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In a recent essay, Artificial Intelligence guru Marvin Minsky explores problems in mathematics education. Minsky’s first distinction is between Arithmetic and Mathematics — grade schools emphasize the former, leaving grander concepts of Mathematics for later education. Minsky points out that if students are bored by Arithmetic, they may be turned off by math in general (that was certainly the case for me!).
Here’s a bit from Minsky’s essay:
Why do some children find Math hard to learn? I suspect that this is often caused by starting with the practice and drill of a bunch of skills called Arithmetic—and instead of promoting inventiveness, we focus on preventing mistakes. I suspect that this negative emphasis leads many children not only to dislike Arithmetic, but also later to become averse to everything else that smells of technology. It might even lead to a long-term distaste for the use of symbolic representations.
…
Anecdote: I asked a younger child “how much is 15 and 15″ and she quickly answered, “I think it’s 30.” I asked how she figured that out so fast and she replied, “Well, everyone knows that 16 and 16 is 32, and then I subtracted the extra 1’s.”
Traditional teacher: “Your answer is right but your method was wrong: you should add the two 5’s to make a 10; then write down a 0 and carry a 1, and then add it to the other two 1’s.” The traditional emphasis on accuracy leads to weakness of ability to make order-of-magnitude estimates—whereas this particular child already knew and could use enough powers of 2 to make approximations that rivaled some adult’s abilities. Why should children learn only “fixed-point” arithmetic, when “floating point” thinking is usually better for problems of everyday life! More generally, we need to find out more about how each child regards each subject. How might it answer questions like “What am I doing here, and why? “What can I expect to happen next?” “Where and when am I likely to use this?
What do you think? How did your Mathematics education affect you — do you love or hate math? How would you have changed your Mathematics education? Share your experience in the comments! Also check out the whole essay, and AI fans will dig Wikipedia’s page on Minsky himself.
I loved math from a very early age. My parents taught me to play cards early and there was a lot of math involved. I also learned that there are many ways to solve math problems. Sometimes teachers would give me grief about not showing enough work or the fact the I did it a different way. The problem with math is once you start to fall behind you will have trouble in school. Any detours are treachorous(sp). I got the best feeling last night when I walked into my first graders room and she was counting her money. She had 76.85. She was right too. Lots of math there.
posted by gpo on 3-14-2008 at 4:17 pm
My American Public School Education was less than great in many ways, math being the most notable. I vividly remember being in “math” classes and asking, “Why?” over and over again to no avail. No one, through my entire education, explained the basics behind math, the why of math, the fun, the exploration, the freedom that manipulating equations can provide. Today, I have a deep love for science and technology, but preventing me from realizing my dreams of discovering the next big thing, is my paltry understanding of the nature of their cores: mathematics.
posted by bounce on 3-14-2008 at 5:10 pm
I totally agree with gpo, sometimes you will love or hate math depending on how you were taught it. I hated math in high school, especially agebra. But I once took geometry and aced it! Then I realized that I math was okay and I wasn’t dumb, I just had a terrible algebra instructor, who only helped the students that were already good at it. Educators often lose track of the influence they have over students - and the fact that different people work out problems differently. All it takes is one lousy teacher, and you could be ruined on a subject for life!
posted by kellyS on 3-14-2008 at 5:24 pm
I love math. Hate arithmetic.
Same with language. I love literature, but hate spelling. I do enjoy grammar, but that has a more mathematical feel to it. There are rules, but not right way to express something; just wrong ways to punctuate it.
posted by mungley on 3-14-2008 at 6:32 pm
My eldest son loves math (or arithmetic) and is able to do some amazingly difficult math problems in his head. Notwithstanding, he often comes home from school with B’s (or 2’s) on tests and home work, as he “did not show his work.” He gets the answers correct, but gets marked down nonetheless.
