Response to the New York State Common Core Math Curriculum

To Whom It May Concern (New York State Common Core Administrators?),

Upon returning from a walk, I found my doorman speaking on the phone with a very concerned and frustrated tone. With little instigation on my part, he approached and showed me a math problem, the nature of which I have never seen before. His six-year-old son had been stuck on such problems for the previous three hours and my doorman and his mother were trying to help him to no avail. As I looked as the problem, I kept reading and rereading the problem with the eager hope that each subsequent reading would reveal something new to me about what it was asking, but my efforts proved useless. The picture and “math sentence” at the bottom only added to my confusion, and served no purpose to my understanding whatsoever.

After 10 minutes, with my husband’s (MIT dual degrees in Math and Computer Science) joined efforts, and numerous internet searches, we came up with what we could only guess was an acceptable “solution”. I write this letter, not so much out of anger which would ultimately serve no purpose, but more so out of SERIOUS concern for what and how our children are being educated. If we turn something which has been around for millennia and serves as the most basic building blocks of modern society into something obscure, difficult, and frustrating, what can we expect further down the road? The primary goal of an educator, especially at this level, is to instill a love of learning in his/her pupil and to make the wonders of the universe more transparent not more opaque. Surely, the Gausses, Newtons and Einsteins of this world evolved from a very different medium of learning and education.

Please reconsider your methods of education. Basic addition and subtraction, in whatever method you choose to teach them, should not make a mathematics PhD think 10 minutes about what the problem is asking and how “to make ten from 12-4”. The frustration and anger these kids feel now very much paints their understanding and sentiment of mathematics further down the road, when it matters even more. We need to bring as much light into math as possible, not add to the darkness, especially during such a formative and impressionable age.

Sincerely,

Doris Dobi, Math PhD

Four-Factor Model for Economic Prosperity

Have you ever wondered what makes some countries rich and others poor? Why is Croatia poorer than Germany despite their geographical proximity? How is it that countries blessed with ample geographical resources lag so much behind their geographically impoverished counterparts? How can it be that countries with old, deep, intoxicatingly interesting cultures lag so much behind cultural newbies such as the US ( e.g., Iran)?

Such questions were deeply interesting to me growing up. Could it be that German people are just better? Could it be that Americans are just that much cleverer than Middle Easterners? What about Africa? Why is it lagging so much? Why is Western Europe so much wealthier than Eastern Europe? Is it in the food? The drink? Are they just smarter? What accounts for the vast disparity in living standards in Asia?

It must be brains. It must be right? Well, not so fast. What about the myriad math and science olympiad whizzes coming out of (relatively) poor countries such as Romania, Moldova, China, India and Greece. I refuse to succumb to the simple, and braggadocio “it must be brains” conclusion. The human spirit strives to achieve, wherever it lives. Could it be that some countries inhibit the creative and productive forces of their citizens more than that of other countries?

Possibly. But how do they do this? And, do they know they are doing it?

A guiding light for me in understanding these questions has been William Bernstein’s book The Birth of Plenty: How the Prosperity of the Modern World was Created. In this great read, Bernstein offers four factors as accountable for the prosperity (or lack thereof) of countries. Via the establishment of capital markets, rule of law/civil liberties (esp. property rights), a scientific method of thinking, and finally through efficient methods of communication and transportation a country creates the genetic blueprint for a thriving economy. If any one of these factors is missing then it poses a great impediment in the creation of wealth.

It is a consequence of historical outcome (and a topic much broader in scope than this post) that by mid-19th century certain Western European nations, in particular Holland and Great Britain, succeeded in being the first in the world to have all four factors in place; albeit via much turmoil and initially at very primitive levels. What do I mean by “historical outcome”? Isn’t this just a way of posing the same questions but in a 19th century backdrop? Why were these countries the first to succeed?

If one carefully, as Bernstein does, compares the happenings of different regions of the world during this period it is not so obvious that Western Europe was destined for dominance and prosperity. There were clear winners, at least in hindsight, but it is not obvious at all, for example, that the Ottoman Empire was any weaker than the West. In many ways it was much stronger, and almost ended up conquering all of Europe.

It is due to the establishment of each of these four factors, and no sooner, that a country can truly prosper. There are very good logically sound reasons as to why each of these factors is a necessity. Each factor works synergistically with the others to create a robust system for learning, creation, distribution, and wealth-generation. It is no coincidence that those countries most Westernized enjoy higher levels of prosperity. It is in the adoption of the same institutions supporting the West that other countries, intentionally or not, create the framework for the four pillars of prosperity.

