# Why can’t our students engage in mathematical research?

The Samizdatmath Library

Every Thursday morning for the past 5 years I’ve been meeting with a 4th grade class to work on a variety of “puzzlas” to stretch their mathematical thinking. I pull these puzzles from a variety of sources, which is not hard as I have a library of math materials that I’ve been collecting for over 30 years. Truthfully, I don’t know if I’ve had an original mathematical thought in my life, as my library provides more than enough inspiration to cover me for several lifetimes.

Every once in a while, an interesting puzzle comes across my desk and this puzzle inevitably leads me to start looking at the problem in a bigger way. That is, the specific puzzle leads to a much bigger question. Such was the case of the  problem below:

On the face of it, the problem has a single unintuitive answer which requires constructing a set of floor plans for 4 apartments that no one in their right mind would want to live in. However, it does beg a bigger question: suppose we got rid of the kitchens and bathrooms, and just had to divide up the square into 4 equal sections of 9 tiles each where each shape was congruent?

I started with the simplest solution possible: 4 squares of 9 squares each. I then started by moving one tile over from each of the squares:

Moving a single tile, as is done to create tesselations, creates new solutions to this puzzle.

I’ve been working on this puzzle for the past 8 months and have come up with over 40 different solutions, but my fear is that I’ve missed out on some. Here’s what the results of my research looks like:

….so here’s where you come in, dear teacher: visit my online store and you can download this activity for free! Print it up, try it out with your kids and let’s see if they can come up with variations that I have yet to discover. I’ll publish them here and then submit the results to some mathematics journals to see if we can get it published. Let’s get kids involved in making mathematical discoveries and show them that far from being a dead subject where everything is known, mathematics is alive and quite well.

# If Martha Stewart taught math…

Lovely place for a school, no?

…she would definitely do this activity.

It was back in the late 90’s (remember them?), and I was working in a school located in a nondescript part of Brooklyn. Whereas most schools seem to be located “in” a neighborhood, this one defied classification: to the south was Green-Wood Cemetery, 248 bucolic acres of greenery which holds the final remains of everyone from the composer Leonard Bernstein to the notorious artist Jean-Michel Basquiat. The north side of the building overlooked the Prospect Expressway, a 6 lane highway which funneled traffic into central Brooklyn; to the east and west, the school was bounded by auto repair facilities and a garage where old bed springs were recycled. Depending on which corner of the building you were standing, you could be in four different neighborhoods.

I was teaching 8th grade, 5 classes, 30 kids per class: they were 75% Hispanic, with families from all over Central and South America, as well as various Caribbean islands. The rest were immigrant families of Asian and West African heritage. Their only commonality was that they were all desperately poor and came from semi-intact homes. One of my favorite students, a very cheerful 12 year old kid named Angel, disappeared for 6 weeks after he was put into foster care when a neighbor turned in his mother for leaving him home alone while she went shopping down the street. He returned to my class, and I continued to follow-him as he delivered food in my neighborhood while attending high school and college. I always gave him big tips, even when the kitchen got my order wrong.

This was not the most motivated group of students and they were several years below grade level in mathematics. Each day I committed myself to creating an activity that would draw students in, to make learning mathematics “irresistible.” I was thinking about my introductory lessons about geometry and started playing around with the idea of something in between a compass and a protractor: the compass allowed one to move around in a circular motion, creating angles of various sizes, while a disk would mimic the measurements found on a protractor:

The design couldn’t be simple enough: 2 arrows and a circular disk, a couple of holes and a paper fastener. I introduced the lesson by explaining that mathematics is about classifying and measuring. We discussed how there are different types of numbers (odd, even, primes, composites), shapes (triangles, quadrilaterals, pentagons, etc.) and operations (addition, subtraction, multiplication and division.) We also discussed all the different ways you can measure different properties of a shape, including length and volume.

With this “preview” finished, I give my students free reign to move the arrows around, and try out the device to measure and sketch angles they find around the room, as well as on the hallway. I keep a stack of old magazines in my classroom (which is why the librarian is my best bud) where they can look for pictures of things with interesting angles. My students record these angles using a ruler to make the rays. At the end of 20 minutes, we have a pretty good collection.

