Here are a couple articles from the local paper:
Math Tests Carry Shock Factor
Preliminary Math Test Failure Rates
Basically, 90% of the high school students are failing middle school algebra. There seems to be some controversy about how those tests were administered, but 90% is simply too large to be merely a statistical outlier. As a department, we agree that this is pathetic and that we want to do something about it, but we have no plans yet (we also have no funds -- but there's some work being done on that side). Our plan is to address this issue at the middle school level because we think high school is too late. Elementary school level would be better for getting students interested in math in general, but we don't think it will have as much of a lasting effect. I think we're going to target some middle school teachers, but this has to be thought out more carefully.
Showing posts with label Rant. Show all posts
Showing posts with label Rant. Show all posts
Tuesday, April 15, 2008
Sunday, August 5, 2007
"Opposites" of numbers
I'm taking a little break from writing lecture notes to complain about things that textbooks do that are really stupid. I'm going through the Math 097 textbook, and I have reached the third or fourth time where I just shake my head at the textbook and hope that somehow either the textbook or the author will feel my frustration and it will be changed in the next edition. I suppose I could write a letter to the publisher or the authors, but I'm not going to do that now.
Why does it matter? I have an expectation that the students will read the textbook. As a consequence of this expectation, I am reading the textbook as well to make sure what they read actually makes sense.
As an aside, in what classes do students pay $100+ for texts that they have no expectation of reading? As far as I know, it's only the math classes.
Anyway, something that really bothered me was the discussion of negative numbers. They talk about 1 and -1 being "opposites". This random terminology that math people would never use reminds me of some terminology I ran into in high school with the CPM program. Apparently, somewhere along the line someone felt that a "common denominator" should be called a "fraction buster". If you don't believe me, just Google "fraction buster." Anyway, I suppose the idea is that when solving an equation involving fractions , you can "bust" the fraction by multiplying by the right number... or something like that. Besides the absurdity of nonstandard terminology, it also means that when you add fractions with different denominators that you would have to introduce the "common denominator" as a separate concept, even though it's the exact same thing.
But back to "opposites". The standard terminology for this is "additive inverse" for the simple reason that the relationship between them is additive, namely that if you add them together and you get zero (4 + (-4) = 0), and zero is the special number that doesn't change the result if you add it to something else (10 + 0 = 10). Then there's a link between this idea and the "multiplicative inverse" of a number. The relationship between a number and it's multiplicative inverse is that when you multiply them together you get one (4 * (1/4) = 1), and one is the special number that doesn't change the result if you multiply it by something else (10 * 1 = 10).
It just got worse as I pushed onward into the following section. Here is a quote from the text:
Furthermore,
If you talk to anyone who does math (this includes chemists, engineers, physicists, and many others), and pointed to -x and told them to read it, they would all say "negative x." So while it would make sense to a student brought up in this way, to the rest of the world, it's nonsense.
But it gets worse. Consider the following sentence "Multiply x by the opposite of x." If given that sentence outside of the context of this subject, I would consider "opposite" to be related to "multiplication" and end up with x * (1/x) instead of x * (-x). The book will likely introduce a different word, "reciprocal" for 1/x. Unfortunately, all this does is introduce more words for students to memorize, and those words are not particularly informative to the nature of the relationship. Why is "opposite" additive and "reciprocal" multiplicative? It just is. Memorize it. (At least "reciprocal" is a common terminology to people who use math...)
I think that's enough of a rant. I need to get back to writing up lecture notes. I won't get into the "Rules for Addition of Real Numbers" table right now...
Why does it matter? I have an expectation that the students will read the textbook. As a consequence of this expectation, I am reading the textbook as well to make sure what they read actually makes sense.
As an aside, in what classes do students pay $100+ for texts that they have no expectation of reading? As far as I know, it's only the math classes.
Anyway, something that really bothered me was the discussion of negative numbers. They talk about 1 and -1 being "opposites". This random terminology that math people would never use reminds me of some terminology I ran into in high school with the CPM program. Apparently, somewhere along the line someone felt that a "common denominator" should be called a "fraction buster". If you don't believe me, just Google "fraction buster." Anyway, I suppose the idea is that when solving an equation involving fractions , you can "bust" the fraction by multiplying by the right number... or something like that. Besides the absurdity of nonstandard terminology, it also means that when you add fractions with different denominators that you would have to introduce the "common denominator" as a separate concept, even though it's the exact same thing.
But back to "opposites". The standard terminology for this is "additive inverse" for the simple reason that the relationship between them is additive, namely that if you add them together and you get zero (4 + (-4) = 0), and zero is the special number that doesn't change the result if you add it to something else (10 + 0 = 10). Then there's a link between this idea and the "multiplicative inverse" of a number. The relationship between a number and it's multiplicative inverse is that when you multiply them together you get one (4 * (1/4) = 1), and one is the special number that doesn't change the result if you multiply it by something else (10 * 1 = 10).
It just got worse as I pushed onward into the following section. Here is a quote from the text:
A symbol like -x, which has a variable, should be read "the opposite of x" or "the additive inverse of x" and not negative x," since to do so suggests that -x represents a negative number.
