I've used the term innumeracy fairly often on this blog, and I've had a few people write to ask me what it means. It's also, I think, a very important idea.
Innumeracy is math what illiteracy is to reading. It's the fundamental lack of ability to understand or use numbers or math. And like illiteracy, true innumeracy is relatively rare, but there are huge numbers of people who, while having some minimal understanding of number and arithmetic, are functionally innumerate: they are not capable of anything but the most trivial arithmetic; and how anything more complicated than simple basic arithmetic actually works is a total mystery to them.
It's frightening to realize just how widespread functional innumeracy is. The first time that it really struck me was during high school. I went to school in what was supposedly a very good school district in New Jersey. My older brother was planning to be a musician at the time. He hated math, but got adequate grades in it. My guess would be that on a percentile basis, he was probably better at math than something around 70% of the students in his graduating class. We were out shopping, and there was something he wanted to buy that was on sale for 20% off. My dad asked him how much it cost. The only way he knew how to figure out what 20% off meant was to sit down with a piece of paper and do cross-multiplication: (20/100)=(x/price), therefore 20×price=100×x, therefore x=20×price/100, and sale price = price-x.
My dad totally freaked out at this, and asked: "What does percent mean?" My brother could not answer. He really couldn't. He had no idea - he'd just been taught to mechanically do that cross-multiplication, but he had no idea what it meant. He couldn't even say what 20% of $100 was without writing it out as the cross-multiplication.
I don't mean this post as an insult to my brother - but rather as an example of how poorly our schools teach people to actually understand math. He's a bright guy, and he's capable of doing amazingly complex stuff. When he was in college, I went to visit him, and watched him doing music theory homework - and was amazed, because there were some very strong parallels between one kind of analysis he was doing on 12-tone serialist music and the kind of matrix algebra that I was learning at the time. So we're talking about a guy who, in the right setting, was entirely capable of learning matrix algebra. He - and the vast majority of people here in America - are perfectly capable of understanding basic math. But our school systems - except for the classes aimed at those of us who make our careers in mathematical fields - are either incapable of teaching math, or simply do not care that they do not teach most of their students the most basic mathematical literacy.
Innumeracy comes at a high price to our society. When people can't understand math,
that means that they're going to be making all sorts of important decisions based on
something even worse than ignorance. They don't just lack knowledge of the relevant facts that affect their decision - they lack the ability to even acquire the knowledge that they need to be able to make the decision.
If you can't understand basic math, how will you decide between different kinds of mortgages? How many people in America today have "interest-only" mortgages? How many of those people actually understand what an interest-only mortgage is, and what kind of risk they're taking with it?
If you can't understand basic math, how can you even do a household budget? How can you keep your spending within the bounds of your income, if you can't understand the meanings of price variations, if you can't understand the interest that you earn in a bank account and the interest that you pay to a credit card?
If you can't understand basic math, how are you going to evaluate the different tax and spending plans being proposed by politicians? How are you going to figure out when they're lying to you? (Well, okay, that one's easy: if their lips are moving, then they're lying. But still, how can you figure out what they're lying about?)
When some creationist liar comes along and quotes a bunch of math at you as a "proof" that science is all wrong about evolution, how can you recognize their lies? How can you distinguish between the people who say global warming is real, and the ones who say it's a lie? How can you evaluate their arguments, when they're based on math, if you have no real grasp of how math works?
The answer is: you can't.
"If you can't understand basic math, how will you decide between different kinds of mortgages? How many people in America today have "interest-only" mortgages? How many of those people actually understand what an interest-only mortgage is, and what kind of risk they're taking with it?"
Or worse, a mortgage with *negative amortization*.
Preach it, brother!!
No, wait ... let me put that another way ...
Couldn't agree more. While recognizing the weakness of anecdotal evidence, let me give you a shortened version of a recent anecdote anyhow. Purchase of $25.72 ... submitted 2 20's. Change offered of $24.28.
"No, that's not right. it should be $14.28 change."
"Not what the register says."
"Could we speak to a supervisor?"
Supervisor: "Well, we have to go by what the register says."
It's not just America.
It seems that schools today are basically schools from the nineteenth century, where a priveliged Ã©lite was tought in small groups by people who knew what they were doing, dumbed down. And it's taught - especially, it seems, when it comes to maths - by people who don't understand the subject matter. (I remember hearing of a girl in second or third grade, being tought multiplication, and asking of the result could ever be smaller than the factors, being told "no", and being very confused because she'd extended the concept to negatives in her head).
The problem lies in the way it has been dumbed down: subjects are reduced to a list of things people should know, like addition (i.e., how to solve five sequential pages of problems like '5 + 3 = ?'), and quadratic equations (i.e., how to solve five sequential pages of problems like '5xÂ² + 3x - 2 = 0, what is x?', rather than a list of things they should understand. I submit that understanding the concepts of percentages and basic probability theory are much, much more important than being able to formulaically construct x = (-b Â± â(bÂ² - 4ac)/2a, or to do long division, or even being able to multiply single-didgit numbers mentally. I'd argue that this was the case thirty years ago, and that it's compellingly, overwhelmingly the case in an age where vast encyclopÃ¦dias are at our fingertips... but no-one's listening to me.
