Friday, November 21, 2008

my own experiences learning: part 2, TAing at MIT.

The other big experience that shaped how I view educational systems was being a TA at MIT. On the whole it was a really disheartening experience. I believe that is much harder and more worthwhile to construct a healthy system -- be it educational or otherwise -- than critique an existing, failing one, and so I haven't thought much about TAing at MIT in the past few years. However, I'm currently living at an MIT house (I'm the Resident Advisor at pika!). Watching people struggle with a system they're unaware of has made me think that these observations are still of use.

After coming back to MIT from Germany, I felt completely unwilling to participate in the educational system. My overall feeling was that the whole system could be run so much better and had so little thought put into it, and yet it didn't particularly bother anyone. I had an amazing resistance to taking classes where I felt like the instructor wasn't trying or caring much and consequently dropped nearly all of the classes I signed up for my first term back.

I took 4 classes my last two terms at MIT, and dropped my former plans of becoming a mathematician mid-way through applying for a Ph.D. My interest in math fell apart as I realized that there were many more important things to work on, especially in the worlds of learning and education. Of the four classes I took, was being a teaching assistant (TA) for 18.02 -- multivariable calculus. It satisfied my last graduation requirement and with my rising interest in education it would be a neat thing to do.

For background, 18.02 is a required course at MIT -- all students must take it to graduate. It's the second math class in the introductory one-year calculus sequence. At many universities, this is a two-year sequence. The class itself was about 200-300 students, and I taught a 1-hour recitation twice a week with 20 students and also graded their homework.

1. My own growing perspective on education

I'd started reading John Holt's How Children Fail and How Children Learn that semester, which were really shaping how I thought about teachers, learners, and educational systems. Those books began to give me an articulation for the intuitions I'd developed in Germany, and made me look at grades as unnecessary and something that simply got in the way of people's learning processes. Instead of giving students the message "You haven't fully mastered this concept yet," grades gave students the message "You haven't fully mastered this concept yet and now you're being punished for it." I began to become aware of how grading cuts into someone's natural feedback process when learning -- changing an intuition like "Man! I wish I understood lift better, it's so mysterious and interesting" to "Man, there's no way I can get an A in this class now." Thinking about Holt in terms of my experiences in Germany made me understand how toxic grades and similar judgements are and how one can thrive when rid of them.

I also saw how liberating it was to be free of notions like "I need to work harder" and "I'm not working hard enough" in Germany. I began to recognize that kind of thinking as symptomatic of a poor educational system -- putting the student in a place where despite their natural curiousity, they felt both overwhelmed (I can't get everything done) and unsatisfied (I need to be doing something differently...) Since students fundamentally can't change their learning environment -- they don't have control over their own learning process -- the only place they have to turn is inward. This leads to the very common spiral of plaguing, nagging thoughts of self doubt: "Maybe if I worked harder, I wouldn't be behind," or "Maybe if I slept enough, I wouldn't be so tired," etc. My experience in Germany, suddenly free of my own self-doubt once I was happily learning all day, has led me to view claims like these, or similar ones like "I'm just not good at this" or "I just haven't tried enough" as signs that the system isn't working, not the student.

2. The first exam

The first few weeks of TAing were pretty ordinary. Myself and my students got used to the rhythm of the class, I got used to balancing presenting material and taking questions in recitation, and so on. The first exam we had was the first event that began making me suspicious of the class.

The test grades were what you'd expect -- some students did well, some did ok, and a few failed. TAs were asked to email any student who failed, letting them know how they could make up a test. I remember looking at one student's failed exam and his related homework and seeing clearly that of the three weeks of material covered, he'd understood the first two weeks fine and not the third week. As far as using the test as objective feedback goes, this meant he had a week's more of learning to do for the course.

As far as the metric used by the class though, this meant he'd failed his first test: a deeply demoralizing event. On top of that, he'd have the spectre of that failure for the next two and a half months: there was no way from him to makeup that failed grade and get a high mark in the class now -- his average would be too weighed down. So while the message ought to have been "Ah! You haven't understood cross products yet -- you have these ideas to catch up on!", it becomes a stigmatizing failure that lasts for the whole term: "You could pass this class if you work extra hard."

I emailed my student asking him to meet. I was eager to show him that there were just 2 big ideas he was missing and then he'd be on par with the rest of the course: the looming feeling that not only did you fail a test, but most everyone else in your class didn't makes the event that much more defeating and confusing: why can't I do this but most everyone else can? It took a bit of pushing to get him to meet with me -- I think he politely declined the email, and then after recitation one day I got his attention in the hallway and asked him if he had time to go over the material right then. He did, and a half hour later he was really relieved I'd pulled him aside. As I suspected, he'd viewed the failed test as a sign that he was really far behind, and by the end of our session he was much more relaxed.

