Why word problems might be difficult for elementary school students

When I was in elementary school, I was puzzled by classmates who had trouble with word problems in math. After all,  they’re not only giving you a math problem, they’re giving you a backstory that will allow you to develop some intuitions about what’s going on. I recall that in fifth or sixth grade, we were taught some tips and tricks for converting word problems to stripped down arithmetic problems in preparation for standardized testing. The teacher told us that problems of the form “Has x things, and then takes y more things” were addition problems. This confused me, and for a while I thought it had to do with reading comprehension.

  While reading this article, I began to formulate an alternate hypothesis. Perhaps it is not that the students don’t have the reading comprehension, but instead that they don’t know how the arithmetic system connects to reality. And instead of teaching the connection, they taught more symbol manipulation tricks, teaching that “more” means “+”, “less” means “-“, and so on. This would explain quite handily the issues many young students had with word problems.

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Why word problems might be difficult for elementary school students

When I was in elementary school, I was puzzled by classmates who had trouble with word problems in math. After all,  they’re not only giving you a math problem, they’re giving you a backstory that will allow you to develop some intuitions about what’s going on. I recall that in fifth or sixth grade, we were taught some tips and tricks for converting word problems to stripped down arithmetic problems in preparation for standardized testing. The teacher told us that problems of the form “Has x things, and then takes y more things” were addition problems. This confused me, and for a while I thought it had to do with reading comprehension.

  While reading this article, I began to formulate an alternate hypothesis. Perhaps it is not that the students don’t have the reading comprehension, but instead that they don’t know how the arithmetic system connects to reality. And instead of teaching the connection, they taught more symbol manipulation tricks, teaching that “more” means “+”, “less” means “-“, and so on. This would explain quite handily the issues many young students had with word problems.

Of Monkeys and Traditions

     If you look at many of the traditions we have today, sometimes you wonder about the sanity of humankind. In some cases, it is quite clear what the behavior in question accomplished, but it is equally obvious that it is no longer useful. So why do we keep doing it? In 1967, professor G.R. Stephensom published a paper titled “Cultural acquisition of a specific learned response in rhesus monkeys”. This study explains a great deal about how human culture originates and perpetuates.
    The study was simple. Five rhesus monkeys were placed in a cage. In this cage was a ladder with bananas at the top. Whenever a monkey climbed the ladder to get the bananas, the researchers sprayed water into the cage. The monkeys, being reasonably bright animals, put two and two together and figured out that their current dripping misery was caused by the monkey who tried to climb the ladder. They attacked the banana-grabber, and thus the group of monkeys learned not to go up the ladder.
     The scientist then replaced one of the original monkeys with a new monkey. The first think the new monkey did was attempt to get the bananas. The other monkeys apprehended and beat the newcomer, who quickly learned that the bananas were off limits, despite never having been sprayed with water.
     Once again, the researcher replaced one of the original monkeys, and once again the new monkey headed straight for those tasty bananas. The other monkeys, /including the other replacement monkey,/ attacked the new monkey. Remember that the replacement monkey did not know that grabbing the bananas led to being sprayed with water.
     One by one, the scientist replaced the remaining monkeys. When the fifth original monkey was replaced, its replacement attempted to grab the bananas. The other four monkies, /none of whom had experienced the spray of water,/ attacked it. The group, now entirely composed of new monkies, still avoided the ladder.
     This research illustrates how it is that traditions continue on after they stop being useful. We humans, however, have an advantage over rhesus monkeys. Actually, we have two. The first is language: we can enquire as to why an action that is forbidden by tradition is bad. The monkeys did not have this ability, and thus only knew “the bananas are bad” instead of “the bananas are bad because we get sprayed with water when someone tries to get them.” Our second advantage is critical reasoning. We can tell when a reason for doing something is no longer valid. In this way, we can stop following traditions when they stop being useful.
     I come from a Jewish background–I have heard much talk about the value of following the days of fasting, and of keeping Kosher. What is not mentioned is that the former was a way of stretching out food supplies just a little bit longer, while the latter prevented a great deal of disease from infected pork. These traditions were not a way of building the spirit, they were a means of survival. Children who did not follow these traditions were punished, because they could endanger their lives and the lives of their family. These children grew up, and when their children did not obey the traditions, they too were punished, and so on through the generations. Over time, the original intent was lost, but the behavior continued.
     The next time you are tempted to talk about the value of following traditions for their own sakes, ask yourself what you hope to accomplish by following that convention. Then ask yourself if that was the original intent behind the practice. If the answer is no, see if you can think of a better way of accomplishing your goal.

How unlikely is uploading

      Based on what evidence I have been able to gather, and assuming Moore’s law holds, computers today are not at all far from human level. A human brain runs approximately 100,000,000,000 (10^11)  neurons, each with an average of 1000 dendrites, at a maximum rate of about 100 impulses per second. This comes out to about 10^16 operations per second, or a rather impressive amount of raw power. According to the almighty Wikipedia, HPU4Science is the cheapest currently available source of computing power, at $1.80 per gFlops (billion operations per second). So we take 1.80×10^16/10^9, and come up with a cost of about 18 million dollars to run a human brain. In simple terms, this means we  are technologically capable of doing an upload now.