Most subjects are tought this way, as a list of facts rather than a field of undestanding, at pre-university level, and it's wrong for most if not all of them, but it seems to be worse with maths. As long as this goes on, we - in the broadest and most inclusive sense -Â will be a culture dominated by ignorance.
The mainstream media doesn't help, either: this guy is a total ass.
I remember seeing a study somewhere that said ~60% of people who are teaching in elementary schools in the US got a D or lower on their last math class. I remember how terrified of mathematics my 2nd grade teacher was. Without my parents, I think I would have been set back years.
I am so delighted to see your statement and several of the comments mention the key word: understanding. I have been ranting about this problem on SciBlog comments, including this one, I think, for a year or so. I taught math for 15 years - a long time ago. Last year after retiring, from 25 years in mainframe computer stuff, I volunteered as a math tutor at an elementary school. I thought that if I worked with one kid at a time at the second or third grade level, I could teach "understanding". It turned out that I couldn't because the teacher didn't want me to. They don't. she said, have time for that. They have to teach manipulation so that the kids can pass the state mandated tests as required by the "no child left behind" act. This despite the fact that she had all the physical tools necessary for building understanding: cuisinaire rods, etc. Another example of Bush admin acting agains science?
What fascinates me most about the situation is that it's OK not to "do math" even among educated people. One of my economics professors said something like, "If I told people, 'You know, I have a PhD, but I really don't read. Sure, I read enough to get through school, but I always thought it was hard, so I really never stuck with it,' people would look at me like I was from another planet. Somehow, it's perfectly OK for one of my colleagues to ask me to do some basic fraction work for her because she doesn't 'do math' though."
She was also the one who described teaching her introductory microeconomics course as "The art of hiding calculus from freshmen."
Did Paulos coin the term Innumeracy for the title of his book or did it exist before?
And I assume you have heard of this travesty
as a scale model builder, I am always astounded (and dismayed) at how many of my fellow hobbyists are incapabable of converting one scale to another. I also find it amazing that there are countless scale conversion programs available to cater to this.
Karl said, "I volunteered as a math tutor at an elementary school. I thought that if I worked with one kid at a time at the second or third grade level, I could teach "understanding". It turned out that I couldn't because the teacher didn't want me to. They don't. she said, have time for that. They have to teach manipulation so that the kids can pass the state mandated tests as required by the "no child left behind" act."
Sadly, I have run into the exact same thing!
On my own end, I started off mathematics under the New Math, where the focus was on understanding. I also had a couple of teachers who understood mathematics AND the New Math teaching process. Life was great. (I still think in sets.)
Then we had one of those reversals in teaching approaches, and I next found myself in classrooms with math-phobic teachers who taught by drills and memorising mechanical processes, and I floundered in the maths until college. I was absolutely shocked the day I heard my college algebra teacher announce, "There are several ways you can solve this; pick whichever one works best for you." Not only did I finally have a teacher who liked the subject and could explain it in different ways (so it made SENSE), but she also gave us the "radical" idea that mathematics was a tool that we could use in ways that made sense to each of us.
Once something makes sense to you, you not only own it for using it, but you can also figure out how to apply it to novel situations.
And novel situations is where the memorise-this-process approach falls apart. Ditto story problems. Most math students are terrified of story problems, because they have to be able to understand both the story and the tools and hand, and be able to figure out how to construct equations (& inequalities, etc) to solve the questions.
Shockingly, real life doesn't hand you a page of problems all lined up, with answers to the odd-numbered ones in the back of the book. And that's where the process approach to teaching math fails.
There's been some very interesting historical studies on the spread of literacy (it's worth noting that historians, who study the past, think of it in terms of the spread of literacy, not combatting illiteracy: near-total literacy is a very recent phenomenon, historically speaking), but almost none on numeracy. There's some great stuff on the history of mathematics -- who discovered what, who read who and didn't give credit, etc -- but not on the history of everyday mathematics. At some point we're going to need to seriously study the spread and uses of numeracy, because, as you point out, it's pretty fundamental to economic behavior.
TroublesomeFrog: I completely agree with the last part of your comment. I'm currently pursuing a BA in econ & math and an MA in econ. I've taken the entire core econ sequence at my university from the intro to master's level, and math courses up through the master's level. Back at the intro level, there was a lot of focus on hiding the calculus (even in the honor's version of microeconomics). We used basic differential calculus at the intermediate level, but we didn't use higher-level math until master's macro theory (multivariate & dynamic optimization, primarily). Considering what you need to go on for a PhD in the field and what modern research is like, it's somewhat sad.
Economics is somewhat odd as a result; as a general rule, an undergraduate degree in the field will not be sufficient to go on for a PhD, even though most programs are post-bachelors. You really need a dual-major in math (or just a BA in math) to go on for graduate studies.
