Education

You are currently browsing the archive for the Education category.

Scott Brown, why do you hate America?

Tuesday 25 October 2011 in Education, Politics, Society | 2 comments

Recently the Democrats introduced a bill to support teachers, police, and firefighters. It’s really short, so you should read it for yourself here. This law would help out the poorest people in the country. And do you know who they are? Children. Children are the largest demographic living in poverty in America today. They are not lazy. They are not riding welfare. They didn’t lose their jobs. They are children. And this law would help children.

And who would pay? According to this bill, only people earning over a million dollars a year. And the tax would only apply to the monies made over a million dollars. Joe Biden explains how easy the math is. If you’re only making $999,999 this year, don’t worry. You wouldn’t pay a cent to fund these teachers and first-responders. And if you do make more than a million dollars, you’d only see an increase of one half of one percent. That’s half a penny per dollar over a million. The average income of the people who would pay this surtax is 3 million dollars; they’d pay on average $500. That’s enough for 400,000 teachers, 18,000 police, and 7,000 firefighters. I only wish I could pay this tax!

But no one will. It got voted down, pretty unilaterally according to party lines in the Senate. The New York Times posted the tally here. All blue and independent voted yes; all red (and two blue!) voted no. Because I’m so baffled, I decided to write my senator to ask what he was thinking. I hope you will ask your senators why they think half a penny for every dollar over a million isn’t worth the stability and safety of our families and children. If you live in Massachusetts, please contact Scott Brown through his website or on the phone, (617) 565-3170.

Dear Senator Brown,

I am troubled by your recent vote against S.1723: Teachers and First Responders Back to Work Act of 2011. Please explain why you chose to cut jobs that serve obvious public good, promote American innovation and prosperity, and keep families and children safe. Teachers, police, and firefighters from the poorest communities will lose their jobs first because the poorest communities are those least able to support their salaries. Your vote is strictly un-American and it clearly acts against the best interests of Massachusetts’ citizens.

Please explain why you turned on American families and children, Senator Brown. Why did you deny poor American children access to adequate education; why did you deny working American families continued public safety? As one of your constituents, I believe you should make the reasons for your action clear.

Sincerely,
Joshua Reyes
A concerned citizen

Please, leave a comment if you contacted your senator!

Privilege (Part 1): Feminism takes women for granted.

Monday 9 May 2011 in Education, Society | No comments

As a resident tutor at Harvard, I get tagged to help students prepare for all sorts of things. Giving mock interviews is my favorite student service. Firstly, I honestly believe that students who master this form of conversation will be better prepared to navigate the system after graduation. And that’s important to me. Plus I love playing the part of a disinterested panel member who mercilessly questions a student about deeply meaningful issues. As a bonus, I usually learn something new about my student and the world in the process. Earlier this spring one student asked me to quiz her before an interview for the Women’s Leadership Award. (Sadly, she didn’t get it.) It was my job to ask difficult but thoughtful questions as a representative of the liberal left to a self-proclaimed traditionalist from way over on the right. And so I did.

I started off with what I thought was a low-ball question, “How has your leadership advanced, strengthened or otherwise benefited the causes of women on campus?” After a few long pauses, I reframed the question. “What is feminism and how is your leadership aligned with your definition?” Now I should admit that I have no formal training in women or gender studies, or, for that matter, in any other field that qualifies me to judge her response with any authority. I was just genuinely curious to know her answer and thought her interviewers might be interested, too. So I wanted her to have an answer ready at her fingertips for the real thing. But when she started to define feminism in terms of women, I thought her description might be too limiting.

As I see it, feminism is a misnomer. The field studies the general mechanisms of oppression and societal priority and, if done well, tries to figure out how to redirect and/or obliterate those forces. Without knowing anything about them, I’d say the same goes for African American studies, queer theory and the like. They might differ in the details, but the spirit is the same. That’s why WGS departments routinely host courses on masculinity, reality TV, urban sprawl, colors, and boats. The point is to figure out what society thinks is important, how those prioritizations are implemented in daily behavior, what the consequences of those choices are, and then to do something about it if necessary. Each of us makes lots of choices everyday—from what to eat, where to be seen, what to wear, and who to smile at—and all of those choices communicate information. Consequently, we should rename women, gender, and sexuality studies to something more general, less political, and more honest. How about ‘social dynamics‘ or ‘advertising’?

Because it deals with the fundamental question of how large groups of people value things in society, gender theory is widely applicable wherever large-scale disparate treatment exists. Since taking that course this winter at the Ed School, I’ve decided that urban education tries to tackle fairly identical problems. Our urban public schools are flooded with poor students of color, while their privileged white (putatively all-male) counterparts escape to a protected life in the suburban lands of milk and honey. How did this happen; how can we fix it; and what do those questions even mean?

The rhetoric in class focused largely around some vague thing called privilege. As far as I can tell, privilege in this context amounted was code for having money and the things that money can buy. And there’s compelling evidence that at least some differences in education are simply a matter of money. The international PISA test results are pretty damning when disaggregated by poverty rate. It goes without saying that the physical conditions of many urban schools are deplorable. What these kids endure is heartbreaking, criminal. But to write off the problem to a matter of resource allocation is lazy, a little self-righteous, and doesn’t help kids. What’s money good for if you can’t keep it? The staggering poverty of some former NFL players provides a good case in point. What we need is to think carefully about the systems that generate and maintain wealth for some but not others. As a clumsy first step, we need to unpack that elusory notion privilege.