      Let me reiterate that. With today’s computing power, we have the theoretical capability of running a human mind in real time on a computer. Moore’s law states that computing power per dollar doubles every two years, so by 2031 it should cost around $20,000, or the price of a new car. Even now, though, our limitation is not computing power. Neither is it memory, not by a long shot. In fact, the amount of memory needed to hold a human mind, neuron structure and all, would cost somewhere around $10,000 today.

    Our real barriers lie in our understanding of neuroscience and our mastery over nanotechnology. Currently, we are only able to emulate circuits of around 25000 neurons (at least, according to IBM, who also created Watson), or a cat’s brain. It takes enormous amounts of work to do this, as each neuron must be analyzed and a circuit diagram must be created for it. This has the unfortunate effect of destroying the neuron structure that was being copied. While we should have the capability of diagramming and simulating a human brain by 2020, actually being simulated would involve being killed, and the first few attempts would doubtless have problems and not run properly.

     This is where nanotechnology comes in to save the day. In theory, it should be possible to create tiny machines that are attracted to the electromagnetic fields of neurons, and can thus line up along dendrites and pass along the same signals as the dendrites themselves do.  These nanomachines would “learn” from the neurons when to fire, and would eventually match to arbitrarily high levels of precision. Basically, the skull would at that point contain two brains, on made of neurons, the other made of nanomachines, both doing exactly the same thing.

      Now for the good part. The brain made of nanomachines would now be functionally identical to the one made of neurons. The structure and function of each of these machines could be uploaded to a computer, at which point a circuit diagram would be trivial to produce. The computer could then optomize from this diagram, leading to a significant  increase in speed. Even if this step were not taken, computer processors run at somewhere around a billion operations per second. Assuming each group of 1000 neurons shared a processor, this would mean a 10,000x speedup over the original brain. This is nowhere near the theoretical maximum speedup (neurons are about 100x bigger in all dimensions and 1,000,000x slower  than the laws of physics say they have to be). Nonetheless, it is a good start.

      As for the timeline, predictions for when uploading should be available range from IBM’s optomistic 2019 to some guesses as late as 2100. I personally would put it sometime in the 2030 to 2050 range.  If all of this is starting to sound like science fiction, just remember that predictions of the future tend to undershoot reality. Even twenty five years ago, nobody would have predicted how ubiquitous the internet would be in our lives.

Thoughts on Evolutionary Psychology

 A lot of people lately have been criticizing evoltionary psychology, and with reason. Many of the ideas expressed are nothing more than stories made up after the fact to explain very specific behavioral phenomena. However, not all evolutionary psychology is worthless. Since a lot of evolutionary psychology has to do with gender and reproduction, particularly the parts that get criticized, I will focus mainly on that subject area.
 A gene is more likely to be passed on to the next generation if it results in a trait that confers a reproductive advantage over those lacking the trait. Over time, this results in a change in the gene pool, known as evolution. One of the most potent reproductive advantages among sexually reproducing organisms is mate selection. This is particularly true among humans, as we do not have any natural preditors or competition for resources besides humans, and among modern humans the competition for resources is not that fierce. This leaves competition for mates as the primary evolutionary force among us.
 It is important to keep in mind what can and cannot be passed along geneticaly. A lot of amateur evolutionary psychologists confuse memes and genes, which are two very different things.
 A gene is a snippet of genetic information that codes for a specific trait, and is spread through reproduction. Its rate of spread is limited to the reproductive advantage it yields, which means that a gene that gives a large advantage will spread far faster than a gene with just a small advantage. A gene that gives a 10% reproductive advantage to those possessing it (which is huge, think dietary adaptions to agriculture) will be 1.1 times as common in each new generation than the generation before, on average. This means it will take approximately 7 generations, or about 150 years in humans, for this gene to double in frequency. From the time the gene is introduced to the time it is expressed in half of the population will take 20 to 30 generations, or in the case of our hugely advantageous gene, 30,000 to 50,000 years. This means that it is unlikely that any significant evolution occurred since the invention of agriculture, except among genes that were already fairly well established before agricultural times.
 A meme, on the other hand, is an idea that can be spread from one person to another. Memes be adopted very quickly in a society, and can give a huge advantage to those who adopt it. One good example of this is agriculture. It is important to remember, however, that the adoption of a meme is not a genetic adaption, despite the fact that is improves the fitness of the individuals who adopt it. Humans who adapted agriculture were gentically the same as those who didn’t but those who did adopt it had a huge reproductive advantage and perpetuated the meme.
 Most reproductive adaptions in humans probably occurred before the introduction of agriculture.This means that any evoltionary psychology focusing on modern situations as leading to adaptions should be scrutinized very carefully to see if they would confer enough advantage to become prevalent in the population, taking into account how common that gene would likely be before agriculture and thus how many generations it would take to become common.
 One last thing to consider is how the gene would be passed down. If the gene confers a reproductive advantage, you need to consider that it is probably not on the sex chromosomes (as that is not where most mutations would occur) and that it is therefore probably shared among both genders. This means that it would most likely be triggered hormonally by testosterone or estrogen/progesterone, and therefore would express in females with high testosterone or males with high estrogen/progesterone. If this is, in fact, observed, the hypothesis may well be true. If it isn’t, the common trait is probably memetic and not genetic.