Furthermore,
As you read mathematics, it is important to verbalize correctly the words and symbols to yourself. Consistently reading the expression -x as "the opposite of x" is a good step in this direction.
If you talk to anyone who does math (this includes chemists, engineers, physicists, and many others), and pointed to -x and told them to read it, they would all say "negative x." So while it would make sense to a student brought up in this way, to the rest of the world, it's nonsense.
But it gets worse. Consider the following sentence "Multiply x by the opposite of x." If given that sentence outside of the context of this subject, I would consider "opposite" to be related to "multiplication" and end up with x * (1/x) instead of x * (-x). The book will likely introduce a different word, "reciprocal" for 1/x. Unfortunately, all this does is introduce more words for students to memorize, and those words are not particularly informative to the nature of the relationship. Why is "opposite" additive and "reciprocal" multiplicative? It just is. Memorize it. (At least "reciprocal" is a common terminology to people who use math...)
I think that's enough of a rant. I need to get back to writing up lecture notes. I won't get into the "Rules for Addition of Real Numbers" table right now...
Friday, April 13, 2007
The importance of thinking
I was up late talking with a friend the other night. He lived with me last year, but spent this year in China teaching English and computer skills in a rural town somewhere. We had a nice conversation about a number of topics, and one in particular was relevant to this blog.
It turns out from our collective experience that most undergraduate students don't know how to think about math at all (ourselves included). This doesn't mean that they are incompetent or stupid, it's just that they have not ever developed the skills of critical thinking and self-reflection.
Before talking about math, let me point out a specific example in real life. Over by the Mandeville center, down where the art people have a space to do their thing, there used to be a painting on the ground (someone painted over it). It was a picture of what I've always supposed was Mary and baby Jesus (woman, child, halo). Next to the figure was painted "Do not push your beliefs on others" (or something like that). I am not sure whether the person who painted that appreciates the irony. As a personal philosophy ("I will not try to force you to believe what I believe"), it works just fine. However, it cannot possibly be given as an instruction to someone else in an internally self-consistent manner ("You should not impose your beliefs on me" -- by making this statement, the speaker is trying to impose HIS beliefs on the listener, thus doing what he says one should not do). A lack of critical analysis leads to nonsense.
The same internal inconsistency exists with bland, naive relativistic statements such as "all religions are the same" (different religions have different claims to truth, and they are obviously incompatible) and "you should accept everybody" (the person doesn't accept you because you don't accept someone, but then that person is himself not accepting of everyone). Most of the people I know who talk that way haven't ever really spent time thinking and reflecting on their system of beliefs -- it's much like how Christians who never study the Bible come to false conclusions about how God "should" behave (really, how God "does" behave).
But what does this have to do with math? I'll grant that philosophical reasoning is a bit stickier than mathematical reasoning. At least with mathematical reasoning, we (the math community) have a set of mostly agreed upon fundamental beliefs (axioms) upon which we build mathematical structures. There also isn't a whole lot of room for "personal interpretation" when it comes down to the formalism of mathematics (there is actually room for personal interpretation when it comes to trying to understand math... but that's a different story).
Where was I? Oh yeah, what does this have to do with math? A lack of critical analysis leads to nonsense. The nature of mathematics for a lot of students is that you find some numbers, or a formula, and plug stuff in, and move some terms around, and get some answer. It doesn't matter where the problem came from and it doesn't matter how silly the answer is. It doesn't matter that it's a word problem computing the terminal velocity of a falling object, and that the final answer they got is "t = 10 minutes." The problems have no intrinsic meaning to the students, so their answers often have no intrinsic meaning.
Here are a few examples of relatively simple mathematical processes that a lot of people know they're supposed to do, but don't actually know why it's the right thing to do (other than they were told to do it that way):
I was fortunate that my school and my teachers were willing to put in extra effort for the bright students. There was a weekly thing in the library for the more proficient math students (I remember learning about exponential growth there by the chessboard and grains of wheat problem).
My fourth grade teacher borrowed books from the library to keep me occupied while she taught arithmetic to the rest of the class (she got me books on finance, but since I didn't have a good conception of money at the time, the real value didn't sink in until much later... for example, if I understood no-interest government student aid loans and all that stuff, I would have realized that when I started my undergraduate studies, I could have taken out a maximal student loan and earned 5% interest in a CD somewhere and make free money off the government. Even at 3.5% APY on a $10,000 student loan, because I was in school for 9 years and the loan would be without interest would have given me an extra $5500 for free. Then taking the $5500 and continuing to earn 5% APY on it from age 27 to age 65 is an extra $35000 in retirement funds for doing essentially nothing. Of course, I probably could have qualified for more at the start, and probably could have gotten better than 5% return on average (I think the average market growth rate is around 8%), but I think this makes my point about the gap between being able to compute things and being able to critically analyze what the math is saying.
