Algebra for All: Educators Challenge Idea That Math Skills Must Come Naturally(washingtonpost.com)
washingtonpost.com
Algebra for All: Educators Challenge Idea That Math Skills Must Come Naturally
http://www.washingtonpost.com/wp-dyn/content/story/2009/05/15/ST2009051503494.html
7 comments
RE: "Well, I was never good at math either!"
Even more upsetting to me (as a horrible math student in K-12) was: "Well, you'll never really use this in the real world anyway. But you should learn it. Now stop asking 'Why?'"
It is almost impossible to learn something if you have no interest in it, and it's difficult to develop an interest in something very complicated without at least a basic understanding of WHY you should be interested in it. Calculus is incredibly interesting and useful, but not to someone who is having it rammed down their throat via rote memorization without any explanation of the overarching principles or reasons for its existence. "Plug and chug" was the only instructional approach my HS teachers understood.
In high school, I loved science but hated math. I was actually classified as being math-phobic. I learned a bit of statistics for my science projects (Westinghouse, ISEF, etc.) because I needed it, but didn't really learn any other math until I taught myself in my 20s. I just skipped any math that wasn't absolutely mandatory in high school, and even the mandatory classes were very poorly taught. I basically just soaked up enough facts in my short-term memory to pass tests, but I didn't understand how anything related or why you actually did anything. And I didn't worry about it, because not a single teacher I had could ever give me a simple example of why you needed to know any of it or what you could use it for. When I asked what calculus was good for, for example, I was told "passing tests." They said if you became a doctor or an engineer you had to know it, but otherwise you would never use it. I asked why doctors and engineers needed to know it if it was so useless, and they usually just shrugged!
I only became interested enough to actually learn math once I saw practical uses for it in my own life (in finance and programming, mostly.) Self-instruction was a struggle, because I lacked any sort of basic foundation. I almost had to start from scratch. I still have relatively poor math skills, but I can get work done and actually enjoy doing calculations now. A big turnaround from being completely math-phobic. I've easily spent $5K on self-instructional math materials (which sounds ridiculous, I know, but I like video instruction and it's usually expensive.)
I'm angry that I avoided math and was scared of math for a decade because not one teacher I had could provide a reason to open the book beyond "you have to learn it." If I hadn't developed an interest in software and finance, I would probably still be as ignorant of math as I was when I graduated high school.
Even more upsetting to me (as a horrible math student in K-12) was: "Well, you'll never really use this in the real world anyway. But you should learn it. Now stop asking 'Why?'"
It is almost impossible to learn something if you have no interest in it, and it's difficult to develop an interest in something very complicated without at least a basic understanding of WHY you should be interested in it. Calculus is incredibly interesting and useful, but not to someone who is having it rammed down their throat via rote memorization without any explanation of the overarching principles or reasons for its existence. "Plug and chug" was the only instructional approach my HS teachers understood.
In high school, I loved science but hated math. I was actually classified as being math-phobic. I learned a bit of statistics for my science projects (Westinghouse, ISEF, etc.) because I needed it, but didn't really learn any other math until I taught myself in my 20s. I just skipped any math that wasn't absolutely mandatory in high school, and even the mandatory classes were very poorly taught. I basically just soaked up enough facts in my short-term memory to pass tests, but I didn't understand how anything related or why you actually did anything. And I didn't worry about it, because not a single teacher I had could ever give me a simple example of why you needed to know any of it or what you could use it for. When I asked what calculus was good for, for example, I was told "passing tests." They said if you became a doctor or an engineer you had to know it, but otherwise you would never use it. I asked why doctors and engineers needed to know it if it was so useless, and they usually just shrugged!
I only became interested enough to actually learn math once I saw practical uses for it in my own life (in finance and programming, mostly.) Self-instruction was a struggle, because I lacked any sort of basic foundation. I almost had to start from scratch. I still have relatively poor math skills, but I can get work done and actually enjoy doing calculations now. A big turnaround from being completely math-phobic. I've easily spent $5K on self-instructional math materials (which sounds ridiculous, I know, but I like video instruction and it's usually expensive.)
I'm angry that I avoided math and was scared of math for a decade because not one teacher I had could provide a reason to open the book beyond "you have to learn it." If I hadn't developed an interest in software and finance, I would probably still be as ignorant of math as I was when I graduated high school.
