Wednesday, July 30, 2014

A Tale of Three Students

Most normal people (a.k.a. non-statisticians) tend to think in linear terms. The trouble is most normal behavior doesn't tend to be very linear. As a rule, you're better off thinking in terms of of U-curves (things go either up or down then come back) and S-curves (things are level, they move either up or down, then they level off again). These are still approximations, but they are usually more reasonable approximations. 

This is particularly true in education. Arguably the best way to model learning is with a series of S-curves. We work and study with little progress, then we have a period of improvement, then we hit another plateau.

If we think in terms of straight lines, ranking students is relatively easy.

But if we think in terms of S-curves, which is a great deal more realistic, things get more complicated.

The x-axis doesn't actually mean anything (this is made-up data), but let's say it represents months studying a language and the lines represent daily test scores. Now, who's the best student depends on when you ask, and that raises some troubling points. 

We tend to put too much faith both in the metrics we use to evaluate students and in the linearity of human behavior. We are not straight -line animals but we have a bad habit of making straight line decisions. In this case, think of what would happen if we made a decision on who to drop from a program after two months. 

Monday, July 28, 2014

The Power and Peril of Positive Thinking

Having hammered away at the importance of student self-confidence and positive attitude as a condition for success in math (part of the larger discussion of applying Pólya's teaching principles), it's important to step back and point out that a lot of people have made horrible, costly mistakes thanks to positive thinking and the influence of motivational speakers (for example).

With fantastic successes on one side and horror stories on the other, it is tempting to call this a wash, but if you think like a statistician (and you should always think like a statistician) and start breaking things down, you'll find that a few common sense rules can tell you when to assume the best and when to prepare for the worst.

Being pragmatic about being positive

For the purposes of this discussion, let's decide on a fairly precise definition of what we mean by positive thinking:

To apply positive thinking to a task, you act under the assumption that, given  reasonable and intelligently applied effort, the probability of success is close to one;

Furthermore, this assumption will not be reassessed unless there is a major change in the situation.

The advantages to this approach are: we can waste a great deal of time and energy worrying; overestimating risk can cause us to prematurely abandon projects; thoughts of failure can cause us to "flinch," to hold back and not give the task our best effort. Avoiding these things can allow positive thinking to create self-fulfilling prophecies.

The disadvantages are that underestimating the probability of failure can cause us to waste resources on projects with negative expected value and, more importantly, failing to pay attention to warning signs can leave us vulnerable to otherwise avoidable disasters.

We could have a general discussion at this point about the relative weight of these advantages and disadvantages but it wouldn't be very productive because neither risk nor reward are evenly distributed. In many if not most situations, a fairly clear case can be made for either positive or cautious thinking. To determine which approach is best for a given situation, think about these rules of thumb:

Before you commit yourself, try to think realistically about the expected value in terms of other people's success rates;

Never bet more than you're willing to lose;

Consider collateral damage (are you putting your spouse and children at risk of hardship?);

Is there incremental payoff? This last one is extremely important. If you decide to start a restaurant or move to NYC to make it on Broadway, and you fail, then you will probably have very little to show for the effort. If, on the other hand, you decide to lose forty pounds through diet and exercise or to go from being a C student to an A student, then there is incremental pay off for your hard work even if you fail to achieve your goal.

All of this leads us back to the original point. We often associate positive thinking with business and entrepreneurship where it is, more often than not, a bad idea, while in education, where we have every reason to encourage positive thinking, we are constantly hearing people like Michele Rhee complain that we spend too much time building up kids' self-esteem.

Don't let the posturing and tough talk fool you. Self-esteem is good for kids and you should do everything you can to convince them that they are capable of doing every problem their teacher gives them, as long as they put in the effort.

Thursday, July 24, 2014

Recess was always my best subject

And according to this article from Tim Walker, I might have been onto something.
Like a zombie, Sami—one of my fifth graders—lumbered over to me and hissed, “I think I’m going to explode! I’m not used to this schedule.” And I believed him. An angry red rash was starting to form on his forehead.

Yikes, I thought. What a way to begin my first year of teaching in Finland. It was only the third day of school and I was already pushing a student to the breaking point. When I took him aside, I quickly discovered why he was so upset.

Throughout this first week of school, I had gotten creative with my fifth grade timetable. Normally, students and teachers in Finland take a 15-minute break after every 45 minutes of instruction. During a typical break, students head outside to play and socialize with friends while teachers disappear to the lounge to chat over coffee.

