Students need a motivating context, a reason to learn mathematics (or any topic). It's called situated learning (also situated cognition, situated action) - with techniques like problem-based learning, challenge-based learning, service learning, project-based learning, learning by design, learning through games, simulations, modeling, programming, etc.
OK, I know a bit about pedagogy, and a lot of it is crap. There's an awful lot of relativism in most of it. Many of the people who teach teachers simply don't care about what works in classrooms, because empiricism is an evil capitalist plot (or something like that). When the Wikipedia page for Education has gems like "Based on the works of Jung", you know you're not in a field which cares about hard facts. Not that it's a real problem - the statistics show that first year teachers are pretty poor (regardless of the amount of pedagogy training they have), but they learn on the job and are pretty good after about 2 years (unless they are just doing a year of teaching to pad their resume, but that's another debate).
I've got a lot of respect for cognitive science though, and that seems to be what you are recommending.
Note that Jung was one of the first people to invetigate the use of words in schizophrenia (in his thesis, as a matter of fact). Secondly, he created the conception of extroverts and introverts that we typically use to describe people today.
Seriously, go read some of the works of Jung before you sound off on his issues. I did, and I was pleasantly surprised at just how sensible he was.
I completely agree that much of modern educational training is quite poor, but take aim at (for example, the Myers Briggs or visual, auditory, kinesthetic learning) rather than poor old Jung.
Except the examples you've given like extroverts vs introverts are not scientifically rigorous and the terms already existed well before him all around the world in different cultures (outgoing vs shy/retiring etc).
I consider science to be the art of making correct predictions. So you need a hypothesis (in the form of equations or computer programs) that can be tested on observations leading to correct predictions.
Extroversion/introversion might be scientifically rigorours by that definition (they might have done quantitative studies) but I'm not sure Jung did those, and even that it makes sense to create such a classificaton - people are a lot more complex and I don't really see the need for use of such classifications.
That's a problem I see with psychology, sociology, economics, and other "soft" fields. The quantification of things which are far more complex than the simplistic models created to the point that they are essentially meaningless. It's meaningless to quantify people into races or skin colors and that's about the level that those sciences are at.
Jung did his thesis in 1903. In 1903, regression had been invented, but was in England and may not have spread very far from there. Fisher was 13, and had not yet invented maximum likelihood and the application of statistics to experimental design that we all benefit from today. Karl Pearson was working on statistics, and had invented the chi squared three years previously. He was in the process of generalising regression analysis in 1903.
Multiple regression (OLS) had not yet been invented. I'm not sure how you expect Jung to have used an apparatus of statistics which really wasn't developed (till Fisher) into a coherent whole when Jung would have been in his forties, having already written a number of books.
Statistics in the sense of looking at populations did exist, but the whole apparatus of modern statistics was developed in the early part of the 20th century, concurrent with Jung.
I agree with your definition of science. Incidnetally, Hans Eysenck developed a physiological test for extraversion/ introversion in the 1950's involving the amount of stimulus required to become noticeable to a person. This has been comfirmed by further research.
Speaking as a (soon to be) psychology PhD (all going well...) I would argue that the "soft" sciences have the exact opposite problem, in that, jealous of all the cool theories of the physicists they have attempted to jump straight to the theory building without the benefits of hundreds of years of observation.
I agree that models in the soft sciences are somewhat simplistic, but I actually think that they're not simplistic enough. We (as a species) need to figure out some invariants if we're ever going to do successful science on people and the systems we create.
Back to Jung, while he didnt use the statistics that we would today, he did spend an awful lot of time attempting to figure out why we are the way we are, without resorting to sex sex sex (like the inimitable Freud). Personally, at this point he's probably better read as a philosopher and student of human nature, but he is well worth reading in that capacity.
That being said, he's an awful writer so it is a bit of struggle. Well worth it though, in my opinion.
Replying to myself as I believe we have triggered the algorithm for shouting matches (though I don't believe either of us were shouting). Dangers of fully automated approaches (black boxes), I suppose.
Funnily enough, I agree with you on psychologists and psychiatrists (to a certain extent). We understand so little, and claim to know so much. Our sample sizes and cultural range is quite poor (anthropologists are good at this, but they tend to lack even a basic understanding of statistics). I do believe that algorithmic approaches to predicting humans have potential, and the reason I now work in the private sector is to get access to some of this data as I believe that masses of data are the only way we'll get invariants to form useful landmarks towards understanding of people.
I would note, however, that I suspect you are classing pychologists (or cognitive scientists, as some of the hip american departments have rebranded themselves) as therapists, which although a common misconception is about as accurate as saying that computer scientists are software engineers (i.e. sometimes, but its not a one to one relationship). Thanks for the discussion, I enjoyed it.
I see where you're coming from, it's maybe useful to psychology practitioners. (I'll come across as a troll if I started airing my view that psychologists and psychiatrists are fraudsters so I'll stop there!).
"OK, I know a bit about pedagogy, and a lot of it is crap."
Any hard empirical evidence for this assertion?
"There's an awful lot of relativism in most of it."
Well, yes, people are not Turing machines or well defined physical systems. As Peter Medawar reminded us, science is the 'art of the solvable'. We need an 'art' for the things that are not solvable in the way he meant.
"...you know you're not in a field which cares about hard facts..."
Yes, teaching and learning is not a field that depends on hard facts alone. The 'soft' factors are important as well. Student teachers need some pointers, models, methods, examples of how to deal with the soft factors. Situated learning big time.
"...but they learn on the job and are pretty good after about 2 years..."
