The following relates mainly to the work of developing an approach for issues suggested in your concluding paragraph: > > But what of thinkers who don't feel drawn to this > style? Let's say a student comes to me who has a hard > time in math. (Like my current student.) And they > reveal some of the their learning style and it is not > an adaptive style for math (like my current student). > Is it my job, then, to help them do math from their > current perspective, or is it my job to introduce > them to a much more powerful perspective, even if > they only get a little bit of it, long enough to pass > math class? > The two major factors relating to your (current and potential future) success in 'math tutoring' are (in my opinion):
1. You really loved math yourself 2. You have a patient and nurturing side (Both are crucial attributes to your potential success as a 'math tutor').
The animation you have described to help illustrate multiplication seems interesting - and as it helped the student grasp the concept of multiplication, it is evidently useful. In brief, that particular animation "CONTRIBUTED TO" the student understanding the concept of multiplication. (Obviously, there could be many other useful animations - and this would well be a field worth exploring to help develop your 'tutoring skills').
You've suggested that "a lot of tutors say that they adapt to the style of the student (different students have different learning styles)" - I believe they are essentially correct in trying to do that. It is not easy to do at all. (I would not suggest that you should try to make all your students think the way you do - though of course you should try to make all your students become excited by math and to love it as you do).
However, your idea that "mathematicians think and learn in ways that make math easier" (for each of them) is (I believe) entirely correct. This idea in fact strongly supports the approach noted above that a lot of tutors learn to adapt to the specific style of the student: every good tutor has to do this, I believe. The 'tutoring' is (or should be) as much of a learning experience for the tutor as it is for the learner.
[I observe that Robert Hansen has correctly remarked that
> And I adjust my style depending on which goal it is
though I believe he is in some confusion in other parts of his responses to you and to Joe Niederberger.
[Professor Wayne Bishop has remarked that you could have found yourself a less time-consuming way of discussing issues in multiplication: he is correct in a very limited sense; however, what you've done is potentially very significant indeed - because it could be the basis of your developing a whole 'system' to help people learn difficult/tricky math concepts of various kinds].
Essentially, your job as a tutor is (I believe) to find out what makes each specific student of yours 'turn on' to mathematics - and then to encourage and enable that process. (You know, of course, that the 'traditional by-rote teaching of math' has led to [and still does lead to] the great majority of students fearing/loathing math).
If you generally 'buy into' the philosophy that underlies the above argument, then you may like to explore how the relationship "CONTRIBUTES TO" may be involved in the learning processes of your various students. (I know that you have only one student at this time). The basic idea that I'm suggesting you may like to explore is the following:
For each student of yours: "The effective understanding of concept 'A' CONTRIBUTES TO the effective understanding of concept 'B'" (and so on and so forth through ALL the concepts of math).
There will of course be a lot of commonalities between several of your students.
The relationship "CONTRIBUTES TO" is 'transitive', i.e.
If understanding concept 'A' CONTRIBUTES TO understanding concept 'B', AND If understanding concept 'B' CONTRIBUTES TO understanding concept 'C', THEN Understanding concept 'A' MUST CONTRIBUTE TO understanding concept 'C'.
The above algorithm enables us to create simple (but powerful) visual graphics that can help us trace out 'route maps' of how each individual may arrive at an effective understanding of the hundreds and thousands of concepts in, say, math to develop competence in the subject.
A powerful, but very simple 'systems tool' (developed from Warfield's approach) to help apply the sophisticated concepts of systems science to any issue of interest to you called the 'One Page Management System' (OPMS) is described in the attachments to my post heading the thread "Democracy: how to achieve it" - see http://mathforum.org/kb/thread.jspa?threadID=2419536 . (After you've looked at this material, if you find yourself to be interested, the required modeling software is also freely available on request. I do caution you that there is a [very] little learning involved - along with a fair bit of 'unlearning', in order for you to be able to apply the process to issues of interest to you - it will take you about 2 months to convince yourself that the process will and must work in all situations).
The only specific 'math-related' case study I have is of a freshman student in math who had NEVER gotten above 45% in math right through his school career - and who was doing very much worse in college. He successfully applied the OPMS to his 'Mission':
"To understand thoroughly all the topics of my math syllabus and THEREBY to improve, very significantly indeed, my results in my math exams, tests and quizzes".
I helped him, over a period of about a couple of months, to develop (AND interpret) his models for the above Mission. I then had to leave on a long-term assignment, so I gave him plenty of 'homework' to do on his OPMS. About 8 months later, I received a letter from him telling me that he was now REGULARLY scoring over 75% in all his math exams etc.
I point out that there were actually TWO Missions involved in the process:
A: Paul's Mission (as noted above) B: My Mission: "To help Paul learn how to apply the OPMS to his Mission".
No doubt you would, if you were to try this approach, develop your own Mission to help you with your understanding of how to make an outstanding success of yourself in 'math tutoring'.