By Samantha Gougher

There is quite a bit I could write about my past four years at Council Rock High School North. During my time here, I’ve published several Indianite pieces satirizing, criticizing, and even blaming this place for the problems I’ve faced. I’ve bemoaned the dress code, suggested that North ought to better respect students’ days off for mental and physical health, and poked fun at the cages recently installed on the stairwells.

I even joined with three other students to publish an open letter to the superintendent himself in reference to an incidence of student intolerance this year. From my snarky comments in real life to my short-lived deadpan Indian-NOT column in the paper, I certainly lead an audience to assume that, come graduation, I’ll be the first one to burst out of this school’s front doors.

Well, in the past year, I’ve actually cultivated a strange fondness for Council Rock North. I suppose that it’s just something that happens during senior year, an anticipatory sense of pre-nostalgia that motivates a student to pay a different sort of attention to her surroundings. For my first three years at North, I had viewed this place as a mental and physical prison—yes, it’s hyperbolic, but it’s always been difficult for me to just be in one place. And although many of those feelings remain on the back-burner of my brain, they are now joined by an odd appreciation for the walls that had seemingly trapped me inside.

Council Rock North, by itself, is not the most architecturally astounding building out there. We’ve got a pool, which beats South, but we’re also older, the original Council Rock High School. Our walls are white and our floors are tiled. There are windows in maybe 10 percent of our classrooms. There’s a framed poster of the periodic table in the third floor East hallway because apparently we don’t have enough already (that’s sarcasm—I’m pretty sure one in each science classroom is enough). The style and influence of the honeycomb windows remain a mystery to me, and probably always will.

What makes North special, what I know I will miss, is not even North itself—it’s the people working and living inside of it.

​North is an exceptional school, and not for the reasons one would expect.  Certainly, we reside in a privileged area, we have many abundant resources, and we rank high on some Newsweek list that I keep seeing on the CR website. But what I care about, what really matters to me as a graduating senior, is the school community that those resources and privileges have allowed us to create.
          
The bond between a North student and another North student, or a student and a teacher, is something that I will never be able to experience again. We all experience degrees of unspoken mutual struggle, benefit, or amusement inside our little bubble, to the extent of developing our own vernacular, expectations, and even memes.

To North kids, it’s totally normal to refer to the time with only the minutes and not the hour, because it only really matters when the bell is going to ring. To North kids, the morning song is either a hit or a miss, but always a topic of conversation. To North kids, certain teachers that we’ve all had are hilarious for one reason or another, and to teachers, there are certain class clowns that we all reluctantly enjoy.

These moments of community are what I’ll miss most about North. Of course my college will probably work the same way, and then my job, and my family, but nothing will ever mimic the exact strange world that I’ve lived in for the past four years.

Whether you’re a part of the music program, the athletic sector, the honors societies, or whatever other activities you stay past 3:00 to enjoy, you know dozens of inside jokes that will last years in your camera roll and head. But eventually, this stuff will go away, and that’s kind of sad. Back up your photos and hold onto your weird drawings and lame poetry, guys. Someday you’ll be happy to have them.

And lastly, despite all its shortcomings, don’t forget that North can have its moments of beauty—little snapshots of wonder, or amusement, or mutual understanding, ingrained inside your brain because they managed to break through the monotony somehow. The four seasons’ shining through the honeycomb windows at 7 AM; your friends’ doing dumb things in the hallways during after school activities; the journey of finding yourself, your passions, and your place within this school community. Those are the things you’ll want to remember.

Included with this article are several images I have captured throughout my past four years at North—and for that strange, wonderful reason, they have made this place beautiful to me. I encourage you to find those moments for yourself and keep them in your pocket as you journey out into the world.  

It’s been an emotional, mental, and at times even physical roller coaster, North (you know what, maybe it’s for the best that we put cages on the stairs).

Thanks for being beautiful in your own strange little ways. 

By Margaret Zheng

If you read my previous article on the need to change our perspective on mathematical learning (and if you haven't, I suggest doing so before continuing), it may have struck you that I did not provide any concrete suggestions for improving how we learn mathematics.  Rather, I painted rhetorically with a broad brush and sparkly colors, talking of magic and educational ideals and pulling theory out of the psychological hat.  I could say, hey, I'm a child of the abstract.  And isn't that what math is - abstraction?  

Well, yes.  Just as a rainforest is its canopy.  