The rest of the semester went fine for him and he remained very grateful for that intervention. It's a nice story because it wraps up cleanly, but of course it's not the norm. It got me thinking: why was it such a big deal for him to learn how to handle cross products right here, right now? Beyond the test -- the artificial environment -- he had no need for them. When we'd met, he explained that he'd been busy with his other classes, having had several exams that week, and just hadn't gotten to calculus yet. This seemed reasonable enough to me: he was doing his work as best as he could, and the idea that he hadn't learned how to do cross products by an arbitrary date causing so much stress just seemed preposterous to me. What was the point?

3. Copying

About midway through the term, I began to notice that a lot of the problem sets handed in were duplicates of other problem sets. From my point of a view as a grader, it would be amusing to see the changes people would put in their problem set -- like substituting a "y" instead of an "x" throughout a problem to attempt to mask an otherwise 6 pages of calculus that was line-for-line identical. On one of the weeks almost half of the work was copied. Some of those may have been the originals, but it was still a lot! Another undergrad TA had noticed the same thing; graduate TAs didn't grade their own section's homework, so it was only a few staff that noticed this trend.

I set out to look at copying objectively -- past the usual moral claim that it's a wrong thing to do and to discern what it meant about the class. It first occurred to me that copying was a literal waste of time: instead of spending an hour or two trying to understand the material at hand, one spends it transcribing equations line for line for pages. Beyond that, there's the mental space that someone is in when they're copying. By copying, someone is acknowledging that they can not or do the requested homework or that they don't want to, yet they still feel obliged to appear to have done so. It's like saying "I can't or don't want to do this, but I have to."

The point of homework is to give the student practice with the skills covered in a course, and to give the instructors feedback as to what the students are understanding. What's happening here is that a student who is copying feels that the practice offered is either not doable (the student does not understand, feels too exhausted, or both) or not useful (the practice does not seem worthwhile or the material does not seem worthwhile.) If the material isn't doable, that feedback is of critical importance to the instructor. Whether it's because the student is too exhausted to learn, as is often the case at MIT, or is just lost in the class, the student would benefit from the class slowing down and addressing this.

It's also valuable feedback if the student sees no point in the exercises. Learning German was effortless because there was a natural context for the skill -- communicating in German. Likewise, when there is no natural context for a skill, learning is a struggle. If you don't see a reason to learn a skill, why would you learn it? Again, this is huge feedback for an instructor to have.

In either of these cases, instead of this feedback going to the instructor through a simple conversation or email, the student feels there is no use in being honest about how they feel, so much so that they spend an hour or two doing something pointless in order to appear as if they did indeed do the homework. This feedback gets masked because to give that feedback honestly -- to have a conversation instead of handing in a copied problem set -- would likely be poorly received, and worse, one would be graded harshly for it if the conversation didn't go well.

4. Bibles

Taking this view and applying it to 18.02, where at its peak half or so of the problem sets were copied shows a pretty bleak picture -- almost half of the students are pretending to do the homework and feel bound to do so. And yet this is no quirk particular to this class. I remembering touring fraternities as a freshmen, and one of their perks was their collection of "bibles": collection of all the notes, homework, and tests from previous years. This idea of copying problem sets is such a normal one that it's well established in MIT's residential culture. The idea that a class isn't working for you in some way and yet there's no way to change that is an accepted one.

I came to realize that part of why students' bibles work is because professors also have bibles. The course my professor was teaching was handed to him by the previous professor. It was the complement to the student's bible: the lectures, problem sets, and exams were all in the course package. It felt on one hand ridiculous (the professor hands out problems he hasn't thought about, and the students hand in answers pretending to have thought about them) and on the other offensive. The class was on a rigid track: there were so many set lectures, homeworks, and exams. There was no room for deviating if the class was stuck on one idea, understood another quicker than expected, or if there was an insight as to where the class should go instead. The class was like a train -- steady and immovable. Knowing how much effort students put into accomodating a class' assignments, it bothered me to realize that it was already pre-determined that the class wouldn't respond to the student's needs.

This begins to make sense of common student questions and concerns like "Why do we have to do this problem when we haven't covered it in class yet?" or "I'm not sure what's going to be on the exam since on the last exam there was material we had only just started..." If a class' homework/test structure is predetermined, it's easy for a lecturer to be out of sync with the questions he's assigning because he simply hasn't read them over or thought them through. While I can't tell you how common this in courses at MIT, I would expect it to be the case in the majority of introductory classes.

Another post-doc in the math dept. -- one of my favorite teachers at MIT, Emma Carberry -- told me as she was applying for professorships that she felt quite frustrated with the high-pressure academic system. She was applying to liberal arts colleges to teach at because she wanted to be in an environment where she was rewarded, or at least acknowledged, for putting a lot of time into her teaching. She said that at MIT and in this tier of academia, the only metric was how well your research was going, and so putting time into teaching well was something that you were implicitly punished for professionally. This anecdote still amazes me; I often wonder why MIT bothers having classes when they don't value them.