Who is to blame for the inability of students to think critically about their math? It's not the fault of any individual, but it's a faulty system. Teachers teach math this way because this is basically how they learned it. Students learn math this way because they have no other model for learning math. The students grow up, and this way of thinking is never corrected, so it propagates itself to the next generation.
Fortunately, I do believe change is in the works. There are a number of educators who are looking into the failing mathematics system, and while I'm not involved there, I do hope they find true solutions and not false ones like having teachers teach to standardized testing... that just reinforces the same problems.
It turns out from our collective experience that most undergraduate students don't know how to think about math at all (ourselves included). This doesn't mean that they are incompetent or stupid, it's just that they have not ever developed the skills of critical thinking and self-reflection.
Before talking about math, let me point out a specific example in real life. Over by the Mandeville center, down where the art people have a space to do their thing, there used to be a painting on the ground (someone painted over it). It was a picture of what I've always supposed was Mary and baby Jesus (woman, child, halo). Next to the figure was painted "Do not push your beliefs on others" (or something like that). I am not sure whether the person who painted that appreciates the irony. As a personal philosophy ("I will not try to force you to believe what I believe"), it works just fine. However, it cannot possibly be given as an instruction to someone else in an internally self-consistent manner ("You should not impose your beliefs on me" -- by making this statement, the speaker is trying to impose HIS beliefs on the listener, thus doing what he says one should not do). A lack of critical analysis leads to nonsense.
The same internal inconsistency exists with bland, naive relativistic statements such as "all religions are the same" (different religions have different claims to truth, and they are obviously incompatible) and "you should accept everybody" (the person doesn't accept you because you don't accept someone, but then that person is himself not accepting of everyone). Most of the people I know who talk that way haven't ever really spent time thinking and reflecting on their system of beliefs -- it's much like how Christians who never study the Bible come to false conclusions about how God "should" behave (really, how God "does" behave).
But what does this have to do with math? I'll grant that philosophical reasoning is a bit stickier than mathematical reasoning. At least with mathematical reasoning, we (the math community) have a set of mostly agreed upon fundamental beliefs (axioms) upon which we build mathematical structures. There also isn't a whole lot of room for "personal interpretation" when it comes down to the formalism of mathematics (there is actually room for personal interpretation when it comes to trying to understand math... but that's a different story).
Where was I? Oh yeah, what does this have to do with math? A lack of critical analysis leads to nonsense. The nature of mathematics for a lot of students is that you find some numbers, or a formula, and plug stuff in, and move some terms around, and get some answer. It doesn't matter where the problem came from and it doesn't matter how silly the answer is. It doesn't matter that it's a word problem computing the terminal velocity of a falling object, and that the final answer they got is "t = 10 minutes." The problems have no intrinsic meaning to the students, so their answers often have no intrinsic meaning.
Here are a few examples of relatively simple mathematical processes that a lot of people know they're supposed to do, but don't actually know why it's the right thing to do (other than they were told to do it that way):
- Why do we "carry the 1" in addition? (Even more fundamentally, why do we start on the right side when we add? -- This one isn't about being right or wrong)
- Long division: Divide, subtract, multiply, bring down, repeat... What does this process actually do?
- x^n * x^m = x^(n+m) -- If you're not familiar with typing math, x^n is x to the nth power.
- Kindergarten - Learn to count
- 1st Grade - Learn to add and subtract
- 2nd Grade - Learn to add and subtract in multiple columns
- 3rd Grade - Learn to multiply, introduction to fractions
- 4th Grade - Learn to multiply in columns and do long division
- 5th Grade - I don't even remember what I was supposed to learn...
I was fortunate that my school and my teachers were willing to put in extra effort for the bright students. There was a weekly thing in the library for the more proficient math students (I remember learning about exponential growth there by the chessboard and grains of wheat problem).
My fourth grade teacher borrowed books from the library to keep me occupied while she taught arithmetic to the rest of the class (she got me books on finance, but since I didn't have a good conception of money at the time, the real value didn't sink in until much later... for example, if I understood no-interest government student aid loans and all that stuff, I would have realized that when I started my undergraduate studies, I could have taken out a maximal student loan and earned 5% interest in a CD somewhere and make free money off the government. Even at 3.5% APY on a $10,000 student loan, because I was in school for 9 years and the loan would be without interest would have given me an extra $5500 for free. Then taking the $5500 and continuing to earn 5% APY on it from age 27 to age 65 is an extra $35000 in retirement funds for doing essentially nothing. Of course, I probably could have qualified for more at the start, and probably could have gotten better than 5% return on average (I think the average market growth rate is around 8%), but I think this makes my point about the gap between being able to compute things and being able to critically analyze what the math is saying.
Who is to blame for the inability of students to think critically about their math? It's not the fault of any individual, but it's a faulty system. Teachers teach math this way because this is basically how they learned it. Students learn math this way because they have no other model for learning math. The students grow up, and this way of thinking is never corrected, so it propagates itself to the next generation.
Fortunately, I do believe change is in the works. There are a number of educators who are looking into the failing mathematics system, and while I'm not involved there, I do hope they find true solutions and not false ones like having teachers teach to standardized testing... that just reinforces the same problems.
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