At the end of the day, privilege is the ability to avoid hardship. Money can get you out of a lot of difficult situations. That’s why I think that people often regard privilege as a heritable good like money or real estate. But its power really derives from interpersonal interaction. No man is an island; it takes two to privilege. As I see it, privilege is different depending on who originates the interaction. In one case, privilege is granted—like when a car stops to let a pedestrian cross the road. In the other, privilege is exerted—as when a family moves from one neighborhood to another that suits them better. Of course the two types are intertwined. In the first example, the driver noticed the pedestrian and stopped, granting the right of way. The pedestrian had to recognize the situation and then exert her privilege to cross the street. Like I said before, it takes two to privilege. More complicated feedback can and certainly does exist in real life. In this post I’ll focus on granting and leave exerting privilege for another time.

In education, it’s useful to know the sources of privilege so that we can teach our students how to leverage what they’ve got. If a student has good teeth, her smile is going to gain her, largely through the accident of her birth, an upper hand over her peer with a crooked smile. People will trust her more. She’ll get more attention from teachers and bosses. As a result she’ll be rewarded for the success others, in part, have granted her. If she learns how to exert this privilege, she can cash these chips in for more success down the road. Once you prime the pump, success flows more easily. (Anyone who’s been job hunting knows well that a good resume begets a better resume.)

And so positive feedback will reinforce the small but noticeable advantages she started out with, like her winning smile. We might believe her success comes from dispositional traits, like hard work and politeness—and to be sure, those pieces need to be in place as well. That’s the American Dream, after all. But we need to step back and see the system for the trees. People respond to visible traits like speech, posture, height, hair style, skin color, gender expression, and dress. They read social cues and reply in turn. When lots of people all respond to these cues in a sustained and coherent way, they generate full institutions of systematic advantage and disadvantage. That’s what urban education is trying to understand and rectify. But it’s hard not to judge a book by its cover. If you know that, it’s easier to game the system. I’ve personally benefited immensely from misread social cues, which I’ve gone on to exploit.

On paper I fit the urban stereotype, minus the urban part. I’m Mexican, from a poor, single-parent family in a rural suburb of Boston. My friends and I drank in the woods to the light of homemade bonfires on the weekend. But I look white. My body type is slender; I have blue eyes, fair skin and brown hair. People, even Europeans, regularly assume that I’m British or Scottish. (I’m not, but thank you.) In college I learned to play squash, developed a taste for sherry, and started wearing bow ties to formal events. When I walk into a room, people grant me all the privilege a white man could want, even if my family were homeless for a time while I was away in college. What’s on the outside matters first, because that’s what people experience first. In my case, my exterior purchases me privilege because that’s what people are willing to pay to someone who looks and acts like me.

Changing what society thinks is worthy of reward is a long, hard seed to sow. And we need to continue on that front. It goes without saying that the same positive feedback that has helped me time and time again could easily have hurt if I looked different, talked differently, or dressed differently. Systematic disadvantage is real and consequential. But as educators, we can hasten the process of useful change if we teach our students the rules society plays by explicitly. We can give them generative tools to acquire and maintain privilege for their own ends. It doesn’t require expensive smart boards or computers or even books. That’s why I love giving mock interviews, write tutorials for paper writers, and askers for letters of recommendation. The best part about this approach is that the rules themselves can help to suggest actionable points of intervention. (Yes, I know I haven’t pointed any out yet. I’ve been setting up the system. The opportunities for change are coming, I promise.) But we’ve only read half the the story. Students need to be able to generate, identify, and exert privilege when the circumstances are right. And who better to teach them than educators?

The Rewrite

Saturday 30 April 2011 in Education | No comments

Recently I promised to follow up on just how I rewrote that final paper for the course I took at the Ed School this January because I think the strategy I used probably works for many of the classes at HGSE and, to my dismay, many other classes all over the place. In the tutorial below, I walk through how I wrote my paper to show you exactly how these methods might take form in practice. After all, worked examples bring principles to life.

I couldn’t dedicate a lot of time to redraft my paper. Although I worked hard to come up with a new idea that synthesized what I had learned during the course, to demonstrate more fully that I had drawn on the sources explicitly for my inspiration would require a lot more thought and time than I had to spend fixing up a paper for a pass/fail course that I took for fun. I already came up with new ideas and understood old concepts differently. Now I needed to sit down for a few hours, write fast, and be done with it. So I decided start over from scratch. Here’s what I did:

  1. Don’t pick a thesis topic. Instead claim that the situation at hand—in this case, the historical blight that continues to trouble urban education—is complicated and that the solution must therefore be complicated, too. If you say anything more specific, then your grader can argue with you. Eschew statements of substance. It’s impossible to wrestle with a ghost.

    To combat these [aforementioned] structural evils, we must take a multi-pronged approach to balance the differences in access to and use of quality educational resources (including but not limited to adequate school buildings, textbooks, well-trained teachers and supportive administration) that have accumulated throughout the history of our nation.

  2. Next summarize each of the readings one at a time, one paragraph at a time as they appear on the syllabus. Restrict your attention primarily to the readings you did. If the syllabus is split into sections with obvious themes like “Segregation” or “Finance Reform”, preface those summaries with a paragraph explaining why that theme is important.