I didn’t see the point of these frequent pit stops. As a teacher in the United States, I’d spent several consecutive hours with my students in the classroom. And I was trying to replicate this model in Finland. The Finnish way seemed soft and I was convinced that kids learned better with longer stretches of instructional time. So I decided to hold my students back from their regularly scheduled break and teach two 45-minute lessons in a row, followed by a double break of 30 minutes. Now I knew why the red dots had appeared on Sami’s forehead.

Come to think of it, I wasn’t sure if the American approach had ever worked very well. My students in the States had always seemed to drag their feet after about 45 minutes in the classroom. But they’d never thought of revolting like this shrimpy Finnish fifth grader, who was digging in his heels on the third day of school. At that moment, I decided to embrace the Finnish model of taking breaks.

Once I incorporated these short recesses into our timetable, I no longer saw feet-dragging, zombie-like kids in my classroom. Throughout the school year, my Finnish students would—without fail—enter the classroom with a bounce in their steps after a 15-minute break. And most importantly, they were more focused during lessons.

At first, I was convinced that I had made a groundbreaking discovery: frequent breaks kept students fresh throughout the day. But then I remembered that Finns have known this for years; they’ve been providing breaks to their students since the 1960s. 

Monday, July 21, 2014

My preferred approach to teaching high school math -- the last twenty minutes

[Dictated to my phone so beware of homonyms]

Note: Though the connection may not be immediately obvious, the following thoughts on teaching will eventually tie in with a larger piece on George Pólya.

Though it varied someone from class to class and situation to situation, my preferred method was to reserve the last part of the class for students to work individually while I went around the room and checked each student's work. Generally, I would give the students a couple of worksheets to be handed in at the end of class. After completing those worksheets, they were instructed to spend the rest of the hour working on their homework. I wasn't always able to get to every student every day, but I came close, and I never let more than a couple of days go by without making sure that I had personally observed a student doing problems in my class.

If a lots of the students were having trouble doing the assignment, I would sometimes interrupt the routine, go back up to the board, and reteach some of the material. That was fairly rare. Most of the time, two or three students would need real help and the rest only needed either a couple of quick suggestions or simply confirmation that they were doing the problems correctly.

The personal help was important, as was the knowledge on the students' part that if they needed help in the future I would be there. This approach also let me make sure that neither the class or any of the students got into a death spiral where confusion and failure started causing a cascading effect. By personally watching students successfully completing assigned problems, I could make sure that everyone was keeping up. Grading was also an important part of that process but for assessment there is no substitute for actually watching how a kid going through a problem.

In some cases, particularly with advanced classes, I might stray from this approach, but if we are talking about at-risk kids in tough environments who need to make up ground academically, I believed then and believed now this is the best way to teach high school math.

If I sound a little over emphatic with that last sentence and perhaps even a little bitter it's because I am more than a little bitter about the direction our schools have headed. I enjoyed that kind of teaching and I got excellent results with it, but if I were to go back into the profession now, there is almost no way I could give that kind of personal attention nor could I take the same level of accountability for students' success. Class sizes have simply gotten too large.

On a completely unrelated note...

Detroit Public Schools EM shifts funds from classroom
By Dr. Thomas C. Pedroni

Many of us are shocked to learn that DPS plans to cut costs in the coming year by further increasing class sizes. Already at an unmanageable target of 38 per classroom in grades 6 through 12, Emergency Manager Jack Martin’s fiscal year 2015 budget allows class sizes in those grades to expand to 43. 

Thursday, July 17, 2014

Resources for SAT self-study -- The Khan Academy

I don't want to be too negative here. This is a decent resource and for most students there will be some value in going through these problems. If that sounds like faint praise, that's because it is. What we have here, while nice, simply isn't that much to get excited about. The Academy, it should be noted, is promising a much more complete and sophisticated set of tools next year, but for now, we pretty much have to limit ourselves to the "better than nothing" standard.

I would recommend that you go to the Khan Academy SAT site and do some exploring on your own.

In the meantime, I have included a few examples to get the discussion started.

So what are my concerns here?

I The Medium is the Message

Ideally, we'd see videos that made better use of the medium (more on that later) but the bigger, more immediate concern is is that they have simply chosen the wrong message here. What we have in these two clips is basically a standard problem session with most of the good and important parts left out.