Funny thing is, when I try to explain certain mathematical concepts to him, he gets frustrated as the teacher says that he has to solve the problem in a particular manner. The way that she showed in class. While my method was just as correct (and more logically intuitive I think), my “help” turns out to be a hinderance.
posted by Maki on 3-14-2008 at 7:02 pm
The bigger problem in elementary education is the belief that all children (and particularly all children of the same age) learn the same. Some children will learn better with the traditional methods and some kids will learn better with the method described by Minsky (which sounds like the “New, New Math”). Unfortunately the public schools have to decide what methods their teachers will use to teach (determined by the books the school system buys). Either way, the current system of grouping children by age instead of ability and learning “styles” will leave a portion of children unserved.
posted by zantimisfit on 3-14-2008 at 7:55 pm
I hated math because I although I ususally had the right answer, my methods weren’t correct….and when in high school you get one point for the correct answer one point for the correct method, 50% is your highest chance, assuming you get every answer correct, which I never did! Regardless I made it into University, and dreaded the statistics course I had to take….but….the prof didn’t care so much about showing your work, he wanted to know that you grasped the concepts and got the correct answer….and I aced the course. The Prof commented on my excellent mathematical mind!! (go figure!) I now know I just don’t think like many other people….but I am still very bright! And I now love excercising my mathematical mind!!!!
posted by Sandie on 3-14-2008 at 7:57 pm
I’ve always hated math since sixth grade when I threw my pencil across the room in frustration after erasing a hole in my notebook paper trying to find the correct answer to some problem. That’s when I first thought of myself as being awful in math and it just went downhill from there. I consistently got points taken off problems because I didn’t understand and hence didn’t follow the exact formulas. My father is an electrician (no stranger to math) and always tried to help me with my math homework, but would end up just as frustrated as me when we would work out the right answer using an alternative method, but struggle to get the right answer showing the appropriate “work.” Whatever math class I had to take always messed up my GPA. I hate math to this day.
posted by FT on 3-14-2008 at 9:06 pm
Oh! I totally agree! I can still remember drilling those multiplication tables OVER AND OVER AND OVER…mind numbing!
And in 2nd grade,because I had figured out a slightly different way to do arithmetic, I so frustrated a teacher that she knocked me out of my chair in anger. (I was smart enough to know that my way worked as well as hers and let her know that I knew. ;-)
And so, yes, I did develop an aversion to math and for a long time wouldn’t touch anything–including a computer–that might even remotely smack of it. I believed I was just really bad at math. Wasn’t until many years later I realized I was actually pretty good at it.
Good points made!
Ranee
posted by Ranee on 3-14-2008 at 9:08 pm
I didn’t hate math, I was terrified by it!
Somehow I had the rotten luck of getting very mean and sadistic math teachers throughout my grade school years. One actually made fun of me as I took longer than most to complete a math test, and another had me at the board in tears as she more or less called me stupid in front of my peers, and encouraged them to tease me. I also (and still) have a strange habit of switching numbers around, so when we took the standardized tests, I’d always get confused if there was an answer like the right answer but flipped!
Because I never had a problem reading or writing, and was recognized to have higher than average reading abilities, they never considered the possibility of a learning disability.
After a while, if I was confronted with an even slightly difficult math problem, I would begin to panic and sometimes even hyperventilate or just burst into tears. It was the reason I opted to drop out of high school altogether and just get my GED, even though I was getting A’s in everything else. It became such a deep-seeded fear and shame for me.
It wasn’t until my husband, and in college a very awesome math professor (who later became a good friend), patiently ‘reconditioned’ me towards different methods of arithmetic and problem solving. It was very similar to that ‘floating point’ arithmetic mentioned in the above quote, where they worked with the way I could solve the problems instead of forcing me to problem-solve in the ’standard’ ways. And you know what? I found that with their methods, I was actually pretty good at figuring problems out!
It’s still hard to shake the panic sometimes, and occasionally my mind just puts up a block if I can’t figure something out right away, but I’m working through it.
Thank you for posting this! When my husband and I finally start our family, I’ll be sure to remember this and keep aware so that my children won’t have the same issues I did!
posted by Oshi on 3-14-2008 at 10:18 pm
Excellent comments! Thanks for weighing in, everybody.
