Saturday, December 4, 2010

My Thoughts on "Race To Nowhere"

Last Thursday I got the opportunity to watch Race To Nowhere, a documentary on increasing level of academic pressure and stress in American schools made by producer Vicki Abeles, a lawyer and a mother of three school going children. The screening followed a question answer session with a panel consisting of the producer herself and a few other advisors. The main points made in the documentary have been summarized, praised and critiqued in many excellent online reviews, including one at NY Times, another one at The Huffington Post, a level headed post by a blogger that I particularly liked and a CNN news report, so I would limit this post to my own thoughts.

As I listened to the narratives, the main thought going around in my head was not one of complete agreement or of violent disagreement with the points made, but one of "whatever is true is also not true" about a topic as complex as education in the large and diverse body of students, teachers, parents, schools and policy makers. Yes, some students get burdened with more work and higher expectations than is healthy for them but is that true for all or even a majority of students, at all or even a majority of schools, in all or a even a majority of households? The documentary itself does not include any statistics or scientific research in this area, and relies mostly on individual testimony by parents, students and educators. I have two daughters, one in elementary schools and one in middle school, have known many parents with school going children and interact with a class of middle schoolers on a regular basis, but have never heard anyone complain about burnout or stress due to academic pressure.

There is no denying that we live in a society where success and achievement means a lot, perhaps much more than inner happiness. But that is the world we live in. Thankfully, we also live in a world where individuals can decide what they want from life and make choices accordingly. No one compels a parent to expect his or her child to get A+ in all subjects, do 3+ hours of sports practice, participate in innumerable after-school clubs and, on top, to do social work while completely ignore play, socializing with friends, experimentation and entertainment. In fact, this was the main takeaway for me: there are a lot of things to do out there but it is up to the students and parents to decide what areas and how much effort is appropriate for them.

Thursday, September 9, 2010

Commentary on "Understanding Divisibility"

Divisibility, the property of an integer being a multiple of another integer, and many other related concepts such as primality, Greatest Common Divisor (GCD), Least Common Multiple (LCM) etc. are central to the learning of Number Sense and form the basis for working with fractions and many other higher level math topics. The notion of divisibility and related operations are usually introduced in elementary schools over a number of years, but are rarely presented as a collection of related ideas. Also, they tend to skip certain kind of reasoning in favor of rote memorization. For example, kids might learn that if the sum of digits of a number is divisible by 3 then the number is also divisible by 3, but may not know why this rule works.

Advanced math texts focus more on formal and structured treatment full of lemmas, theorems and formal proofs under the general subject of Number Theory.

So I planned to start this year's MATHCOUNTS club with an overview of Divisibility. As there is rarely enough time in the class to go over all the topics in sufficient detail, I thought it will be good to point them to relevant stuff on the Web. However, my search drew blank. Wikipedia articles are good but too advanced and disjointed for an average middle school student. Relevant articles at other sites also didn't stand up to my expectations.

So I resolved to roll my own tutorial: Understanding Divisibility. It takes somewhat unconventional approach to introduce ideas and provide insight on mathematical reasoning by stating them in plain English and avoiding too much use of notations that may be a put-off for the younger crowd, such as modulo arithmetic for explaining rules of divisibility.

The current version at the time of this post is still rough at edges but I plan to polish it over time based on feedback here and discussion in the class.

Sunday, January 31, 2010

Unjustified Criticism of The Way Math is Taught

I have come across many unjust criticisms of US education system in general and the way math is taught in particular, but this one takes the prize: the author, who himself is a math teacher, criticizes a particular problem statement as being unrealistic, the solution to be artificial and the laments that the problem doesn't teach any problem solving skill.

It is not that I see no need to improve the US education system or the way math is taught. There are many valid criticisms and many more ways to improve the system. Carefully chosen and well worded problem statements are certainly one of them. But I found the acerbic reaction to this particular problem statement to be misguided. Let me explain myself.

Let us first take a look at the problem
A youth group with 26 members is going to the beach. There will also be 5 chaperones that will each drive a van or a car. Each van seats 7 persons, including the driver. Each car seats 5 persons, including the driver. How many vans and cars will be needed?
The problem can easily be solved by representing no. of cars and vans as two variables and writing equations satisfying the constraints as stated in the problem. However, the author feels that this particular problem is not worthy of being introduced to students for following reasons:
  1. "One, is the problem realistic? Would a real person need to solve this problem?
  2. "Two, is the solution realistic? Would a real person solve the problem using a system of two equations?
  3. "Three, in what ways does this problem help our students become better problem solvers?"
The author does elaborate on each of the points and you should read the blog post to understand his point of view. What follows is my reaction to his points:

Is the problem realistic? My answer: yes, it is as realistic as it can get. The domain of the problem statement is certainly familiar to most students and they understand the role of chapreones as drivers, the difference between cars and vans and the need to optimally use the vehicles. A real life scenario may have additional constraints in some aspects such as the need to seat friends together, availability of specific types of vehicles only and less constraints in others such as no need to fill all vehicles to their full capacity. But simplification or idealization is routinely done in scientific or engineering problems and I see no reason why it shouldn't be done in a math problem.

