Do what you don't have to

This morning, a student came by during office hours and asked if he could discuss a private matter.  He told me that he was nearly done with all his courses in engineering school, but, to his disappointment, he didn't feel passionate about any of the subjects he had taken. He asked me what was the secret to finding something one has passion for.

I thought about it a bit, aware of the awful weight of responsibility, the need to give a good answer to this bright young man's question, which clearly came from his heart.

I offered him this thought, that I believe passion comes from "doing things no one else has required you to do". I talked with him about how this impulse is in fact thwarted in classrooms with their many assignments, quizzes, exams and canned projects (yes, even mine.) I encouraged him to tinker and do some projects on his own, engage in independent study.

I also told him to try and be mindful of his own inner-voice, to find out for himself what activities he finds rewarding, and seek ways in his life to do more of that. I told him about my own motivations for seeking an academic life --- my realization that I really, really enjoyed teaching my classmates the night before exams what we should have learned in the weeks prior.

Later in the evening, I was still thinking about the conversation. I was thinking to myself how it is that when you feel passionate about something, work or learning doesn't feel like a chore -- it can be exhilarating, as meaningful, as play -- and how the truest form of learning only takes place when you are fully engaged in and enjoying the experience overall. The crux of it is having a clear sense of autonomy --- that you are choosing to do this, you are doing this because you want to, that you could and would stop doing this if it wasn't worthwhile or interesting.

This article by Psychologist Peter Gray makes nearly the same point, in encouraging us all to let our kids play more.

He writes:

The reason why play is such a powerful way to impart social skills is that it is voluntary. Players are always free to quit, and if they are unhappy they will quit. Every player knows that, and so the goal, for every player who wants to keep the game going, is to satisfy his or her own needs and desires while also satisfying those of the other players, so they don’t quit. 

The kind of play he is talking about is not merely light-hearted diversion, though it includes that too, but also the kind of intense immersive play that requires one's full concentration, skill, and forces one to constantly build and improve on one's abilities. Plenty of exposure to this kind of play as a child is indeed essential to self-motivation, creativity, and passionate immersion in learning and work as an adult.

Teaching Mathematics as a way of Thinking

I came across today a brilliant blog today about learner-centric mathematics education, called Doing Mathematics.  The writer of this blog, Bryan Meyer, is a high school math teacher in San Diego. He writes that:

Most people think that mathematics is a body of knowledge and/or procedures that must be passed from teacher to student. For many students, this is both irrelevant and unexciting. My belief is that mathematics is a way of thinking and sense making. It is creative, beautiful, individual, and dynamic. It is being curious, asking questions, figuring out why things work, breaking problems apart, seeking regularity, making predictions, and creating logical arguments.  These things we all engage in on a daily basis and have relevance in our lives outside of a mathematics classroom. My goal is to create an educational environment in which mathematical thinking is not only the core of what we do, but runs through ALL of what we do. 

Bryan is inspiring because he does things many teachers wouldn't think of doing, but that strike me upon reflection as a very good thing to do. For instance, in one post, he describes his class working on and discussing a probability problem, at the end of which some students thought one answer was right, while others felt that another answer was right. But he never jumped in to tell them which one was the right answer, even at the end.

This is unnerving to read at first because, of course, the instinct is to say --- wait, shouldn't the teacher announce the correct answer so no one is left with the wrong impression? He chose differently. He writes:

I'm curious what you think and what you would do in this same situation. I let it go. I felt I did my job by helping students test their ways of thinking, not by telling them what to think. 

Bryan's point is that in the long term, showing his authority by presenting the right answer could potentially do harm to their ability and confidence to test out their own ideas. As he writes in a different article,

I am inclined to think that my distaste is not actually for right answers but rather for the students' lack of authority in deciding that answer. As it stands now, students' ways of thinking are always subject to some greater authority (teacher, textbook, video, etc.). As Schoenfeld puts it, students:

"...have little idea, much less confidence, that they can serve as arbiters of mathematical correctness, either individually or collectively. Indeed, for most students, arguments are merely proposed by themselves. Those arguments are then judged by experts, who determine their correctness. Authority and the means of implementing it are external to the students." 

