How many Books

My 8 year old asked out of the blue, the way kids do, "Dad, how many books are there?" 


After some discussion, it emerged that he was looking for a count of all the books ever published. 


Using rough back of the envelope techniques I teach engineering freshmen, I conservatively estimated an upper-bound of about 500 million. 


Then I looked it up online. Turns out Google did a count just a couple of years ago, and found that there were about 174 million known unique works.

Failing by Solving

The upcoming February 2012 issue of the Notices of the American Mathematical Society contains an interesting essay titled "A Modest Proposal" by Alan H. Schoenfeld, a Professor of Education and Mathematics at UC Berkeley. He writes of some "horror stories" of how kids being taught mathematics respond to questions that don't match the usual pattern:

Reusser [2] asked ninety-seven first- and second-grade students the following question:  

There are 26 sheep and 10 goats on a ship. How old is the captain?  

More than 3/4 of the students “solved” the problem, obtaining their answers by combining the integers 26 and 10.

( Note: The cited paper is K. Reusser, Problem solving beyond the logic of things, Instructional Science 17 (1988), 309–338. )

Schonfeld concludes that kids are not being taught how to think correctly in classrooms and advocates teaching in such a way that they come to view mathematics as providing  "a set of sensible answers to a set of reasonable questions".

This is certainly commendable, and I like the example he gives of how to make the arc length formula simple and intuitive. But, as I've argued repeatedly on this blog, the problem runs deeper.

Over-schooling, by its very nature, so constrains young children's thinking that they are unable to see outside the patterns that they have been drilled on and made to repeat. They need to learn in a more carefree environment, ideally one where they have plenty of time to think of many "reasonable" questions and possible solutions on their own through their natural habits of play and curiosity-driven exploration, free of the overwhelming pressures of adult expectations.

Here is a relevant piece from John Holt's foreword to his classic "How Children Fail"; the italicized words in brackets are mine, connecting the quote to the above setting:

...there is a more important sense in which almost all children fail: Except for a handful, who may or may not be good students, they fail to develop more than a tiny part of the tremendous capacity for learning, understanding, and creating with which they were born and of which they made full use during the first two or three years of their lives.  

Why do they fail?  

They fail because they are afraid, bored, and confused.  

They are afraid, above all else, of failing, of disappointing or displeasing the many anxious adults around them, whose limitless hopes and expectations for them hang over their heads like a cloud. [Every question posed to them at school has a clear and definite answer; and failure to answer disappoints the teacher. So they know once they are asked a question, they *must* set out to solve it.] 

They are bored because the things they are given and told to do in school are so trivial, so dull, and make such limited and narrow demands on the wide spectrum of their intelligence, capabilities, and talents. [Face it. At the end of the day, solving word problems about addition are simply not as interesting as any of the dozen other things related to playing and exploring and being with friends that little kids would rather be doing.]

They are confused because most of the torrent of words that pours over them in school makes little or no sense. It often flatly contradicts other things they have been told, and hardly ever has any relation to what they really know — to the rough model of reality that they carry around in their minds. [If much of what is presented to them in their workbooks or by the teacher doesn't make sense anyway, most of the time,  why should this puzzle be any different? What all this confusion sadly does is to breed a special form of intellectual laziness --- it's easier just to play along and mindlessly apply some known formula, rather than say over and over again, "I don't understand".]

I asked my eight-year old the same question. The amazing school he goes to leaves him to his resources almost entirely. I do teach him a bit myself from time to time, mostly because of my own interest in mathematics, but I try hard not to over-do it. And when we do talk about math, our discussions are very much in the spirit of Schonfeld's "reasonable questions" approach. So I was quite sure how he would respond.

Nevertheless, it was with some relief that I heard him say, "Wait, that doesn't make sense! What does that have to do with anything?"

***

This post seems as good place a place as any to put in a pointer to one of my favorite songs:

The pseudo-science of testing

A brilliant article in yesterday's NYTimes by Michael Winerip, titled "10 years of assessing students with scientific exactitude" describes the ups and downs experienced by the New York school system over a decade of No Child Left Behind-inspired Testing.