Ultimately, one can still argue whether or not the West is in some way “superior” for having been the first to achieve this state. But, I think, it is important to pause and reflect on the meaning and fruitfulness of this debate. Whether or not the West is inherently better is largely irrelevant. What is very much relevant and very much hopeful lies in this four-factor description for the foundations of prosperity. Only once a country has given the four-factor model a try and its results are clearly visible, can we begin to have a meaningful conversation about the underpinnings of progress. Only once we have leveled the playing field, and have done our best to atone for the seemingly random and at times brutish outcomes of world history can we begin to abandon myopic discussions of superiority and have a chance at global prosperity as one race.

MODULATOR!

In fourth and fifth grade I loved playing this game called 24. Inspired by the experience I had playing 24, I designed a game (my husband did the programming) over the summer with a similar feel, but many twists. Take a look for yourself! It is called MODULATOR.

When you first start you’ll see a screen which prompts you for either instructions or play:

start_screen

Click on instructions:

instructions

The idea is that you will have four integers between 0 and 9 and three operators, chosen from +, −, ×, or ÷ either operator or integer can appear more than once, and you need to combine all four integers with all three operators to get to a final answer. If your final answer is on a closed and crossed out pipe, try again, you need to get an answer that is on the pipe which is open.

For example, if the following puzzle appears:

example_2

Your final answer needs to be a number which is congruent (read equals) to 2 mod 4, e.g., 2, 6, 10, 12, …, 142, etc. If we try:
2÷3 we get 2 (because 2×3 is 2 mod 4), then we can do 2−3 which is -1 which in turn is 3 mod 4, and finally we can do 3−1 to get to our final answer of 2. Bingo!

Let’s try another. Consider:

example_1

Our mod here is 6, and we need the final result to be 4 mod 6. If we try: 5÷5 which is 1, then 3÷1 which is 3, then we can do either 3−1 which gives us 2 and is not the desired answer, or 1−3 which gives us -2 which is 4 mod 6 and is what we’re looking for!

The game is free so play it for yourself and see how good you can get!

Ultimately, everything is molecules—Part 2

——————————————-PART 2————————————————-

But how well can we really know ourselves? How much information about our molecules do we need in order to be able to predict future behavior?

In 1931, Kurt Godel demonstrated a remarkable, very interesting and provocative fact about mathematics. Without getting too technical, it says that certain mathematical statements can never be proven (nor disproven) within the logical framework of mathematics itself. Loosely speaking, a closed system can never know everything about itself; it always has to go outside of itself to gain further information. Hence, by doing so it creates an even more complex system with yet more unknowns. The Incompleteness Theorem has been taken to mean that we will never fully understand ourselves, since our minds, like any closed system, can only be sure about what it understands about itself by relying only on what it understands about itself.

One can argue that people are not rigid closed systems that operate on a fixed set of axioms, and hence Godel’s Incompleteness theorem has limited applications when it comes to us understanding ourselves. But there are other compelling arguments as to why we cannot fully know ourselves. For one thing, our brains alone have something like 10^50 neuronal connections. Understanding all of these connections takes more time than any human has…a generous calculation shows that it would take much longer than the age of the universe to count so many connections.

Furthermore, knowing ourselves requires knowing what molecules we have where, how many of them we have, how fast they are moving and where they are going. The last two requirements are where we can see Heisenberg’s uncertainty principle coming in. Heisenberg’s uncertainty principle states that it is physically impossible to exactly know both the location and the velocity of a particle. On top of it all, there are also random molecular perturbations and movements due to heat inside of our bodies. Hence, from this randomness and from physical law, there are limits to how well we can understand ourselves and to how well we can predict future behavior based on a current understanding of our molecules. In at least this sense we are also limited in our knowledge of the future.

What can we gain from this knowledge?

If we accept a molecular basis for all things, then we must realize that there is always a molecular framework on which all psychological and environmental influences act. By better understanding this framework we can better handle adverse influences. We are also limited in our understanding of ourselves and there is a limit to what we can explain. Some behaviors happen for no other reason than random movement of molecules due to thermal jumps. As hard as this may be to accept, sometimes we just have to realize that some things are truly random (the ethical and regulatory implications of this are really too long and unclear to go into here).

My vision:

Provide people with a way to better understand their biological needs. This is no easy or cheap task at all, but it will revolutionize health and humanity as a result. Imagine a loving parent finally having a tool that they can use to better understand their child’s molecular predisposition, to better understand nutritional deficiencies before they turn into major life altering behaviors, to better understand what kind of an imprint the environment their child grows up in leaves given the greater backdrop of molecules.