The book by John Allen Paulos mentioned in a comment above is titled "Innumeracy: Mathematical Illiteracy and Its Consequences". It has a sequel called "Beyond Innumeracy".
I enjoyed them both immensely, and I learned two or three things I didn't know. They're 10 bucks at Amazon.
It's truly frightening how many people have no understanding of the most basic concepts of math. I've been tutoring calculus students all through college, and I can't believe the number of them who don't understand the most basic concepts of arithmetic. How are they supposed to do calculus if they persist on saying (x+2)^2 = x^2 + 4, or worse, can't even multiply fractions?
What's worse is that our professors, who are GREAT for the higher-level math they teach, all have a terrible habit of allowing the calculus students to submit their answers with simplifying them. Basically, the students can get away with having none of these basic skills, as long as they can apply derivative rules to get a right answer (even if it takes two or three lines to express).
Innumeracy occurs at all levels, and teachers at all levels allow it to go uncorrected. I intend to teach at a college level once I get my PhD, and I can assure you I will do everything in my power to get every one of my students to understand what they are doing!
Also, in regards to economics, I now understand why I had so much trouble with economics in high school--there was no math in it!
I once worked for a guy who owned several small businesses. He had learned a rote method for calculating markups from someone who'd been taught a rote method himself.
The problem was, the method didn't work. We had electronic calculators (with paper tapes for checking results), but the boss couldn't understand that a 20% markup of a $30 item was found by multiplying $30 by 1.20.
He insisted the correct method was adding one number and then subtracting another number, he just couldn't remember which two numbers they were offhand.
I tried in vain to explain adding one number and subtracting another number is the same as adding only their difference, and that any 'solution' for one value would be wrong for any other value.
He never did get it. But his son talked him into letting me program the computer to calculate markups for him.
The boss didn't trust my work until he showed it to business friends at a liquid lunch and they all agreed it was correct.
Sometime, would you kindly explain why n^0 = 1 ? I've yet to hear an explanation that made sense ...
In this case, I don't think that the NCLB stuff is part of Bush's little war on science.
In fact, I think it's something which is almost worse than that. I actually think that NCLB is actually a well-intentioned program, and that he really genuinely thinks that making kids better at rote test-taking is giving them a better education.
As a person, Bush is an example of profound mediocrity. He's a person with no curiosity whatsoever, no interest in actually understanding anything. To him, going to school was just a matter of showing up, and putting the right answers down on tests. I think that he doesn't believe that there is anything more to education than getting kids to write down the correct answers on tests. I think he doesn't understand the concept of understanding at all. So when he's confronted with the problem of many kids getting a lousy education, the only thing he can understand is making them do better on tests - because in his mind, tests are all school is about.
Yeah, I've heard (and been appalled by) the Verizon call. I even posted something about it here.
I think that innumeracy was a real (if not common) word before that book. I'm pretty sure that I remember my father using before the book was published.
"... because in his [George W. Bush's] mind, tests are all school is about."
With all due respect, and agreement with the paragraph in which this was embedded, I see it another way.
To George W. Bush, Yale was where his father went (and was the pitcher on the baseball team) and where his grandfather went (also a sports star). The most important institution within Yale was Skull and Bones.
To someone in Bush's class (Old Money) the purpose of school is to build a network of personal relationships from a nucleus of nodes in one's parens' and grandparents network of relationships. In old terms, building up your Rolodex.
The people you meet in school are the people whom you will later hire, and the people who will lend you money, and the people who will pull strings for you.
This requires a level of understanding of people and networks. George W. Bush is incurious and stupid by your academic perspective, but NOT by the people/network perspective.
On first introduction to 50 members of Skull and Bones, George W. Bush immediately was able to point to each and rattle off their name and title from memory.
It was classes that didn't matter. It was books that didn't matter.
This goes well beyond innumeracy, or maybe orthogonal to it. George W. Bush was the dumb one of his siblings, but still able to learn in school what his family empire expected of him.
The people that he appoints also seem like idiots to you and me. But they pass his kind of test, which depends on loyalty and Class and personality.
Of couurse Bush knows this. His network believes profoundly in a estoeric startegy, and an exoteric one. No Child Left Behind is exoteric. Skull and Bones is esoteric.
Remember that over much of 1,200 years, the Egyptian Empire has an illiterate innumerate Royal class, and massively parallel arrays of literate numerate slaves doing the data processing for surveying (the Nile washed away real estate boundaries each year) and taxation.
The Bush Administration openly advocated turning the USA from a Republic into an Empire. This they have done. They just forgot to specify WHICH empire.
They should worry, though, when "wage slaves" become illiterate and innumerate.
We have two different sorts of innumeracy. Discalculia (sp?) is the math equivalent of dislexia, and is probably caused by an organic brain problem. But this only affects perhaps 5% of the population.