It starts with:

In the last decade, we have emerged from the Education Stone Age. No longer must we rely on primitive tools like teachers and principals to assess children’s academic progress. Thanks to the best education minds in Washington, Albany and Lower Manhattan, we now have finely calibrated state tests aligned with the highest academic standards. What follows is a look back at New York’s long march to a new age of accountability.

After reading the full chronological listing, I didn't know whether to laugh or cry.

***

Perhaps the single hardest (worst) part of my job is making and grading tests. I never know in advance how a particular class of students is going to do on a particular test I make up. Well, the students that do extremely well might do well in any variant of it. The students who do extremely poorly may not do that much better in some other variant of it. But for the vast majority of students, I find there is great sensitivity to every aspect of the exam, from the choice of topics, to the subtle deviations of these questions from what has been covered exactly in the notes, the book, or the homework assignments, to the length of the exam, and even the ordering of the questions.

Even in a seemingly objective field of study like Engineering, there is a lot of subjectivity in how one grades (going beyond the obvious subjectivity inherent in the choice of questions to put on an exam). We try our best to be consistent and fair across all the students for the same class; but for the same question and answer, unless one uses shallow multiple-choice questions (the approach adopted by many standardized tests), it is certain that no two instructors would grade the same way. While there may be one way (or relatively few ways)  to get the answer right, there are exponentially many combinations of errors that trip up students. Particularly if one wishes to go down the road of offering partial credit, the art of grading requires one to differentiate between these and place a value judgement on them: do you give a student that got the right numerical answer through incorrect reasoning some credit? Do you give a student that took completely the wrong approach to the problem but applied that approach correctly albeit to give the wrong answer more credit than one that tried out something new and original but failed with it and gave (if it is possible) an answer even further from the correct one? What if you discover upon grading that a question that seems perfectly straightforward to you has been misinterpreted by number of the students to be quite different from what you had intended? How large a number does this have to be for you to factor the possible ambiguity in the wording into account when grading? Does it matter if the misinterpreted question is easier or harder than the originally intended question?

Unfortunately, the politics of public K-12 education and the economics of higher education dictate that we must always have assessment and grading. Testing is a necessary evil that we cannot wish completely away. Let's continue to strive to be as fair as possible in making and grading tests, but let us not pretend that test scores and GPA's are objective, noiseless, measures of a student's intellectual capability (or, in the case of public schooling, of the effectiveness of a system of education).

No mistakes on the Bandstand

As someone who appreciates Jazz, I highly recommend this video. Stefon Harris talks about the importance of paying attention to the teammates when doing improvisation. He gives a great illustration of what it means to go with the flow, and how that's different from commanding the team to do something specific that one has already set one's mind to. 

This talk is also a great metaphor applicable to many interactive activities that an academic is involved with. Whether it is working with Ph.D. students, research collaborators, or even in class while teaching, there has to be a lot of give and take, and mindful awareness combined with a certain letting go of the ego makes for a richer and more rewarding experience. Even seemingly discordant notes are an opportunity to go someplace new. 

I have been experiencing this increasingly in the classroom, myself. I am finding that the more open I am to new ideas coming from the students through their comments and questions, the more willing I am to digress from a pre-set path, the more interesting, the more creative, the classroom experience is for all of us. This allows us to stray away from well-trodden paths of textbook exercises to occasionally discovering entirely new problems. This semester, for instance, based on student questions in my wireless networks class, we formulated and solved an interesting new variant of the problem of power allocation across parallel channels to maximize total rate (a classic Information Theory problem that is solved using the so-called ``waterfilling" algorithm). This variant was similar enough that we could use the same approach, but different enough that we could appreciate resource allocation at a deeper level. And because it was motivated by questions the students themselves had asked and clearly something new to all of us, I think it might just have made a more lasting impression at least on some students compared to the usual routine. 

In light of the ongoing debates about online education, it also occurs to me that this kind of improvisational interactive classroom experience is precisely what cannot be replicated in mass-marketed pre-packaged instructional videos.