Students often have a few great classes at MIT. With an instructor who carefully thinks about the interaction between all of the course's components: homework, tests, lectures, and so on,
and integrates feedback as it comes, a lecturer can create a good learning environment for a student at MIT. These professors are unfortunately rare because MIT's professional system selects against this, as was the case with Emma Carberry. The professor I was working for told me that he wanted to be teaching grad students in his field of research, but was assigned the introductory class and could do nothing about it. I thought the professor was doing a decent job too (he did lots of things well -- taking feedback from TAs, worked to create good materials for recitations) and yet at the end of the day, it was clear that this was not a project the professor was interested in. The lack of choice the professor had ("I don't want to teach this class, but I have to") led to the automatic production of the class from the bible, which in turn led to most of the students having the same reaction ("I don't want to take this class, but I have to.")

5. Tests

Midway through the semester, as I began to notice all of the copying, I began to really see the class for more of a charade: both the professor and majority of students were there because it was a required class, not because they thought it was worthwhile. Exam grading only furthered this along.

The professors and the TAs graded the exams, and each staff member got a question to grade for 3 hours or so with a partner. It was a pretty mind-numbing experience, grading 150 or so of the same exam question again and again. On this exam, the question that I graded was one where 2/3rds or so of the students made the same mistake. I remember thinking that I wished I had a stamp for how many times I wrote down "can't used Green's theorem when the line integral isn't closed!" It began to dawn on me as I did this over and over again, that this was a clear sign the class did not understand the concept in any intuitive way. This was great feedback (the course pace should slow down and go over this again, ideally approaching it in a way that develops more intuition), but feedback that had no place in a predetermined class. The results of the tests got turned into numbers, the numbers into a distribution, and that was the feedback that the course received. The details of which concepts had been mastered by the group and which concepts hadn't been understood at all -- the feedback that mattered -- was left behind in the wake of lots and lots of exam scores.

This exam just added on to the feeling that the class was a big, stressful game of pretend. The students didn't know how to do the problem, and the mistake stemmed from trying to match patterns: there are 4 big ideas being tested, 6 questions on the test, and if you match them up right the test will turn out fine. The feedback from the test that students didn't understand Green's Theorem didn't fit into the course's structure and so was ignored. The staff gave tests because they had to, the students took tests because they had to, and the course went on, staying on track.

6. Context

There were about three weeks left of the class at this point, which covered material like Stokes' theorem. Stokes' theorem is something that everyone at MIT recognizes -- having had to take 18.02 -- and hardly anyone knows, including math majors. I myself never had a strong intuition for why Stokes theorem was important, and tried to find a good context to present the theorem in.

I came across Schey's Div, Grad, Curl and all that , which explained vector calculus and the material we were covering oin the context of electricity and magnetism. The book was great -- it was very simple and answered my own personal questions about why this material was valuable.

It didn't help for presenting it well though. I tried once, and realized that I was trying to elucidate one abstract concept -- vector calculus -- by putting in context of another abstract concept -- electricity and magnetism (E&M). This was particularly weak because many of these students were taking E&M at the same time as 18.02 -- there was no guarantee that E&M was something these students had any familiarity with, and furthermore, any intuition for.

I then asked the simple question -- how many of these would actually use this material, based on their declared major? I surmised that of the 20-some majors at MIT, the ones that would use 18.02 extensively (something more than just adding a week's worth of a material to a course to explain a needed mathematical tool) were mathematicians, physicists, and anyone who studied flow (so mechanical and civil engineers,) I was willing to bet another two majors used the material in ways I couldn't think of, but that for the rest of them, beyond those six, their disciplines were not reliant on this material in any way. That quick estimate puts the number of students who would use 18.02 later on to be somewhere between 1/4 and 1/3 of the student body. Yet everyone was taking it!

I also thought about my friends who were physicists -- who in E&M used vector calculus all the time. Most of them told me that the way they'd learned vector calculus was by learning E&M. This made sense to me too: you learn something by using the knowledge, not by preparing to use it. It made perfect sense to me: needing to use vector calculus and would create a much more powerful context for understanding and remembering the material than the artificial one of 18.02. Only 1/4 or 1/3 of the students who actually learned this material, and it seemed clear that they were better off learning it in the context of their discipline anyway. What was the point?

6. Feedback

One of the things that was so powerful in Germany was having complete control over my learning environment: being able to fold in the feedback from each thing I did into my daily process. One of the reasons that 18.02 felt like such an ineffective class to me was that the feedback loop was too large to change quickly and too fragmented to understand what it should change.