The real problem is the teaching, especially at elementary and junior high levels. This is a tough nut to crack. As long as salaries and the level of esteem accorded teachers is so low we will largely get those who had no understanding of math, faking it, by teaching rote cookbook rules. Then of course when a student exhibits a creative way to solve a math problem, they get slapped down for not following the cookbook. Of course the problem was that the teacher was completely uncomfortable with anything beyond the cookbook, and protected herself, by outlawing creativity.
When I was a toddler, I had a really neat scale for learning addition. The different numerals had weights proportional to their value, so say 4+2=6 was a matter of selecting equal weights. Now with calculators, and multiplication/addition tables you just get rote memorization, boring, and not a very useful basis for greater progress.
And of course if you talk politics, most issues are not clear cut.
Do X and it helps some over here, but hurts over there. But if the voter has no clue how to determine which effect is dominant, the odds of rational decision making become pretty low.
"Then of course when a student exhibits a creative way to solve a math problem, they get slapped down for not following the cookbook."
Very true. Once in high school my friend (who admits that she isn't good at math, but at least she makes an effort, and understands things if she takes them slowly!) came to me and told me what had occurred in her math class that day. Her teacher had asked her to solve an algebra problem, and she solved it in such a way that she only needed to use her calculator at the very end. Her teacher insisted that this was the wrong way to do it and demonstrated, using a method that required making one calculation, writing down the answer (with rounding!), then using that to make the final calculation. As a result, he had a roundoff error, and his answer differed from that of my friend. She and I confronted the teacher and tried to make him realize why her solution was the correct one, but he was extremely resistent and finally said that math shouldn't be done by being "creative" but by following the rules. At that point we just gave it up, and I had the very discomforting feeling of not only knowing more math than the math teacher, but also that the math he was teaching his students was dead wrong.
On the other side of the coin, once in a while a student is capable of understanding things without the teacher's help! I once participated in a program in which we tutored 3rd and 4th graders in math, and I worked with a girl who was definitely of the "math is boring" type. She had a page of addition tables to do, in which the numbers from one to ten were laid out on each axis of a grid, in random order, and she had to fill in each square of the grid with the sum of the numbers in the row and column. She started off thinking hard to remember each sum, but pretty quickly recognized a pattern: the numbers in the 7 column were always one less than the corresponding number in the 8 column. So she proceeded to fill in the 7 column by subtracting 1 from each number in the 8 column, then did the same for the 6 column and so on. Maybe she wasn't practicing her memorization of addition tables, but she was starting to understand the kinds of patterns that occur in addition, and all on her own! I was thrilled!
I have been teaching precalculus and other math to Education students in remote areas in Canada. At the start of the course I've given them an assessment test and it's not unusual for a third of the class to be unable to add 1/2 + 1/4. And they're supposed to have Grade 8 or 10 in Math. It's been a real eye-opener to find out how little many people know. Try asking people who do not have a math background what a reciprocal is and you'll see what I mean.
Math is clearly very stressful for many people. I think this is probably because anyone taking a math test is often fully aware that they are floundering and may not realize that they have 80% of it correct. On the other hand, for most other subjects they could be happily writing away not knowing that they have forgotten the most critical parts, or that most of what they are writing is rubbish.
I've also taught introductory statistics to the same students and interestingly they seem to have less trouble. I don't know why - perhaps they haven't had bad experiences of it.
Sometime, would you kindly explain why n^0 = 1 ? I've yet to hear an explanation that made sense ...
If we use the example of n = 3: 3^4 = 81, 3^3 = 27, 3^2 = 9, 3^1 = 3. Each time the exponent decreases by one the result is divided by 3 (the value of the base). We just continue the same pattern, so 3^0 = 3/3 = 1, 3^-1 = 1/3 and so on. I hope that helps.
(Remember. When it comes to math, there are three kinds of student - those who get it and those who don't.)
"I was absolutely shocked the day I heard my college algebra teacher announce, "There are several ways you can solve this; pick whichever one works best for you." Not only did I finally have a teacher who liked the subject and could explain it in different ways (so it made SENSE), but she also gave us the "radical" idea that mathematics was a tool that we could use in ways that made sense to each of us."
Unfortunately, I'm actually seeing less of of this kind of teaching in my college math classes than I did in my highschool math classes. I was gifted with a phenomenal math teacher in highschool, but alas, I'm not seeing anything near that quality in my college math experience. I'm struggling in math, for the reasons almost everyone is pointing out, because I'm not gaining an understanding. I'm trying to understand trigonometric substitution, and it's not making sense because I'm not really seeing what's going on. I eventually figured it out, but I can't really give credit to my math prof for that. What I learned there was my own teaching.
I don't think anyone can emphasize enough the effect that a teacher can have on the learning of a student. While I was with that highschool math teacher, the excellent teacher, I seriously considered entering into a Math degree program. I found it to be one of my most interesting classes. University has been the complete opposite. I find myself planning my program around minimizing the calculus I will take, simply because I don't trust the faculty to give me the education in the subject I want (As a side note, I know i'm a slow learner, and find the time constraints difficult. I'm very interested in math, but I really do wish there existed an extended program to give more time to the course.).