Creating a helping relationship: questions to ponder

Carl Rogers, 1902-1987

In a 1958 essay titled "The characteristics of a helping relationship", Carl Rogers, the noted humanist psychologist, formulated the following ten questions, which I present in his own words. He crafted them in the context of being a therapist, but they naturally apply to all kinds of relationships, including the relationship between a teacher and his/her students:

1. Can I be in some way which will be perceived by the other person as trustworthy, as dependable, or consistent in some deep sense? 
2. Can I be expressive enough as a person that what I am will be communicated unambiguously?
3. Can I let myself experience positive attitudes toward this other person --- attitudes of warmth, caring, liking, interest, respect? 
4. Can I be strong enough as a person to be separate from the other? Can I be a sturdy respecter of my own feelings, my own needs, as well as his? Am I strong enough in my own separateness that I will not be downcast by his depression, frightened by his fear, nor engulfed by his dependency? ...[for] then I find that I can let myself go much more deeply in understanding and accepting him because I am not fearful of losing myself. 
5. Am I secure enough within myself to permit him his separateness? Can I permit him to be what he is --- honest or deceitful, infantile or adult, despairing or overconfident? Can I give him the freedom to be? 
6. Can I let myself enter fully into the world of his feelings and personal meanings and see these as he does? Can I step into his private world so completely that I lose all desire to evaluate or judge it? 
7. Can I be acceptant of each facet of this other person which he presents to me? Can I receive him as he is? Can I communicate this attitude? 
8. Can I act with sufficient sensitivity in the relationship that my behavior will not be perceived as a threat?
9. Can I free him from the threat of external evaluation? In almost every phase of our lives --- at home, at school, at work --- we find ourselves under the rewards and punishments of external judgments: "That's good"; "that's naughty", "That's worth an A"; "that's a failure." "That's good counseling"; "that's poor counseling."... in my experience, they do not make for personal growth and hence I do not believe that they are a part of a helping relationship. Curiously enough a positive evaluation is as threatening in the long run as a negative one, since to inform someone that he is good implies that you also have the right to tell him he is bad. 
10. Can I meet this individual as a person who is in process of becoming, or will I be bound by his past and by my past?

I like this list. A faculty member at another institution pointed me at it, saying that he likes to consider these questions before teaching class, as a way to reflect on how much "regard, empathy, and genuineness" he could "experience (and express clearly) in every class."

For me too, many of these questions resonate with the way I like to interact with students, particularly my Ph.D. advisees. I have learned over time that it is most rewarding to view them not as resources to be exploited, nor as clay to be molded and shaped according to my inclination, but as individuals to be supported in their own personal journeys towards realizing their potential. 

These are not easy questions, and I believe that answering them all in the affirmative represents an ideal that may not always be achievable in practice. For instance, question 9 poses a conflict for a faculty member given that a significant part of their official responsibility when teaching a class is to provide an evaluation of the student's performance. But these questions are certainly worth pondering if we aim to be helpful to others.

Let kids play on mountains

For amusement, late this evening, I took up pen and paper to show my 8-year-old how lines have equations to go with them. You know, simple things like y = x, y = 8-x, y = 5... He patiently helped me identify a few points on each line, and watched me connect them. He was particularly amused  to see the equation approximating the relationship between his age and his younger brother's.

He then asked, what about circles? Do they have equations too? So I showed him x*x + y*y = 4, focusing on the positive quadrant, carelessly mentioning there was something called square-root, and showing him where (sqrt(2), sqrt(2)) was on this circle.  This concept he had never heard of before intrigued him.

He then asked, does every number have square roots? I showed him sqrt(4) as another example. He could then figure out square roots of 0 and 1, whose consistency pleased him.

He then asked, what about negative numbers, do they have square roots too? I didn't launch into a lengthy explanation, but pointed out simply that square-root of -1 is nowhere to be found on the number line he is familiar with.

I was thrilled, of course, at the sheer effortless-ness of this conversation, which took all of ten minutes, and touched on such a wide range of mathematical topics, from analytic geometry to surds to imaginary numbers. But my point is not at all to show off my son as a genius of some sort. Like many kids his age, his biggest interests and activities revolve around video games, spinning tops, cartoons, and hanging out with friends.

I believe what this little incident truly exemplifies is that any kid can make creative jumps and connections and ask great questions quite easily when they're simply curious, when their learning is not being controlled via rigid structures, when it is not boxed into an arbitrarily fixed place and time and dragged in lock-step with everyone else. It helps to have an environment that supports this kind of freedom and stress-free exploration. I am grateful for his amazingly unique school, Play Mountain Place, which lets him follow his own interests at his own pace. Learn more about it at playmountain.org

Inclusive education

At its best, education must be inclusive, and focus on helping everyone grow their abilities.