    To understand why my and many public urban classrooms around the country have a disproportionately high representation of poor and students of color today, we must look into political choices of yesteryear. While it is common knowledge that it is illegal to legislatively mandate segregation, other perfectly legal social forces can still institute de facto segregation silently and efficiently.

    Be sure not to inject your own thought. Summaries should not introduce new ideas or material. Tow the party line. The readings were selected because they are important. Demonstrate that you understand how important the readings are by rephrasing their main points. Again, try to leave out substance whenever possible. Justification and nuance only give the grader something to disagree with. You can get the details wrong even if your summary is correct.

    For example, if you ask a kindergartner the shape of the world, she’ll invariably respond, “It’s round.” That’s what we teach kindergartners, after all. And her answer is correct. If you push a little further, though, and ask her to draw the shape of world, she may very well draw a pancake. Round, but wrong. What’s the moral? Elaboration is dangerous. Simply restate generalizations mentioned in class without backing them up. The teaching staff will assume you know what you’re talking about.

    Multicultural education offers a hope for real change in the lives my students. [...] By presenting academic heroes and ideas from a diverse range of ethnic and cultural backgrounds, we validate the identities and experiences of our students. Students from marginalized groups will be able to see themselves in the narratives of a host of historical figures and interpret their stories into realizable, personal action.

    Aside: It pains me to offer the above quotation without qualification. Saying that students will automatically learn from role models just because role models are around is shortsighted. Such a strategy is just as effective as locking up kids learning to read in a library because being around books confers kids with ability to read. Sadly, that’s just not how the world works. Using resources is tough stuff. That’s why teachers exist, at least in part. Otherwise we could just shut kids up in prison cells outfitted with the newest and best educational tools and expect them to emerge smart and successful. But when you’re writing a paper, it doesn’t matter if your argument makes sense or lacks evidence, as long as it sounds good.

  3. Sprinkle in numerical examples whenever possible. The earlier you can unload a statistic into the paper, the better. Numbers always make arguments more authoritative. You can easily pull numerical examples from sources you didn’t actually read ahead of time. They can be used without presenting appropriate context and they quickly add to your paper’s reference count. In the following example, the statistic I quote has no direct relation to my claim. But the words sounds more or less similar, so the numbers do their job: my argument carries force. Notice how I follow up with another another sweeping, data-free assertion? It doesn’t follow from anything I’ve mentioned, but that’s beside the point. Numbers are really powerful. Use them to your advantage.

    Racially and economically segregated neighborhoods immediately translate into racially and economically segregated schools. In 1997, for example, Public School District 65 in the Bronx enrolled only twenty-six of the eleven thousand elementary and middle school-aged children who were white. The result was a legal segregation rate of 99.8 percent (Kozol, Shame of a Nation, CP p. 4). As a matter of course, such segregation persists in public urban university classrooms for much the same reasons.

  4. Acknowledge your own personal failings. Without being specific, produce broad mandates that will make up for your human frailties. Papers in education need to draw on personal experience and exhibit emotional appeal. Try to make your writing obviously smack of social justice, even if you end up being backhandedly racist. Any sentiment backed by good intention is ipso facto valid in this sort of class. The image of the noble savage was an especial favorite in this class. Of course you’ll need to swap out the word savage with something more suitable like “diverse”, “urban”, or “disadvantaged”. Do not define your terms. It’ll only make you sound condescending and/or prejudiced.

    As an instructor at an urban public school it is vital that I understand the social forces that have shaped my institution and positioned the students in my classroom so that I can be sensitive to their uniquely “urban” needs. To this end, as an urban educator I must celebrate the diversity of my students and expose and repurpose the mechanisms that have nurtured systematic inequality in order to level the playing field for my students.

  5. To illustrate the validity of your thesis, include at least one source to disagree with. If you can show that someone else is wrong, then you must be right. This tactic has been enormously successful in all sorts of political campaigns. (E.g., if evolution can’t explain the origin of life, then God must exist.) Moreover, there’s a fantastic chance that your professor will indicate which of the readings is wrong in lecture or during a class discussion. Just reiterate whatever the class decided was mistaken. Ostensibly, the course prepared you to write your paper. Don’t do any extra work. It’s already been done for you.

    In fact, [Hirsch] goes further to say that to merely understand the cultural circumstances of knowledge is not enough; he believes the truly literate person can wield this background information in her writing and that this incorporation is necessary if she wants her voice to be heard by other literate people. Of course, cultural literacy is really a polite way of signifying membership in the dominant power structure. Students from diverse backgrounds have learned to think, communicate, and live differently, not worse.

  6. Lastly, address some of the shortcomings of your approach. You merely need to acknowledge their existence and provide a slightly refined restatement of your thesis as the solution.

    If you can’t come up with a chink in the armor of your framework, then try to answer the following interview question: What is your greatest weakness? Your answer should pretend that a strength is actually a weakness and then explain how in reality your weakness is a strength. You know, “My greatest weakness is that I work too hard.” “I’m a perfectionist.” Straw men always fall easily.