 I am very excited about discovering Bryan's blog, because his thinking and most admirably, his doing, is very much along the lines of what I myself noted as the ideal in discussing the dangers of the classroom:

The learning process should leave the learner with an innate, ever-improving 'filter' of quality, to be able to evaluate for oneself, when encountering some piece of information, whether it is reliable, new, significant, important, or useful... [however] what is taught is often only what is already well understood, processed and neatly packaged. This makes it harder for students to think outside the box or understand the current limits of our knowledge, let alone how they can be extended...  The loss of autonomy and the presentation of processed information deprives students of the confidence to develop their own filter...

Bryan Meyer also has a nice set of recommended readings I plan to go through. I know of a few of them, but am unfamiliar with most.

A Professor as a Student

Rebekah Nathan's "My Freshman Year" should be required reading for all college professors. This is an account of a middle-aged college professor who decided to spend a year at a large public university in 2002-2003, living in the dorms and experiencing everything a first-year student would. As a trained anthropologist, she gives an exceptionally insightful and vivid account of American undergraduate culture.

What I found most original about the book is it offers an account that undergraduates themselves could not articulate so well. For it is hard to critically examine one's own culture, or recognize the ways in which it may appear peculiar to outsiders. And although all of us who are professors now were students once, we did not observe so keenly the various aspects of student culture, or synthesize them as she does in this book. Also, the perspective of faculty members, who were often (almost by definition) among the more academically-inclined students in their classes as undergrads is biased, and not representative of the experience of the majority of students going to college. (Hence the routine lamentations heard in the halls outside faculty offices: "What is wrong with these students? Why don't they study as hard as I used to? Why don't they seem to care about what I'm teaching?")

Rebekah Nathan (which turns out to have been the pen-name used by Cathy Small, from Northern Arizona University) offers us a more nuanced, balanced view, of how college appears from the students' perspective. Based on her observations as well as interviews with her fellow students, she draws a number of insights:

1. For many students, "the most compelling reason for staying in college was 'the college experience' --- the joys and benefits of living within the college culture rather than in the real world." Classes and learning and immersion in the intellectual life were a secondary objective for many. "Classes, in fact, were described in multiple instances as the 'price one has to pay' to participate in college culture, a domain that students portrayed in terms such as 'fun,' 'friendships,' 'partying,' 'life experiences,' and 'late night talks.' "

2. One of the barriers to learning in many traditional classes is the perception of their formality and disconnectedness from the everyday experiences of students. In contrast, what makes a course popular to a majority of students, is if it can subsume "formal academic content within an informal, largely, social world characterized by equality, informality, intimacy, and reciprocity, while at the same time [providing] a context for learning that [is] 'fun,' irreverent, and separated, both geographically and ideologically, from the formal aspects and authority of campus."

3.  She writes that time-management is a huge and very challenging part of the college experience for undergraduates. She writes, "Going to school, I found, was a time-management nightmare; ... in a single semester, there were eight different people who made rules or created structures that I had to respond to as a student. Each wanted us to access readings, or prepare papers, or communicate with him or her according to a different protocol. As is typical in a large state university, none of the instructors coordinated assignments or schedules with one another or even with a master university schedule." She writes that the most successful students figured out that the way to manage time was not merely to make to-do lists and be efficient in their use of time, but also by making smart choices about which courses to take. "The key to managing time was not, as college officials suggested, avoiding wasted minutes by turning yourself into an agent of your day planner. Neither was it severely curtailing your leisure or quitting your paying job. Rather, it was controlling college by shaping schedules, taming professors, and limiting workload."

4. Shaping schedules: In particular, she writes that even mature students do want to take a few "easy A" / "low workload" courses to balance out the "harder" courses, which they knew to be necessary. And sometimes, they just take courses because they are offered at the right time so that they can fit it into their schedule. She describes this moment of epiphany: "It suddenly became clear to me why, as a professor, I had had a number of students enrolled in my basic cultural anthropology course who had no idea what anthropology was. My course was likely the last piece in their scheduling puzzle, and frankly, they didn't care what anthropology was."

5. Taming professors: Mature students also learn to play the college game in ways that don't promote independent learning. She describes a university-sponsored presentation in which a section of the talk was devoted to "figuring out your Profs." "What do profs want? The speaker told us: 'They think the world revolves around their subject, so they want you to get it. They want to see effort, and they want you to voice an opinion. So give them what they want and you'll get what you want too!" There is something instrumental and transactional about this student-teacher relationship that is far from the ideal learning-centered relationship that faculty crave. She asks a mature and competent senior for tips on success and gets this response: "I take the information I need from the professor --- how they're going to grade you and what they think is important --- and I use it. If you write what you want to that prof, you're gonna end up with a bad grade. Whereas, if you write to them, you win --- you can still have your own mindset and say, hell, I know this isn't the way I feel, but I'll give them what they want."