I already discussed the example of homework: where students hide their honest feedback in bowing to the system. And the example of testing: where the staff, prioritizing grading, ignores the real feedback generated. Beyond this was just the hierarchical mess of having 1 person responsible for 300 people's learning.

I remember one example where a problem set contained an unusually difficult problem. The TAs had trouble doing it and couldn't figure out how to do it. Some TAs unwittingly gave out false solutions in office hours, not realizing they didn't know how to do it correctly. There were tons of questions that week about the problem, and their was nothing illuminating about solving it. It was supposed to be practice for a calculus idea, but it was really an exasperatingly long geometry problem.

This problem easily added on at least 2 hours of work to each student's problem set or furthered the "I can't do this so I better copy it" mindset. It would've taken the professor or the course admin maybe two hours to do the problem set, find that problem, and throw it out. But, since they were just handing out problem sets handed to them, this problem was kept in and a group of students wastes a good 300 - 600 hours.

Letting one person control 300 people's time is the core structure of this and other lecture-based course. It's pretty tough to give a good hour-long presentation, let alone three on a week on something you don't particularly care about, as was this professor's lot. There's virtually no room for feedback: most students, when confused or stuck, hold in their questions because it's quite difficult to have a conversation in a 300-on-1 environment, proceed to forget their questions, and struggle to follow the rest of the lecture.

There are good lecturers at MIT. However, it seems to be akin to being a good performer: lecturing, or performing, well is a rare skill and one that takes work. Giving a good lecture means having a good command of your voice, your blackboard, and having an intuition for how to present your material in a way that is engaging and not confusing. Looking at feedback from the lecturer's point of view, it is harder to get feedback the larger your class is. A lecture to a 300-person audience has to be one where the instructor is already familiar with most of the pitfalls and confusions. It takes a lot of work to become experienced and knowledgable enough to be a good lecturer. It can be done, but given MIT's stance of prioritizing research in an academic's career, a good lecturer is going to be the outlier and not the norm.

Despite this, the lecture is regarded as the most important part of the course. In my term in 18.02, students attendance was highest in lecture, second highest in recitation (a 1 on 20 environment, where at least one or two questions per student could be fielded), and lowest in office hours (a 1 on 3 conversational environment.) I was fascinated by this; my students knew they could come to my office hours for anything, even a recap of the lectures that they'd decided not to go to. Very few did, despite it being the most active us of their time, the environment where they could ask the most questions and address their own confusions. The worst opportunity in this sense -- lecture, where they had no way to engage but to passively listen and hope they didn't get lost too quickly -- was the best attended. It seemed to me that by putting students, in an environment that prioritizes the lecturer above all, they will unwittingly waste their time trying to make use of the lecture, absorbing 15 - 20 minutes of material for the hour they spend there.

7. Demoralizing students

Midway through the class, one or two female students told me, independently "I used to think I was good at math until I came to MIT." It made me crazy to hear -- I wanted to explain to them, as succinctly as they told me their self-doubts, all of the systemic things I'd been noticing and explain that they should by no means take this class and its grades personally. The impact of struggling in a class is huge -- the take home message is not "I have not understood as much vector calculus as some other people in this class." It's "I'm not good at math." I always wondered if it was just coincedence that the students who told me this were female, or if it was the result of carrying the weight of the stereotype "woman aren't as good at math as men" around, and finally they had an experience which confirmed it.

I remember wondering one weekend why we (the 18.02 staff) were stressing people out so much about things like finding the volumes of arbitrary shapes. I got to a point where the whole class seemed preposterous, and eventually even repulsive. For the final, one student told me that his plan was to get no sleep the night before the test because if he studied, and then slept, he'd forget all the equations he'd just learned at wouldn't be able to use them on the test. Another final came back stained in Pepto-Bismol because the student had brought it with him to the exam, trying to calm his stomach down from pre-test anxiety and vomiting. What was the point of all this? To help people learn how to calculate flows and volumes?

I came to see the course as something that the students didn't need and not designed in their interests. It was taught and it was taken because it was required of both parties. It's main result wasn't to get people excited by these ideas, or even to understand them, but mostly to pretend that they knew what was going on and wait for the class to stop. In the meanwhile, it was a thoroughly demoralizing experience for them, one with repercussions -- making math seem impossible -- that go exactly against the point of having the class in the first place.

As I finish writing this, my great hope is that this analysis will help someone understand, a little more closely, what is happening to them in their own struggles in college. I think that this kind of system is incredibly difficult to see when you are inside of it, and I hope that my different perspective from being a TA, and from thinking about education non-traditionally, are of use. I am really happy to talk more about this if it strikes anyone -- just leave a comment or send me an email.

Monday, November 17, 2008

my own experiences learning: part 1 -- Germany.

