Too often, I’ve seen the worksheet and / or drill as a substitute for conceptual understanding. Yes, students have to know their mechanics and practice makes perfect, but mathematics is not just about the mechanics! A mechanically sound student is still weak since that student has no idea about the context. And it’s not just about the concepts either. An overemphasis on concepts still leaves the student weak, just in a different way. Conceptually knowing how to solve a problem and then actually solving it are two different things. Torturing students into endless drills without context, is a first sin in teaching mathematics.
2. The formalities come too early. Exclusive-Or (XOR), for example, can be formally defined as follows: If \(1\) is True, and \(0\) is False then \(A \mbox{ XOR } B = 1\) in the following cases: \((A = 1, B = 0)\), or (haha) \((A = 0, B = 1)\). In all other cases, \(A \mbox{ XOR }B = 0\). This is incomprehensible if this is the first time someone is exposed to logic (and this is the mild version, I’ve presented). This is where it is important to ease into the formalities. Mathematics is a new language and there are new concepts that students are being exposed to; they have to adjust to both of these things simultaneously. First solidify the concept. Pick simple, stupid things to demonstrate what XOR is, for example. Then, translate that into the mathematical symbolism. Once that’s squared away, feel free to work with the symbolism. Too many classes, too many teachers, push the symbolism too early. This detracts students from mathematics, because now, not only are there new concepts to understand, there is a new spoken language, and a new written language. Too many moving parts all at once! Sin number two.
3. Teachers need to figure out their instructional style and ought to be reasonably versed in other methods of knowledge delivery. There are a lot of instructional styles and teaching methods: direct instruction (the whipping boy of the ed reformers), project-based learning, inquiry-based learning, yadda yadda yadda. We’ve all sat through many, many classes, and many, many lectures. All of us have at least one story of a professor who gave great lectures and all of us have plenty of stories of terrible classes. This too comes down to, “the teacher had better know what he/she is doing”. There are many delivery mechanisms and each requires a certain level of responsibility on the teacher’s part and on the student’s part. Each delivery mechanism can be effective. Mathematics isn’t a subject that “either you get it or you don’t”. Everyone gets it and can get it. Sticking to how the teacher wants to teach is sin number three.
4. The student has a responsibility as well. It’s easy to rail against the education system especially when practically all of us have had virtually the same experience of “ugh, school.”. If the student doesn’t want to practice, won’t practice, doesn’t want to think, and believes all the time that he/she isn’t learning because “the teacher didn’t explain it well”, then that’s sin number four and it’s on the student.
5. This is mathematics we’re talking about, not mysticism. I remember when I was just a wee boy and in one of my math classes, the teacher wrote \(\frac{1}{a} + \frac{1}{b} = \) and asked what that equals. I blurted out \(\frac{2}{a + b}\). The teacher responded with, “This is mathematics, not mysticism.”. Cold? Harsh? Maybe. But he was right. This isn’t some form of black magic. What my math teacher did afterwards though was more helpful. He asked if I believed that \(\frac{1}{2} + \frac{1}{2} = \frac{2}{4}\). As I clearly did not believe that, he asked why I would believe that \(\frac{1}{a} + \frac{1}{b} = \frac{2}{a+b}\). And so, things became clear. Mathematics is logical, don’t teach it as if it weren’t. Students will make mistakes, it’s natural. In fact, the reply of \(\frac{2}{a+b}\) is a natural mistake because we’re applying one pattern for addition to another object where it doesn’t apply. This is where practically all the “common mistakes” come from. Ending the conversation with “That’s just the way it is.” is sin number five.
6. Mathematics isn’t about a gigantic set of rules and algorithms that have to be memorized. We memorize plenty of things. We have the alphabet memorized. We have the route to work / school / friends’ homes memorized. But there’s a difference in memorizing the quadratic formula the way an actor would have to memorize it if they were playing a movie mathematician versus the way a student has to “memorize” it. A student will, with enough (proper) practice (see sin number one and four) simply remember the formula. And if it is a standard formula it’s easily retrievable from a book, the interwebs, via theory, etc. Having students memorize formula on top of formula is sin number six.
7. Grading and grading policies. This is less a sin of the teacher and / or of the student than it is a sin of the system in place. To be realistic, six quizzes, eight homework packets, and three exams, all glommed together into one number via some “weighted average” is not a measure of learning. It never was, and it never will be. Learning is time-dependent. An average, the way it is computed when giving a grade disregards the learning that happens over time. An average is not a “max” nor is it a “recent data point”. How much a student knows at the end of a semester is more meaningful than how well they did on exam 1, homework 2, and quiz 1. Yet this is the traditional system. Why do we have it? What’s wrong with letting a student retake an exam? Because it’s simple, it’s lazy, it serves its purpose from a macro perspective of labeling and sorting people. So a bunch of people are mislabeled as B-caliber students when they should be A-caliber. From a macro perspective, who cares? There are enough students being correctly labeled as A students for our society, in general, to benefit. So depending on the perspective we want to take, how we grade our students is either a sin (if we take the individual’s perspective) or imperfect but still reasonably good way for sorting people (if we take the macro perspective). If you are a student, recognize this disconnect about what you think you are versus what your grade says you are. As a teacher, try to be as fair as possible since you are educating individuals. As an admin, think real hard about what makes sense from the different worlds you’re caught in (the Board sitting above you, the teachers, students, and parents whom you serve, and all the stupid metrics that you have to report). And I have nothing to stay about the education system. It does what it does and it’ll never do whatever it does correctly. For macro systems, an average is a nice, clean measure.