(This passage made me burn at a memory I had long suppressed, from my own Junior/Senior year. It was a humanities course, which I particularly loved as the readings were provocative and thought-inspiring. I spent hours thinking about and writing up an essay for the first paper in that class, which I was really proud of. It came back with a B-, to my utter shock and disappointment. My friend seated next to me got an A+. I asked to see his essay, and gasped when I read it. "But, but..." I stammered out to him, "you are pretty much just rephrasing here whatever he said in class!" "Of course!" he said, and smiled, knowingly.)

6. Limiting workload: In the face of their heavy workloads and scheduling nightmare, students have every incentive to do the bare minimum they can, while still doing well in the courses in terms of grades. "On several levels, students assess what is needed to get by. Depending on the course and the instructor, they decide whether to buy the book, whether to go to class, whether to do the readings in a given week, and how much effort to put into assignments." On absenteeism, she writes, "in classes where attendance is expected but not required, the frequency of absenteeism rises with each of the following characteristics: the class is large, the class is boring, tests are based on readings rather than lectures, grades depend on papers rather than tests, the class is early in the morning, the class is on friday." She writes that cutting class is actively encouraged by peers, who value it as an act of rebellion against authority. She also writes that it is in the students' interests to minimize any additional reading or learning-oriented activity that does not directly correlate to the course grade, or that does not seem essential for their career; such activities are seen as "busy-work", to be avoided.

7. Students rarely speak up in class to discuss and debate ideas. Time pressure is one contributing factor in low class participation. Others, she determined through a survey, were "peer pressure, power of the teacher, and lack of personal interest or purpose." She writes that "I was struck by the realization that, despite official assertions about the university as a free marketplace of ideas, the classroom doesn't often work that way in practice. Ideas are rarely debated, and even more rarely evaluated. Most classroom discussion, when it does occur, could be described as a sequential expression of opinion, spurred directly by a question or scenario devised by the teacher, which is subject to little or no commentary." Even if there were conversations about ideas in class, "the moment we walked out of class... the subject at hand was abruptly dropped, as if the debate had only been part of a classroom performance." Indeed, outside of class, she found that there was little discussion about learning or ideas. Most conversation centered around due-dates, complaints about the progress of the course, work-load, and grades. She speaks of her disillusionment: "Taken together, the discourse of academe, both in and out of classes, led me to one of the most sobering insights I had as a professor-turned-student: How little intellectual life seemed to matter in college."

8. On why student time for learning and classwork is limited and steadily decreasing over time, Rebekah Nathan hypothesizes that this is primarily due to the rising costs of college, forcing students to take on more part-time work than they used to in the past. "A 2004 government report found that increases in tuition and fees during the preceding decade had outpaced both inflation and growth of the median family income. The result has been debt --- a huge amount of debt that college students are incurring for the sake of their education --- and a sharp rise in the percentage of borrowers among full-time undergraduates." I found this very interesting; it answered in part a question I have had on my mind and discussed in a recent blog post: why do students study less these days on average compared to 40 years ago? There is no doubt that this is the single biggest challenge facing higher education these days.

9.  The rising costs of college and student debts is also resulting in a rising tide of career-mindedness that undermines academic traditions. She recalls a popular talk given by a speaker to entering freshman from a nonprofit division of a for-profit company, which she says embodied key features of the student culture and attitudes. "There was the 'fun-party-independence-youth' veneer, which is long-standing in student culture, but the more dominant statement of the presentation was one of pragmatism and careerism. Hard work, forethought and organization were part of this career message to students, but so too was the idea that grades have primacy, that you should join groups with resume building in mind, that a smart student should 'figure out your profs' (the title of one presentation segment), that students are in individual competition with one another for grades and recommendations, and that, above all, college is about positioning yourself for a good job and an affluent future."

10. She warns that as universities become more market-driven, and students become more career-driven, in response to the rising college costs and student debts, new dangers lurk. "Degree programs tightly geared to the marketplace become products themselves, and are likely to bust and boom with the fickleness of the times." She writes that "in the long run, we would not want a university to become so immersed in the world as it is that it can neither critique that world nor proffer an ideal vision of how else it might be. These are purposes of universities that none of us should surrender."

 

Three Things Research-oriented Faculty Do

This is based on something I jotted down in a late night email to a former Ph.D. student from USC, who just has landed a faculty position, and will start as an Assistant Professor this fall.