I then take the pictures and sort them into three groups: those with acute angles, those with obtuse angles and those with right angles. I intentionally leave out straight, reflex and zero angles because they are not “typical” angles. I ask my students to describe the difference between the three types of angles, and as they do so, I have them start labeling their “angle finders” with the angles and their names. One of the things you’ll notice in the photo above is that “right” angles are labeled twice, both as 90˚ and 270.˚ I do this because I want to reinforce the idea that a “right” angle has nothing to do with the direction an angle faces, but with the degrees between the two rays.

Incidentally, the term “right angle” is derived from the practice of carpenters to build everything from door frames to cabinets using 90˚ angles; since this angle was so common in their work, they referred to it as a “right angle.” This is an important linguistic distinction to make, as our students are too often tripped up by the difference between the vernacular and technical language of mathematics.

As we continue to classify angles, I also have students put in the “landmark” angle measurements, which includes 90˚, 180˚, 270˚ and 0˚/360˚. It is important that students “see” that there are two types of uncommon yet still important angles: that is, that a straight line can be classified as a 180˚ angle and that a single ray can also be thought of as the overlap of two rays pointing in the same direction, creating either a 0˚ or 360˚ angle. Even those these are “exceptions,” I find it critical to point these out, as mathematics is filled with all kinds of “exceptions” and these “exceptions” are no exception.

There are many other activities that go along with this little gadget (you can purchase the template for this device as well as a set of clue cards that incorporate problem solving into this activity), and it’s been enormously helpful in keeping my students motivated and proficient in their understanding of geometry. Check it out!

# Legions of Followers

Robert, what’s a “follower?”

A follower? Followers are the “sheeple” who mindlessly flock to whatever shiny object happens to appear in front of their noses. They don’t know good from bad, which is what makes them followers. They have no minds of their own, have terrible taste, and wouldn’t know a Kandinsky from a Klee if it was signed and dated for them with an accompanying letter of provenance.

Robert, I notice that you have over 100 followers on TeachersPayTeachers; isn’t that great!

Oh, are you talking about my followers? They are completely different kinds of followers. No, my followers are the ones who are smart enough to recognize that I’m doing something interesting and unique, that my work outshines all the others, and that they want to encourage me to do more of what I do.

There’s someone on TpT selling activities that are based on the characters from Dr. Seuss books. She has over 9,000 followers.

Yeah, but her followers are idiots.

How do you communicate with your followers?

Communicate with them? What the f•ck are you talking about?

You can send them a message, don’t you know that?

I can?

Robert, are you serious? Don’t you know you can send them a message once a month! You have to reach out to them and tell them how wonderful your store is….

But it’s not that wonderful; it’s just stuff I put together for my teachers, and I figure people will like it enough to buy it. I mean, I love my followers, each and every one of them, and I’m sure someday I will repay their love and devotion with something. I just don’t know what that is, yet.

A word of thanks would probably be a good start…

Oh, yes, of course! Dear Followers: Thanks!

# “What’s a ‘product?'”

I was helping Robert pin some of his activities, when I noticed that despite the fact that he seems to have plenty of time to play concertina and cook lamb chops, he doesn’t seem to post a lot of products on his TeachersPayTeachers store. Okay, 93 products is not a bad number, but the guy has been at this for months and months. You would think he would have hundreds of them, seeing how much time he devotes to other things, like memorizing poetry (if I have to hear him recite “Drowning” by Grace Paley one more time, I think I’ll put my head in the bathtub and fill it up) and leyning Torah.

“Robert, you’ve been doing this for what seems like a long time; shouldn’t you have more products in your store?”

“What’s a “product,” Emily?”

“What do you mean, what’s a product? You know, those materials you publish on TeachersPayTeachers. PRODUCTS!”

“They’re not ‘products’ – what are you talking about? It’s not like I manufacture anything.”

“I know, but people buy them and use them, so they’re products, right?”

“Um, no, because I don’t have a factory, and its not like I have employees who build this stuff and then ship it out. They’re not really products – they’re like books and music and art. They’re something else, but they’re not ‘products.'”

“So you think you’re more like a writer or a musician or an artist? May I just roll my eyes now?”

“No, you may not roll your eyes, and yes, developing curriculum materials is an art, not a manufacturing process. So stop calling them ‘products,’ will you?”

“Robert, if you’re going to make some money, you have to have more products posted in your TpT store!”

“What’s a ‘product?'”

