WyzAnt Wants To Know: The 2 Sigma Problem

We recently passed the 30th anniversary of “the 2 Sigma Problem,” which is the problem of achieving the effectiveness of personalized, one-on-one instruction at a large scale. As a tutor, how do you help multiple students at the same time while retaining the benefits of personalized tutoring?

To me, one of the major benefits of personalized tutoring is that the tutor has the space, time, and flexibility to respond to the student’s needs. If I am tutoring a student in math, we can spend as many sessions as we need on a given topic to make sure the student understands it thoroughly. I can also try a lot of different methods to explain a topic, since not everyone learns the same way. If a spatial or visual learner is having trouble with division, I might bring in a bag of M&Ms and show them physically the process of dividing up a pile into smaller piles. If a student is having trouble understanding probability, I might bring in a set of polyhedral dice and discuss it practically. Those same polyhedral dice might also find a use during discussions of geometric solids, in situations where it is tough to accurately illustrate a 3-D figure on a flat piece of paper.

When working with more than one student at once, I take advantage of the fact that thoroughly understanding a topic often means that you can explain the topic to someone else. When I work with students in math, I focus a lot on the student being able to explain the concepts back to me – so if there are multiple students present, they can try explaining the concepts to each other! Maybe the student for whom the topic has just clicked will have a novel way of explaining it to the other student; a turn of phrase or an analogy that I may not have thought of myself. This is particularly effective when the students are friends outside of class, since they know each other far better than I do and are more likely to understand how the other one thinks or be able to connect the topic to a passion or interest.

Encouraging the students to help each other also helps them to feel as though they “get it,” which increases their confidence later on. I’ve found that a large part of math improvement is simply a question of confidence; if a student doesn’t quite understand the material, it’s going to be that much harder to be consistent. Once the student “gets it,” though, I often see a burst of happiness as they begin to complete problems with little or no help from me. Often when that happens that student will instinctively begin explaining the concept to the other student, subconsciously wanting their friend to feel that same burst of happiness and excitement.

Mathematical Journeys: Zeno’s Paradox

Suppose I place you at one end of a long, empty room. Your task is to get to the door at the other end of the room. Simple, right? But what if I told you that this simple task is actually mathematically impossible?

Think about it – in order to traverse the whole room, you first have to get to the halfway point, right? You’ll have to travel one-half of the way there. And before you can get to that halfway point, you have to travel one-quarter of the way there (halfway to the halfway point). And before you can get to the one-quarter point, you have to travel one-eighth of the way there (halfway to the quarter-way point). Since you have to go half of each distance before you can go the full distance, you’ll never actually get anywhere. The task requires an infinite number of steps, and you can never complete an infinite number of steps since there will always be another one. Furthermore, in order to even start your journey you would need to travel a specific distance, and even the smallest of specific distances can be divided in half, giving you another step before that one. So you’d have to traverse infinity in order to go anywhere.

Make your brain hurt? Don’t be ashamed; it’s supposed to.

In fact, this is a very famous thought experiment called Zeno’s paradox. Zeno’s conclusion was that all motion must be an illusion, since travel over any finite distance can never be either completed or begun.

In practice, of course, this is not really a paradox – at some point your remaining distance will be so small that you cannot practically traverse half of it. Perhaps it would be down to a distance that is smaller than the length of your foot, so you are in effect already standing at both ends. The point of Zeno’s paradox, though, is to illustrate the elusive nature of mathematical infinity.

Infinity tends to cause problems for math students when they expect it to behave like a number. The truth is, infinity is a slippery concept, one that can only really be comprehended obliquely because if you ever try to stare directly at it it will squirm away. The way to understand infinity is to acknowledge that it is a concept that theoretically exists, but that you will never personally see or pin down. Very much like the imaginary number i, infinity has a definition but it cannot be evaluated as a number. This is why we can only talk about a series approaching infinity – we cannot say that it reaches infinity, because infinity cannot ever really be arrived at. Wherever you are on the number line, infinity is always out of your reach.

WWTK: Starting the Year Strong!

WyzAnt Wants To Know: What advice would you give students going back to school so they start the year strong?

This is a great question, and one that I’ve answered before on this blog. In general, I’d say the most important thing for starting the new year strong is starting the new year ORGANIZED. Go back and look through your notebooks from the previous year, but not for content – look at them like a detective. What does your note-taking style say about you? Do you have spiral notebooks stuffed full of handouts with rumpled edges? Are your note pages just solid blocks of hurried scribbles that are all but impossible to read? Did you have to add extra notebooks halfway through the year? And most importantly, how easy is it to find a specific piece of information in one of your notebooks?

Take the opportunity while summer’s still going strong to head to an office supplies store and wander around. Really look at all the organization solutions, and try to imagine yourself using them. Organization is a very personal thing; what works for me won’t necessarily work for you. But once you find that workflow that clicks with you, you’ll be much better prepared for the classes ahead. Here are a few of my favorite tools and tricks concerning the monster of all organization debates: Spiral Notebooks versus Three-Ring Binders.

Ah, the age-old debate – spiral notebooks or three-ring binders? Once again, it all comes down to what you plan to do with them and how you personally take notes. When I was in high school, I liked to use three-ring binders to organize my notes, since I could carry around one large binder that had notes from all of my classes in it. When you’re walking around a high school with only 4 minutes between classes, not needing that extra trip to the locker can make a huge difference. Even more so if, like mine was, your locker is in a block in the basement that is kept locked whenever it’s not a lunch period.

There are down-sides to three-ring binders, though – in particular, my problem with them was always how loud opening and closing the rings was. I never wanted to open them during class, so I was forced to figure out ways to write on the back sides of paper with my wrist craned around the rings, which wasn’t exactly comfortable. I have since figured out a solution to this problem, though – bring a clipboard with your looseleaf paper on it, and write on that. Then transfer your notes to the binder at the end of each class. Oddly enough, looking back now, there was a classmate of mine who did exactly that – but I didn’t realize it at the time. I remember thinking it was a bit odd that she was carrying a clipboard around – but odd is fine if it keeps you organized!

Once I got to college, I realized that I didn’t need to carry all of my notes around with me all the time – I frequently only had one academic class per day, and was already carrying around enough as it was. So I switched to spiral notebooks because I could take only what I needed and lighten my load, plus it was easier to keep information from each class separate in my mind when it was separate in my bag. But I always hated having to rest my wrist on the spiral when writing on the back sides of the pages. During my junior year I solved that problem – I found full-size spiral notebooks with the spiral across the top rather than down the side – stenographer style binding but with regular ruling. It was perfect! I could write on both sides of the paper without discomfort.

The only downside to the steno-style spiral notebooks was that they didn’t have pockets. I had one class where the professor liked to load us down with handouts, sending us home each day with a new stack of paper. I never quite figured out how to handle that, and my perfect note-taking system became a bit unruly with a chunk of loose paper stuffed under the front cover of my notebook.

There is a happy medium between both of these methods, though it requires some specialized equipment. Some companies make a hybrid system that uses a series of specially-shaped rings to create a notebook with re-arrangeable pages. The most famous of these systems is Circa, designed by Levenger, but you can get knock-offs at some office supply stores as well. These systems are easy to use, and the pages come out and go in simply and quietly, removing the loud-snapping-rings issue. The only issue with these is that to really make them work, you have to invest in the hole punch that makes those special cutouts. Armed with one of these notebooks and the hole punch, you can punch your handouts and put them directly into the notebook exactly where they belong, but still have the feel of a spiral notebook.