One problem with teaching mathematics in the K-12 system - and I see it as a major difficulty - is that there is virtually nothing the pupils learn that has a non-trivial application in today's world. The most a teacher can tell a student who enquires, entirely reasonably, "How is this useful?" is that almost all mathematics finds uses, in many cases important ones, and that what they learn in school leads on to mathematics that definitely is used.Things change dramatically around the sophomore university level, when almost everything a student learns has significant applications.
I am not arguing that utility is the only or even the primary reason for teaching math. But the question of utility is a valid one that deserves an answer, and there really isn't a good one. For many school pupils, and often their parents, the lack of a good answer is enough to persuade them to give up on math and focus their efforts elsewhere.
mathematician in residence programs for 8-12
Another possibility to try to motivate K-12 students (actually, in my experience from visiting schools and talking with their teachers, it is the older pupils who are the ones more likely to require motivation, say grades 8 or 9 upward) is for professional mathematicians to visit schools. I know I am not the only mathematician who does this. There is nothing like presenting pupils with a living, breathing, professional mathematician who can provide a first-hand example of what mathematicians do in and for society.
I recently spent two weeks in Australia, as the Mathematician in Residence at St. Peters College in Adelaide. This was only the second time in my life that a high school had invited me to spend some time as a visitor, and the first time overseas - over a very large sea in fact! In both cases, the high school in question was private, and had secured private endowment funding to support such an activity. For two weeks, I spent each day in the school, giving classes. Many classes were one-offs, and I spent the time answering that "What do mathematicians do?" question. For some 11 and 12 grade classes, we met several times and I gave presentations and mini-lessons, answered questions, engaged in problem sessions, and generally got to know the students, and they me. You would have to ask the students what they got from my visit, but from my perspective (and that of the former head of mathematics at the school, David Martin, who organized my visit), they gained a lot. To appreciate a human activity such as mathematics, there is, after all, nothing that can match having a real-life practitioner on call for a couple of weeks.
Thought of on its own, such a program seems expensive. But viewed as a component of the entire mathematics education program at a school, the incremental cost of a "mathematician in residence" is small, though in the anti-educational and anti-science wasteland that is George Bush's America it may be a hard sell in the U.S. just now. But definitely worth a try when the educational climate improves, I think. If it fails, the funds can always be diverted elsewhere.Devlin's Angle
Keith Devlin
MAA Online
June 2007
I've been introducing the idea around here; I know some parents are interested.
"For many school pupils, and often their parents, the lack of a good answer is enough to persuade them to give up on math and focus their efforts elsewhere."
ReplyDeleteBaloney! Eveyone knows math is important, even in third grade.
"...try to motivate ..."
This is quite different. Everyone knows math is important, but it would just be nice to show kids where it leads to. I remember giving a talk about the math in computer graphics and computer games. I described how numbers are used to define [X,Y,Z] points, how points are used to define triangles in space, how triangles are used to define polyhedra, and how you can use geometry to determine visibility, reflection, and shading.
It's all very nice, but kids still have to go home and do their homework. They still have to figure out why the point order doesn't matter when they calculate the slope of a line. It's like the late-night motivational speaker who convinces you to go on THE diet, but then you get up in the morning and eat a half dozen Krispy Kreme donuts.
"...though in the anti-educational and anti-science wasteland that is George Bush's America it may be a hard sell in the U.S. just now."
I'm not a fan of W., but don't blame any sort of anti-educational trends on him. The schools have been doing a great job themselves.
Speaking of motivating with real-life examples, that is what all science teaching is nowadays; hands-on. But if the kids don't get a rigorous schooling in math, then the best they will become are lab technicians; all hand-on and no theory. You can have all the motivation in the world, but if you don't have the math, you are nowhere.
Does it really make a difference to children's motivations if they know the maths they are teaching is useful?
ReplyDeleteI had one of those mind-altering experiences a few years ago when I read in the Institute of Electrical Engineers' magazine a letter from a university lecturer in engineering stating that his university had started trying to attract students into engineering in the 1950s and enrolments had been dropping ever since.
At another time, my boss of the time told us about his experience teaching university maths in Canada, and how he started off the year with a lecture about all the practical applications of the maths he was going to teach in the course and two days later was called into the dean's office and told that he had the greatest number of students dropping the course the dean had ever seen.
This has ever since made me wonder about the motivational benefits of knowing the applications of something.
Does anyone know of any controlled experiments done on this?
there is virtually nothing the pupils learn that has a non-trivial application in today's world.
ReplyDeleteJust thinking about this. I use arithmetic in dealing with money all the time.
When did money become trivial?
Very good questions, Tracy, and I've had them too. Who decreed that immediately perceived relevance is necessary (and sufficient) to spawn interest? I for one always dreaded the math applications (they bored me); I loved the pure mathematics, the theory, the logic.
ReplyDeleteI think this push for "relevance" in all educational arenas might be nothing more than a meaningless "meme" relentlessly propogating itself.
Speaking of motivating with real-life examples, that is what all science teaching is nowadays; hands-on. But if the kids don't get a rigorous schooling in math, then the best they will become are lab technicians; all hand-on and no theory. You can have all the motivation in the world, but if you don't have the math, you are nowhere.