The main epiphany I had early on about life as a faculty member is that you have three main functions as a researcher:
1. do good work and publish
2. raise money
3. network and collaborate

Doing good work and publishing it in respectable venues is, of course, the main thing.

But these three all feed inextricably, recursively, into each other:

• To do good work you need to raise funds to support students, and it helps to network and collaborate with others. 
• To raise funding, you must do good work, and it helps to network and to collaborate.
• To network and collaborate, you must do good work, but also need funding to travel.

Doing a good job as a faculty member at a research university thus means constantly balancing and juggling these three functions. New Faculty must particularly pay attention to 2 and 3, because 1 is the thing they were good at as students or post-docs (or they would not have been hired), but in many cases 2 and 3 have not been as much of a focus for them in the past. 

I would actually encourage senior Ph.D. students seeking academic positions,  and certainly post-docs, to gain as much experience as they can with respect to 2 (for instance by practicing writing proposals for/with their advisor) and 3 (by collaborating with other students and faculty, being engaged and interactive while attending conferences, making themselves known to faculty at other schools, giving talks at other schools). This will both help them land faculty positions, and give them a head-start on life as a research-oriented faculty member. 

It's interesting to note that excellence in communication is crucial to all these functions.  

Teach no thing (but rather, how to think about things)

Marion Brady is a history teacher who advocates teaching students not merely facts, or "content", but rather how to think.

He writes:

Long before corporate America began its assault on public schooling, American education was in trouble. Educators were, however, increasingly aware of the problems and were working on them. When Bill Gates, Jeb Bush, Mike Bloomberg, Arne Duncan, Michelle Rhee, and other big name non-educators took over, that worked stopped. 

What I want people to understand is that the backbone of education — the familiar math-science-language arts-social studies “core curriculum” — is deeply, fundamentally flawed. No matter the reform initiative, there won’t be significant improvement in American education until curricular problems are understood, admitted, addressed, and solved.

His analysis of the crisis in education that we face today at all levels is that "The traditional core curriculum delivers existing knowledge, but adapting to an unknown future requires new knowledge. Obviously, what will need to be known in the future isn’t yet known, from which it follows that it can’t be taught. However, the process by means of which new knowledge is created can be taught."

Going beyond rhetoric, Brady has put together a non-content-based curriculum called "Connections: Investigating Reality" that attempts to teach middle school and high school students, or rather, help them teach themselves, how to identify patterns, how to extract meaning from texts and the world around them, how to reason and organize their thoughts.

I could not agree more with his ideas.

I myself experienced the importance of learning how to think when I took a introductory course on analysis in the mathematics department at Cornell. Though the material that this course was ostensibly about was reasonably familiar to me: real numbers, limits, sequences, etc., what the course was really about was how to prove things, how to express mathematical ideas precisely, rigorously, cleanly. It was about how to logically layer arguments, building only upon what is already known, to come to a certain conclusion. I know that the way I thought about things, not only related to mathematics, but about all manner of things in the world around me --- economics, politics, human relations, even literature --- changed very substantially after taking that course. For the first time in my life, I felt confidence that when I constructed a logical argument carefully from a set of premises, it would not be flawed, and that, given time, I could spot a faulty argument when I came across one. Without this confidence, I would never have made it through the Ph.D., to a faculty position, and past tenure.

Now, I try to make sure that all the Ph.D. students I advise take the equivalent course at USC (Math 425a, if you are curious). And from my experience over the years, by comparing my impressions of them before and after they took the course, I can vouchsafe that nearly all these students dramatically improved their ability to think rigorously because of it.  (Interestingly though, I find, this appears to happen not during or immediately after the course, but a few months afterwards, as though it needed some time after that course for their neurons to make all the connections, during which perhaps their subconscious was hard at work applying what they had learned to various problems.)

Some colleagues and I were having a dinner tonight, downtown, at the somewhat quiet but always enjoyable Thai restaurant downtown --- Soi 7. Our conversation turned to educational philosophy.

A senior colleague mentioned that he too believes that what is important in a classroom is not merely conveying the lecture material, or even showing students how to solve a particular set of exercises after having them read or watch the material on their own (as is done in the increasingly popular flipped classroom model of teaching), but rather get them to think in class collaboratively, de novo, about the concepts and problems that the material is about before leaving the students to master the content and do exercises on their own. This way the emphasis is on fresh, critical thinking, rather than on content and drilling. He jokingly called it the "re-flipped" classroom.