    [I]t is not straight-forward how to assess a very bright student who does not display her intelligence according to outmoded though prevailing measures of standard success. Strict rubrics and standards seek to normatively define achievement, rather than to democratize education (Slater, p. 20). Therefore, it is important to expose the individuality of each student by providing [assignments] that engender creative and personally meaningful expression and accompany such exercises with opportunities to explain the thought process that help to generate the end product.

  7. In your conclusion, don’t be afraid to paint that better world we all could live in. That is, if only people took what you learned in your class seriously. Again, infuse your words with the spirit of social justice for a strong finish.

    While multicultural curricula, as I have outlined here, will not immediately change the large-scale structures that have created a landscape of imposed segregation and inequality in our urban environments, the long-term, coordinated efforts to celebrate the diversity in our student populations will eventually change the way students of privilege think about and understand their diverse classmates. As an urban educator, it is my hope that I will instill the values typified by multicultural curricula in my students so that they will choose to improve society over the natural but selfish inclination toward individual gain.

Good luck and happy writing!

All animals are equal, but some are more equal than others.

Thursday 28 April 2011 in Critical Thinking, Education, Society | No comments

This winter I ventured over to a part of campus that I hadn’t explored in the ten years I’ve been hanging around Harvard. In January, I took a two-week boot-camp style overview course on the ‘Foundations of Urban Education’ at HGSE. As many of you may know, I’ve always had a sweet tooth for teaching and learning. As an undergraduate I helped teach in the Mathematics for Teaching graduate program at the Extension School. After graduation I ended up at a publishing company where I wrote the chapter exams for middle school math books. (If you’re a sixth grader in California, I’m sorry.) At the same time I started in a spunky, free-thinking masters program in education at UMass Boston before getting whisked away to teach introductory computer science elsewhere in the university. Along the way, I hung around an urban charter school in Dorchester as part of my coursework. To be sure, education is really important to me.

So at first I wasn’t sure what to make of my HGSE course. These days I spend most of my time in the lab with delicate scientific instruments, goofy and less delicate scientists, and large, slippery, and quick-moving frogs. The pressures of real-time classroom conservation in a field that I’d been away from for so long with people who live and breathe this stuff everyday was, to be perfectly frank, intimidating. We had received our course pack and reading list in early December. Because time was at a premium, all of the lectures were prerecorded and posted ahead of time. And the teaching staff encouraged us to hit the books nearly a month before the first day of class. Yowsers. I had a lot of catching up to do.

But then the course started. I assumed course discussion would reflect the same sort of openness and thoughtfulness I had enjoyed at UMass. But if I learned anything in those classes, it was that I need to dispense with assumptions, suspend judgment, and, as we say sometimes in biology, let the data speak for itself. And, oh, did my classmates speak.

Never before had I encountered such persistent intellectual bullying. Not just at Harvard, but anywhere. It was shocking to me that some ideas could be heretical; certain topics entirely taboo. The main theme of the course was exactly what I expected: there are large groups of people (mostly blacks and Hispanics) who have been systematically disadvantaged throughout their history in this country. On the other side, another group (of wealthy, white men specifically) has manipulated mainstream social and political structures so that their children are systematically advantaged. To level the playing field we need a swift injection of money and multiculturalism. All of this seems completely reasonable if done reasonably. I want to give a voice and power back to the dispossessed; don’t you?

The sermon was predictable. Course material provided us with sound bites that we could wield quickly in a pinch. But the way I was supposed to think was fundamentally unchanged. In fact, it felt like that was by design. Opinions that weren’t recognizably aligned with the gospel truth were denied flatly. Those ideas that actively disagreed with mantra of the noble but disadvantaged youth—savage is no longer politically correct, but the sentiment is—were silenced. My classmates rode around on the white horse of moral supremacy to quash discussion and avoid making concrete suggestions for fear of criticism. Whenever someone took a definite stance, someone else inevitably asserted their fears that the dominant power structure was secretly creeping in to rob the poor of their humanity. Now don’t get me wrong, many times that was exactly the case.

The extent to which people who extolled the enlightened practice of listening burrowed their heads in the sand to hide from new ideas would have been laughable were these people not actual educators who interact with actual children. In group break-outs, my classmates railed against me (and other heretics) with passion but without evidence. During an exercise on curricular planning, for example, I suggested that mathematics isn’t itself hierarchical and that our classes need not be. Algebra doesn’t need to precede geometry. We just insist it does because of an accident of history that has been frozen into the curriculum. I didn’t mention my experience in math education. I wanted my ideas to stand for themselves. A classmate of mine insisted that math follows a linear order. Basics first. Advanced topics later. And that’s that.

Another time, I pointed to models of inequality that abolished race but were unable to dismantle financial segregation. Consequently I suggested that we should investigate how people acquire and maintain wealth and incorporate what we learn into our classrooms. This time another student, at a loss for words, told me that segregation was about race. It just is. Full stop. My TF consistently commented that my response papers could be stronger if concluded something that I believed contradicted my main arguments. Naturally, her arguments recapitulated the party line: in this case, that honors tracks are categorically bad. Like Lisa Loeb, I was only hearing negative, “No, no, no. Bad.”