Much of my views on learning and education were formed through two powerful learning experiences. They fully changed my mind as how I learn, how quickly I can learn, and how others learn. They helped me value one's own intuition for how the learning process is going above and beyond any system or institution's feedback. Said another way, it is way more important that learning feels fantastic than getting good grades. The first experience was going to Germany my junior year in college, and the second was being a TA at MIT the next year.

Germany: learning how I learn!

1. Getting ready to go.

During my first semester of junior year I was at MIT majoring in theoretical math. I was taking lots of classes and I was putting lots of time into them and yet I was learning very little and retaining even less. I felt like I was never getting enough done and in was in general quite unsatisfied. I'm writing from the personal perspective, but in my experience this has been the perspective of every MIT student, whether or not they're aware of it.

Personally, I knew I was not learning nearly enough for the amount of effort I was putting in, the work felt like work (it was not satisfying), and I began to have the suspicion that not only did I not know why these mathematical tools -- the material I was studying -- were important or worth working on for a lifetime, most of my students and professors didn't know either.

I wanted out pretty badly, and somehow stumbled upon the idea of going abroad for a semester. I talked to my friend and fellow math major Ananda about it at dinner one night, who felt similarly about MIT and was game for trying to go abroad. We chose Germany, for no particular reason beyond an opinion from an advisor that they had good math programs in Germany. We talked to the study abroad officer, who while being puzzled that we didn't know any German and yet still wanted to go to Germany, got things set up for us and we were set to go for the next semester.

We got to work on learning German, starting with Pimsleur's language tapes. Some of the tapes were from the 80's or 90's, and had you repeat sentences like "I love to work on spreadsheets! Do you have the floppy drive?" They were an excellent introduction for us -- with no tongue, ear, or mind for the language -- by using a call and response system that built up a simple vocabulary of tourist-useful phrases ("I don't speak much German"/"I'm from America"/ "Where is the ____?," and so on.)

We got a German tutor who met with us twice a week the month before we went. She gave us a wonderful crash course in all the different cases and tenses, all the rudimentary tools we needed to decipher a sentence, given a German-English dictionary and ample time. She has us read a kids' book, played lots of fun games with us, and was overall really fantastic.

And then we were off!

2. The first weeks in Germany

The first week in Aachen was pretty overwhelming and bleak. Our German was really weak then and we couldn't hold a conversation. We hadn't realized that we were in effect leaving our warm, co-operative home, all the people we knew, and a common language all behind by going to Germany until we got there. This was compounded by the fact that we arrived two and a half months before the school semester started, at the beginning of the inter-semester holiday period. The idea was to give us time to learn German, but it meant the dorm we were staying in was practically deserted , compounding the feelings of being lost and lonely. It felt like we had done something massive and gotten in way over our heads.

Things started to settle as we slowly made sense of the city -- where the supermarket was, how the town was organized, how to use the busses, and so on. The math department at the RWTH, where we were studying in Aachen, gave us an office to use, a kindness Ananda and I were really surprised by and thankful for. We had signed up for an intensive language class that would meet four hours a day, and we'd expected to fall into a rhythm of filling up our days with the class and its homework, building a daily routine around that, and feeling a bit less on edge about not knowing what to do with oneself in a place where you didn't know the language.

We went to the first day of the language class. The beginning of the class was spent with administrative work (the teacher wanted to split the class into beginner and intermediate sections, but no one wanted to leave the faster paced class.) We did a worksheet conjugating strong verbs in the present tense (Ich trete/I kick, er tritt/he kicks, that kind of thing) and took turns individually reading sentences out of a book out loud.

The class felt really ineffective to me afterwards. Watching a Chinese student read out loud, I realized the pronunciations difficulties she was going to have were going to be completely different from mine, which was interesting but not useful for learning German. I had really disliked the worksheet, and while I couldn't articulate why, there was something about it that felt like a trap. I couldn't do all of the exercises in it, but the natural idea that one should therefore do more of it felt wrong to me. The worksheet felt like a mechanical struggle -- there didn't seem to be any path to doing such worksheets other than memorizing and trying again.

The whole experience felt like a really poor use of time, and I told Ananda what I thought. I was really cranky by the end of the class, saying things like "We're not going back!" Ananda and I talked about it and settled on trying it a second day. We both acknowledged that being around classmates who also spoke in broken German was a relief and we tried it a second day. That was our last in the class. I can't even remember if we stayed in the whole class, but it was more of the same and it became clear that this wasn't going to work for us.

3. Learning on our own

The next week I remember being quite nervous. Ananda and I weren't taking the intensive German class, but of course still wanted to learn German. How were we going to do that? My own nervousness was heightened by dragging a friend along with me: while my own dropping out of a class would've been one thing, I had a convinced a friend to do so too and felt responsible for the situation we were in. We wanted to learn German, didn't know how to, and had just jettisoned the plan we'd been relying on to do so.