In the campus bookstore I came across a wonderful book titled "The Power of Neurodiversity," by Thomas Armstrong. In one part Armstrong describes a unique school called William W. Henderson Inclusion Elementary School in Dorchester Massachusetts. In this school, he writes, special education students (about a third of the student population) are included in classes full-time with other students. He quotes former principal William Henderson (now retired):

We challenge students with Down syndrome as well as our most academically advanced students to read as much as they can. We challenge our students with cerebral palsy as well as our fastest runners to exercise as much as they can. We challenge our nonverbal students as well as our most polished speakers to communiate as effectively as they can. We challenge our students with autism as well as our "social butterflies" to interact as positively as they can. The goal for every child at our school is to "get smarter, feel smarter, and act smarter"

Armstrong writes that instead of putting everyone on exactly the same page, this school accepts the diversity of abilities and inclinations and encourages each student individually in academic and non-academic subjects. At the same time, he points out that this approach is much more than having a special ed teacher working with some students at the back of the room while the regular teacher instructs the rest of the kids.

It is clear that such an inclusive classroom experience benefits all students. As a society, we will all benefit from breaking down boundaries that separate people with different abilities.

The above youtube video shows a celebration event at this school.

The founder has written a book titled "The Blind Advantage: How Going Blind Made Me a Stronger Principal and How Including Children with Disabilities Made Our School Better for Everyone," with proceeds going to the school.  I have not read it yet, but it's high on my list.

*** update ***

After I wrote the above, a reader posted a comment recommending that I present some evidence or citations to support the claim that inclusion is beneficial.

First of all, let me say that the idea of inclusion as an educational principle draws its primary support from its appeal to our "better nature", our sense of compassion, our desire to live in a society that draws fewer boundaries between its citizens. The more inclusive a classroom, the more aligned it is with these inherently desirable goals.

Second, I am not sure how much weight I would  place on scientific studies about inclusive classrooms. Like many social problems, there are simply too many variables to control for and it is hard to quantify benefits. No two school classes, in terms of the background, composition, and dynamics between the children and the adults around them, are the same. Inclusion is unlikely to be implemented in a uniform manner, due to varying levels of understanding and support for it. And the long-term benefits in terms of fostering a more open outlook among the children, and improvements in self-esteem are not amenable to ready evaluation.

Having said these, I give below pointers to a few articles discussing research on inclusion in the classroom.

Rebecca Hines authored this 2001 article titled "Inclusion in Middle Schools". It points at research, particularly, as described in the following book:

Kochhar, C. A., West, L. L., & Taymans, J. M. (2000). SUCCESSFUL INCLUSION: PRACTICAL STRATEGIES FOR A SHARED RESPONSIBILITY. Upper Saddle River, NJ: Prentice-Hall.

Hines writes that Kochhar, West and Taymans have identified a number of positive benefits of inclusive education, ranging from higher levels of achievement, greater teacher resources, and improved understanding and social acceptance. She identifies other studies that have shown that inclusive classrooms do not have less instruction time, though they do require greater time spent on coordination among co-teachers. She also identifies organizational, attitudinal and knowledge barriers to the successful implementation of inclusive classrooms. 

Another widely cited book that presents further arguments in favor of inclusive classrooms along with guidance on implementing inclusive classrooms based on collaborative teaching is the following: 

C. Walther-Thomas, L. Korinek, V. L. McLaughlin, B.T. Williams, Collaboration for Inclusive Education: Developing Successful Programs, Pearson, 2000. 

The European Agency for Development in Special Needs Education has written a comprehensive survey of research from fifteen European countries on Inclusive Education and Effective Classroom Practices. 

At the other end of the spectrum, here is a qualitative research account of one particular inclusive teacher's classroom: E. Brown, "Mrs. Boyd's Fifth Grade Inclusive Classroom", Urban Education, 37(1), 2002. 

Criticisms, and a defense, of the education system

A review article by Steven Brint in the Los Angeles Review of Books, titled "The Educational Lottery" is worth a read. He discusses a number of different perspectives on higher education, based on his evaluation of four recent books on the topic.