I have to respectfully disagree with you. It's true that there is a large group of kids who would be better served if their teachers focused more on applications; however, I think that there's an equally large (if not bigger) group of students for whom learning about applications is not helpful. When I was in public school, my teachers always made an effort to highlight how the math we were learning could be applied to real world problem -- the end result was that often the struggling students would say "You use this to build rocket ships? Well I'm NEVER going to do that... I give up."
The most important thing we can do is change our attitude. It's hard to develop an interest in something that you find difficult when you're receiving mixed messages from all of the adults in your life; when adults will demand you get better grades all while telling you it's okay because "math is for nerds" or "not everyone can do math."
I did well (good grades, but it's not like I was Terence Tao or anything) in math in school, and my teachers in other subjects, my coaches, my friends' parents, etc, acted like I was a freak because of it. You can't give a high five with one hand while you're pointing and laughing with the other. This is the attitude that must change before we can raise a generation who take pride in developing math skills.
The most important thing we can do is change our attitude. It's hard to develop an interest in something that you find difficult when you're receiving mixed messages from all of the adults in your life; when adults will demand you get better grades all while telling you it's okay because "math is for nerds" or "not everyone can do math."
I did well (good grades, but it's not like I was Terence Tao or anything) in math in school, and my teachers in other subjects, my coaches, my friends' parents, etc, acted like I was a freak because of it. You can't give a high five with one hand while you're pointing and laughing with the other. This is the attitude that must change before we can raise a generation who take pride in developing math skills.
I agree with you wholeheartedly about the mixed signals thing. I definitely experienced that phenomena because of my science pursuits. I can also understand your point about how certain students would develop a "Well I'm NEVER going to do that..." attitude when shown certain specific examples of math. But a more practical applications approach - no matter how poor the examples - should still interest more students than a "I don't know why you have to do it, but just do it" approach.
I think your disagreement with me comes about in part because you are thinking in more concrete, specific terms than I am. You're correct that "You can use this math to build rocket ships" would probably not have helped me if I didn't have an interest in rocket ships. But a teacher wouldn't necessarily use just a single type of example. Beyond that, I'm saying that my teachers never explained the overarching concepts of math and how they were related or explained - in GENERAL terms - what it could be used to do. Each math 'concept' was presented as a discrete type of chore that you completed in order to satisfy some perverse deity for no apparent reason. Math was not presented as a language of logic and reason that could be used to solve practical problems, but as a completely made up busywork exercise. I might as well have spent my time memorizing Klingon grammar rules. I literally didn't realize calculus was the study of change until college, even after having passed a course in it! Maybe you had a better experience with math instructors, and it is just difficult for you to understand how woefully bad some practicing math instructors actually are?
I think your disagreement with me comes about in part because you are thinking in more concrete, specific terms than I am. You're correct that "You can use this math to build rocket ships" would probably not have helped me if I didn't have an interest in rocket ships. But a teacher wouldn't necessarily use just a single type of example. Beyond that, I'm saying that my teachers never explained the overarching concepts of math and how they were related or explained - in GENERAL terms - what it could be used to do. Each math 'concept' was presented as a discrete type of chore that you completed in order to satisfy some perverse deity for no apparent reason. Math was not presented as a language of logic and reason that could be used to solve practical problems, but as a completely made up busywork exercise. I might as well have spent my time memorizing Klingon grammar rules. I literally didn't realize calculus was the study of change until college, even after having passed a course in it! Maybe you had a better experience with math instructors, and it is just difficult for you to understand how woefully bad some practicing math instructors actually are?
Re: my experience with math instructors, you're probably right. I do think that there's a certain sort of base level of application information that should be imparted with any given mathematical topic (for instance, that calculus is about change! Wow, I'm sorry you had such awful teachers), I just think that focus on applications is a method that's been tried already and just hasn't seemed to improve math education enough.
I've easily spent $5K on self-instructional math materials (which sounds ridiculous, I know
That doesn't sound ridiculous to me, although I don't think I've spent that much on tangible math instruction materials yet for my four homeschooled children. The total budget for my oldest son, including travel expenses to summer math programs and gasoline for travel to local programs, has surely exceeded that by now. I'll open a separate thread
http://news.ycombinator.com/item?id=633540
on the present value of knowing math well, which I have wished to do for a while.
That doesn't sound ridiculous to me, although I don't think I've spent that much on tangible math instruction materials yet for my four homeschooled children. The total budget for my oldest son, including travel expenses to summer math programs and gasoline for travel to local programs, has surely exceeded that by now. I'll open a separate thread
http://news.ycombinator.com/item?id=633540
on the present value of knowing math well, which I have wished to do for a while.