Another colleague of mine at the table then indicated that this is very reminiscent of the famous "Moore Method" of teaching mathematics. This method, referred to as discovery learning, was developed by Professor Robert Lee Moore, who encouraged and guided his students at UT Austin to build on basic axioms and prove everything on their own, with minimal hints and without recourse to a text.

I feel inspired to think through the implications of such non-content-based learning of critical thinking and problem solving skills as well as discovery learning techniques, and plan to experiment with them in my own courses in the future.

Are Students Smarter Now than Before?

The following chart is from a NYTimes article from last year that a colleague pointed me at. It is based on data from the paper Philip Babcock and Mindy Marks, "The Falling Time of Cost of College: Evidence from Half a Century of Time Use Data," Review of Economics and Statistics, May 2011, Vol. 93, No. 2, Pages 468-478.

Source: NYTimes 

First, it shows that Engineering students study more on average now (well, as of 10 years ago, but one presumes it's about the same now) than other majors, though it used to be Health students that studied the most. Second, it shows that students now (again, as of 10 years ago), like all majors study much less than 40 years before. In particular, it shows that engineering students study on average about 18 hours a week outside of classes, where before they studied about 26 hours a week.

In the author's own words:

Using multiple datasets from different time periods, we document declines in academic time investment by full-time college students in the United States between 1961 and 2003. Full-time students allocated 40 hours per week toward class and studying in 1961, whereas by 2003 they were investing about 27 hours per week. Declines were extremely broad-based, and are not easily accounted for by framing effects, work or major choices, or compositional changes in students or schools. We conclude that there have been substantial changes over time in the quantity or manner of human capital production on college campuses.

This is really a quite stunning and fascinating finding. Whatever accounts for this difference? Are students simply smarter than before? Do the reduced numbers of hours they study indicate greater learning efficiency (because of better textbooks, better teaching, new instructional media)? Are there simply more distractions that prey on students' time (such as video games and the world wide web, which was just starting to take off then; I'm very curious to see what the numbers look like today, with the added distractions of facebook, twitter, mobile devices)? Is it that more students are working part-time to pay for increasing tuition costs?  Is it because there are more older students that are working full-time taking continuing education classes ?  Is the average reflective of an increase in overall numbers of departments where the study expectations are lower?

Whatever the cause, I've certainly heard several colleagues who have been teaching engineering for two or more decades complain that students these days don't study as hard and with as much discipline as they used to, whine more about grades than they used to, and in particular, are not as strong mathematically as they used to be. These data seem to suggest that they're not just being cranky.

And what are the implications of this study? Are we being forced to lower our expectations of what college students should be able to do, or handle? Instead of being smarter, do our graduates today actually have fewer skills and abilities than those of yesteryears?  How has this affected the economy?

What do you think? 

Multiplying Negatives

I was talking with my elementary-school-aged child this evening about arithmetic.

"What is 2 x 2?" I asked.
"4," he replied.
"What is 2 x -2," I asked.
"Negative 4," he said, reasoning correctly that it was adding up two "negative 2's".
Hoping to stump him, I followed up with "What is -2 x -2?"
"Negative 4", he repeated.

"Wait, this is different from what I asked you before!", I said, "It's not obvious what it means, right?"
"Oh yeah!" he agreed.

"So what is it?" he asked. We were walking outside, feeling quite chilly in the evening air.

"Well, I'll tell you, but you're not going to believe it", I replied, "it's 4!"

He thought about it for a bit, and smiled at the incongruity of the answer. "4?!"

I could sense the puzzlement lingering in him. I knew I had to say more.

An awful feeling grew slowly in my stomach as I struggled with what to say next. "Well, it's not easy for me to explain why though..." my voice trailed off as I started to think about it more carefully.

"... I mean, I mean, I mean, I can't just say consider adding -2, -2 times", I stammered out, as we entered the warm indoors.

I thought back to how I myself had learned this. A shudder went through me as it occurred to me that I was probably simply told by the teacher that "negative times negative is positive" and made to absorb it simply as a convenient fact about numbers.

Up to this point, over months of conversations, I have been explaining arithmetic to my son using familiar concepts and ideas. Natural numbers correspond to putting up fingers one by one. Adding them is an extension of the counting process. It is easy to illustrate with one's hands why 2+3 is the same as 3+2. The positive numbers line up nicely on a line. Subtracting a smaller number from a bigger number corresponds to removing the smaller number of objects from a set of the larger number of objects and counting what remains. It also corresponds to going backwards on the count, or on the number line.