We have many gadgets that play music and movies and games and all sorts of other entertainment at will. What if we had yet another one that helps us better understand and take care of ourselves via a molecular basis. It may be that an individual’s nutritional needs are as unique as their fingerprints. What if we could each have a way of knowing what those needs are and if they are being met. What if we could have a way of understanding our chemical makeup and its implications? We might not have a say in the choreography of our molecular dance, but we might be able to make it flow better, more smoothly, and with a lot more foresight about where it’s going. Such a device may be available in the future, but it requires tremendous resources and research to achieve. As of now these things are nascent dreams, but we have to start somewhere.

Ultimately, everything is molecules

——————————————–PART 1———————————————-

I recently finished reading Molecular Gods, How Molecules Determine Our Behavior by Philip B. Applewhite, and it provides a very good background for a lot of the things that I have been thinking about, on and off, for a few years now. Namely, in an attempt to understand myself better, I am curious to know the critical factors which help determine important attributes such as temperament and behavior. Which factors determine our physiology? Which factors give rise to those inclinations of certain patterns of thought and being each of us has that pave the road of life’s journey? How much can we change? What can we know about ourselves? How can we change?

I’ll focus on the parts of the books that really captured my interest.

Applewhite, essentially a reductionist, claims that our behavior is controlled by molecules—and nothing else: “Every thought, every movement, and every gesture of ours depends upon the right molecule being at the right place at the right time in our bodies.” He goes on to say, “While they (molecules) shape our noses and mold our feet, they also cause us to be aggressive or passive, fat or thin, creative or dull, homosexual or heterosexual, mentally ill or happy. These molecules act as gods, so to speak, ruling us perhaps more than we imagine.” If molecules are indeed so powerful, then we should reevaluate how much of an influence our environment actually has. Our environment acts on us via molecules; but, to what extent can this interaction alter our behavior, especially if the genetic machinery is not already there?

As an example, consider that many serious diseases and afflictions in the past have been discovered to have a biological basis. They are not caused by identity problems, traumatic life events, society’s faults, faulty interpersonal relationships, unhappy childhoods…etc, as once thought, but rather by identifiable metabolic disorders and microorganisms. More concretely, genetic studies have revealed that if one identical twin becomes depressed, the other twin is far more likely to suffer also than if they were fraternal twins. Adoption studies have indicated that identical twins raised apart by different parents are just as likely to be depressed as identical twins raised together by the same parents. Changing environment does not reduce the chances of suffering from depression. Many other examples show that innumerable diseases are strictly biological, essentially to not having enough or having too much of one molecule or another in a particular place.

The implications of a molecular-based approach to understanding people and disease are huge. For one thing, this approach diminishes the role of environmental impact and greatly enhances the consideration of diet in assuring the availability of certain chemical precursors necessary for proper functioning. Maybe the approach to treating people like molecular machines: if the right molecules are not present at the right location in the right amounts at the right time then there will be malfunctions, can lead to not only a very concrete way of dealing with disease, but also to a near eradication of guilt-induced feelings arising from a misunderstanding of how much influence psychological and environmental factors have. Imagine people liberated from the guilt that they are responsible for another person’s mental or physical illness. Imagine a world with a much clearer blueprint of cause and effect.

I don’t believe that the influence of environment can be eliminated altogether, especially since the environment impacts our senses, which in turn affect our molecules. But it could be very possible that we are either placing too much emphasis on the environment, or if not, we can easily alter its effects by supplementing our bodies with the right molecules or even counteracting the effect of “bad” ones.
The practical applications of a molecular-based approach are numerous: if in a certain region of the world people are genetically inclined to not produce enough of a certain enzyme for optimal functioning, enhance that molecule. If a group of people are subject to molecule-altering stressors supplement them with the correct molecules. By having a clearer picture of which molecules we need, which molecules are responsible for specific functions, and the amounts of molecules necessary, we can provide some insurance against life-altering illnesses by understanding the future consequences of each individual’s molecular dance.

What the heck is meaning?

I was initially going to title this post “what do we mean by meaning”, but opted for the less circular heading above. The search for meaning is ubiquitous. What does that mean? What’s the point? Is it possible for people to live in a meaningless state? For how long?

I think there are as many forms and attributes of meaning as there are ways of looking and experiencing the world. Any attempt to define it will be futile and at best incomplete. I wish to express some of my own thoughts on why we need meaning and what that means (here we go with the circularity again :P).

