Whee, Standard Texts
The following is a review of the text I had to teach from this year, Robert Blitzer's College Algebra Essentials - Third Edition. He has a Master's degree in math, but his PhD is in behavioral sciences, which puts me on the alert. I'm thinking of posting this on Amazon, but I might want to make a new account that's not under my name, just in case. I'm a bit burnt out, so it may be a bit lacking in terms of the creative flourish that I usually try to include with my disdain.
"Whoever wrote [the NCTM Standards] must be a physical education teacher."
- Jaime Escalante, living legend of math education
Blitzer's text in so-called "college" algebra is yet another example
of a continuation into college of the low expectations placed on
students in the American K-12 system. The first impression that I got
from examining this book was that it looked like an extended
advertisement for McDonald's. Anyone who feels a little unsettled by
politicians, businessmen, and things that reek of marketing will feel
ill from teaching from this book. If your higher-ups have you use this
book, you will likely be asked to not expect your students to really
learn how to complete the square. I will list some grievances with
examples:
* Examples that are given are exceedingly arbitrary, come from
nowhere, with no construction, and never connect with earlier
examples. "We pulled this out of thin air using math! Isn't it NEAT?
Look, it's got numbers in it!"
- Example, p. 24: "What is the maximum speed at which a racing cyclist
can turn a corner without tripping over? The answer, in miles per
hour, is given by the algebraic expression 4sqrt(x), where x is the
radius of the corner, in feet." (actual quote!)
- Example, p. 304: "Here's a 4th-degree polynomial H(x). If x is the
age of your dog in ordinary years, H(x) is the age of your dog in dog
years. If you are 25 years old, how many years old is a dog whose age
in dog years is also 25? We set H(x) = 25. And then what? How do we
solve for x? In section 3.3, we tell you some methods for solving some
higher-order polynomial equations, but not this one. But hey, look,
it's a picture of a puppy!"
- Example, p. 374: "This section's on exponential functions. It would
make sense to start this section out with a problem about reproducing
organisms in a population. Anyway, the probability of an O-ring on the
space shuttle failing at the temperature x in degrees Fahrenheit is
given by f(x) = 13.49*(.967)^x - 1. Look, the variable is in the
exponent! Isn't that WEIRD?"
* Proofs, for the most part, are not included. The appendix, "selected
proofs", is barely a page long and consists of proofs of two facts
about logarithms. There are only two proofs, and Blitzer makes certain
that they're in the back where the student may safely ignore them.
* The book is packed full of cookbook-style mathematics. And then the
"critical thinking" exercises are often in the following format: "Here
are 4 problems just like the ones you just did, with proposed answers.
Which one's right?"
* There are some omissions that are very puzzling, and when faced with
two methods of solving a problem, Blitzer always chooses the one that
obfuscates the structure of the problem but is perhaps slightly easier
to perform, for a short-term and short-sighted gain.
- Example, section 1.5: For the math-intolerant, completing the square
is introduced in the context of equations, and is not discussed in the
setting where we merely wish to rewrite a given quantity. So, for
example, finding the graph of f(x) = x^2 + 8x + 8 will be confusing,
since it involves a different application of completing the square -
we're suddenly not out to solve f(x)=0, we want to write f(x) =
(x+4)^2 - 8. This will be confusing when it comes up, and artificially
new to them. Of course, completing the square will be brushed under
the rug with the "h = -b/2a" trick, which will be a little bit
crippling for students who may need to complete the square in later
Calculus courses where completing the square is neccesary to do
partial fractions in any kind of generality.
- Example, p. 51: It deserves to be mentioned that integer polynomials
with rational roots, like 8x^2 -10x -3, can be factored by grouping if
the middle term is split into two parts. The task of factoring ax^2 +
bx + c, it can be shown with a few computations, amounts to splitting
up bx = mx + nx, with the condition that mn = ac. Blitzer would have
you write out pairs of binomials and guessing at the coefficients
until something works. Factoring by inspection quite often has its
merits, but the systematic approach has a certain mathematical
structure to it that's worth studying. Blitzer shuns it.
- Example, section 2.5: In this section, the focus of study is
transformations of functions. That is, given a function f and its
graph, we wish to find a graph of g, where g(x) = af(bx+c) + d.
Blitzer here makes the truly puzzling decision to always have b=1 or
b=-1, skipping over any discussion of how replacing x by 2x "speeds
up" the function. Blitzer tells 3/4 of a story and leaves this
whodunnit mystery of the symmetries between precomposition and
postcomposition unsolved for all time. Sometimes the b term can be in
some way factored out and grouped with a, but the students will
presumably have to reinvent the wheel in later courses where they
study the graph of, say, sin(3x).