I think that $5K of stuff is nothing compared to what public schools spend to fail to teach math :)
(Per student, but over the course of several years, and taking into account that he was buying stuff just for himself, whereas at a school each video could be shown to 30+ people.)
(Per student, but over the course of several years, and taking into account that he was buying stuff just for himself, whereas at a school each video could be shown to 30+ people.)
There is a strong mindset about math that "you're either good at it or not". It's a nice application of the research by Carol Dweck about people's views of intelligence, popularized in this article: http://www.scientificamerican.com/article.cfm?id=the-secret-...
I now do a lot of stats for my work (I'm a psychometrician), and as such, people see me as being good at math, but in school, I was a mess.
Here is the problem with math education: The way we teach it guarantees that only those with a cognitive predisposition to it will develop an interest. Math is just a tool. You can use it to do all sorts of really cool things, but you never even get to see those things in school. What you see is an entire evening shot solving quadratic equations for no reason whatsoever.
I actually enjoyed geometry and trig, and the former is still useful many times a year. I also enjoyed physics, which just just math.
Basically, we approach teaching math the way people used to approach teaching language (full disclosure: I teach foreign language--I develop standardized language tests, which is where I use a lot of math): memorizing conjugations, etc., rather than focusing on tasks that one can complete using what you know already, but which will stretch you and make you develop new skills. Language teaching has realized that the way to keep people who may not be predisposed to language (another very difficult, largely-left-brain activity that I actually did like in school) interested is to keep the application of skills front and center at all times. Not everyone will be a great mathematician or become highly proficient in a foreign language, but many more people than do now could do very, very well if classes were more focused on application than "here are 100 problems; they're due tomorrow."
Here is the problem with math education: The way we teach it guarantees that only those with a cognitive predisposition to it will develop an interest. Math is just a tool. You can use it to do all sorts of really cool things, but you never even get to see those things in school. What you see is an entire evening shot solving quadratic equations for no reason whatsoever.
I actually enjoyed geometry and trig, and the former is still useful many times a year. I also enjoyed physics, which just just math.
Basically, we approach teaching math the way people used to approach teaching language (full disclosure: I teach foreign language--I develop standardized language tests, which is where I use a lot of math): memorizing conjugations, etc., rather than focusing on tasks that one can complete using what you know already, but which will stretch you and make you develop new skills. Language teaching has realized that the way to keep people who may not be predisposed to language (another very difficult, largely-left-brain activity that I actually did like in school) interested is to keep the application of skills front and center at all times. Not everyone will be a great mathematician or become highly proficient in a foreign language, but many more people than do now could do very, very well if classes were more focused on application than "here are 100 problems; they're due tomorrow."
[deleted]
I still think some people have dyscalculia. (http://en.wikipedia.org/wiki/Dyscalculia)
I think I am one of those people. Even after spending 8-10 hours on one problem, with tutors, teachers, and the willingness to learn, I am usually unable to grok it.
And then the minute I learn it, and then move on to something else, the old math knowledge is forgotten.
I do not have this problem with language. I scored an 800 on the verbal of the old SAT (the best you can do).
For some -- and I believe I am one of them -- higher math is just impossible.
I think I am one of those people. Even after spending 8-10 hours on one problem, with tutors, teachers, and the willingness to learn, I am usually unable to grok it.
And then the minute I learn it, and then move on to something else, the old math knowledge is forgotten.
I do not have this problem with language. I scored an 800 on the verbal of the old SAT (the best you can do).
For some -- and I believe I am one of them -- higher math is just impossible.
Can you post a quick example of the sort of problems you were trying to solve, and a hit at why you might have had difficulty with it?
My favorite quotation on math learning (used as my tagline where I first started using this screen name):
"The proper thing for a parent to say is, 'I did badly at mathematics, but I had a very bad teacher. I wish I had had a good one.'" W. W. Sawyer, Vision in Elementary Mathematics (1964), page 5.
By the way, the cited book is indeed very good for learning math.
"The proper thing for a parent to say is, 'I did badly at mathematics, but I had a very bad teacher. I wish I had had a good one.'" W. W. Sawyer, Vision in Elementary Mathematics (1964), page 5.
By the way, the cited book is indeed very good for learning math.