I think in general, something must be changed to create more passionate math teachers, in elementary, in secondary, and post secondary.
Richard Simons wrote, "If we use the example of n = 3: 3^4 = 81, 3^3 = 27, 3^2 = 9, 3^1 = 3. Each time the exponent decreases by one the result is divided by 3 (the value of the base). We just continue the same pattern, so 3^0 = 3/3 = 1, 3^-1 = 1/3 and so on. I hope that helps."
Damn! Beautiful. Lovely. It makes perfect sense. So how come my junior high math teacher couldn't explain it that simply? It's really not a complex concept ...
Economics is somewhat odd as a result; as a general rule, an undergraduate degree in the field will not be sufficient to go on for a PhD, even though most programs are post-bachelors. You really need a dual-major in math (or just a BA in math) to go on for graduate studies.
I really enjoyed my time as an economics major. I double majored in computer engineering and economics. I was most interested in DSP, so I took some extra math courses. On the economics side, my university actually offered a math track for its BS in economics. Unfortunately, the track is no longer offered as it was extremely unpopular (one of the classes required for it was only offered as an independent study course as nobody ever had the prerequisites for it). It was a blast while it lasted, and it was the track that the faculty pushed as the most useful way to get into serious research. Now they're back to recommending a double major in mathematics and economics.
I can't say that the econ degree is quite as useful as the CE degree as an embedded systems guy, but I'm convinced that the ability to think quantitatively about wealth and resource allocation has made my life better. It seems like such a fundamentally important thing that it's a shame more people don't get some serious exposure to it. Of course, I imagine that classics majors say the same thing, so maybe it's just my perspective on things.
I'm a sophomore in high school (who doesn't come close to understanding most of the topics here), but I recognize this topic all the time. I try to go for understanding, and often get some math questions wrong when I put, say, 83.47 instead of 83.5 (I long ago gave up arguing against that..)
I tutor an 8th grader in pre-algebra. No offense to the kid, but a lot of the time he just doesn't get it. He didn't understand that a number and it's opposite add to 0. This led to a whole lot of troubles with other things. He's so trained around using formulas when he needs to derive something before using the formula, he's stuck. If he's given the diameter of a circle, he'd forget to take half in order to find the area. When given 20% he'd use 20 instead of .2. A few weeks ago we were doing the areas of simple polygons. First thing I did was ask him to explain why base*height was the area of a rectangle. He could barely do it. It sort of clicked with him that a triangle of half a rectangle. And he didn't know the areas trapezoids and parallelograms past the formulas (I guess he teacher never did the cut-outs and moved the pieces around..) I didn't even bother with circles and volumes of cones and pyramids, mostly because I don't really understand them myself.
I really don't know who to blame though. It might be that he just can't understand it, or it could be his teacher. I don't know. Let's say there was x+6=23 (overly simplified example). I'd say to subtract 6 from each side, and he'd write out x+6-6=23-6. Then he's cross out the +6-6 and on the next line put x=17. Why? Because that's what his teacher wants to see. But I understand all that (not really). I'm currently in Algebra II and Precalc. I was aiming to skip Algebra II, but apparently it needs to be on my transcript. =( It's not that I don't like Algebra II. I like Algebra. Just that the teacher I have teaches from the book and says to show all work. I've already gone through trigonometry in Precalc and we are just starting to covering Degrees-Radians conversion in Algebra II. I see 45Â°, I put Ï/4. It's already engraved in my brain. I seriously hope tomorrow she doesn't expect to see me multiply by Ï/180Â°. (Also, to quote her "You won't need to memorize these angles, but you need to know them.") She explained how to use a Casio calculator to simplify that (45/180) and keep it in fractional form. I complain (jokingly, most of the time) when people can't do 5/6 in their head. *sigh* Sorry, I think I'll stop now that I've digressed to babbling about her teaching.
But then again.. there was that person that asked me if sound was faster than light and didn't believe me when I used seeing lightning before hearing thunder as an example showing light being faster...
Innumeracy and Beyond Numeracy are both great books. While I often smile or and sometimes chuckle when reading a good math or science exposition, John Allen Paulos makes me laugh out loud. To pick just one example, look at the "Prime Numbers" chapter of Beyond Numeracy, p. 184:
The first dozen prime numbers are 2 (the only even prime—why?), 3, 5, 7, 11, 13, 17, 19, 21 (just kidding), 23, 29, 31, 37.
Andrea - I'm glad it helped. As far as I know, that is the way it is usually explained in books but perhaps your math teacher wasn't too clear in his/her own mind. Sometimes the problem seems to be that the concepts come too fast and more time is needed to properly assimilate them.
I find some people understand one explanation while others prefer a different approach. At the beginning of each course I usually tell students that it is important to understand as that avoids the need to remember large numbers of formulae. I also tell them different explanations make sense to different people, so I suggest that if they can't understand from me, try books, other students or one of my colleagues. I once had a student turn to her neighbour to explain a statistical concept. He understood when she explained but I could not hear any difference from what I had said. But there must have been something.
perhaps a quibble.