Gradually, over the week, we found ways to fill our time up with German. Each activity seemed like a stroke of genius, and then quite obvious afterwards. We went to the library and checked out lots of kids books and some audiobooks. We found a few international student groups and would go to pubs to meet up with them: while we couldn't talk to Germans yet beyond the beginner's exchange of pleasantries -- "Es geht mir gut, und dir? / I'm good, and you?" -- international students would happily talk to us at length, despite how slow our German was. We started watching Hollywood-produced movies in German (which was the majority of what played in theatres,) which was a surprisingly immersive experience: without all of the foreshadowing and clues about what's coming next, movies that would have utterly bored me in English were spellbinding in German.

We began to fill our days with these sorts of activities, happily rotating between them. We intermittently had a German tutor for two hours a week who would answer our grammar questions and clear up confusions we'd had but couldn't find the answer to. As our German got stronger we bought a TV and watched the news. We also found tandem partners: native speakers who would practice with us for an hour in German in exchange for an hour in English. As time went on we stopped seeking out tutors and directed all of our questions to the tandem partners.

The days went by quickly and happily. The learning was so fast and clearly palpable -- every week a new skill or a new idea would be mastered, practiced, and in our active usage. With so many different ways of approaching German: reading books, listening to audiobooks, watching TV or movies, speaking to people, and a bit of writing, we were able to round out practicing all the various parts of a language, and never got bored.

The most amazing test of our German knowledge came with taking math classes in German. (though not all of our classes were in German. One prof. lectured in English solely because Ananda and I were in the class. Most Germans have a conversational command of English from their schooling, so it wasn't crippling to his students, but to give an entire course in a foreign lecture just for two guests blew me away. What kindness!) The first month of classes was sluggish -- lecture was often spent writing down new words to look up after class. By the second month we began to understand our classes in German without too much effort. In the last month, I was able to converse with other students about our homework. Understanding lectures on partial differential equations was astonishing to me. It was very empowering!

4. What made it go so well?

This experience has been a personal reference point to me -- comparing how I feel in a given situation to how I felt learning German is a way I check to see if I am learning as best as I can. The whole physical and mental feeling and experience of being there has been an invaluable tool to me over the past 4 years.

This has led me to analyze the experience deeply and to break it down into the individual factors that made the whole situation so effective, so that when a current situation isn't going as well, I can look concretely at what factors are different and what potentially could be changed.

Learning the parts of a skill in an order determined by using the skill itself: I mentioned above not trusting the conjugation worksheet but not being able to put my finger on why. I found that as my German progressed, the order I was learning the language in was quite different, and in some cases inverted, from the order it was traditionally presented in.

For example: the traditional method has one learn how to say a short, simple sentence perfectly, like "I want to borrow books from the library." To say this sentence perfectly, you need to be acquainted with the accusative and dative cases (for direct and indirect objects) and how to decline plural nouns. You'd first have an introduction to the dative and accusative cases and how to decline plural nouns, be presented with charts for memorization, and then do a lot of exercises that amounted to memorization practice.

In real usage -- i.e. talking and understanding German -- we found that the most important thing for being understood was the idea of putting verbs in 2nd position. Basically, this means that the verb needs to be your second grammatical unit in the sentence -- you have one grammatical unit before the verb (like the subject, object, or adverb), then the verb, and then your sentence goes on. Your verb in the second position is what the rest of your sentence pivots around. This is different way of thinking about sentences than in English -- so much so that I had trouble with English for a day or two on my return, still being in this habit.) And, unsurprisingly, if you don't do this, no one can understand anything you say in German. It's like saying "Tomorrow work I late." in English instead of "I work late tomorrow." The word order is one of the first big things that needs to be right in order to communicate successfully.

This rule -- one that is regarded as an intermediate rule in a normal German class' sequence -- was one of the very first things we learned. And there was tremendous feedback from learning it: by constructing sentences this way instead of translating word-for-word from English, people instantly understood us much better. It was a sudden, magical shift. With the feedback being so strong, the idea stuck permanently. I can't imagine lapsing into thinking "was it 2nd or 3rd position that the verb goes in?", though this would be unsurprising had I memorized this rule out of context.

In this same light, learning the details of when to use what case and what gender (German nouns have three genders -- male, female, neuter -- which come up pretty much whenever one uses a noun) was one of the last things that fell into place for us. It actually started happening on its own around our last month in Germany -- Ananda noticed one day that her cases were coming out of her mouth correctly, with no particular effort having gone into cases or declinations beyond the general daily effort of communicating in German.