He writes that

The American education gospel is built around four core beliefs. First, it teaches that access to higher levels of education should be available to everyone, regardless of their background or previous academic performance. Every educational sinner should have a path to redemption. (Most of these paths now run through community colleges.) Second, the gospel teaches that opportunity for a better life is the goal of everyone and that education is the primary — and perhaps the only — road to opportunity. Third, it teaches that the country can solve its social problems — drugs, crime, poverty, and the rest — by providing more education to the poor. Education instills the knowledge, discipline, and the habits of life that lead to personal renewal and social mobility. And, finally, it teaches that higher levels of education for all will reduce social inequalities, as they will put everyone on a more equal footing.

He identifies various classes of critics that attack these pillars:

• Critics he refers to as new "restrictionists", who feel that a college education is wasted on those who are not prepared for it, who argue that college education has been watered down to support this universal access, who in fact argue against allowing/encouraging everybody to go to college, because it is not even ultimately useful to them.
• Those he calls "educational Romantics", who believe that much of schooling is a form of servitude and limits creativity and rail against all systematized forms of eduaction.
• He identifies some who rail against the "fool's gold" idea of education as paving the way for social mobility and equality. 
• And finally he discusses the sect of "true educators", for whom the purpose of education is inner transformation, not sociological. As an exemplar of this opinion, he cites Philip W. Jackson, who has written a book titled "What is Education?" 

Brint writes about Jackson's work:

For Jackson, the good teacher strives for perfection, leavened by a loving outlook. Striving for perfection is important to Jackson. Like religion, education aims high. It wants to tell the truth. It is a moral enterprise, because it seeks to make everyone it touches better than they currently are. Because education is about truth, it is also about correction. The possibility of elevation, as Hegel knew, requires the negation of error. Yet because they must kindle an interest in the spirit of learning, teachers are inclined to downplay students’ weaknesses while applauding their efforts. Because she wishes students to remain on the path of improvement, the teacher makes more of their contribution to knowledge than perhaps is warranted. 

To transform people, education requires a particular attitude of students too: the attitude of receptivity and affection. “We feel close to those objects toward which we profess love,” Jackson writes. “We identify with them … We possess them. They become ‘ours’ … These forms of attachment reduce the separation between subject and object. They bring the two closer together, which is the principal goal of education.” Because education is a relationship between teacher and student, students need to be known by their teachers. The possibility of identification with the subject matter flows as much through the teacher as through the course material. If students do not feel known by the teacher, it will be difficult for them to identify with the course materials, and even to feel the desire to know more about the topic. They must want, at some level, to be like the teacher. For the student, learning involves both grasping and shaping. Students grasp course materials and the spirit of the subject through appreciation, and they shape them for their own uses through assertion.

Ultimately, though, Brint, who I would term a pragmatic conservative, a defender of the status-quo, dismisses all these criticisms as being limited in their perspective, hopelessly idealistic, or worse, dangerous.

Heretics often offer penetrating insights about the flaws of dominant doctrines. They are usually less perceptive about the limitations (or dangers) of the alternatives they favor. The new restrictionists run the risk of forgetting about the problem of inequality and further privileging the privileged. Romantic dissenters do not often require the complement of deep knowledge and discipline on which adult creativity also rests. The “fool’s gold” school has no concrete plan for erecting a just social system in which workers are paid a living wage, non-corrupt labor unions are encouraged, and the wealthy are taxed enough to support decent public services. And “true educators” live in a rarefied world of one-on-one tutorials and private education, one that, however inspiring, is utterly divorced from the contexts in which most teachers actually work.

Brint argues that the system as we have it may be inefficient and under-performing, but may be the best possible compromise for our society. He advocates instead more incremental changes such as cutting down class sizes, stiffer entrance requirements, use of clickers to improve participation, performance-oriented evaluations, and somehow (magically?) ending the "attitude among under-motivated students that 'the only thing that matters is the credential'."


I'm left dissatisfied and underwhelmed by his conclusions.

Joi Ito's Views on Learning

Joi Ito is the new director of the MIT Media Lab. He is a highly accomplished, very well known technologist. He has been the founder of many Internet companies, and chairs the board of directors of the Creative Commons. According the Wikipedia entry about him, he first went to Tufts University for an undergrad in Computer Science, and dropped out. He then went to the University of Chicago for physics, and dropped out of there too. He spent countless hours playing world of warcraft online and credits it with teaching him valuable lessons about management and leadership. So he certainly has had a non-conventional educational path for someone who now heads one of the most visible, most prestigious academic centers.