These explanations pile nicely, building on each other. From the above, it is intuitive and therefore easy to explain to a child why subtracting a bigger number from a smaller number requires us to put some more numbers on the line before 0, and that we call these negative numbers. It is easy to explain also on intuitive grounds that though we don't encounter them when counting objects, this could sometimes be a useful way of thinking of numbers (such as when we consider scenarios involving borrow and lending money). Building on the analogy of debt, it's possible to explain intuitively why negative 2 plus negative 2 is negative 4 and then note that this is the same as negative 2 minus 2, so that adding a negative number is the same as subtracting the positive counterpart of that number.  It can also be pointed thereafter that 3 - 2 is not the same as 2 - 3, so that in this respect subtraction is quite asymmetric. Multiplication can be introduced as a shorthand for iterative addition: that when we say "3 times 4," what we mean is to add the total number of objects in 3 collections of 4 objects each. Pictorially, through rows and columns of objects, it can then be explained why "3 times 4" is the same as "4 times 3". And then, building on the prior understanding of negative numbers, it is not a stretch to explain to a child what it means to talk about "4 times -2", and how this is the same as "-4 times 2".

But "-2 times -2" is a qualitatively different animal. It does not follow directly from the above explanations. Because iterating a process a negative number of times is not meaningful, the earlier intuitive understanding of multiplication does not readily generalize*.

Determined to keep things intuitive, I put down my bag, took off my jacket and sat right down in front of the computer. I searched the web for an explanation suitable for a child. I found one immediately at
http://www.mathsisfun.com/multiplying-negatives.html:

("common sense" explanation  from http://www.mathsisfun.com/multiplying-negatives.html)

Reading this worthless bit of chicanery made me feel physically sick. It is insulting the intelligence of a child to even try and explain a mathematical concept through unrelated verbal shenanigans.

To be fair to that site, this junk is soon followed up by an alternative explanation, that is not bad. This second explanation equates -2 x 3 to walking backwards two steps on the number line three times, and 2 x -3 to turning around first (so that one is facing backwards) and then walking forward two steps three times. Finally, -2 x -3 is explained as a combination of these, turning around first, then walking backwards two steps three times, ending up at the right answer of 6.

As explanations go, this is more plausible. At least it builds a clean map between the operations in question and a physical process in such a way that the outcome in the physical process is consistent with the arithmetic operation. In this, it is in keeping with all the previous intuitive explanations I myself had been offering my child. But I couldn't help feeling that the analogy is still rather forced and somewhat lacking in elegance. Why should the negative in the first number correspond to walking backwards and the negative sign on the second number correspond to turning around?

One problem is that all of the above "intuitive explanations" are not rigorous. They are simply suggestive analogies. Because they are grounded in experience, they offer a scaffolding, helping children slowly build familiarity with numbers and what can be done with them. But this approach has its limitations.

Mathematics in its pure form is in fact not a description of our world as it really is. It is fundamentally about abstractions, axioms and deductive inference. We are fortunate that pure mathematics can in fact be applied; that elegant logical, axiomatic reasoning about abstract objects turn out to be useful in solving real-world problems in science, engineering and economics, once a connection can be established through the art of mathematical modeling.

I want to give my child some feel for this, pure mathematics, so I outlined for him a solution based on the following.

We use three statements that he currently  understands and accepts on intuitive grounds:

A. a + (-a) = 0.


B. a x (b + c) = (a x b) + (a x c)

C. -a x b = - (a x b)

The explanation proceeds as follows:

• Start with the expression  -2 x (-2 + 2). Since what is in the paranthesis is 0 (from A), we have that
• -2 x (-2  + 2) = -2 x 0 = 0  ---- (1)
• But also, from B, we have that
• -2 x (-2 + 2) = (-2 x -2) + (-2 x 2)
• And from C, we have that  (-2 x 2) = -4. This yields that
• -2 x (-2 + 2) = (-2 x -2) + (-4)  = 0, the last equality following from (1)
• From A, we then get that the term (-2 x -2) must be 4. 

His eyes lit up at the last step when he could infer for himself that -2 x -2 must be 4.

When we finished, I could tell him that "pure mathematics is all about axioms that you start with and the things you can prove from them." Though the hour was late, I think, I hope, it left some impression.