Since we are talking about building understanding, I would go further with the explanation of why n^0 = 1.
Something like this:
We have this symbolism - 3*4 means that you add four 3's, i.e. 3+3+3+3
3^4 means that you multiply four 3's, i.e. 3*3*3*3. This is a definition of the synbolism. It only holds for positive integer exponents.
From there we DEVELOP the rules for multiplying and dividing using exponents, 3^a *3^b = 3^(a+b) and consequently 3^a / 3^b must = 3^(a-b). (I'm leaving out all the steps that you go through to show why this is so.)
From there you get 3^n / 3^n = 3^(n-n) = 3^0. But 0 is not a positive integer so that symbol is not defined. We observe that any number divided by itself is 1. So, in order for 3^0 to have any useful meaning, we must define it to = 1. And then I would go on to your explanation. This provides a broader background - more understanding. It also introduces an idea that will be important later - that of defining new kinds of numbers and inventing symbols to represent them - useful for fractions, decimals for irrational numbers, and especially imaginary and complex numbers.
Innumeracy is to math what illiteracy is to reading.
This problem can strike at all levels of math education. The last few college math courses I took, I got a B.S. in math, I came to the realization that I was just following the steps. I had no idea what I was doing or why. I think I had a feeling akin to that which younger students get when they just don't understand math concepts. It was very frustrating and really turned me off to math as a whole (since college I have forgotten pretty much everything I learned). Maybe I just didn't work hard enough, or wasn't smart enough, to fully understand the concepts, but it was the first time I really empathized with those who feel revulsion at the mere mention of math.
Damn! Beautiful. Lovely. It makes perfect sense.
Better yet: plot it. Pick a number n, plot n^x, and you will see with your own eyes that the graph intercepts the y axis at 1. Pick another n, rinse, repeat. This is what convinced me that n^0=1 is not an arbitrary definition, but the truth. (We did it in the equivalent of highschool, though a bit late.)
I'm not a math person, but a speech pathologist. But I work in an elementary school and a high school, and I see kids getting A's when they cannot tell me or show me what they are doing, never mind application(at least in 'practical level' classes). It seems to me the NCTM (math teacheers) are on the comprehension train, but the Powers that Be, namely No Child Left Behind, and its multiple guess tests, encourages learning of discrete facts and algorithms without comprehension or application. Chalk up another one for Bush. And it is true that elementary teachers, who are generalists, but still... dont understand or like math themselves, on the whole. I did poorly on math in school, but I find it is in there anyway, and with the tiniest of reviews, it's up front and useable. Maybe somehow math for grownups should be snuck into... I dunno, crime shows on TV or something.
I've certainly had my share of dealing with math teachers uninterested in developing understanding. One of my greatest mentors was a high school math teacher who actually had us write papers such as "What is a line?" and "How does compound interest work?". We also had to physically construct the geometric interpretation of solving a 3-by-3 system of linear equations, using cardboard boxes and yarn. It was really tedious, but it's an insight that has stuck with me ever since. Unfortunately, the other math teachers did not agree with his approach, and he no longer teaches math.
One thing I wonder, however, is whether the degree of innumeracy has significantly increased recently. It seems so, but then again it might be that we just have more exposure to the kinds of people who haven't had good math education (the internet does wonders). Anyone have any thoughts?
I don't think it's increased recently. I'm basing that on some of my own experiences, so it's purely an anecdotal argument, so take it for what it's worth.
When I was in high school, I worked as a cashier in an auto parts store. At the time, I was 17-18, and I dealt with customers of all ages, and with managers who ranged from about 25 to about 70. (We really did have a 70 year old manager.)
Customers of all ages couldn't understand the most trivial math. I think that a sizeable majority of the customers really genuinely didn't understand what a percentage was - I know that every time there was a sale, people would be asking questions about how much things would cost. "All tires are 20% off. If I get these tires, how much will they cost?" "They're $30 each, so that would be $24 each with the discount." "What about these ones?" "They're also normally $30 each, so that's $24 with the discount". "Ok. What about those?" "Normally $40 each, so $32 with the discount." and so on.
These weren't people my own age. We're talking about a complete cross-spectrum slice of the (male) population. (The number of female customers was astonishingly small - less than 5% by my guess.) All ages, wealth levels ranging from people on welfare to rich doctors buying tires for their Jags; from high school dropouts to MDs/PhDs.
There comes a time in the life of a reasonably intelligent student when he (or she) recognizes that he is smarter than most of his teachers. This is usually the start of a truly dangerous spiral, in which the student begins to learn that the world is a much bigger place than he ever expected and that most of the things that people tell you are either so narrowly focused as to be useless for any meaningful purpose or completely wrong. This, in turn, leads to questioning authority and the formation of independant ideas.
The resulting person does not reflect the best interests of society, which much prefers that people remain passive sheep, who believe anything that they are told.