This type of instruction, done properly, is contextualized and highly interactive. Putting problems in context is essential because, even more than drudgery, the thing that drags kids down in math is not knowing why they're doing what they're doing. Working a problem for a class should be a conversation that both frames the question and gives the students a chance to participate in the solution. (For example, ask them "If this length is x, what is this other length in terms of x?" and let them come up with the 2x.) Without the conversation, there's not much point in presenting this information in a video because...

II You've already got the message in a better medium

These clips are basically an instructor reading out loud the kind of solved examples that can easily be found in books or online. The videos add little to the printed version and they arguably lose quite a bit. This will strike many as blasphemous, but for certain tasks, print actually is a superior medium. It's good at conveying precise information and it gives the reader a level of control that is difficult to achieve with pausing and rewinding a video. It offers tremendous variety and availability. It is easily searched and database-friendly. What's more, the ability to pick up a book or go to a website and teach oneself is a valuable skill for everyone and an absolutely essential ability for anyone in STEM.

Is the spoken word ever better than print? Sure, much of the time. If you're going for a conversational voice; if the gist of the message is not dependent on any single word or phrase; if you want a particular emotional tone... Unfortunately, these clips not only do things that print would do better; they leave out the things that their medium would do better than print could.

III Rush, rush, rush

There's a choppiness here both in tone and structure, combined with a hurried quality. The approach is very much "Here's a problem. Do this. Do this. Do this. Here's the answer." No time spent orienting the student. Worse yet, no time spent checking the answer. In terms of  Pólya's How to Solve It, they tend to jump directly to "carrying out the plan" and stop there. That's very dangerous because, even if the students do manage to follow what you're doing, they won't have a firm grasp on why you were doing it and that's going to be a problem when you're not there to hold their hands.

IV Lots of trees, very little forest

If you want students to understand what they're doing and, just as important, if you want them not to feel confused and anxious, you have to give them a sense of the overall problem.

The carton question above is a perfect example of how to screw that up. The instructor jumps directly to the exact variables and equations needed, skipping a number of steps that are not at all obvious to the students. In fact, it is those skipped steps that trip up most of the students who miss these problems.

If you tried this with actual students, they would say "How did you know to do that?" and that's an extraordinarily good question. The answer is that we start with translation from English to mathematics (a faithful but not necessarily idiomatic translation, though I probably wouldn't mention that to the students). "[H]ow many of the $50 cartons did she buy?" suggests number of $50 cartons should be a variable (let's be original and call it x) and it makes sense that the number of $30 cartons would be another (y). "Sarah bought 20 cartons" becomes x + y = 20 and so on.

If students understand the why behind the what, they will be happier and learn better.

I don't want to be too harsh here. This isn't exactly a bad resource and I very much believe that the Khan Academy's heart is in the right place, but when it comes to the SAT, they still have a long ways to go.

Note: I made a few minor wording changes after first posting this.

Tuesday, July 15, 2014

From Scientific American -- "Don’t Take Notes with a Laptop"

Back in graduate school, I was a TA for one of the most effective college level instructors I had ever seen. She was also, perhaps not coincidentally, one of the best and most self-disciplined students I had ever seen. She was finishing her Ph.D. when I was getting my master's, so I had a chance to observe her process up close.

Her notes were particularly valued by other students. At the beginning of every semester, there would be a rush to her office to borrow the appropriate binders. The notes were compiled through a two-step process: during class she would attentively take detailed notes. She would also make audio recordings; after class she would make a second set of notes based on both her original notes and the recordings.

I was reminded of this when I came across article from SA which looks into the effectiveness of different methods of note-taking. It also raises some important questions about the role of technology in the classroom.

A Learning Secret: Don’t Take Notes with a Laptop

When it comes to college students, the belief that more is better may underlie their widely-held view that laptops in the classroom enhance their academic performance.  Laptops do in fact allow students to do more, like engage in online activities and demonstrations, collaborate more easily on papers and projects, access information from the internet, and take more notes.  Indeed, because students can type significantly faster than they can write, those who use laptops in the classroom tend to take more notes than those who write out their notes by hand.  Moreover, when students take notes using laptops they tend to take notes verbatim, writing down every last word uttered by their professor.