My meeting during office hours with the professor was the most surprising example of this multiple-ways-of-knowing, except-in-this-class philosophy. Initially I had scheduled time with her to talk about careers in education, but by the time our appointment rolled around it was clear that our conversation would focus on my final paper and its subsequent rewrite. I should admit two things about my final exam. First off, I was confused about what a semi-reflective, semi-analytical paper ought to look like and my first guess was bad. My paper was disjointed and poorly written. Second, I put a lot of original thought into it. The version I submitted contained what I believe to be a thoughtful proposal for urban educators that integrated, if indirectly, most things we had read, discussed, or otherwise touched on in class. In lecture, our professor asked, “how can we best respect the diversity in our classrooms?” To come up with an answer, I defined respect and diversity, drew meaningful connections between them and proposed a framework for thinking about diversity which differed usefully from those found in our readings. But my response wasn’t “recognizable” to my TF or professor. And more, importantly, it seems, my paper didn’t explicitly retell the history of inequality in American schools. They needed a book report narration to prove that I had done the assigned readings. These were important, after all. At one point during our meeting I asked directly if I should just parrot back the readings one at a time for my redraft. At this point the professor responded that she would not usually want to sound so “anti-intellectual”, but that yes, that would indeed suffice.

The point of my favorite reading, one by Hirsch, argued that in order for a marginalized voice to be heard, it needs to speak the same language that those in power speak. The class had universally dismissed Hirsch, because they claimed (incorrectly) that he privileges rich, white viewpoints. In doing so, they proved his point: if you don’t sound intelligible, no one will treat you intelligently. So figure out how intelligent people sound and talk like them, but say what you think. The conversation I had with my professor, who specializes in civic education, marginalized voices, and social justice, did just the same. My point wasn’t recognizable, so it didn’t exist. (Like my TF, she also decided that my paper was about the necessary evils of tracking.) I’ll tell you how I rewrote my final paper in case you ever take a class at HGSE. There’s a recipe you can follow. It doesn’t require much thinking but guarantees success.

Race is Different than Money

Saturday 8 January 2011 in Education, Philosophy, Policy, Politics, Society | No comments

This winter I’m taking a course on urban education. Our first topic: segregation and desegregation in schools.

Firstly, what do we mean by segregation? As a working definition, I’ll offer that segregation is the spatial pattern of people across some attribute. So we could talk about segregation by race, by income, or by favorite ice cream flavor. Once we pick something to measure against, we find that every city is segregated according to this definition. What matters is in what way the segregation manifests and the consequences on the populace the pattern has. Segregation patterns can be uniform, with all groups distributed more or less evenly within a region, or clustered. Likewise, we could also calculate the extent to which subpopulations are isolated from each other—which also gives a rough estimation of how often members of one group is likely to run into someone outside of their group. I think when we talk about ‘segregated’ groups, we typically mean highly clustered populations that are isolated from the other groups in the city.

I don’t think that clustered, isolated groups are necessarily bad on their own. I love visiting the North End and Chinatown. Because they’re both T-accessible, it’s easy for me to get there. (Though, both neighborhoods have had rough pasts.) And Harvard Square is the nicest place I’ve ever lived. Score one for segregation!

Moral judgments aside, self-selection can have a big influence on patterns of segregation, at least it can in models. The positive feedback loops reinforce small, individual choice to generate large-scale patterning. Schelling’s model of segregation is a classic, good first example of what I mean. In this model individuals exhibit only a slight preference to have neighbors that are similar to them. The individuals in this model are not racist. (Or maybe they are. I don’t have a good functional definition of racism yet.) When individuals find themselves in a neighborhood that is too unlike themselves, they move somewhere else at random, possibly to a neighborhood more dissimilar from themselves than the last. Even with this mild, partially blind behavior, a totally segregated structure emerges.

In more relaxed models that completely ignore race, even more realistic patterns of segregation form. In this class of model, individuals simply choose to live in the nicest area they can afford. As if by magic, isolated poor and rich neighborhoods form. Depending on the details of the model, wealthy suburbs appear spontaneously. If we use socioeconomic status as a proxy for race, it’s the same old story. Except this time, we have a systems-level mechanism that generates isolated, poor communities that lack the power to advocate for equitable resources and very rich communities with disproportionately high share of public goods insulated by a buffer of middle class individuals. Race was not the cause; money was.

When was ask whether it’s morally justified for a white family to send their kid to a predominantly white school, I think it’s important to know what about the school is so attractive. Do all parents value differentiated cultural and social understanding across many kinds of experience? Are they likely to value it more than a pretty campus or reputation of success by its graduates? Sure, in some cases the choice may be motivated largely by racism. But I’d expect that in many cases, it’s mostly a matter of ensuring access to the most and best resources possible for their child. It just so happens that low-resource groups aggregate, even in the absence of race.

I believe that diversity (of background, experience, perspective, and the like) is important in schools because, as has been mentioned a few times by others, students learn how to navigate social situations outside of school from the people they meet in school. But when we talk about diversity, do we really mean racial diversity? As an example, imagine that an elite, wealthy, mostly white college in the Northeast has recently been chastised for admitting a student body that is not sufficient diverse. Consequently, the school begins recruiting wealthy black students from Africa, some of whom attended the same boarding schools as students already enrolled in the college. In time, the student body comes to be half white, half black with an even mix in all classes and housing situations. In what sense, if any, has the college increased diversity on campus? Do you think the college has produced the diversity they were previously lacking?

