To some extent the problems with mathematics learning and teaching must have something to do with the quality of mathematics teachers.
Yes, mathematics is difficult, perhaps it is the most difficult school subject of all for most people. And it may be that Lockhart is largely right in that mathematics teaching has become stuck in a tradition and a routine that is not at all true to the real nature of the subject. It is not easy for the individual teacher to break loose from this.
If it is the case that many (most) mathematics teachers are not up to the task of teaching mathematics is such a way as to minimize the inherent problems in understanding the subject and maximizing the learning, then this is not a problem that can be blamed on individual teachers. It is a system error.
Who’s not doing their job? Could it be the teaching schools? Lockhart certainly says that. He calls teaching schools a complete crock.
I went to teaching school myself (in Sweden – remember I’m now writing about the Swedish situation). Not the long version, but rather the short version, adding on a year of studying pedagogics ten years or so after earning my civil engineering degree. Long time ago this was the standard way into the teaching profession. First you studied the subjects, then you added on a year of teaching theory and practice. It is really hard to see what’s wrong with this. There is a focus on mastering the subjects you are supposed to teach and then you learn the basics of the practical craft of teaching. Granted that you have some talent for your subjects and some talent for teaching and you are interested in teaching, you should stand a good change to become a good teacher.
But at the time I did this, twenty-two years ago, it was considered as an aberration, something that was tolerated because of the need to recruit science and technology teachers to the gymnasium. This was not said explicitly, but you could feel it if you had any social competence. It came out in one explicit, and quite funny way. There was a requirement that we, the students, should get practice “in society” as it was phrased, by a three weeks section of practicing at a work-place. Can you believe it! Here we had folks in their thirties and forties who had been working in industry, in the public sector, as teachers, as researchers and civil engineers in many different jobs – and they were required to go out and get practice “in society”. Of course we refused to do it, and it never came about since the school itself had no contacts “in society”!
But this was long ago. For many years now the standard way into teaching has been a shift of focus from the subjects to the theory of teaching itself. Pedagogy, didactics, psychology, sociology, child development theory et cetera has moved up front and the subjects are sort of mixed in piece by piece here and there during up to five years of study.
To highlight the difference: before you were a mathematician teaching mathematics. Now you are a pedagogue teaching mathematics.
In the individual case there may be no difference. Most likely both systems has produced good mathematics teachers. But statistically, on average, which system is most likely to produce the greatest number of good mathematics teachers?
This is a semi-scientific question that perhaps can be answered by some cleverly designed empirical study, perhaps across times, perhaps across different countries, taking into account other cultural and historical aspects et cetera …
I know what I believe, and what the colleagues around me believe.
Let’s apply some logic. Take an extreme case. Suppose a person knows almost no mathematics beyond what’s given by common sense and basic schooling but has studied a lot of pedagogy. Is this person likely to be a good mathematics teachers? Suppose on the other hand that you have a person well schooled in mathematics, broadly and deeply (in some areas at least) but hasn’t that much pedagogy. Is that person likely to become a good mathematics teacher? Yes, I think so, unless he or she is uninterested in teaching and/or is a social disaster that cannot communicate and relate to other people. Any lack in “didactics” can easily be picked up if the need and interest is there.
Isn’t it obvious that it is impossible to teach mathematics without broad and deep mathematics knowledge? It is not enough to know only what you’re supposed to teach. You must know much more, and in particular you must have had time and opportunity and motivation to think deeply about the mathematics you’re teaching. Such a lack of knowledge and lack of thinking cannot be painted over by any amount of pedagogy and didactics paint. Indeed, I believe that in particular humanistic aspects of mathematics require much more time to assimilate than any skills in manipulating symbols.
The counter argument here is often that people who know a lot of mathematics are in some way “weird” and socially inept and generally uninterested in teaching – a kind of nerds not suitable for teaching who rather prefer to sit in attics and do pure research . Well you could perhaps find such examples, but I don’t think it is common.
It is not that I’m adverse to research and scientific knowledge about how we learn. Quite to the contrary. I did go to teaching school and I enjoyed most of it. But the pace was slow and much time was wasted in us students having to sit around in a circle and discuss things that we had already discussed over coffee in the morning. Sometimes we had to fool a teacher into telling us about his, undoubtedly broad, knowledge in pedagogy rather than us relating our “experiences”. When I heard about Ference Marton’s research into deep and surface learning (it was mentioned in passing – possibly by mistake), I was electrified with interest and asked if we could have a session about this. But it did not come about. Can you believe that? The man himself was sitting in the same building. And some years ago I took the pedagogy course for university teachers out of pure curiosity. I may be one of the few in the whole country who has done that without having to. So I’m very interested in pedagogy. Still I don’t think pedagogy is a remedy for lack of mathematics knowledge and understanding.
The mix is wrong, the focus needs to be shifted.
At the same time, I do think that we as university teachers in mathematics, ought to be more interested in learning theory and research based approaches to teaching and in discussing teaching in a systematic way.