Encouraging young people to think is only going to cause trouble in the long run. This isn't a new or novel concept. They got Socrates for it a very long time ago.
There's a very real danger that when the reasonably intelligent student realizes that he (or she) is smarter than most of his teachers, he (or she) will metamorphose into what psychologists call an "arrogant prick". Intoxicated by their newfound intellectual superiority, an AP may forsake the religion of their parents and community, for example, but instead of adopting a pragmatic approach where all claims are open to empirical inquiry and criticism, they effectively replace a god named Jehovah with a god named Reason, whose pantheon might include a lesser deity called Free Market.
I think this happens when the intellectual in question doesn't have any friends their own age. Four years at MIT gave me the opportunity to meet several people who met that condition, and they were not very happy about their lives. I wonder if the unpleasant experiences of their youth have poisoned them forever.
In general and on balance, this is a better time to be an intellectual child than ever before. You can get more resources at less cost than at any previous time, if you know where to look, and interacting with real-life academic folk gets easier every day.
While some people, enamored of hierarchy and authoritarianism, think this is a bad thing, those who appreciate a pluralist society can only be grateful.
As a student, prior to college, I was constantly frustrated in math classes by the lack of material, slow pace, and unavailability of the teachers to help me follow my curiosity, as well as lack of time to study anything in depth or step back and look at a topic from a different angle. As a student, in college, I was suddenly deluged with material, the pace was insanely fast, and the teachers wanted me to master everything in depth but were unwilling or unable to help me do that. I couldn't do it and crashed and burned about four semesters later. Sorry, just couldn't do it. Not with all those years of rote and trivial pre-college math.
Then, as a teacher, NCTM advised me to devote a lot of nonexistent time to teaching the children to think about complex things while the approved curriculum insisted on hammering the skills to calculate absurdly simple stuff, and I found a small group of poorly socialized children (and whose parents were no better) devoured everyone's time by requiring constant behavior correction, while my other students sat there starved for material and attention. The only thing I could do is try to identify any student with any interest at all and spend time during lunch, before school, after school, whenever (as an aside, imagine that a few poorly socialized people rendered your entire working day nonproductive, and your only option is to come in early, work through lunch, stay late, and work for free on your own time in the hopes of getting your job done), trying to fan that spark. Most kids fell through the cracks and probably went to college and bombed out of math after a couple of semesters. The cycle continued.
There has to be a better way. We can't spend 12 years trying to hold people back in one large low-achieving group, and then suddenly say to the ones who miraculously survive that process, o.k., you finally get to run with it, while the rest of you all suck, why don't you study anthropology or something. It is no wonder that most people hate math. That drive, desire, talent, and circumstances somehow conspire to produce a few mathematicians in this system is the true miracle.
We've created a social darwinist nightmare and called it math education. And we're surprised it produces a handful of good mathematicians, a lot of casualties, and widespread innumeracy.
I'm surprised nobody has yet mentioned John Mighton, the Canadian playwright and mathematician who has developed simple ways of teaching math to ALL kids, including the lowest-achieving ones. Basically, the approach boils down to very step-by-step explanations, at least at the beginning, and building the kids' self-confidence. Mighton's book, The Myth of Ability is well worth reading.
The psychological side of learning math needs far more attention than it currently gets. In school, I usually understood math concepts but messed up actual calculations. Somehow, I survived a year of undergrad calculus but it took an excellent mathematical ecology professor and a friend willing tutor me to build up my confidence and teach me that my intuition was valuable. I then went back and relearned the things I had been messing up on from seventh grade onward. End result? I'm now doing graduate work in network ecology and my first two midterm scores in a differential equations class were 100% and 103%.
Why is my comment above anonymous? Trying again.
For some reason, TypeKey swallows your user name. If you post while logged into TypeKey, you can use more URLs in your comment without triggering the spam filter, but your comments show up with the byline "Anonymous". (The website you specify in the comment's URL field still appears, oddly enough.)
Jane, I'm in a similar boat, but alas, i've yet to meet my mentor.
It always sucks to get stuck with bad teaching, it's one of those things that's just luck of the draw.
I don't know if it will help you feel any better, but often the calculus classes, like all intro classes, have bad teaching just because few professors have them as their first priority, and so the jobs get rotated through the department, often landing in the hands of professors who don't want to be there (upper level classes are assigned by what the professor specializes in, but intro classes are moved around to share the burden).
One thing you could try, if you haven't already, is find out if your school has any academic support services---my school provides two different places to get free math tutoring, for example.
The first reaction of everyone who's having trouble is to take the "easy" classes. But that works only if the material itself disinterests you, or you have true difficulty with it. If it is bad teaching, and boring work that is making it hard to learn, I have found that the "harder" classes can actually be easier to succeed in, because they're often led by good, motivated professors, filled with good, motivated students, and assigning challenging but interesting work. It is a risk though, because the work really is harder, and interest and motivation aren't everything. The "harder" math classes in college, once you're past the first couple calculus classes, tend to be harder because they're focused more on concepts and proofs than calculation. If that's what you prefer, you may just be happier there.