***

Graeme McRae has a nice page with a very accessible discussion of the axioms of integer arithmetic, that presents a minimal set of axioms from which all the familiar rules of arithmetic with integers can be derived. I am thinking this could be a good resource for a deeper discussion about integer arithmetic and axiomatic reasoning in the future.

Technically, the matter is actually a bit more complicated and interesting, it turns out. The axioms by McRae presented in the above link define what is called an Integral Domain, which is in modern algebra terms a commutative ring without zero divisors. Treating them as an Integral Domain is sufficient to derive most familiar rules of arithmetic with integers, but in fact it is not enough to characterize them completely and uniquely; what is needed in addition are some axioms pertaining to the strict ordering of integers (less-than), and one pertaining to Induction. A nice explanation of these technicalities is provided in these excellent notes on Axiomatizing Integer Arithmetic from a senior-level Applied Logic computer science course at Cornell.

We are so close to the heart of one the greatest (albeit negative) findings in pure mathematics and logic that I cannot resist mentioning here it at least in passing. Kurt Goedel in 1931 proved that in fact there are no set of axioms that can be used to prove all the properties of natural numbers (known as the Incompleteness Theorem). This doesn't mean that the axiomatic approach is totally useless, but shows that it has its limitations. His finding famously undermined Bertrand Russell and Alfred North Whitehead's ambitious joint attempt to put on firm axiomatic grounding all of mathematics in their Principia Mathematica.

***

Though I was exposed to Euclid's axioms of geometry and the idea of proving theorems as early as middle school, I think I only truly got what it's all about for the first time when I first took a senior-level undergraduate math course providing an introduction to Real Analysis, at Cornell. I remember one day early in the course working on a homework problem, struggling to deduce some seemingly trivial property of elementary integer arithmetic, and saying to myself, "how could I have gotten this far without understanding that all these 'facts' I took for granted can in fact be proved from much simpler axioms?!"  One reason it took me so long to figure this out is that virtually all of the mathematics taught to engineers is from the perspective of "here are some useful facts and techniques that you can use to model real systems and analyze those models," and axiomatic development is consistently under-emphasized. This is perhaps as it should be, since abstract axiomatic development tends to be slow, painful, and of relatively little utility when it comes to applied mathematics, but it's a pity nonetheless.

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*Update: Marc in a comment below points out a Khan academy video that gives a great intuitive explanation of the multiplication of negative numbers. In fact, if one models multiplying by a negative number as iterative subtraction, one can build an understanding of why -2 x -2 is 4 using intuition alone. I wish I had seen this sooner, but then I would have missed out on the opportunity to discuss with my child how deductive reasoning works.

The Importance of Grace in Education

Francis Edward Su, a Professor of Mathematics at Harvey Mudd College, has made public the text of a stirring talk he recently gave upon accepting (a clearly well-deserved!) teaching award, in which he expounds on the "Lesson of Grace in Teaching".

He summarizes the "lesson of grace" in two statements: "1. Your accomplishments are NOT what make you a worthy human being. 2.  You learn this lesson by receiving GRACE: good things you didn't earn or deserve, but you're getting them anyway. "

His definition and exposition of "grace" crystallizes beautifully what I have felt and experienced myself but could not articulate so well. I will never forget the mentors in graduate school who offered me warm, unconditional, support and advice. I am grateful to have the chance to pay it forward in my own interactions with students, and do my best to treat them as worthy individuals deserving of encouragement, advice, help, or as they most often need, a friendly, non-judgmental, ear.

We live in an achievement-oriented society, particularly in academia, with its pervasive emphasis on competition, external evaluation, recognition. Much harm is done by this system, not only to those students who do poorly, but counter-intuitively, even to those who do well, because it reinforces a fundamentally flawed connection between self-worth and achievement.

I wrote about something related a while ago in the context of the "impostor syndrome" (the name given to the feelings of inferiority that many students suffer from while doing the Ph.D.).

I wrote in that post that "having given the matter much thought over the years, I feel that what exacerbates the impostor syndrome, or perhaps even gives rise to it in the first place, is adopting the world view that ties one's sense of self-worth to one's achievements and treats achievement as the goal of one's efforts." I argued that it is important for students to cultivate a mindset that places a greater emphasis on learning and discovery as ends in themselves rather than as a means to success through achievement.

Francis's talk calls upon educators to think differently, and to treat the individuals they encounter with grace, regardless of their academic performance as measured by tests and exams. It is a call that, if heeded, would have a tremendously positive impact on the lives of many.