I supposed quadratic equation walking calculator count as mastery of mathematics!
Maybe just maybe, that 30 questions or dressed up word problems doesn't teach anything beyond how to follow steps to solve problem?
Maybe just maybe, that 30 questions or dressed up word problems doesn't teach anything beyond how to follow steps to solve problem?
In many countries, math has long been recognized as a tough subject that can be mastered through hard work, said Tom Loveless, a scholar at the Brookings Institution who analyzes international math test results and cultural differences. Efforts to make math more fun or dress up textbooks are not the answer, he said.
OK, yes, but the state of American textbooks is TERRIBLE. They don't need shiny graphics or examples using social cliques. They don't need personality quizzies. (Ex-Winnie Cooper, I am pissed at you.)
In the lower grades, some schools are subsidizing textbooks by purchasing ones from companies that use brand names in the problems. E.g. "If Tommy has 3 Oreos..." That is not the kind of dressing up they need, no.
But if you compare your basic Silver Burdett piece of shit to a book like "Mathematics: A Human Endeavor" or Geometry by the same author (Harold Jacobs), you'll see the difference.
I have the Geometry book. It uses Escher, Alice in Wonderland, and other great and useful devices for explaining math. (Hofstadter uses the same devices - not a coincidence, I am sure.) It uncovers math in everyday life and unexpected places. It reveals. It pulls back the curtain. That is the magic of learning.
It does not pander, which is what all the other textbooks do. And, in fact, the entire school system does.
OK, yes, but the state of American textbooks is TERRIBLE. They don't need shiny graphics or examples using social cliques. They don't need personality quizzies. (Ex-Winnie Cooper, I am pissed at you.)
In the lower grades, some schools are subsidizing textbooks by purchasing ones from companies that use brand names in the problems. E.g. "If Tommy has 3 Oreos..." That is not the kind of dressing up they need, no.
But if you compare your basic Silver Burdett piece of shit to a book like "Mathematics: A Human Endeavor" or Geometry by the same author (Harold Jacobs), you'll see the difference.
I have the Geometry book. It uses Escher, Alice in Wonderland, and other great and useful devices for explaining math. (Hofstadter uses the same devices - not a coincidence, I am sure.) It uncovers math in everyday life and unexpected places. It reveals. It pulls back the curtain. That is the magic of learning.
It does not pander, which is what all the other textbooks do. And, in fact, the entire school system does.
the state of American textbooks is TERRIBLE.
Fully agreed. And most textbook evaluation processes used in American schools ensure that they will remain terrible. A favorite author of a favorite series of math textbooks once commented, "I would like to make one comment here. Some of my American colleagues have explained to me that American students are not really accustomed to thinking and working hard, and for this reason we must make the material as attractive as possible. Permit me to not completely agree with this opinion. From my long experience with young students all over the world, I know that they are curious and inquisitive and I believe that if they have some clear material presented in a simple form, they will prefer this to all artificial means of attracting their attention--much as one buys books for their content and not for their dazzling jacket designs that engage only for the moment. The most important thing a student can get from the study of mathematics is the attainment of a higher intellectual level."
http://math.berkeley.edu/~wu/Gelfand.pdf
Fully agreed. And most textbook evaluation processes used in American schools ensure that they will remain terrible. A favorite author of a favorite series of math textbooks once commented, "I would like to make one comment here. Some of my American colleagues have explained to me that American students are not really accustomed to thinking and working hard, and for this reason we must make the material as attractive as possible. Permit me to not completely agree with this opinion. From my long experience with young students all over the world, I know that they are curious and inquisitive and I believe that if they have some clear material presented in a simple form, they will prefer this to all artificial means of attracting their attention--much as one buys books for their content and not for their dazzling jacket designs that engage only for the moment. The most important thing a student can get from the study of mathematics is the attainment of a higher intellectual level."
http://math.berkeley.edu/~wu/Gelfand.pdf
"In many countries, math has long been recognized as a tough subject that can be mastered through hard work, said Tom Loveless, a scholar at the Brookings Institution who analyzes international math test results and cultural differences. Efforts to make math more fun or dress up textbooks are not the answer, he said."
It's absolutely shocking to me how many educators I've encountered (even and especially at the undergraduate and postgraduate levels) who dismiss a lack of basic mathematical proficiency by saying "Well, I was never good at math either!" If this article is an accurate representation of where American math education is headed, then for once I might feel optimistic about it.