Why Math is Hard

There has been a lot of buzz around using "technology" for math education. Specifically, I have heard more people talking about using computer programming to teach math. While I think there are some real benefits to this, there are some major drawbacks if we just take the old methods and apply them to programming. Consider the standard algorithm for adding two positive integers:

• Start with the ones place. Add these. The result will be one or two digits.


• Take the ones place for the result and make it the ones place for the sum.


• Move to the tens place and add the digits. If the result from adding the ones was two digits, add this as well.


• Process the results for the tens place in the same way the result for the ones place was processed.


• Continue this process for each digit in the two numbers.


• Don't forget to add the carry digit at the end, if there is one.

WOW! No wonder kids hate math!

If you don't believe me, this is a snippet of the 75 lines of JavaScript code needed to add two numbers using this algorithm (this uses a bunch of helper functions and a 10-digit addition table):

function add(a, b) {
var digit = 0;
var carry = 0;
var sum = 0;
while (digit < numberOfDigits(a, b)) {
  var result = addSingleDigits(getDigit(a, digit), getDigit(b, digit));
  if (carry > 0)
    result = add(result, carry);
  sum = setDigit(sum, digit, ones(result));
  carry = tens(result);
  digit++;
}
if (carry > 0)
  sum = setDigit(sum, digit, carry);
return sum;
}

OK, so this could be trimmed with some clever programming tricks, and it should be made more efficient in special cases, but it is still a monster algorithm for just a simple operation. I'll happily send the full code to anyone who really wants it. By the way, binary calculations done by the most simple electrical circuits work in almost the exact same way.

Other arithmetic operations are no better. Think about the algorithms you use to multiply, divide or simplify fractions. Then add decimals and negative numbers.

If education is to move in this direction, we should think about how to simplify calculation algorithms into more logical, less comprehensive systems. Do people really need to add $473991 and $47838, or do they just need to know whether this means $500,000 instead of $1 million? Of course not!

Is it even important for kids to be able to describe and apply such an algorithm in plain language?

I think the answer is yes, due to the complex nature of the systems in modern society, but we should be providing tools to allow students to do this without the unnecessary complexity of the medium provided. I envision a sort of computerized graphic organizer that can allow students to express their thoughts through a path of least resistance. This is an education issue, not a technology one (the technology is easy).

As with so many of these initiatives, more work is needed on developing thoughtful learning tools before they are technologized so that kids can spend more time staring blankly at a screen.

You've Got to Keep 'em Seperated

School is not just a facility for learning. It is a social environment that children use to learn how to navigate the world. How unfortunate this arrangement is!

We have endeavored as a society to link academic achievement with social promotion (yes, I said those two words) for far too long. Why should these be related at all? The dire state of public schools in this country seems to indicate that, in attempting to perfect both, we have mutilated both beyond "reform."

Note: This is a commentary primarily of secondary schooling, but the implications thereof are applicable to other grades.

The easiest way to survive high school is to follow the social norms expected of both peers and teachers. Peers will respect anyone who can boost their egos, while teachers will appreciate a student who quietly struggles without making trouble. A smart student will learn to disengage during school, only to spend hours struggling on homework just to get by academically. The real question is why this type of student would go to school at all.

When students do not follow the social expectations of their peers, they are damaged horribly by bullying. Society's response? Anti-bullying campaigns! What is so amazing about this is that the schools often are creating the underlying conditions and pressures that cause this bullying in the first place. While the schools must react to bullying when it happens, the proactive measures necessary to prevent it are never even discussed. This is because bullying is based on a competitive model that the education system promotes and that schools rely on for efficiency.

The few students who thrive in high school do so despite the system. Intrinsic motivation and a well-adjusted home/social life keep these students moving in a positive direction. If no diploma existed to reward their efforts, they would create one. It is too bad that these students have to put up with school.

Why do we bother?

There is a belief in society that everyone has, at the very least, the ability to graduate from high school. Academically, this is likely true with very few exceptions. Socially, this is likely true with very few exceptions. Together, the statistics speak for themselves. In some areas, nearly half of all students drop out. Within the half that graduates, many attain a diploma by the skin of their teeth, barely accomplishing the minimum to warrant a piece of paper. They do this because colleges, jobs, families and other societal pressures deem it necessary for success in life.

Education is seen as a necessary component of modern society due to its ability to efficiently train people to perform complex job functions. Yet, when most people are asked how often they use the skills that they learned in high-school, college or post-doctorate studies in their jobs, they will readily admit that 90% of the material is lost. It is the experience of education, not the content that society places a value on.

What can we do?

The experience of school, the social aspects, the emotional supports, the rites of passage, and the growth of personal qualities can be partially separated from the academics. These skills should be taught explicitly and not by sitting in rows of desks. A large component of the value that society places on education is based on these experiences. Instead of pouring resources into flushing out the "best" wording for academic standards that everyone has a general agreement of, why not create new standards for learning how to manage time, resolve conflicts and work independently? Why not completely separate these classes from the development of essential academic skills? Academics will work themselves out. Personal qualities will not.

For all of those who loathe the state of academics in this country, this would seem to be heresy. Taking valuable time away from math and language to give to organizing calendars and learning to get along with others is simply unacceptable. That is, unless we change the entire model of school to fit a more realistic approach. Students learn very little in a classroom with 20-40 students where a teacher is lecturing as it is. It would seem ridiculous to argue that this model needs to be expanded for the sake of academics.

An implementation of this would include online coursework, but that is not a silver bullet. Simply making texts, lectures, assignments, etc. available for free to students and guiding them through an appropriate timeline is half of what students need. Teachers could actually have increased capacity if in-person lectures were removed from the model. The role of a teacher would be closer to coach and tutor than to public speaker. While this is already philosophically adopted by educational institutions, it is not adopted in practice.

School buildings could operate more like community centers, giving students access to resources, rather than imprisoning them. Schools have workshops, science laboratories, experts in multiple fields, AV equipment, theaters, gyms, computers, books and a host of other resources that teenagers would love to get their hands on. This could be an exciting place for teenagers, rather than the range of experiences that is currently anything but.

What about "accountability?"

How about "achievement" instead? A high school diploma is currently an explicit award for academic achievement and an implicit certificate of socialization. The meaning could be shifted to be certificate of both achievement in both academics and personal qualities based on rigorous standards for both. This is not difficult for most people to agree with. The difficult part is that, in the current implementation, the only way to do this is to severely compromise the quality of both aspects. By changing the underlying structure of schools away from a time-based in-person approach organized by academic subject, it would be possible to address both in a thoughtful way.

Time, age and grade would all need to take a back seat to achievement. This is a major shift from the current model. Course completion would be based on performance against standards, not seat time or assignment completion. Tests would almost definitely be used, but their impact would be limited to their ability to accurately measure achievement.

It is already happening! MIT's Open CourseWare and other, similar programs have opened the door to open access to educational resources. Online high-schools are emerging and there is growing interest in explicit instruction in personal qualities. In many ways, however, the implementation is preceding the philosophy in this realm. Most resources are purely academic in nature with little concern for the physical student. There must be a balance here, and there must be a recognition that this is not the future for a few, alternative settings. This is the only viable alternative for a 22nd-century education.