In the UK, most teacher training is based around a supervised teaching practice already, and our government is planning to make it more so. You do not become qualified until after your probationary year is complete (that is on top of the minimum one year training). Another strand is reflective practice, actually looking at your own teaching, its impact (or lack of impact) on students and how to change things. That means a student teacher can 'learn on the job' more effectively.
There's no hard empirical evidence that Scientology is crap, but I'm comfortable making that statement too.
Statistical studies showing little to no positive correlation between level of training (not including practicums - which the UK is obviously using extensively, good on them) and student outcomes is pretty damning though. There's some correlation between doing a masters level education course, and English teaching ability, but OLS shows this is just because students with good English ability (high writing scores in high school) are more likely to do a masters level teaching course.
> Yes, teaching and learning is not a field that depends on hard facts alone. The 'soft' factors are important as well. Student teachers need some pointers, models, methods, examples of how to deal with the soft factors.
For example, they should be learning psychology that's not so outdated that the Cosmopolitan magazine should be embarrassed to write articles on it (Myers-Briggs is totally discredited). I'm not saying I'm a hard reductionist. Education should be based on fundamental psychological foundations if possible, and while that's not always possible they should at least try to either teach stuff which is fundamentally sound, or actually works in the classroom.
"There's no hard empirical evidence that Scientology is crap, but I'm comfortable making that statement too."
I tend to agree. We both accept that there are grounds for belief that do not depend on empirical evidence or a scientifically articulated theory subject to strict Popperian falsifiability. I would extend that acceptance to include the idea that works of psychology, psychoanalysis, sociology, management theory and aspects of cultural studies and history may be useful to training teachers when trying to make sense of what happens in lessons!
"For example, they should be learning psychology that's not so outdated that the Cosmopolitan magazine should be embarrassed to write articles on it (Myers-Briggs is totally discredited)."
They don't in the UK. Gardner is quite popular though as a guide to lesson planning. Totally is a strong word though. Have a look at...
It does sound like teacher training in the US requires some looking at if the courses are not based around a supervised teaching practice. In the UK, during the one year PGCE course, the training teacher must teach for 150 hours under the supervision of a placement mentor (i.e. an actual teaching member of staff in the school or College). The University tutor conducts developmental teaching observations, and the placement mentor conducts the summative assessment (i.e. is person ok to let loose in my classroom). Our dear government is going even further and cutting the University element. There are doubts as to the wisdom of that.
Myers-Briggs as a psychological test is basically useless. MBTI indicates bimodal distributions of the attributes (people are either introverts or extroverts), when they are more like a normal distribution.
Gardner also lacks any real credibility, unless you throw out everything except "different people can be good at different things". It's good to encourage students to study a range of stuff (art, music and interpersonal stuff, not just math and english), but that's not really what Gardner is about. Students who are good at music aren't likely to learn math better if you sing it (OK, maybe I'm exaggerating).
Individualization isn't a great method. It's hard to do, and has a fairly small impact on student outcomes. Most students work best if you show them a diagram, explain everything, give them a chance to try it themselves, and demonstrate things in a number of ways. It's not about "different strokes for different folks"; most students like a bit of variety.
The reason many students struggle is because they lack the right foundations. They didn't learn it last year (for whatever reason) so they struggle this year and can't catch up. This is much more common than students simply not having the right talents.
I'm not from the US, but Australia. From what I can tell, the US varies from state to state. The problems with the US aren't really related to pedagogy, but social issues (large black / hispanic communities going to black / hispanic schools, where everyone is trapped in the poverty cycle) and the way their standardized testing dominates everything. Every year the teacher is mostly concerned about the students passing the end of year test, at all costs, and the end of year tests seem to be badly designed.
"Myers-Briggs as a psychological test is basically useless."
But as a story around the camp fire their theory (along with Honey and Mumford's development of it) might be useful. Stories, ideas, ways of thinking through. Basis for action, then evaluate the action. Kekule (benzene) and his opium.
"Gardner also lacks any real credibility"
But as a way of getting training teachers to think about the sensory modes they stimulate, there might be some value.
Action based research, situated theory, stories. Not science. Definitely not Science, depending on what time of day it is (before teaching, I did research)
Thanks. I'm a physics major, and I've done quite a lot of research into physics education, but I haven't looked into math education. I'll have to take a look into your references. I wrote this article a few months back when there was a flood of "why do we need math?!" articles after some editorial said it's worthless; I only got around to putting it up Friday. So it's aimed more to why we need math, rather than how we should teach it.
I've got a whole separate article on physics education which I'll have to put up soon.
"with techniques like problem-based learning, challenge-based learning, service learning, project-based learning, learning by design, learning through games, simulations, modeling, programming, etc."
I'm glad you mention games, because currently this field is mostly all about teaching arithmetic and other elementary subjects. Notable innovative exceptions are DragonBox and a few others in development.
It's probably a clichè to say that mathematics education is archaic, but it's true in a lot of cases. I've been to lectures where a simple 3-D program like Blender a few minutes of simple 3D work would have significantly helped countless students to imagine the mathematical entities being discussed. But alas, such things seem to be too advanced for much of academia - instead they build plastic models that only 1 person can use at a time.
Students need a motivating context, a reason to learn mathematics (or any topic). It's called situated learning (also situated cognition, situated action) - with techniques like problem-based learning, challenge-based learning, service learning, project-based learning, learning by design, learning through games, simulations, modeling, programming, etc.
That's one side of it. Another is embodied learning. That's how we understand math (and other) concepts. See for example work by Rafael Nunez http://vislab.cs.vt.edu/~quek/classes/aware+embodiedinteract... http://www.cogsci.ucsd.edu/~nunez/web/FM.PDF