Over at Pharyngula, PZ Myers has commented on one of the periodic issues that occurs in mathematics education and that is what mathematics should form part of the general education of everyone. This time the discussion is over whether algebra should be a requirement for a basic general education. Those who argue for its removal say that it is not a skill that most people need in everyday life and that in addition, students seem to find it very hard and fail in large numbers.
PZ argues that “Algebra is a kind of minimum standard for elementary numeracy” but despite my love for the subject I am not convinced, mainly because that statement depends on how much algebra we are talking about. While some algebra can have practical benefits in daily life, at what level of depth is really the contentious issue. I suspect that most people get through life without using any algebra at all or just minimally. The question of whether the manipulative techniques that one learns to use in algebra are generally applicable is also an open one. I am also leery of the argument that students should learn some subject purely because it ‘strengthens the mind’ in some vaguely defined way. Those arguments were used formerly to require students to learn Latin and Greek but we no longer do so.
But that is a different debate. I want to focus on a different issue because it has long puzzled me and that is this question of why algebra is seen as so hard. I can understand people struggling with calculus because the concepts of infinitesimals, limits, differentials, integrals, and why dy/dx is a single entity and cannot be reduced to y/x by canceling the d, are all kind of tricky. The same is true for trigonometry. One starts with the concepts of sine and cosine and tangent defined in relation to right angle triangles and that seems fine. But these three things then break free of their triangular roots and take on a life of their own and go on a wild spree, forming many complicated relationships with each other that seem to have little concrete meaning. While I loved manipulating trigonometric identities (I viewed them as essentially logic puzzles that I enjoyed solving) I can see why their appeal could be mystifying to those who did not love puzzle solving.
But algebra seems straightforward to me and I am sure it was so for some readers of this blog too. Is that because we were weird nerds? Is the teaching of algebra in the US really lousy so that only a few spontaneously grasp it? Note that schools in Sri Lanka start teaching algebra in the 6th or 7th grade, a much younger age than they do here, and I don’t recall that the general student body found it so hard that it created some sort of crisis. Were students to the left of me and students to the right of me falling away in large numbers and I just did not notice? I don’t think so but memories of long ago can be notoriously unreliable. Even if many of them did not ‘get it’ quickly, the fact that it was taught at such a young age suggests that it was not seen as being a particularly difficult subject to learn, given the right teaching. So while the merits of whether it needs to be a basic component of general education can be debated, arguing that the level of difficulty is an issue is surprising to me.
It should be noted that the system in Sri Lanka was to teach all the math and science subjects every year so that we got it regularly in smaller doses over a long period, like vaccines, unlike in the US where there is a tendency to teach a topic in a single year. The latter requires much faster processing of new information.
There are many problems that students encounter in learning algebra. One is the common belief that for any function f, f(a+b)=f(a)+f(b). Hence students tend to think that (a+b)2=a2+b2, sqrt(a+b)=sqrt(a)+sqrt(b), and so on. You would think that by simply inserting numbers into those equations as a test, they would quickly realize that those relationships are not true. But that does not happen, because many students do not instinctively check to see if their reasoning is correct but march along. Some also lack the kind of intuitive sense of numbers that will enable them to realize when they are doing something wrong. While it may well be true that the lack of a number sense is due to the widespread use of calculators, there are three things that the educational and cognitive science literature has exposed that compound the problem, and all are based on the idea that the brain tends to take the intuitive easy way out (System 1 thinking) when learning something.
The first is that given any number of alternative schemas for explaining something, people seize the first plausible one that comes along without stopping to weigh possible alternatives that might be better.
The second is that the human brain has a tendency to take any plausible belief that is learned in a specific context and then disregard the context and apply it universally as a general rule. In this case, students learn early the distributive rule that c(a+b)=ca+cb where a,b,c are numbers. This makes intuitive sense and is simple. They then think that this also applies for f(a+b) even when f is a function and a+b the argument of the function, maybe because a function is a more complicated beast and it is easier to treat f as if it were just a number too. After all f and c are both just letters, no?
The third is perhaps the most problematic and that is that even showing in a few instances that these simple beliefs lead to manifest contradictions is not sufficient to change these entrenched beliefs because it is so much easier to retain an incorrect belief, especially if it works on a few occasions, because that reinforces the idea that it is correct (the familiar confirmation bias problem).
This is a problem in physics education too. Students tend to acquire an Aristotelian view of dynamics early in life because it makes so much intuitive sense. They absorb it based on their everyday experiences so that one does not need to be even taught it. Even when they learn Newtonian dynamics, misconceptions about (say) the Third Law are very hard to dislodge. Similarly, the idea that electric current is ‘used up’ when a battery powers a device makes so much intuitive sense that getting students to realize that the amount of current leaving an electrical device is the same as the amount entering is difficult, even if they learn to manipulate the laws of electricity.
One needs a systematic educational strategy for combating these features of the brain that lead to entrenched wrong ideas.
I personally did not find algebra easy at the beginning but there were specific reasons in my case. I left Sri Lanka in the middle of 6th grade for medical treatment in England and missed a whole year of schooling due to being in hospital. But despite this, when I returned, I was promoted to the middle of 7th grade. I was able to catch up fairly easily on the missed year of work in all subjects except for algebra that they had started teaching in 7th grade and thus I had missed the first six months of it. My algebra teacher was also terrible, simply assigning one problem after another from a huge textbook named Hall’s Algebra as I recall. I remember struggling and being baffled by this mysterious entity called ‘x’ that we were asked to ‘solve’ for and being tremendously frustrated at my inability to grasp the subject. But I do recall that at some point, maybe after a month or so, it dawned on me what the basic idea of algebra was and after that I quickly caught up and loved it.
So I can sympathize with people who have difficulty with algebra at the beginning. My point is that if I had started with the rest of my peers, the dawn of understanding would likely have come earlier, and having a good teacher would have helped a lot. So is the problem just bad teaching of algebra that makes it so hard for so many students here? What was your experience with learning algebra? And how much algebra do you use in your life?