Today at a conference that I am attending (Emtech 2011), he was being interviewed. I got the chance to ask him the following question after his interview:

What are your views on the University as we have it today and how that's going to change as a lot of changes are happening in technology and society?


Here is his response:

I just was talking to the Ph.D. students yesterday and I said, first of all, if I were at the Media Lab, I probably would have graduated. But, the Media Lab is great, because it is very much an interest-driven learning model, they are focused so much more on learning than some institutions. I think MIT is obviously better than most, but a lot of institutions I feel that you're in the university to try to get out. Because what you're looking for is the degree, and then once you get out... the focus is on the degree and not learning. And I said this as a joke, but I kind of almost mean it, I said to the Ph.D. students, look, if at the end when you are graduating, and I go "psych!" and take away your degree, I want it to be that you still are completely able to get a job as though you had the degree and the degree doesn't matter. I want you to have learned so much and it become so interesting that the degree is something [only to show] that we're happy you are a graduate... the degree isn't the reason why you're here. 

...thinking about the degree as a byproduct of doing great learning and great work is the way I'd like the Media Lab to become... But I really hate classes and education getting in the way of learning. I mean, that to me was why I left, because it was getting in the way of me learning. I think that focusing on learning, and focusing on creativity is a key element of the Media Lab. 

I think there are certain types of characters. Like my sister is a double-Ph.D., Magna cum Laude, Harvard, Stanford... The difference is that when she was 5 years old, 6 years old, she could plan her life. She said this is what I want to be when I'm 25 years old, 30 years old. I couldn't think past the next day and so if it wasn't interesting and useful for me to do today, I was on to the next thing. I think there are more kids like me than there are like my sister, so you have certain kids like my sister who make it through the formal education process, no problem, and they learn tons. But there's a bunch of people, misfits like me who can't fit in... it's not like I want to change all of higher education, but there is a role for this kind of learning through doing, learning through tinkering, interest-driven learning, because I've had, not necessarily the best example, but I'm exhibit A of somebody who was able to stitch together their own learning, without very much formal help, thanks to the Internet in large part. And so, and it's funny because my sister studies education, and after she had become quite well known in the field of education, she looked over at me and said "wait a second, how did you? how did that work?"... and now she is focused on this informal learning stuff, so I sit around and try to explain to her what I'm thinking, and she sort-of rigorously tries to explain it in education-anthropological terms... 

For me, the Media Lab, I'd like it to be a prototype for a new kind of higher education and that we don't become just one side-show, but that the DNA of the Media Lab starts to infect all kinds of other universities and institutions and companies about new ways to think about innovation, think about learning.

I find myself in near-total agreement with his emphasis on interest-driven learning, and the primacy of learning over formal coursework and degrees. This has been the primary theme of many of my posts here...

(The full video of the interview and the Q&A afterwards is available online.)

In Memoriam

He was not a religious or spiritual leader. He did not bring a nation or a people freedom from oppression or slavery. He was not a general. He never held any political office. But there was something about what he did and how he did it that has clearly commanded the respect of millions around the world.

Steve Jobs gave this commence speech at Stanford, titled "How to live before you die":




Here is the transcript of his speech.

Two inspiring quotes from this speech summarize his attitude towards work and life:

" Sometimes life hits you in the head with a brick. Don't lose faith. I'm convinced that the only thing that kept me going was that I loved what I did. You've got to find what you love. And that is as true for your work as it is for your lovers. Your work is going to fill a large part of your life, and the only way to be truly satisfied is to do what you believe is great work. And the only way to do great work is to love what you do. If you haven't found it yet, keep looking. Don't settle. As with all matters of the heart, you'll know when you find it. And, like any great relationship, it just gets better and better as the years roll on. So keep looking until you find it. Don't settle. "

"Your time is limited, so don't waste it living someone else's life. Don't be trapped by dogma — which is living with the results of other people's thinking. Don't let the noise of others' opinions drown out your own inner voice. And most important, have the courage to follow your heart and intuition. They somehow already know what you truly want to become. Everything else is secondary."