I never excelled in math (arithmetic) in grade school — language and other subjects came more naturally to me, and at some point I sort of decided I wasn’t “good at” math. But at the same time, I loved computers — a domain that was heavily associated with math in the early 80’s.
Today I consider myself slightly math-disabled. At this point I don’t remember how to do long division by hand, though I often think about the “desert island problem” in which I’d be stranded on a desert island and forced to use math to build stuff, navigate, etc. That seems really exciting to me, for some reason. Perhaps I just need to suck it up and do some math in my real life to find that excitement. :)
posted by Higgins on 3-14-2008 at 10:37 pm
I was very lucky, because my elementary school recognized I did math very well, albeit differently, and put me in the gifted & talented enrichment program (which was not the best thing for my social development, but that’s besides the point). It saddens me that they’ve cut that program now. Getting out of class once a week to actually use your brain and not be bored stiff by the teacher explaining things you already know to people who may or may not even want to learn was awesome.
I still do my math in a “weird” way, which makes it hard to explain things to my friends. As my friend once said to me when she looked at a paper I was turning in, “You put a 5 over there and a -7 over here, and somehow you ended up with 23 as the answer. Which is actually what I got too…”
My problem with math is abstract concepts. If I don’t see how I’m going to use something in every day life as a normal human being, my brain won’t click and let me learn it.
I also suck at factoring equations, which is why I failed my reducing equations quiz in Algebra II yesterday :(
posted by S on 3-15-2008 at 1:06 am
As a fourth grade teacher, I find that teaching math is a struggle, especially when trying to help students through division. I use “floating point” arethmetic, and have also tried teaching that to the kids- showing them that it’s ok to solve the problem in more than one way. However, that often leaves the little ones all confused. They are not used to there being more than one way to solve a problem. They have been taught -by fourth grade- that you have to do something in only one way. It’s sad. Math should help students build their critical thinking skills, not help them memorize multiplication facts.
posted by lynn on 3-15-2008 at 8:50 am
I think there are two significant problems plaguing America when it comes to math education:
#1: To be successful in mathematics, you have to be willing to memorize certain facts and terminology. Some of the criticisms I’m seeing in the comments are quite valid but they are primarily focused on methods and algorithms. When I work with students who are struggling in math, I often find they don’t know what many words mean and they can’t give me basic facts which fit into the logical framework of a problem. But then, even if the students learn basic facts and terminology, we have problem number 2…
#2: Many elementary schools are filled with teachers who neither like nor understand mathematics. While they care genuinely for their students, these teachers are poorly prepared for teaching math and often let the kids know they don’t much like it themselves. They struggle to get through the lessons and if a child does it an activity differently than the way the “Annotated Teacher’s Edition” says, they are not capable of adjudging whether the student’s methods are appropriate.
Fixing elementary education teacher programs at the college level would help long term but the corrective movements are slow and only somewhat effective. The best we may can hope for is a change in perspective over the course of a generation.
posted by Tom on 3-15-2008 at 11:01 am
Tom is right on the mark!
I taught math at university, and the math for elementary school teachers I taught once in the college of education was the worst class I ever had. About half of the students couldn’t even perform adequately at a secondary school level. I even had a student get up and scream because she couldn’t understand why I was using so many letters. It scared me that these people would be teaching our children! I’m sure they are very caring individuals, and truthfully, I don’t think I could do their job, but I’m sure they are passing on their own distaste for and fear of mathematics to their students.
posted by Jim on 3-15-2008 at 1:35 pm
I have dyscalculia, which is like the mathematical equivalent of dyslexia. Rather than actually taking the time to consider that I might have a learning disability, my teachers accused me of “not trying” and tried to force me to learn exactly the way all the other children did. Hence the fact that I made straight A’s in every other subject but failed math for two years in a row.
Over the years, I have finally managed to come up with some alternative ways of solving problems that (usually) work for me. I still hate math, however, and probably always will, because of my terrible experiences with it.
posted by JJ on 3-15-2008 at 2:07 pm
I teach high school math, and have also taught it in the elementary and junior high schools. I have found that math is primarily a ’skills’ course, whereas most other courses are ‘facts’ course; the former involves solving problems by learning procedure, and the latter involves learning pieces of information, with some skills instruction occasionally mixed in.
