Tony's mailings were interesting to me as, having had a great time pre-18 with maths, the next leap forward happened at 31 with research. The disaster was the undergraduate curriculum. We were presented with definitions theorems and proofs, without engaging with the concepts which made the definitions menaingful, or awareness of the patterns which the theorems described. Without the concepts or the patterns, the proofs were hollow. I do not doubt that when things are working well, and problems are being addressed, the psychology and the logic interact.
I have spent much time with intending primary teachers, for whom the perception of pattern and the growth of concepts are at the core of what they are about. Skills and technique then follow, and that is where a lot of secondary schooling operates. I only leant to grapple effectively with undergraduate maths and its teaching when I realised that the patterns and concepts were the stuff that the logic and the imagination must work with. Bob Burn University Fellow, Exeter University Sunnyside Barrack Road Exeter EX2 6AB 01392-430028 ________________________________________ From: Post-calculus mathematics education [MATHEDU@JISCMAIL.AC.UK] On Behalf Of Jonathan Groves [JGroves@KAPLAN.EDU] Sent: 05 June 2010 20:14 To: MATHEDU@JISCMAIL.AC.UK Subject: Re: What Is Mathematics For?
Tony and others,
This is a good question to consider: How many students can become interested in mathematics (and I mean "real mathematics" as opposed to "school math," which is essentially a collection of rules)? I don't know, but I do know that we cannot make students learn to enjoy mathematics just as we cannot make them become interested or learn to enjoy any other subject. If our teaching does help a student learn to enjoy math (or whatever other subject it is), then whatever we had done is something that appeals to the student. And whatever appeals to one student doesn't appeal to another. Even different mathematicians have different interests and different reasons for enjoying math, as we already know.
But few teachers try more than one or two approaches to math, which often are approaches that appeal to only a handful of students in a class, and then falsely conclude that only those students can become interested in math. Of course the teacher will probably not voice that conclusion, but it doesn't mean that the teacher isn't thinking it. Not all teachers do that, of course; some may not try other approaches because of lack of time, because of laziness, because of high pressure from administrators to "cover" all the material in the curriculum, or even because these few approaches they have tried are the only ones they know. In short, the fact that there are widely varying interests even among mathematicians and others who enjoy math and are widely different reasons for why people find math fascinating or at least worthy of study does not seem to be adapted well into K-12 and even early college teaching.
Fluency and computational skills and other basic skills cannot be ignored. They are essential to success in mathematics. But traditional teaching focuses so much on these basic skills that math becomes to students nothing but basic skills and computations and a bunch of rules to learn. Much of the beauty is missing. And I know that something is wrong with the way math is presented in school when I see lots of people who think that mathematicians are nothing but number crunchers and human computers. Unfortunately, few of these students end up learning these skills, and even fewer learn how to transfer them to new situations. And worst of all, few students have any idea why they are learning all these skills. And that happens year after year for many years, and frustration and fear and hostility build up in most students. By the time they reach adulthood, they are determined to be through with math and avoid it as much as possible. At that point, it is almost impossible for most teachers to find ways to help them become interested in math or at least to find ways to help them see that studying math is worthwhile for themselves (and not simply for the math geeks or scientists or computer wizards).
Lack of motivation and lack of reasoning (reasoning for why the math works) behind these basic skills for fluency in mathematics are the main reasons that the vast majority of students learn to detest mathematics. And the lack of emphasis on problem solving guarantees that only a relatively few number of students can learn to transfer their skills to situations that are new to them. There are ways to motivate this stuff and to go through the reasoning with them while including problem solving. Problem solving can be a significant tool for motivating skills and techniques and even for practicing computational and other basic skills. Rote practice is boring and turns many students off from mathematics; this is especially true for those who are highly creative or at least desire creativity in their studies and work. Of course, we will not see all students learn to be fascinated by mathematics, but I believe we can do a much better job than we are currently doing in America in helping students learn to find mathematics enjoyable or at least worthwhile for themselves.
As for my comments about my experience with math, let me clarify a bit: Before that time, I did enjoy math. But later in college and in early graduate school when I had the chance to learn how mathematicians work and how they view math and when I had the chance to work as a mathematician, I then had a major revelation that this new view of math offers a lot more satisfaction and excitement than my previous view of math, which was based on the view of math promoted by school math. So math for me went from being good to incredibly fascinating.
On 6/5/2010 at 6:24 am, Tony Gardiner wrote:
> > I know my view of what mathematics is and why it is > fascinating had > > changed late in college and early in graduate > school. > > Does anyone think this experience is *rare* among > those > (a) who reach the age of 40 > and > (b) who claim to have an inkling that mathematics is > indeed a > remarkable discipline? > > I suggest not. > > I have had the privilege of 30+ years working with > many committed > teachers and with many wonderful young mathematicians > (through > activities on five continents with talented > youngsters) . > > But I have met 20-50 "late developers" (as in the > opening quote) for > every one 18 year old who claimed to "know" already > that mathematics > was fascinating as a result of his adolescent > experience. > > And those I have met who seemed to know this at age > 18 mostly exhibit > either just the first, or BOTH of the following two > features: > > (i) they have an inner urge (independent reading, a > delight in > grappling with hard problems, etc.) which sets them > apart from > typical students in a high school class; > > (ii) in addition, if they are lucky, this inner > commitment has been > encouraged at a key moment by a remarkable teacher. > > A good teacher on his/her own can get students to > master and to enjoy > school mathematics, and to perform at a relatively > high level. > But they don't seem to be able to create the crucial > ingredient (i) > *in adolescents* if it is not there. > [Experience suggests that a "good (secondary or > college) teacher" can > sow seeds which may give rise to this ingredient in > adulthood. That > is, they can prepare the ground for the student's own > inner drive to > begin to develop and to bear fruit much LATER. But I > have no > evidence to suggest that an excellent secondary math > teacher can > produce 18 year olds who have much of an inkling > about the > fascination of mathematics. > That seems to require an inner commitment - since it > emerges from > *personal engagement*: a commitment which is either > there already, or > at best may develop later.] > (I realise that this observation carries within it > the seeds of its > own destruction! For one might then imagine a > parent, or a primary > teacher sowing a seed which bears fruit during > adolescence. All I am > saying is > * that a high school teacher cannot be expected to > work this alchemy > from scratch > * that even where it occurs, it is very rare - and > maybe we should > accept this.) > > > In short, what makes mathematics such a fascinating > subject is not > > shown to students. > > I suggest it cannot be effectively "shown" - because > "shown" is > passive and "dirigiste". You cannot make me > interested in > mathematics; you cannot make me musical - but you can > get me to sing > better than I otherwise would, which may much later > bear more > profound fruit. > > What one needs is > > (a) good basic instruction for as many as possible, > so that more 18 > year olds have a basic fluency and some understanding > of how the bits > fit together > > (b) some well-timed input to awaken something much > more subtle - > namely the inner desire to make sense of this > remarkable world (which > is not a very adolescent thing, so naturally arises - > if at all - in > early adulthood). > > Once we have got (a) in better shape, we can begin to > work on (b). > Meantime, we should avoid rhetoric which claims > anything will improve > if we try to burden the average high school math > teacher with (b) as > though this might allow us to by-pass our failure to > deliver (a). > > Tony