Obviously it is advantageous to draft more complete notes that precisely capture the course content and allow for a verbatim review of the material at a later date.  Only it isn’t.  New research by Pam Mueller and Daniel Oppenheimer demonstrates that students who write out their notes on paper actually learn more.  Across three experiments, Mueller and Oppenheimer had students take notes in a classroom setting and then tested students on their memory for factual detail, their conceptual understanding of the material, and their ability to synthesize and generalize the information.  Half of the students were instructed to take notes with a laptop, and the other half were instructed to write the notes out by hand.  As in other studies, students who used laptops took more notes.  In each study, however, those who wrote out their notes by hand had a stronger conceptual understanding and were more successful in applying and integrating the material than those who used took notes with their laptops. 
What drives this paradoxical finding?  Mueller and Oppenheimer postulate that taking notes by hand requires different types of cognitive processing than taking notes on a laptop, and these different processes have consequences for learning.  Writing by hand is slower and more cumbersome than typing, and students cannot possibly write down every word in a lecture.  Instead, they listen, digest, and summarize so that they can succinctly capture the essence of the information.  Thus, taking notes by hand forces the brain to engage in some heavy “mental lifting,” and these efforts foster comprehension and retention.  By contrast, when typing students can easily produce a written record of the lecture without processing its meaning, as faster typing speeds allow students to transcribe a lecture word for word without devoting much thought to the content.
To evaluate this theory, Mueller and Oppenheimer assessed the content of notes taken by hand versus laptop.  Their studies included hundreds of students from Princeton and UCLA, and the lecture topics ranged from bats, bread, and algorithms to faith, respiration, and economics.  Content analysis of the notes consistently showed that students who used laptops had more verbatim transcription of the lecture material than those who wrote notes by hand.  Moreover, high verbatim note content was associated with lower retention of the lecture material.  It appears that students who use laptops can take notes in a fairly mindless, rote fashion, with little analysis or synthesis by the brain.  This kind of shallow transcription fails to promote a meaningful understanding or application of the information.
These findings hold important implications for students who use their laptops to access lecture outlines and notes that have been posted by professors before class.  Because students can use these posted materials to access lecture content with a mere click, there is no need to organize, synthesize or summarize in their own words.  Indeed, students may take very minimal notes or not take notes at all, and may consequently forego the opportunity to engage in the mental work that supports learning. 
Beyond altering students’ cognitive processes and thereby reducing learning, laptops pose other threats in the classroom.  In the Mueller and Oppenheimer studies, all laptops were disconnected from the internet, thus eliminating any disruption from email, instant messaging, surfing, or other online distractions.  In most typical college settings, however, internet access is available, and evidence suggests that when college students use laptops, they spend 40% of class time using applications unrelated to coursework, are more likely to fall off task, and are less satisfied with their education.  In one study with law school students, nearly 90% of laptop users engaged in online activities unrelated to coursework for at least five minutes, and roughly 60% were distracted for half the class.

Monday, July 14, 2014

The first wall you expect is the last one you hit

[As you may have already guessed, I'm going to be tying this in with the ongoing Pólya discussion]

This is another reason why it is important for math teachers to work so hard on building your student self-confidence. Whether we are talking about calculus or golf or playing the guitar, failure usually comes when we hit one of the following walls:

Lack of ability;

Lack of time;

Lack of patience and self-discipline;

Lack of resources.

It is often the fear of these walls that prevents us from investing the time and effort into mastering a skill we would very much like to have. That is a perfectly rational attitude. As mentioned before we always judge the time and effort needed to do something against the expected returns. Unfortunately, people often have a very unrealistic concept of these returns, particularly when it comes to the placement and order of these four walls.

Students generally expect "lack of ability" to be the first wall that they hit and yet this almost never happens. After years of teaching and working with students, I honestly can't think of an example where this was the case. They run out of time; they run out of patience; they run out of resources. These are things you see all the time, but I don't know that I have ever seen a student who simply put not handle the material. I am not saying that this does not happen but I am saying that it is extremely rare.

This does not mean that the "ability wall" isn't out there somewhere. It's important to realize that even with a tremendous amount of effort and support, some goals will still be beyond you. For the extraordinarily (or perhaps more accurately, obsessively) driven, this can be a problem. I'm sure instructors at Julliard encounter this all the time. However, for those teaching math on the primary, secondary, even undergraduate level, this is probably not something you will ever have to worry about.

Kids will not be receptive to instruction, they will not respond to incentives, they will not focus on material, and they will not put forth serious effort unless you can convince them that they are not about to hit the ability wall. This may not be the most important part of mathematics instruction, but it is the first part.