# Robert’s Stochastic Month: March Madness on TeacherPayTeachers

I’m always bugging Robert to try to earn more money on TeachersPayTeachers, primarily by making his products a little “cuter,” as well as just making more stuff. I know he loves to cook, and my mom thinks he’s quite handy in the kitchen, but this OBSESSION WITH LAMB CHOPS HAS GOT TO STOP!

I’ve asked Robert about how much money he makes each month on TeachersPayTeachers, and the only concrete response I’ve gotten is reference to an old Jewish joke about an old man who gets mowed down by a car and, as he is lifted into the ambulance, is asked by the EMT, “sir, are you comfortable?” The man replies, “Comfortable, eh, I make a living.”

I don’t understand this joke. In fact, I find it kind of offensive: this old man gets run down in the street, and he’s probably going to die and all he could do is make this comment? If you can explain why this is funny, please tell me.

Robert wouldn’t tell me the exact state of his financial activity on TpT, but he did offer me this: a graph showing what the sales during one month looked like. Here it is:

The question is, did Robert have a lousy month on TpT, a good month on TpT or a great month on TpT? I don’t know what to make of it: it looks like he had some really good days and some really lousy days. Robert did reveal that this was the best month he’s ever had, but he only started taking TpT seriously during last summer, and since he’s so busy doing things like cooking lamb chops, writing blog posts that decry charter schools, the common core AND Go Math!, while also running his various math programs around the city (not to mention playing that horrible concertina), I guess he’s pretty happy where he is, wherever that happens to be, because he won’t tell me how much money that is!

What Robert will say is that he likes selling on TpT because the sales graphs are very interesting. When he looks at the graph above, he sees something called “stochastic variability,” which means that his sales are very unpredictable. Some days he does really well, and other days it seem like his sales collapse completely. He says that one of the findings he find most interesting is that instead of having steady sales over the course of the month, there are “highs” followed by “lows” and that the highest highs are also followed by the lowest lows. I don’t know if that’s good or bad, and while other people might find this vexing, Robert assured me that this is pretty much how life works. In fact, if the graph showed his sales going up steadily, he would be very worried.

This is known as the “Bernie Madoff property” – you probably all know how Bernard Madoff stole billions of dollars from thousands of clients by doing something very simple (and statistically impossible, according to Robert, who seems to know these things): he showed that their investments were growing at a very steady rate of about 8% each year. This doesn’t seem like a lot, but when you compare to the 1 – 2% that most banks offer, it appears to be astonishing. Of course, it was astonishing, because it wasn’t real. See his graph below?

Basically, if it looks too good to be true, it is: if you’re hoping your sales go up at a steady rate each day, or even stay steady for more than a day, you’re definitely deluding yourself. The world just doesn’t work that way, says Robert (and a whole lot of statisticians – actually, every statistician who is really a statistician.)

I asked Robert if he get worried when he has “bad days” on TpT. “That’s a good question,” he told me, “which reminds me of another joke,” and believe me, I’m not going to repeat this one.

Good luck to all of you and go with the stochastic flow.

# “No, I am NOT an ‘entrepreneur…'”

I was eating dinner with Robert & my mom, and they were  talking about a new project he was working on, which was advising a school in the Bronx on their math program.

“So, Robert, what does it feel like to be an entrepreneur?” I asked.

Robert almost choked on his celeriac schnitzel.

“A what?” he sputtered.

“An entrepreneur: you know, one of those people who sets up businesses and they’re really successful and independent and make lots of money?”

“Yes, only all but three of those things you named are true of me: I’m not all that successful, I am constantly at the beck and call of whomever I’m working for, and do you see a Gulfstream anywhere on this block waiting to jet me off to Tahiti?”

“Yeah, but you’ll make it someday, don’t you think?”

“Um, Emily, I don’t know if you’ve noticed, but I’m 54, count ’em, 54 years old. Unless there is a “genius grant” for people whose primary accomplishment is knowing the difference between a vinculum and a vowel, I doubt I’m going to be considered an entrepreneur anytime soon.”

That took care of that. But I have to say, I do admire what Robert does, because I think it’s somewhat important and he’s into it, and hey, if your goal is to do more than build some sort of app that allows people to crush candy or fling birds at pigs, then maybe you’re not really an entrepreneur. Or maybe he is, but doesn’t want to admit it.