There are always good teachers and bad teachers in a department. Students who have been there a while will probably know who they are. Ask someone, and ask why they are good or bad. Trying to pick the better teachers can make all the difference.
Lastly, someone once told me that the best thing about studying math is that everyone else will assume you're smarter than you really are.
Whatever you choose to do, I wish you the best of luck (and hope that you take my advice, but only with a grain of salt).
Any comments on Neville Holmes, IEEE Computer Mar 2007 (PDF) Digital Technology and the Skills Shortage suggestion of using computerized drill from an early age, with a radical redesign of education?
I've heard much of the same advice before. Its a sad situation, but one I plan to fight through. I'm going to be in a CS degree soon, so I'll be needing more math, and that's all I can do. Fight through the terrible intro classes, and then hopefully excel later in with better teachers.
My own personal horror story (and yes, I'm still pissed off about it, as you can no doubt tell):
One of my brothers was tasked with babysitting me, when he was 15 and I was about 3. In order to keep me quiet, he used to sit down and do his math homework with me (apparently I loved it; early geek tendencies?). As a result, though, I had mastered -- and actually understood, at the level of being able to internally visualise how it all worked -- all the concepts between 2+2=4, and solving linear equations.
It sounds like bragging, but at that stage it was actually fairly easy for me to picture how numbers interacted, and I attribute that to the fact that I was just learning it all at the same time as I was absorbing language and how physical objects in the physical world could be expected to behave (like, if you push a door open too hard and it bounces off the wall, it might hit you, sort of thing). Math concepts just slotted into "this is how the world works and this is how you talk about it." If you stack up 2 apples against 2 negative apples -- two apple-sized holes -- then it simply makes perfect sense that you end up with no apples, and that works for any type of "2", and that's how I reasoned. I picked up on patterns fast (like the patterns you get in multiplication, which of course simply work in reverse in division), fractions didn't seem to present much of a challenge once you learned to picture how they interacted, and graphing things on a 2D grid was trivial.
Then I started elementary school.
My teachers -- universally, in one grade after another -- were absolutely horrified when I did the entire math workbook out of sheer boredom. I was repeatedly punished for "not following along with the class." Then when I *finally* got to a grade where we started division --"at last, something marginally interesting!" I thought, but no. I had an unfortunate tendency to do short division, and just jot down a few numbers in corners. No, this was unacceptable. I HAD to do long division, "so that my teachers could see that I understood it". I was willing to go along with this for the first month or so, but I genuinely thought (little naif that I was) that they would eventually see that I knew what I was doing and would let me start using techniques that I knew worked. That was, of course, silly of me. I remember knock-down, drag-out arguments with the teachers where I insisted that I could sit there in front of them and work the problems, and they would be able to see that I was not cheating, and I could explain every step as I went, and that surely the fact that I got the answers right no matter what problem they threw at me should be enough proof for them that I understood what it was that I was doing. If I could show that I knew exactly how division with 2 digits worked, perhaps they would let me use shorter calculations when we moved on to division with 3 digits or more.
No. Absolutely not. I was *NOT* to use short division, or any of the factoring tricks I had learned, or any of the fast internal methods of calculation that I had developed -- it all had to be written exactly the way everyone else in the class did it (the "x+6-6=23-6" example that Josh C used resonates all too painfully), over and over and over, every single bloody year -- and I was penalised by the teacher for "skipping steps" too. In grade after grade. In math subject after math subject. It became nothing more than grinding tedium.
Result? An absolute hatred for mathematics, of course, which really only started to fade a few years ago. As soon as I could stop taking the subject in high school, I did.
The problem with the school system is that, in common with every other human system on the planet, in order to achieve a desired goal (in this case a basic understanding of maths) a bureaucracy makes rules; then hires people who will follow the rules, rather than who have individual ideas about achieving the ultimate goal; with the result that, systematically, the rules become far more important than the original goal.
Luna: I had a similar, though not as dramatic, experience as you throughout elementary and middle school.
I got very lucky in high school, though, as I had several teachers who recognized my skill. They were so sure that I knew my stuff that they let me program my calculator to do it all for me (after all, if I knew it well enough to program my calc, I obviously knew it).
I ended up creating a mathematics suite that was shared across the school. I was still the only one allowed to use it on tests, though. ^_^
n^0=1 is a great example of algebraic number theory in action. Every number is itself times an infinite number of 1's.
To put it more specifically: n is always equal to the product of n and an infinite number of 1's. A square is n times n times an infinite number of 1's. Cubes and higher powers are constructed in the same manner.
So it follows that an infinite number of 1's NOT multiplied by any n is equal to the product of an infinite number of 1's.
The same is true for negative exponents. 10^-2 is 0.01, which is equal to the product of an infinite number of 1's DIVIDED by 10... twice.
By the way, any number is also the sum of itself and an infinite number of zeros. The trick in algebra is to know how to construct zeros and ones of various forms.