While I think that racial segregation is a problem, I don’t think race is necessarily the capital-C cause. In a world without racism, economic segregation will still exist. But I’m willing to bet that in a world with no financial disparity, a lot of the troubles we associate with racism would evaporate. And so, I think race will play a secondary part in the solution to segregation. In fact, I think that race may even obscure the issue of access to equitable education for all. (I’m not sure if that’s what we’re really trying to achieve, but I think it’s a good start.) Instead, I believe that the struggle of the American education system is one of power and status. As such, I think we should talk about resource allocation (including strategies that move students to resources as well as bringing more resources to students), causes and effects of socioeconomic segregation, and cultural and pedagogical practices that systematically discourage/motivate students to learn the skills required to become an informed and capable citizens.

Summer Informal Seminar

Tuesday 29 May 2007 in Education, Mathematics, Personal, Physics | 2 comments

No, I haven’t been off on vacation in the Caribbean—but I have seen the third pirates movie. It was great. But I wanted to let anyone out their who is interested that I’m going to lead an informal seminar on general relativity at UMass/Boston this summer starting June 4th. Here are the details.

What: An informal seminar on differential geometry and general relativity
When: MTh 5–6:30pm
June 4th — early-August
Where: Taffee Tanimoto Conference Room
Science Center
Third Floor, Room 180
Website: www.gsd.harvard.edu/~jreyes/GR

Games: a Ludic Structure for Problem-Solving

Wednesday 28 March 2007 in Computer Science, Creativity, Critical Thinking, Education, Mathematics, Philosophy, Society, Sport | 3 comments

Today I’ve decided to post a journal together with a longer paper about games. You hear all the time that we need to inject more play into education, that we need to return to childhood, etc. But why? You don’t as frequently hear why play is useful in education. People claim things like “If learning is fun, children will learn better.” I’m not sure of the connection. I suppose that if kids are engaged in learning, then they have a better chance of actually picking something new up than if they’re not trying to learn at all. That’s like saying if you look for something you have a better chance of finding it then if you don’t look at all. Sure, I buy that. But why play? By the same argument, we could just as easily pay kids to go to school and do their homework.

Of course some people do give reasons why play is useful. In these two papers, I’m building on some insights found in a 1933 paper by Lev Vygotsky entitled Play and its role in the Mental Development of the Child. (Vygotsky, you may well know, is one of my current heroes.) I remind the reader that in play, you can find all sorts of higher-order thinking skills taking place. Imaginary play is a very natural, distilled, abstractly difficult thing to do. Yet kids seem to do it on their own anyway, and before they even step foot in a classroom. If taught effectively, I think play is a useful vehicle for transfer of skills and tons of that ever-so-hot interdisciplinary work that goes on nowadays. (Wait until I get my genetic algorithmic music up and running.)

Journal 4 Journal 4: Methodological Doubt, Belief, and the Structure of Play

Paper 2 Reflection Paper 2: Decision-making as Game: A Mode of Prediction and Solution

Peter Elbow introduced concepts of methodological doubt and belief in his book Embracing Contraries: Explorations in Learning and Teaching. They’re central to his believing game and doubting game. Traditionally, doubt has been used as the primary tool in critical thinking. This unbalanced attention really makes a lot of analysis blind to new insights that can be gleaned from a moment of pure, suspended disbelief. (My ego won’t let me pass up an opportunity to say that both games show up automatically in my coffee mug model of classroom education.)

In my first paper I remark that all games require its participants to engage in the believing game—they have to believe that the rules imposed by the game are real and that the game itself is real. There are no consequences in any game if you don’t except them. You can always pick up the ball with your hands in soccer, unless you firmly believe that you can’t. For this reason, we might frame any situation as a game.

In the second paper, I extend my ideas to show that framing a situation as a game can greatly improve your power to predict behavior and arrive at winning strategies by simply considering the acceptable moves in your game. To illustrate my point, I work through a problem of the type sometimes given in consulting or computer science job interviews. The example shows, additionally, how mathematical reasoning (which I believe is no different than plain, old, vanilla reasoning) can be used to solve a problem without once using “math.”

As always, please comment freely. I’d love to get some feedback on this stuff.

Technorati Tags: , , , , , , , , , , , , , , , , , , , , , ,

To Infinity, and Beyond! On a Llama.

Friday 23 March 2007 in Education, Mathematics | 1 comment

I have a few other posts saved as drafts, and I want to get to them, especially one on technology, but I can’t pretend to have finished talking about that alternative decimal representation of the number one (0.999···) that carries with it an infinite chain of nines. To say that I’m comfortable with that representation of the number is almost as brash as claiming to have solved Zeno’s Paradox. Infinity is a funny thing. In fact, it would be more honest to say, infinities are funny things. After all there are lots of them. And there’s no reason they should all be the same—in fact, they are not.

But before we dive off the deep end, we should pause to think about what it means for finite numbers to be the same. Have you ever given much thought to statements like “5=5″? So, now, before you read on, turn to the nearest 8 year old you can find and ask her, “Does five equal five?” [That's the easy part.] Now ask her, “How do you know?” and listen for a response. [That's the medium-hard part.] If your 8 year old doesn’t sufficiently convince you that five is, indeed, equal to five, explain to her why it’s true in plain terms that no one could dispute. [That's the hard part.]