The thing is, what mathematics publishes itself as -- numbers and symbols and abstract manipulations -- is really only the visible upper surface of a complex, creative, concrete process.  Physical, usually visual, models become your tools to branching into clearer thought.  Once mathematicians have grasped one concept using pictures or 3-D models, they refer endlessly to the concrete tools to extend their concepts and construct more mathematics, so much so that mathematician James Gleick remarked, "It's masochism for a mathematician to do without pictures" (link).  Thus came such wonders as graphs, number lines, modeling clay, crayons, markers...and fingers.  

Does the last one surprise, or even disgust, you?  Perhaps you were told that only little kids just beginning to learn their numbers use fingers in math.  Perhaps you were directed by your parent or teacher (or Kumon instructor) to stop counting with fingers and get on to the "real" math of memorizing your addition and subtraction.  

Yet brain research has shown that even beyond basic counting, mathematics - and mathematicians themselves, mental calculators or not - depend on those fingers.  When calculating without finger-counting, the area of our brain that "sees" fingers lights up with activity, as if representations of fingers were still used for arithmetic.  In one study, six-year-olds who were better able to identify without sight which finger was touched by an investigator were more likely to see future mathematical success than those who simply scored high on cognitive tests.  Even finger-savvy college students have been found to have higher calculative skill, and those who improved finger awareness also improved arithmetically (link).

It appears that if you brushed up your piano skills or practiced your typing, you might kick-start your brain into thinking more comfortably as a mathematician.  In fact, Jo Boaler, the educational researcher I mentioned in my previous article, suggests finger coordination activities for classroom use, which can be found on www.youcubed.org/visual-math .  But just as important as finger-representation in the mathematician's brain are visual and spatial processing, which along with many other areas of the brain, "light up" when we reason mathematically.  
 
Clearly, there is no one "math skill" that American culture could identify its imagined "math people" to be born with.  By creating visual tools like diagrams or digital models or even "acting out" mathematics with friends (for according to mathematician Keith Devlin, mathematical ability may evolutionarily be a mere extension of the social calculations humans make), we can make even the seemingly most difficult mathematics intuitive.  
 
The usage of concrete tools to aid understanding of mathematics even applies to the Common Core Standards for Mathematical Practice, which mention graphs, diagrams, and technology under "Make sense of problems and persevere in solving them" and "Use appropriate tools strategically," giving additional motive for us to explore these representations of mathematics in class.  All of us, even those historically troubled with math, have the potential to grow our ability to manage the abstract by bridging from the concrete.  

To give an example of gaining mathematical intuition through modeling, I say that instead of pressuring ourselves to quit finger-counting and start mindlessly memorizing, we should explore the idea of adding 6 + 8 by playing with blocks or dots on a page and finding that 4 + (2 + 8) = 4 + 10 and other simpler calculations give the same result.  The method generalizes to large, "scary" sums like 385 + 218 (that is, (385 + 15) + 203 = 400 + 203) and also to algebra (including "completing the square" problems like x2 + 4x - 1 = (x 2 + 4x + 4) - 5, which can also be done with literal square diagrams).  Might I say that in geometry you also can find areas by moving pieces around, and that calculus is a mere extension?  Everything is connected in mathematics, conceptually and neurologically.  Learning, at its most basic, is association - that is, making connections between what one knows already and what one must now grasp.  

So I have now provided some concrete suggestions for mathematical learning to bridge the gap from the persuasive poetry of my previous article.  Now that summer is to begin, you (students and adults) must assign yourself a mission.  Not "homework."  A mission.  You are to be motivated to engage with math.  
 
Short of poring over textbooks while your family and friends enjoy the beach, go pick up seashells in the sand and marvel at the frequent appearance of the golden ratio, which also appears in the Parthenon and in Renaissance art.  
 
Try the Rubik’s cube (I have yet to do so myself!), play math-related computer games like Dragon Box or even the Creative Mode of Minecraft, learn basic computer programming on Khan Academy, solve mathematical and logical puzzles, read books like The Number Devil (a children’s fantasy fit for all), and exercise your finger coordination on a musical instrument.  Seek out math in your everyday life, or even better, try to avoid it completely.  I bet you can’t!  Mathematics is not a chore of school, rather a never-ending game of the mind, infused in all we do.  
 
The end of the school year calls for reflection on how we have contributed to the learning process and how we can learn better.  Even senior students share this obligation; as a senior friend said to me, do you stop caring about something just because you are leaving it?  We reflect upon ourselves, as individuals and as a cooperative community.  Students, when you next see your math teachers, thank them for helping you grow as a mathematician, and tell them in person or in writing what you would like them to do more in class.  That could be something the teachers already do and should continue or something you want them to start.  Teachers, thank in turn your students for their insights and their energy, and tell them what you want them to do more for their continued education and growth.  
 
Working together, we grasp the abstract ideals we dream of and unite them with the concrete ground.