You’ll never confuse Robert for this guy; for one thing, this guy’s got much better hair!

What is an entrepreneur, anyway? I hear about them all the time: they’re the ones that came up with the big idea, developed it into something popular and then sold it for a gazillion dollars, like Instagram or WhatsApp. And then I look at Robert: he’s got this traveling math exhibit that goes around to schools and brings families together doing really neat math. He also does these workshops with snarky names like, “Fractions: You’re Teaching It Wrong.”  And then there’s this “store” he set up on TeachersPayTeachers where he sells the materials he develops for his teachers and students. And, if that’s not enough, he’s got this idea for computer chips that you can implant in the brain and change people’s personalities for the better – seriously, he actually built a website for it! Do I have to go on?

Well, you decide: Robert – entrepreneur or just some strange man making a life in Brooklyn? I’ll let you figure it out, because I sure can’t.

# What happens when Robert gets up on the wrong side of the bed….

What everybody knows….

# In Defense of Cutesy…

I DON’T DO CUTESY.

In addition to my duties as Robert’s ghost-blogger, I also spend about 15 minutes each day pinning his products onto various Pinterest boards. It’s pretty mindless work, and it gives me a chance to wander around the TeachersPayTeachers store and see what all the other sellers are up to, which gives me ample time to advise Robert on what he should be doing to improve sales and generally beef up his store. Here’s what the conversation usually sounds like:

Robert, you need to make your things more cutesy.

I don’t do cutesy.

But if you made your things cuter, you’d sell a lot more of them.

I DON’T do cutesy.

But don’t you understand, if you did, then…

Emily, whose the boss? Who signs your paycheck?

You haven’t payed me yet…

You get the point: Robert is one stubborn nut, and asking him to budge is a waste of my time and wits. With this being among the many points of contention between us, Robert decided I should use this blog post to defend the cause of cutesy.

What “cutesy” does to Robert…

Before I can defend anything, I decided I had to get to the root of Robert’s antagonism towards all that is cute. Robert, being the person he is (that is, a white, 54 year old male), did not have much to say about subject. I pressed him over and over again, and the closest thing I could get to a detailed reason was “I don’t like cutesy. It bothers me.”

Robert’s idea of “cute”

My only familiarity with psychology is based on what I remember from Saturday morning cartoons, but I think I have some inkling of why Robert maintains an aversion to cute. As I said before, Robert is a man, and while he is far from being snobbish (good heavens, he buys \$4 wine by the case and serves it at dinner parties!), he is also somewhat insecure about his masculinity. Yes, he does “manly” things like participating in endurance swims in the Hudson River and constantly reminds me not to show up at a knife fight with a curling iron, but at the same time, he was raised around 2 sisters and a mom, and chose to work in a profession that is female dominated. Add to the fact that he has 2 daughters and lives with my mom and her two daughters, and you can see that he clearly needs an outlet to assert his manliness.

Robert maintains that “cutesy” is discriminatory and is often used by teachers to “dress-up” products that are “thin” on content. He finds them overly-feminine, and, that by portraying children as little adorable angels with big eyes and cutesy smiles, these products pander to the female teachers’ tastes, rather than elevating that of the students’. Of course, this is Robert’s philosophy, which he admits is based on a freshman year class called “Introduction to Feminist Semiotics” that he took at Brown University back in….wait for it…. 1977!

Robert’s idea of “cutesy”

At the same time, Robert thinks all this discussion of cutesy is designed to drive him batshit crazy: he loves the visual jokes he puts into his product. His latest prank was to place a photo of the U.S.  Secretary of Education into the top left corner of one of the product covers, identifying it as “Arne Duncan Approved.” What you have to know is that the only person Robert detests more than Michelle Rhee is… Arne Duncan! And so it goes…

You want my advice? I personally think cutesy has some kind of place in the classroom. How else could we get girls to become interested in math? What do you think: do you like cutesy? Do you use it to motivate your students?

Robert replies: Really, Emily? Must I read this? Can you just tell my fan(s) about the excellent quesadillas with mango salsa I made last night?

# Got Math Anxiety? Welcome to the Club….

Having gone to a high performing math and science high school (not to brag), I’ve taken many a math exam.

The ghost’s humble alma mater..