It made sense instantly in light of looking at how important it was to communicating: Germans could understand us fine when we swapped "den Tisch" for "dem Tisch" (different cases), or even "die Tisch" for "der Tisch" (different gender.) If the rest of the sentence was constructed correctly, the meaning would be clear and it was just a simple sound that was off. On the other hand, without putting verbs in second position, our words were in the wrong order and we were practically incomprehensible.

So looking simply at the rules of German -- we needed to learn the rules that had the largest impact first. Putting words in the right order was much more important to being understood than the details of genders and cases, and so it made sense to learn word order first. This is a completely different sequence from how German is traditionally taught -- where attention is paid to cases and genders first in order to construct perfect simple sentences, before more "advanced" ideas about word order are covered. I think it's because this order is out of sync with what's most important to communication that tables and memorization are used to teach this material. Simply using the skill -- using German -- is a natural way to sort out what's important to learn when.

Having a learning partner: One of the big things I've taken away from being in Germany is the tremendous value of having someone on relatively equal footing to think with. 90% of the time, when Ananda or I couldn't understand a sentence or a grammatical idea, we could figure it out together. The rest of the time we had a tutor or a tandem partner to save the question for.

This meant a few things. One, we would save our own questions for each other -- we had an instant resource whenever we were stuck. Since we had a similar level of language comprehension, understanding each other's mistakes was quite natural. I once asked a tutor why she kept using the verb "to cook" (kochen) all the time -- it seemed to mean something like look, and she used it all the time, like "cook something in the dictionary" or "cook something over there". I figured that this usage was simply a colloquial use of kochen that didn't make sense literally. The tutor had no idea what the hell I was talking about. Through discussion, Ananda and I were able to generate enough examples for her to recognize that I meant "gucken" (which indeed means to look.)

This kind of mistake is easy to puzzle out with someone who has a similar understanding as you do (in this case, a sufficiently untrained ear to hear g's for k's) and find different ways to articulate the question in a way that an expert can understand: without several examples, I think the tutor would have just been baffled. Upon arriving at the answer together, we realized not only that a different verb was being used, but more importantly that we couldn't tell hard g's and k's apart.

In general, questions for native speakers came ready with examples and a clear explanation of what we were confused about. When Ananda and I would get stuck, we would generate examples and clarify the question as we tried to figure it out. This made our use of the expert's time very efficient and receiving the answer for us very exciting -- finally understanding something we'd been wondering about for days. It also helped us remember the answer: it's hard to forget something you've put so much personal effort into.

Finally, there's the structure that two people can create for each other. We informally agreed to work on German from around 8 am to 6 pm every day -- going to our office to do so -- and would occasionally do more in the evening if we felt like it. Having another person really kept me to this structure: while it took getting up a bit earlier than I wanted to, our overall progress was so rewarding that I was eager to keep the going. The human aspect of having a partner -- someone to bounce off of when feeling confused or tired or just out of it -- also makes the whole process much smoother: learning German was more like hanging out with someone than a force of will.

Having lots of time to practice and a high "practice time:instruction time" ratio. This is of course the whole theme of what I'm saying here -- by using German and finding ways to make it work with our skill level, we learned quickly, easily, and happily. In particular though, we had one or two hours of time with an expert (a tutor) for about half of the weeks we were there (we didn't have a steady tutor.) Teacher-student exchanges often fall into the rhythm of "You should learn this skill: let's introduce it" on a first meeting, and then on a second meeting, with the learner having had very little time to practice or use the skill, a repeat of the ideas of the first meeting with some sort of drill-and-practice routine.

In our situation, the lessons were largely spurred by our own questions, and the ratio of hours of practice to hours of instruction was at least 20:1, and usually more like 30 or 40:1. This meant that in the time that we were given a new idea or insight into how the language worked, we'd had time to try it and practice it, often to the point of mastery. Ananda and I would discuss any questions that came up, and usually work on figuring out a new skill. Our next lesson would start with resolving any last questions about the old skill and quickly move on to questions about a new skill we were working on. This kept the instruction from getting stale and made the instruction invigorating for both teacher and learner -- rapid progress is exciting to experience and to work with. It also kept us out of the trap of "I just didn't have time to do this/If only I'd worked harder" of traditional student/teacher interactions. I believe this happens when learners haven't had enough time between instruction to try something out on their own. This can derail a learner's path from making their best efforts on learning something to wondering why they aren't able to do something, taking their attention away from their learning process and into self-doubt.

I have found looking at the ratio of practice time:instruction time to be a valuable one when understanding why classes are working well or not. I believe people need more instruction at the beginning of learning a skill, when they have no foothold from which to practice and to experiment, and less and less as they find more and more to explore and question on their own. This is one reason that I think the Pimsleur tapes were so effective -- they were a self-directed way to get a lot of instruction on listening and speaking, via listening to phrases and repeating them, and could be done on one's own, whenever you wanted. They really helped us get an ear for the basics of the language, which becomes an entry point into deeper activities like listening to audiobooks or movies or having introductory conversations. One of the great things about Pimsleur is that while being able to listen to a phrase, repeat it, and listen to it again is a great way to begin to understand unfamiliar sounds, it's something that would be really tedious for a German speaker to work on with a beginner and a difficult resource to find otherwise. The tapes were an ideal format for that kind of instruction.