When I was taking an abstract algebra course, my professor asked the class what the number three is. We all knew it was a trick, so we waited in nervous silence, each hoping that he wouldn’t start dead-calling members of the audience. After all, college math concentrators should know what the number three is. We were studying math, and numbers, I’m told, are an integral part of mathematics. So what was the Fields medalist‘s definition? He stole his answer from a six year old: the number three is three fire trucks without the fire trucks. Hold up, what? That’s a surprisinglyl useful way to think about number. I think that’s how Frege and Hume viewed number, and by some standards, they’re famous. So maybe this little kid is on to something.

Let’s pretend for a little while. You didn’t know this, but I have a pack of llamas. Every evening I feed them each one carrot for dessert at dinner time. The problem is, I can’t count. I have a bunch of carrots and a pack of llamas. How can I know if I brought out the same number of carrots as the llamas—without counting?

Well, I could feed the llamas each one carrot. If I had more carrots than llamas, then I’ll have carrots left over at the end. If there were more llamas, then I’ll run out of carrots before I finish feeding—the llamas hate that. But if the number of carrots and llamas equal, then after feeding time, each llama will have had exactly one carrot and no carrots would remain. That is, I would be able to put all the carrots in a one-to-one correspondence with the llamas. If I had more carrots than llamas, then some llamas would get the left-overs. That relationship is not one-to-one because some llamas get more than one carrot.

But we’ve got a problem, one-to-oneness is certainly necessary for there to be the same number of carrots and llamas, but it is not sufficient. If I had fewer carrots than llamas, every llama who gets a carrot gets only one. But because I run out of carrots before I run out of llamas, some llamas are left out. In order to know whether there are the same number of carrots and llamas, every llama needs to get exactly one carrot and there can’t be any carrots left over. This is tricky business.

If every llama gets at least one carrot, then we say that the pairing of carrots to llamas is onto. Ontoness is also a necessary condition, but like one-to-oneness it is not sufficient. Having more carrots than llamas leads to a pairing that is onto. Every llama gets one carrot and some get more. What we’re looking for is a matching of carrots and llamas that is both one-to-one and onto. Then for every carrot there is exactly one llama. Likewise, for every llama there is exactly one carrot. The size of the pack of llamas and the size of bunch of carrots is the same. Mathematicians like to have a standard way to talk to one another, so they call the number of elements in a group the group’s cardinality.

So, we’ve done it! We’ve found a way to determine whether two collections of things are the same size, if they have the same cardinality, without resorting to counting. In fact, we’ve secretly discovered a very powerful way of thinking. One thing we can do with our new-found friends, the one-to-one and onto functions, is define what it means to count. I don’t have the room to do it here. In fact, I don’t think we’re even going to get to infinity in this post. Instead, I’ll cop-out and refer you to some of my set theory notes. (That’s what we were doing: set theory.)

Set Theory Lesson Plans Set Theory Lesson Plans

In my notes, I’ve asked just a number of questions. I wrote these questions for a practicum for a class it took this fall. And I used them on real, high school juniors in at Codman Academy. We had a little bit more time to go over each of the questions carefully and resources that allowed dynamic, colorful diagrams—which the students largely produced for me. I was only asking questions, it was up to them to answer them. But please, don’t wait until 11th grade to feed the llamas.

Technorati Tags: , , , , , , , , , , , , , , , , , , , ,

Inductive Proofs, Constructive Understanding

Monday 19 March 2007 in Education, Mathematics, Personal, Travel | 1 comment

I remember when I first learned that advertisers will often use glue instead of milk in breakfast cereal commercials. The whole thing blew my mind. Initially, I felt confused. Why would they do something like that? Of course, because glue looks more like what people expect milk to look like than milk does [on camera]. Even after my personal revelation, I still felt confused. Except now my confusion came from within: why had I assumed that things on TV were what they looked like? Even now, I still feel a little uncomfortable thinking about it.

Again in the the seventh grade, another discovery left me feeling the same way: the infinitely repeating decimal 0.999··· is the same as the whole number one (1). I know that this must be true; my math teacher Mr. Heleen proved it to us. First, let’s hide the infinite string of 9s under a clean variable name, say x. Then we can distract ourselves long enough to arrive at a meaningful conclusion. Here’s what Mr. Heleen did:

10x = —9.999···
—x = —0.999···

Subtract the two lines (something that is hard to do in HTML) and you’ll get

9x = 9,

Or, as I claimed earlier, that x = 1. Even now, I find that fact a little bit mysterious. And this is one of my central problems with algebraic methods in general. They’ll tell you that a statement is true, but they seldom lend themselves to obvious readings of just why a statement is true.

In fact, this reminds me of a frequent difference between inductive proofs and constructive proofs: inductive proofs often accompany theorems which speak only about existence—what you’re looking is out there, the proof guarantees it, but you have no idea where; constructive proofs, on the other hand, actually give you what you looking for. Constructive proofs are usually more useful than inductive proofs because they automatically satisfy existence by virtue of demonstration. (Imagine what economists would do with a constructive proof of the Brouwer fixed point theorem; and I’d understand Sard’s lemma a lot better if the proof I learned didn’t rely so heavily on induction.)

So, is it any wonder that I gravitated toward geometry over algebra in college? Geometers use inductive arguments, too, to be sure. However, problems are usually cast in ways that are about as tangible as mathematical problem can be. Some of them even have straight-forward physical interpretations. It’s not (too) hard to imagine that soap bubbles could represent minimal surfaces, for example. However, what does a Dedekind domain look like? (If you can help me visualize a Dedekind domain, I’d be very grateful. Had you helped me three years ago when I was taking algebra, I’d've been even more grateful.) Like I said, algebra is hard.