I’ve endured the PSATs, SATs, math APs, and teacher created assessments that lasted me through calculus. The subject itself was never my strong suit, but on top of everything else, I panic when it comes down to the wire. Heck, I usually draw a complete blank when I take math exams, not because I don’t study, but because the fear of what may be on the exams just freaks me out! Who knows, maybe if I didn’t have that problem I would have aced all my math classes and exams (poor me, I had to settle for “most.”) I am confident that I could have done significantly better had I learned some practical skills to cope with my fear and anxiety, which would have no doubt been more effective than the exhortation to “work harder.”

Robert, who hoped to start a PhD program this fall, took the GREs a few weeks back. He did a few practice tests to prepare and, even with numerous distractions in the background (like earning a living and feeding himself), it appeared he was going to score pretty well, despite not having taken an actual math course in over 20 years. In the days leading up to the exam, Robert was almost unbearably nonchalant and confident (he’s been in the math game for 30 years, after all), yet at the same time he dismissed GRE prep as a sham and convinced himself that he could “outsmart” the test by inhibiting his anxiety, getting a good night’s sleep, and drinking a giga-dose of caffeine the next morning.

When he came home after taking the real thing, he reported the feeling of “shock and awe” at both the length and complexity of the exam; the math section was much more difficult than he was led to believe and with the clock running, he left several questions blank. To make things worse, he actually started to panic when an additional math section appeared on his screen.

Time’s Passing; Will Robert?

Oddly enough, for a person who spends most of his time training children and teachers to be patient and perseverant when tackling difficult math problems, Robert suffered a minor “freak out” during the final math section of the exam. In the end, he posted spectacular scores on the writing and verbal analysis section (98% percentile), while earning what he considered a “mediocre” rating on the math tasks (that is, if you believe that scoring better than 79% of the other test takers constitutes mediocrity.)

There are only two possible explanations to this outcome: Robert is not as “smart” s we all thought (at least, when it comes to GRE level mathematics) or he really did suffer a “math assessment freakout,” which would really be a hoot, considering that he actually teaches a full day workshop about the sources of math anxiety and how to counter it.

The opening activity in this workshop is “googling” the phrase “math anxiety” (spoiler alert: this search will return a lot of hits.) While many teachers, particularly in high school and college level mathematics courses, dismiss anxiety as an “excuse” for poor performance, there is more than enough evidence to prove that the phenomenon is real and problematic.

So how to minimize the issue…Robert counsels ease off on the pressure, for one. By doing this, the emphasis should be on real learning and understanding rather than memorization or getting the “right answer in the shortest amount of time.” In point of fact, Robert points out,  this is  how “real” mathematicians operate. It’s not like they go into work each day, sit down at a computer and solve 35 problems on linear algebra in 30 minutes. From the dawn of education, mathematics has been seen more as philosophy than a blood sport. We all know that Archimedes did not figure out whether the king’s crown was actually made of solid gold in 5 minutes; it took a warm bath to develop the insight that led to his “eureka!” moment. Would that we could all have the luxury of warm baths to meditate on implicit differentiation and the Taylor expansion…

Besides approaching mathematics at a measured pace, Robert advocates cooperative learning between students as a huge asset—something that, unfortunately, I didn’t come to learn until later in high school. In my experience, it was often the case that my classmates could explain things better to me than teachers, plain and simply because they were more familiar with my language. And finally—and I whole-heartedly agree with this point, as partial credit has always been my best friend—praise students for their efforts, rather than “smarts”.  It shows that you believe in their capacity to grow and develop as budding mathematicians, rather than fixing their place as “good” or “struggling.”

So take a breather, go through problems and concepts step-by-step, and see what students are having trouble with along the way. Build a solid foundation for problem solving instead of diving right in. It’ll make a world of difference.  (Robert’s comment: this is not at all helpful, so I’m striking it out. It’s too reasonable for me, so I’m going to add my own coda.)

Let it be known: it is my opinion (Robert) that mathematics by its very nature is anxiety provoking: let’s accept it as a truth and live with it on those terms. The best we teachers can do is acknowledge it and put into practice techniques that will “turn down the heat.” I actually do a full day workshop on this; it’s not something that you can just read about in a bunch of bullet points and “do.” It requires a major attitude shift, that involves altering our language, expectations and even our very role as professionals.

You can read about Robert’s freak-out and recovery from the GREs at his “other” blog, which is slightly more off-kilter than this.