Being in control of feedback: We were free to make commitments and break them as far as learning German went. The first example was with our intensive German class that we dropped, but this happened with a number of other things. There was an international students group that hosted day-long outings. The first was fantastic: we were dying to hang out with other people and talk to people that would tolerate our slow, fragmented German, and were absolutely thrilled with the trip and level of conversation. By our third trip -- two months and lots and lots of German later -- we were beginning to feel like the general pace of conversation was excruciatingly slow for us, and so we stopped going.

Here's another example: tandem partners became the heart of how we learned German, once we could comfortably hold a conversation in German. The first few times we met with a tandem partner were like using a muscle you've never used before: we were incredibly drained afterwards and our minds were quiet and exhausted. While there was a value in watching ourselves become more and more comfortable learning German, and that first meeting showed us how difficult speaking in German for an hour was at first, it took us a while to really get into tandem exchanges simply because it took our German a few months to get to the level where it was a useful activity (and not just a taxing one.) By being able to control the frequency of these meetings, we were able to make use of them when were ready to.

The point is that we were in control of all of these interactions. What went well we did more of, what didn't we did less of, and we changed paths as we felt we needed to. By having complete flexibility, we stayed on a path of learning German as well as we could, and were very happy watching our progress unfold so smoothly. We never felt the typical institutional trap of "I have to do this", agreeing with the long-term goal but not the short-term methods. When the short-term methods needed changing, we could change them!

It all felt easy: The first week of learning on our own, and figuring out what to do, was tough. From there on out, things got easier and easier, until it began to feel really natural and almost effortless. It was such a rewarding pattern that it felt satisfying to work on in a long-term sense, and it consisted of doing so many fun things that each day was a joy. Gone were the feelings I had at MIT of "I should be working more" and "I'm not getting enough done." We felt like we were getting tons done and were really content with it.

In my experience, this is how every good learning experience feels. It feels good. Starting camp was like this too -- it had the same hallmarks of intense engagement and intense immersion (too busy doing it and loving it to question whether I'm doing it to the best of my ability), and similar daily bursts of understanding -- in this case about how to excite children about making things.

I'm really grateful for the experience -- I had no idea how powerful it would be. It's been extremely valuable to have this experience as a reference point for how to learn and how to live. It helped me trust myself as a learner and doer, and to trust the systems I was in less instead of trusting myself less when learning and doing wasn't going well.

Thursday, November 6, 2008

quick thoughts on tutoring math

A friend of mine wrote me an email asking about my thoughts on teaching math. Having thought a lot about math (that's what I did in college) and teaching, I've got lots of thoughts!

Below is a simple summary of my broad thoughts on tutoring math. I'm also working on making an instructable of a bunch of the math games I refer to below.

"> hey, so how did u get into teaching math to kids? i'm trying to figure out
> what the heck i should volunteer for while i'm stuck at home, and daniel
> gave me the good idea of teaching math, then i remembered that you did it.
> what are your opinions and experiences?

I did some tutoring in college. I think that tutoring is an excellent way to learn how to get used to the rhythm of explaining an idea, watching the person's feedback, learning how to read signs of confusion, and so on.

I also think it gets boring really quickly, because usually it's a matter of helping a student with a class they're taking. My big thing with teaching math is that it's got to have some inherent value for it to stick with a person, which reduces down to it either needing to be practical or beautiful. Tutoring often feels like showing a person the mechanics of a system when they're not ready for it -- they're asking for
the information not because they need it, and not because they're interested, but because someone else is making them.

I taught some group math classes, and those were a lot more interesting to me. It was with kids ages 6 - 8 and 9 - 12. The goal there wasn't to teach computation/arithmetic, but to get kids excited about doing math. We did lots of puzzles and games, and the guideline for me was how interested the kids were. That was really enjoyable. I started with a big collection of puzzles my friend used, and after a few months made up my own. Making up puzzles that kids liked was a very satisfying experience.

The other good thing about doing group classes (I taught 8 kids, sometimes with an assistant) is that you can charge much less -- $10 - 15 / child is much more affordable than a reasonable tutoring fee, and a lot more affordable for the family. I also think the experience is better for the child -- they have other people to play and think with, and better for the adult: in a 1-on-1 scenario, the adult is
ideally giving the child a lot of time to think on their own, and in a group setting they're not waiting idly for a single child.

Those're my thoughts. I'd be happy to share materials/puzzles with you were you interested.