But let’s get back to our infinitely repeating decimal. Why should it be the same thing as one? Well, I suppose we should ask, what is the number one? There are lots of answers—many of them correct. In this case, one is particularly useful: the number one is a label for a point on the number line.

The decimal 0.999… is also a label. But then again it’s so much more. Both 1 and 0.999··· are directions to the points that they label. How convenient! Here’s how you read the roadmap embedded in every decimal. First you need to arbitrarily pick a point called zero. That’s up to you. Next you need to pick another point that’s a unit length away from zero. This choice is also arbitrary. In the metric system you might use centimeters. In the English system, the unit you pick might be feet. If we were measuring something large, maybe you’d choose lightyears. What you choose is really a matter of convenience.

Now the fun part comes in. The first digit d after the decimal tells you to chop up the unit length into 10 smaller pieces of equal length. This smaller distance (1/10) will become the unit you use in the next step. Then you go to walk to the d-th piece. In this case, we chop up the length 1 into 10 equal pieces and walk to the ninth piece.

In the second stage, you chop up our new unit (1/10) into 10 smaller pieces of equal length. That smaller length becomes our new unit (1/100) for the next iteration, so remember it for later. Now walk over to the piece that the second digit in the decimal tells us to go to. In the third stage, we repeat the process, always taking tinier and tinier steps. For an infinitely repeating decimal we have to take an infinite number of steps to get the point the directions describe. Eventually, the steps we take will be so small that for all practical purposes we stop. This is the idea behind a limit point.

Of course I haven’t been terribly rigorous. That’s where the algebra comes in. We already proved that 1 = 0.999··· above, but the geometry is where the understanding is, at least for me. Ideally, I would’ve had some pictures in this post—but modern technology is years behind pencil and paper. But kindergarteners can draw; more importantly they can walk. Maybe limit points aren’t especially useful in most kindergarten curricula, but I think that this shows that they probably have a fair shot at understanding the concepts. And maybe now I can put this demon 0.999··· to rest.

p.s.—Wikipedia also has an entry on 0.999··· with more pictures and deeper, more confusing jargon.

p.p.s.—Now I really need to write up something about infinity. After all, 0.999··· has an infinite number of 9s in it. What does that even mean?

Technorati Tags:, , , , , , , , , , , , , , , , , ,

Critical Thinking Journal/Weak-sene, Strong-sense, and Probabilities

Wednesday 14 March 2007 in Creativity, Critical Thinking, Education, Mathematics, Philosophy, Psychology, Society | No comments

That’s right. It’s time for another installment of “What has Josh been writing for class?” This week I responded mostly to an old article by Richard Paul—who, I think, bears a striking resemblance to Walker Texas Ranger: hold on to that.

He differentiates mainly between two types of styles of problem evaluation: weak-sense and strong-sense critical thinking. To paraphrase, perhaps unfairly, weak-sense is marred by an overly narrow subproblem formulation. It’s atomistic. First you take a big problem, chop it up into smaller problems, and then solve each of the bite-sized pieces one at a time. Paul rightly notes that oftentimes this method misses the larger problem that arrise from the interplay of the otherwise well-behaved subproblems. The mathematician in me has to note that the local-behavior-does-not-imply-global-behavior phenomenon has been a central theme in differential geometry from about its beginning. The same problem creeps up just about everywhere else you look for it. I’ve tried to talk about this before in vague terms relating to urban planning and chaos theory. Maybe I should try again sometime. But for now:

Journal 3 Journal 3: Weak-sense, Strong-sense, and Probabilities

I agree with Paul. Strong-sense thinking is more appropriate for lots modern problems. International conflict, curricular design, and global warming all require strong-sense critical thinking, for example. (Ordering dinner at a restaurant typically does not.) While I like Paul’s network approach to problem solving, I think the primary weakness of weak-sense thinking lies in its absolutist view of truth, not necessarily its divide-and-conquer methodology. Truth, when viewed as a certainty, is rigid and fragile. Today’s demanding social and business landscape calls for something more adaptive, fluid, and functional. (Yes, you were supposed to read that last line with an announcer’s voice.) So how do I amend his framework? With probabilities of course. Really dedicated readers will see that I’ve mostly recycled my blog entry about assumptions. But to keep things fresh, I had to add something. And you knew it would happen eventually. I couldn’t resist.

I center my discussion around a theorem from linear algebra. Gleason’s Theorem tells you exactly what the probabilistic measures on the closed subspaces of a Hilbert space are (basically they’re projection operators). And according to some, it’s central to future research in information retrieval. I use it to show the usefulness of multiple points-of-view with some scientific flare. Of course, my treatment is clumsy—but technically I’m only allowed one page per entry. How thorough could I have been? Maybe later I’ll clean this up and expand it a little. For now, it’s probably okay.

References

Paul, Richard. “Teaching critical thinking in the ‘strong’ sense: A focus on self-deception, world views, and a dialectical mode of analysis.” Informal Logic Newsletter, 1982.

Technorati Tags:, , , , , , , , , , , , , , , , , , , ,

« Older entries