Not wanting to get too philosophical, ultimately, I think meaning is our reason for existing, and it may change with our environment and life circumstances, but it’s there, or at least somewhere
deep inside of us something provokes us to find it again, or to fine tune it until we can hear the music, as we like it, again and again.

People have found meaning in suffering, in their children, in their work, and in a vast number of other things. There has to exist some outlet for our meaning, for our reason of being, whatever that may be to each of us.

Maybe it’s the fact that we can only have so many neural synapses, so many breaths, so many moments of flow, so many moments right after a light spring rain just before sunset when with each passing moment daylight grows noticeably dimmer and the sun smoothly shifts in the sky making room for it’s other familiar companion that makes one of such experiences painfully and beautifully and tangibly meaningful. I am not sure whether it is our mortality that creates an ideal environment for the growth, nourishment, and search of meaning, but it’s a powerful nutrient

For me meaning has evolved (devolved?) and shifted as I change, but it seems that there remains a common thread directing me, however thin at times. It includes a strong need to understand and experience life (maybe even to understand and experience meaning itself). For some people, I think, this thread eventually ends up making the very fabric of the meaning it lead it’s user to experience. In its more recognizable form this fabric makes up our symphonies and art, our textbooks and novels, our architecture and fashions, our wars and hymns, our choice of lovers and personal expression. This fabric shields us from the cruelties and existentialism of life; ultimately this very fabric makes up our lives. For the two seem to be inseparable.

Why matplotlib is an incredible tool

I have become a huge fan of python (I’ll write about this later), but I’ve recently discovered a python library which is so useful it makes me really excited and thankful that some awesome people out there have coded so many awesome things. Together with numpy, these two libraries coupled with python provide me with all of the automation and visualization tools I’ll need anytime soon.

Check out this page for a description of matplotlib. I thought that matplotlib was pretty much exclusively a visualization tool, but as it turns out, there’s a lot more to it. In this post I’ll focus on its finance module (for some reason this module does not show up in the documentation, but it is very useful), since this is most relevant for my research. Explore this module on your own, it’s a gold mine.

To give a sense of where my excitement is coming from, about a year ago I had to write a python script to scrape some data from yahoo finance. I had never done this before, so it took some explaining and help from a certain someone to get my feet wet. Eventually, I wrote it, and it worked. Turns out, I need to do a lot of scraping as a starting point for my research, and I always refer to this first python script whenever I have to do this. I’ll attach the code for the script at the end of the post if you care to see it. Amazingly, this whole process is completely automated in matplotlib.finance. The finance module contains a function called quotes_historical_yahoo that takes as inputs the ticker symbol, the start date and the end date, and returns a record array with fields: date, open, high, low, close, volume, adj_close. This exactly what I wrote my script to do, but after importing the matplotlib package it becomes a one-line job. Here’s an example:

>>> from matplotlib.finance import quotes_historical_yahoo
>>> import datetime
>>> ticker = 'WFM'
>>> start_date = datetime.datetime(2012, 8, 1)
>>> end_date = datetime.datetime(2012, 8, 10)
>>> WFF_List = quotes_historical_yahoo(ticker, start_date, end_date)
>>> WFM_List = quotes_historical_yahoo(ticker, start_date, end_date)

Now, WFM_List is an array of tuples each consisting of the following data: (date, open, close, high, low, volume). Let’s see what the first three entries in this array look like:

>>> WFM_List[0:3]
[(734716.0, 92.269999999999996, 92.689999999999998, 93.609999999999999, 92.0, 1449900.0), (734717.0, 92.0, 92.909999999999997, 93.879999999999995, 91.670000000000002, 1453800.0), (734718.0, 95.200000000000003, 94.599999999999994, 95.980000000000004, 94.409999999999997, 1414800.0)]

Notice that the first entry in every tuple is a gregorian ordinal, so to obtain a gregorian date we can do:

>>> datetime.datetime.fromordinal(int(WFM_List[0][0]))
datetime.datetime(2012, 8, 1, 0, 0)

You can confirm for yourself that this agrees with the data obtained from yahoo finance. There are also some very practical things you can do with the finance module, but I’ll let you explore that on your own. As promised, here’s my original python script written from scratch:

import httplib
from time import sleep
from os.path import join, exists
import csv

def get_symbol(symbol):
conn = httplib.HTTPConnection("ichart.finance.yahoo.com")
conn.request("GET", "/table.csv?s=" + symbol + "&a=07&b=1&c=2012&d=07&e=10&f=2012&g=d&ignore=.csv")
res = conn.getresponse()
print res.status, res.reason
data = res.read()
return data

print get_symbol('SPY')

If you run this code you’ll find that the output agrees with that above.