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 good 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.


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.

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.

Friday, July 11, 2014

Rational students, incentives and expected returns

In my last post, I contrasted George Pólya's humanistic style of teaching with the scientific management approach favored by most movement reformers. The focus of that post was more on the underlying philosophy (I'd contend that George Pólya had a very different view of people than did Frederick Taylor), but there are some extremely practical reasons for adopting Pólya's approach, particularly when it comes to building students' self-confidence.

Every investment involves a comparison of costs vs. expected value. We never know how a decision will turn out, so we have to balance money, time and trouble against expected returns. I realize this point seems to border on too-obvious-to-mention but it's surprising how often people forget there are random variables in their arguments, sometimes with disastrous results.

Take performance-based incentives. Let's say I'm going to offer to pay you a certain sum if you accomplish a task but nothing if you fail. In order for you to agree, your time and effort will have to be valued less than the product of my offer times the likelihood of success. Once again at the risk of stating the obvious, as that likelihood approaches zero, your idea of a reasonable offer will have to approach infinity. Of course, in real life, there are always bounds on the amount of money I can offer but your estimate of the likelihood of success can always get closer to zero.

In a business context, we normally deal with the small perceived likelihood problem by finding someone else or opting for a different compensation plan or simply walking away from the deal. This is yet another reason why it's dangerous to have people who don't thoroughly understand both business and education try to transplant ideas from one field to another (it also reminds us of Pólyas warning that "it is foolish to answer a question you do not understand").

In education, where we should try to reach every student, low perceived likelihoods of success can be deadly. Any reward you offer for an apparently unattainable success will seem worthless; any penalty for apparently inevitable failure will seem brutally unfair. If you want to motivate these students, you will have to convince them that, with reasonable time and effort, the odds of success are pretty good (this happens to be true for the vast majority of students but that's a topic for another post).

Pólya said "If the student is not able to do much, the teacher should leave him at least some illusion of independent work." This was, of course, meant as a last resort, but his point was that students absolutely have to think of math as something they are capable of doing.

Wednesday, July 9, 2014

Pólya was a humanist. That puts him at odds with the education reform movement.

What do I mean when I say that George Pólya was a humanist and imply that movement reformers (at least the ones that matter) are not? All too often, people use 'humanist' or 'humanistic' simply as an impressive way of saying nice or decent. While I certainly believe that Pólya was both a nice and decent man, I have something much more specific in mind here.

Pólya's approach to mathematics pedagogy was humanistic in the sense that it was based on certain assumptions about human nature, viewing students as playful and inquisitive animals who were naturally inclined to learn and to solve problems. That natural inclination meant that the best way to help students was through "common sense" suggestions (a phrase that featured prominently in How To Solve It).

He also focused a great deal of attention on the emotional and psychological state of the student. It was explicitly part of the teacher's job to instill self esteem, self-confidence, and self-reliance in the student, even if it occasionally meant giving the student an exaggerated sense of accomplishment.

"If the student is not able to do much, the teacher should leave him at least some illusion of independent work. In order to do so, the teacher should help the student discreetly, unobtrusively." [emphasis in the original text.]

This humanistic view of learning math lead unsurprisingly to a similar view of teaching with a strong emphasis on empathy and individuality:

"The teacher should put himself in the student's place, he should see the student's case, he should try to understand what is going on in the student's mind and ask a question or indicate a step that could have occurred to the student himself." [emphasis in the original text.]

In order to see how this figures in the larger education debate, we need to introduce the field of scientific management. This field is largely based on the idea that people can be treated like any other component in a complex system. The secret to optimal performance is simply to gather the right data, derive the proper metrics, then use these metrics to put the right components in the right roles and create optimal set of incentives.

The education reform movement with its emphasis on metrics, standardization, and scripted lessons is entirely derived from scientific management. Those scripted lessons in particular represent a complete rejection of Pólya's approach of "getting inside the students head." and personalizing the instruction. Not coincidentally, David Coleman, arguably the intellectual leader of the movement, started out as a management consultant.

Another area of sharp contrast between David Coleman and Pólya is Coleman's strong support of deliberate practice in mathematics education. Scientific management is heavily reliant on reductionist approaches and their are few pedagogical techniques more reductionist than deliberate practice. Pólya was wary of reductionist approaches to teaching. He saw drills as a sometimes necessary evil, but as a rule, breaking down problems for the student was a dangerous habit. For Pólya, the process of problem solving was about taking problems and examining them, restating them, generalizing them, simplifying them, comparing them to other problems, and, yes, breaking them down into sub-problems, but the important part of that process is deciding what to do. To break the problem down for the student is to defeat the purpose.

Many, if not most, of those horrible, multi-step math problems which have become associated with Common Core are not what Pólya would consider problems at all. The problem solving has all been done in the preparation of the lesson; all that's left for the student is the mechanics.

As a side note, Pólya believed that any method you use to solve a problem was acceptable as long as you understood the problem and could prove that your answer was right. This included guessing. Pólya even joked that his initials stood for "Guess and Prove." This approach allowed him to make a string of major discoveries in Twentieth Century mathematics but it would not have gotten him a passing grade in a Coomon Core influenced elementary school.

Pólya's humanistic approach no doubt reflected and underlying philosophy but there was another more practical reason for adopting this approach. Put bluntly, after a long and successful career doing and teaching mathematics, Pólya had a strong sense of what worked. I will get into the practical underpinnings of Pólya's humanism in the next post.

Monday, April 28, 2014

SAT tips -- the opposite of the worst test you ever took

[This is part of an ongoing series of posts designed to help students, parents, teachers and other interested bystanders learn about and prepare for the SAT]

A message for perspective test takers...


No, really. Think back over the tests you've taken over all the years. Now think about the ones that caught you completely off guard. The ones that covered material you hadn't expected, asked questions completely different than what you expected. If you've ever had a test like that you probably still remember that feeling of being surprised, unprepared, even lied to, and worst of all, helpless.

If you're like me, those were probably the worst test-taking experiences you've ever had.

What would the opposite of a test like that be? What about one where the teacher gave you a stack of all of the old tests over that section going back ten years? On top of that, what if the teacher promised to keep the test as similar as possible to all those old tests, same material covered, same format, same type of questions, same difficulty? Wouldn't that pretty much be the opposite?

That's what we have with the SAT. The company that makes the test has been releasing its old tests for decades. You can buy collections of old tests in almost any book store. Better yet, you can get them for free from your school library.

And you usually don't have to worry about the new tests throwing in a lot of big changes. This next year might be an exception (there's been lots of talk about 'reforming' the test), but in general the SAT changes very slowly. This next year might be the exception, but other than that if you get a collection of old tests and work through them, asking your teachers for help whenever you get confused, you'll find that the SAT is exactly the test you expected and prepared for.

And that's a very good feeling.