ReplyDeleteAbsolutely. As a scientist myself, I can attest that good hands are important but they are nothing without the ability to think through the science, design the experiment, interpret the results, make the necessary calculations, and put everything into the proper theoretical framework. Not to mention the fact that interdisciplinary work (which requires a vast store of KNOWLEDGE from many different DISCIPLINES) is where the advances are coming from these days.
Good points, particularly Tracy's about money. Many of the word problems in K-8 grade math books have to do with money. Percentage problems that require figuring ot the price of something that has been discounted at 20%, say. In fact most problems in elementary school math books relate to something familiar--its relevance is obvious as Steve pointed out.
ReplyDeleteIn fractional division they give problems like cutting up some length of ribbon into 1 3/4 inch pieces; how many pieces are there? My daughter found that relevant since the person in the problem was doing it to make something for her dolls. But even apart from practicality, my experience with students I tutor (including my daugher)is that they enjoy being able to know how to do something. In multiplying fractions, my daughter enjoyed the concept of cancelling simply because it was fun for her. It is when students don't understand how to do something that they might ask "the question": What is this stuff good for.
And in all my tutoring, I only had one student ask me that question; it was related to the midsegment theorem for triangles, in a geometry course. He asked what the theorem was used for. I wasn't ready for the question but rather than justify it in terms of daily practicality, I explained that it was a key theorem useful in proving many theorems and problems. During the rest of the course, whenever there was an application of that theorem, I would make emphasize that fact and show its usefulness as a tool to develop and advance more sophisticated mathematical ideas. Which is one of many things mathematics is about. In my opinion.
".. they enjoy being able to know how to do something."
ReplyDeleteExactly.
I used to leave out math worksheets for my son when he was in Kindergarten. He would see the sheets and do them. He loved being able to do the problems and the praise that he would get. When we mentioned this to his teacher, I thought she was going to accuse us of child abuse.
As kids get older, this is still the case. I don't have to justify the basic operations in fractions for my son. He just wants to know how it works and be able to do the problems. It's like a puzzle. He has all sorts of puzzle toys that don't have any justification. In fact, many math reform types really love these puzzles and patterns that don't have any real justification. Can you say tangrams? Can you say Sudoku? It just so happens that learning about fractions is so much more useful than tangrams or Sudoku.
I always feel that reform math just dances all around the subject and never wants to dive right in and get to work. They want you to learn by osmosis.
As a side issue, I just got the latest (2007) student workbooks for Everyday math. My (soon-to-be) sixth grader will actually jump into 7th grade Pre-Algebra (Glencoe) and I wanted to see what he will be skipping over for sixth grade. I also have the Singapore Math grade 6 books coming in the mail. It will be an interesting comparison.
My initial reaction to the new EM books is the same as before. If you open up to any page, what you see doesn't look too bad. Then you realize that embedded in almost every lesson is a review of some material covered before. Distributed practice is good, but there is a big difference going on here. EM never requires initial mastery. It's not mastery followed by distributed practice. It's distributed practice (repeated partial learning) used to achieve mastery. For many kids, it doesn't happen.
They still have to figure out why the point order doesn't matter when they calculate the slope of a line. It's like the late-night motivational speaker who convinces you to go on THE diet, but then you get up in the morning and eat a half dozen Krispy Kreme donuts.
ReplyDeleteI love it!
It was fantastic for me to read the first paragraph - that none of the high school math you learn is useful; you learn it in preparation for learning math that will be useful.
ReplyDeleteI wonder about this constantly as I work my way through Saxon Math.
Obviously, I'm perfectly happy to learn high school math whether it's useful or not -- and in fact learning high school math is highly useful for me since I am expected to "assist with homework."
But I'm always curious: is there something people DO with this particular procedure??
Do people in the Real World factor quadratics a lot?
etc.
I still don't know the answer to that particular question (maybe people in the Real World factor quadratics a lot as an embedded step in a larger process...)
But knowing that high school math is, broadly speaking, a preparation is great.
Does it really make a difference to children's motivations if they know the maths they are teaching is useful?
ReplyDeleteI strongly doubt this.
My one conversation with M.D., the retired writing instructor here in Irvington, was memorable for that idea.
He said that if there was one thing he'd do over again, if he were starting out now, it would be to teach grammar directly instead of as an embedded activity in a larger project so as not to make students dislike grammar.
He said, "Kids will learn what you ask them to."
my boss of the time told us about his experience teaching university maths in Canada, and how he started off the year with a lecture about all the practical applications of the maths he was going to teach in the course and two days later was called into the dean's office and told that he had the greatest number of students dropping the course the dean had ever seen.
ReplyDeleteThat's fascinating!
There's an element of "magic" in academic courses....I always feel it, anyway.
A new course, or a new book, feels like an adventure - all kinds of wonderful stuff might be inside!
I personally don't really want everything demystified & prosaic-ified.
(I don't think this statement contradicts what I've just said...my curiosity about procedures and concepts gets aroused as I go along.)
Just thinking about this. I use arithmetic in dealing with money all the time.
ReplyDeleteI don't think he's talking about arithmetic, is he?
At least, I had the impression he was talking about high school math.
He has all sorts of puzzle toys that don't have any justification.
ReplyDeletegood point
I think that's what M.D., the writing teacher was saying.
ReplyDeleteHis tone was cheerful and respectful; his point was that kids like to learn, and WILL learn what you "ask" them to.