One of the things I tell my students is that I am not a ‘my way or the highway’ teacher–if students discover a new method to solving a problem, and it doesn’t involve cheating, and it has 100% accuracy, I want to see it, to make sure it’s mathematically viable, but by all means, use it.
I also encourage the use of mental math, calculating in one’s head. I’m pretty adept at it, so I show students little tricks, including not necessarily starting in the right column or going straight down a column to add numbers.
I also encourage lateral thinking, solving problems where you sometimes have to get answers from out in left field. Math isn’t just about calculation–it’s about how to think.
Unfortunately, learning math is a bridge from year to year, and all it takes is one inferior teacher to cause possibly irreparable damage to a student’s math education.
posted by Michael on 3-15-2008 at 6:07 pm
I agree with Maki. When I attempt to show another way to solve a problem, my 3rd grade daughter complains, “We have to do it the way the teacher says!”
She’s really good at solving problems, but not always the “right” way. She surprises me with a quick answer, so I’ll ask, how did you solve it? I’m hoping she can get the right answer and then write down the “right way” for full credit. I’m hopeful that colleges will worry more about her correct answers than her elementary grades.
posted by Brent on 3-17-2008 at 9:01 am
How true the statement of mathematics being awfully hard to learn is?
Well, I know I might sound philosphical but mathematics is anything but just the way you look at it.
There I was studying for my Chartered Accountancy exams, when inside I knew that I sucked big time at arithematic/ mathematics skill, right through all of my school life till my college. I was seriously contemplating switching over to a different study stream like arts.
Nevertheless, over a period of time I met some people who seemed to genuinely love Maths. During the course of interactions I happened to discover that they had a way of simplifying everything while calculating - just like the estimation techniques etc.)
At that moment, I resolved that I will take responsibility of the maths menace. After seeking guidance and trying out different sources of learning maths, I begun to make improvement.
Today, I feel so much better and confident of my mathematics/ arthimatic skill. You can well guess my confidence when Im writing here in favour of maths not being that difficult.
P.S-There is a wonderful book by Scott Flansburg by the name of Mathemagic.. You’d like it….
posted by Ashish S. on 3-17-2008 at 10:37 am
Most of the methods I was taught seemed needlessly complex. And as others have said, most of my teachers weren’t interested in explaining the “why”–not only the why of the method but also the why of “why do I need to do the calculation at all.” I think that’s why I always loved “story problems”–I actually had a “reason” to do the calculations!
Teaching the methods and the arithmetic is kind of pointless for most people if they aren’t also taught how to apply those methods to “real life” problems.
posted by frumpiefox on 3-17-2008 at 1:12 pm
As a 1st grade teacher, I struggle with the “Math was not my favorite subject in school and now I have to teach it,” syndrome. I had some really great professors in college who helped me to overcome some of that, and I do try to make math fun for the kids. Fortunately, I can do 1st grade level math. :)
The problem with allowing the little ones to solve math problems any way they want to is, that sometimes they come up with a really creative way to do it, but not a way that will work every time. Also, there are some skills that are dependent on the basic, fundamental skills in math. Also, memorizing the multiplication tables, while tedious, is ridiculously useful both in everyday life, and in higher level math.
I agree that the “why” to math was and is not taught well(if at all) in the US. It always frustrated me when my dad (a French man) would ask me, “but WHY is that the answer,” and I would say, “I don’t know!” And he would make me read the text book again. (he REALLY didn’t get the American system…) I was always baffled by this exchange until one day in 9th grade (we were living in France) when the teacher asked us (in French), “But WHY is that the answer?” I blurted out, “OH!” (and thoroughly confused my class and teacher, but I finally understood my dad!) While not a good math student, when we moved back to the US for my 11th and 12th grade years, I did REALLY well in math class. I would sleep during math tests and still get a perfect score. Can’t say that I was doing all of it in English, but I got the answers right, that’s all the teacher cared about.
posted by greenstrawberries on 3-18-2008 at 5:11 am