If you can't beat them...

My kids have been playing and talking excitedly to me about Minecraft for several months now. They play it every day, alone and with friends in multiplayer mode. And when they're not playing, they spend an inordinate amount of time either watching youtube videos about it (typically tutorials posted by  experienced players, but also some hilarious musical parodies -- check out "Revenge" below, and also "TNT"), or discussing with friends what they are going to do the next time they play.

As a parent, I have to confess this worried me. This much passion for a video game? I wished they would be this excited about reading books, as I recall being when I was a kid. True, computers or video games were not as common when I was their age, but still, aren't books inherently better because they lead directly to a love of learning? Aren't video games mere amusement? Empty calories for the brain?

Yet, there's another part of me that objects to these fears and resists them. At their age, the books I read were comics (mostly Amar Chithra Kathas). Not because they were educational or helped me improve my reading, but because they were fun, entertaining, engrossing. The joy I derived from them spurred me to read more, and eventually broaden my interests when I was ready to do so. I can trace in my mind a clear, direct path from that joy to many of my current interests and abilities. Pursuing one's interests passionately, with joy, at one's own pace, is a big part of what it means to be a child. They must be given a lot of freedom to play, grow and learn in their own way.

My older kid insisted that I play it for myself to see what is so compelling about Minecraft. Under his expert guidance, I got a taste of it for myself. It is a first-person adventure game. There are weapons (swords, axes, bows and arrows --- no guns!) and menacing creatures to fight (some that are familiar from other contexts: zombies, spiders, skeletons, and some that are quite unique to the game: creepers and endermen), and one has to gather various resources (food, wood, stone, minerals) and craft some of them into stronger tools and armor to fight these creatures. But it is also very significantly about building one's own world, brick by brick, (houses, shops, markets, buildings, entire villages), and about inhabiting and exploring this virtual world with real-world friends.

Minecraft is an independent game originally created by a single programmer (Markus "Notch" Persson) in 2010.  Its retro graphics demonstrate an appealing emphasis on functionality, game play, and interactive story creation, over the slick-looks-but-tired-game-play emerging from the large commercial game design powerhouses these days.

As this article, titled "Why is Minecraft so damn popular?", notes:

It is, first things first (and before I am lynched by Minecraft fans), a good game. Maybe even a great game. It's got an iconic look, it's widely accessible, it allows gamers to create their own stories, and perhaps most engrossing of all, has an initial simplicity and ease of play that quickly gives way to a complexity as deep as the mines you'll soon find yourself digging. 

It is a hugely popular game. It is reported that about 8.7 million copies of the game have been downloaded for personal computers so far (this does not include other platforms on which it is available, such as Xbox, mobile devices and tablets).

There is another aspect of this game that accounts for its appeal. It has been made available as an open-source platform (in the form of the Minecraft Coder Package). Anyone is free to see the source code of the game, and modify it to their heart's content. It's all written in Java, so this is relatively easy to do for anyone with a decent exposure to programming. There is a large and growing community of modders that have created enhancements and imaginative alternative versions of the game.

A couple of days ago, a friend pointed me at a particularly interesting mod that has been very recently released for Minecraft, called ScriptCraft. This mod, created by Walter Higgins, integrates JavaScript into the game and allows players to write small programs (scripts) to automate building tasks while inside the game. I told my kids about it and they were instantly thrilled and wanted to see it.

I've previously introduced them to Scratch, which allows kids to write simple games quite easily using a drag-and-drop graphical interface for programming, so they are already quite familiar with the idea of writing code to make simple games. But this promises to take coding fun to an whole new level.

Due to some compilation challenges I spent quite a bit of time fighting to install this mod. They watched my progress (or lack thereof) in tense expectation, which made it very clear to me how much this mattered to them both. It was rather amusing, actually. The older one shed real tears when the compilation repeatedly turned up 47 errors. And the little one, as he reluctantly went in to have his teeth brushed,  was heard muttering to all who cared to listen, "how come there's only one "javac" on the computer?"

To their immense relief, I did finally get it working. (On the odd chance this is read by anyone else finding it hard to install or modify something for MCP involving new packages, a helpful piece of advice: use eclipse!) The kids are now having fun dropping ready-made cottages, forts and castles in the game, and for now, with my help, modifying their size and shape.

In the weeks to come, I think this could be a powerful motivator for my older one to delve deeper into programming and algorithmic thinking, which would not be too bad a thing...