Is the solution realistic? Well, given the simplicity of the numbers involved, a student might be able to solve the problem by simple trial and error and I'll accept that as a valid solution. In fact, that is what I would expect from a 3rd or 4th grader. A system of equations with two variables is certainly a more generalized solution and very realistic for the problem. Historically, the system of linear equations arose in the field of transportation as a way to optimize routes and this particular problem and solution are good substitute for the more generic problem.

In what ways does this problem help our students become better problem solvers? First, students learn how to convert a real life scenario to a system of equations. Based on my own experience working with middle school children I can say that this in itself is no mean feat. Second, they are able to relate the answers back to a real life solution. If the teacher encourages alternative solutions, such as the one using trial and error or the one using a single variable, then the students also get a better understanding of the system of equations. Much better than just solving a long list of equations without relating to any situation, real or unreal.

Yes, we can come up with more real problem scenarios from different fields of Science, Engineering, Operations Research or Accounting but will the students find that more palatable? I doubt. Saying that most people don't face situations requiring solution of a system of equations and hence should not be included in curriculum misses the point about teaching maths.

Saturday, January 16, 2010

Love from Bing and Google

Was pleasantly surprised to find that Bing is returning as the top entry on searching "Pavaki". The site has been public for less than two weeks and I have done absolutely no Search Engine Optimization.

Of course, the real test will be with Google, which currently lists as the third entry from top.

Not that this is going to send horde of people seeking problem solving skill flocking to the site, but at least people knowing the site name can stumble to the right place!

Friday, January 1, 2010

The Story of Pavaki.Net was born as a set of humble scripts I wrote to enter, store, format, and track math problems for students attending MATHCOUNTS coaching sessions at Peterson Middle school where I volunteer as a coach. For each session, I would decide a main topic and type-in problems of varying difficulty in LaTeX, a type setting system developed by Donald Knuth and used widely by mathematicians and scientists. In the beginning, I stored the problem text in a database and used a script to generate nice PDFs. This may seem too elaborate for a simple task but I had my reasons -- I wanted to assign problem sets based on student ability and also to change order of problems within a set to frustrate students who happily copied the answer from the next guy (it was never a gal!). It was my observation that bright students lost interest when the problems were too easy and not so bright ones gave up rather quickly when the problems were out of their reach. Also, a large percentage of the class would take the path of least activity and would simply copy the answer from next guy simply because they could.

Although I wrote the programs capable of generating such problem sets with problems in random order, putting this in practice turned out to be much harder. It was just too much of a hassle to print 30 different practice sheets for each coaching session and discuss the solutions in the class without having a common sheet to refer to. In hindsight, it is quite obvious that the system could have never worked in a classroom setting.

At this point I started thinking of other uses of my programs, who had grown in number in complexity over time. What if I could assign the problems to students over the Internet and they could access and attempt the problems directly on their computers! They would be able to work on problems and review suggested solutions at a time and place nof their choosing and I will have the flexibility to recommend problems based on their performance, something which the computer can help me do. Once in place, such a system could be used by any parent or coach, I reasoned. And why limit it to only students and maths, even technical job seekers preparing for interviews, or those preparing for competitive exams such as SAT could use such a system to test their own knowledge in a particular area of expertise.

Of course, all this needed a much more sophisticated web application than the loose collection of script I had. Also, the problems and the solutions that I have created are probably not going to be adequate for everyone. So there had to be a way for people to create their own problems, suggested solutions and ability to use them in a way I could with my students. It would also be nice for people to share their problems, with all users or just their own friends.

So began my multi-month hobby project to create, a hosted web application that allows a user to enter, organize, share, assign and solve problems. I populated it with problem sets created for the MATHCOUNTS class. Over time, I also added other kinds of problems -- word analogy problems for my younger daughter who sometimes gets stuck on specific words and technical problems in areas of my interest to amuse my own colleagues and friends and also to brush up my own knowledge.

The result, is long way from being complete but, at least in my humble opinion, ready for use by daring individuals.