Also puzzling about 2.5 is that 2.6 is where function composition is
introduced. This is stange, since these transformations are
compositions.
- Example, section 1.8: This section is on quadratic and rational
inequalities. The idea is to rewrite a given inequality as "p(x)/q(x)
> 0" or "p(x)/q(x) < 0", factoring, and using the factorizations of p
and q to conclude where the inequality is satisfied. The method
proposed? Find the zeroes of p and q and test points in between. This
can be awkward and tedious if the zeroes are, say, 4/17 and 8/33. The
alternate method, which favors examining the structure of the problem
over punching numbers into a calculator, is to make a quick +/- table
of the individual factors to determine the behaviour of the whole. The
alternate method is not even mentioned. Where as the +/- table method
is essentially a rigorous proof of the solution, the method of picking
points in between has a "trust us, kid" tone to it.
- Example, section 2.4: The discussion of even and odd functions makes
no mention whatsoever that x^n is and even function when n is even,
and an odd function when n is odd. Also completely missing is an
exploration of what happens when you add together two odd functions or
two even functions, which together yields a method that allows an
instant classification of any polynomial as even, odd, or neither.
* The difference quotient is given a massive hand wave - we're told
that "they play an important role in understanding the rate at which
functions change", and then we have to compute a few from time to
time. A sequence of exercises studying the difference quotient, the
slopes that they define, and what happens for small h, all in
anticipation of the concepts of "limit" and "derivative" would be
nice. No such luck. No investigation ensues ever of what a difference
quotient means or how it relates to linear equations.
* At the end of each chapter, there are review exercises. The harder
problems are consistently left out.
* Little math biographies are thrown in for fun - "Gauss was really smart! He figured stuff out! But you don't have to be smart. Here, have a cook book."
Anyway, this book contributes to the ongoing self-fulfilling prophesy
that our students will, for the most part, be intolerably dense, need
lots of hand-holding, and feel entitled to their feelings of scorn
whenever anyone asks them to think for themselves. Use this text,
write a problem on a test that even trivially varies from the format
of the exercises, and watch what happens!
"Whoever wrote [the NCTM Standards] must be a physical education teacher."
- Jaime Escalante, living legend of math education
Blitzer's text in so-called "college" algebra is yet another example
of a continuation into college of the low expectations placed on
students in the American K-12 system. The first impression that I got
from examining this book was that it looked like an extended
advertisement for McDonald's. Anyone who feels a little unsettled by
politicians, businessmen, and things that reek of marketing will feel
ill from teaching from this book. If your higher-ups have you use this
book, you will likely be asked to not expect your students to really
learn how to complete the square. I will list some grievances with
examples:
* Examples that are given are exceedingly arbitrary, come from
nowhere, with no construction, and never connect with earlier
examples. "We pulled this out of thin air using math! Isn't it NEAT?
Look, it's got numbers in it!"
- Example, p. 24: "What is the maximum speed at which a racing cyclist
can turn a corner without tripping over? The answer, in miles per
hour, is given by the algebraic expression 4sqrt(x), where x is the
radius of the corner, in feet." (actual quote!)
- Example, p. 304: "Here's a 4th-degree polynomial H(x). If x is the
age of your dog in ordinary years, H(x) is the age of your dog in dog
years. If you are 25 years old, how many years old is a dog whose age
in dog years is also 25? We set H(x) = 25. And then what? How do we
solve for x? In section 3.3, we tell you some methods for solving some
higher-order polynomial equations, but not this one. But hey, look,
it's a picture of a puppy!"
- Example, p. 374: "This section's on exponential functions. It would
make sense to start this section out with a problem about reproducing
organisms in a population. Anyway, the probability of an O-ring on the
space shuttle failing at the temperature x in degrees Fahrenheit is
given by f(x) = 13.49*(.967)^x - 1. Look, the variable is in the
exponent! Isn't that WEIRD?"
* Proofs, for the most part, are not included. The appendix, "selected
proofs", is barely a page long and consists of proofs of two facts
about logarithms. There are only two proofs, and Blitzer makes certain
that they're in the back where the student may safely ignore them.
* The book is packed full of cookbook-style mathematics. And then the
"critical thinking" exercises are often in the following format: "Here
are 4 problems just like the ones you just did, with proposed answers.
Which one's right?"
* There are some omissions that are very puzzling, and when faced with
two methods of solving a problem, Blitzer always chooses the one that
obfuscates the structure of the problem but is perhaps slightly easier
to perform, for a short-term and short-sighted gain.
- Example, section 1.5: For the math-intolerant, completing the square
is introduced in the context of equations, and is not discussed in the
setting where we merely wish to rewrite a given quantity. So, for
example, finding the graph of f(x) = x^2 + 8x + 8 will be confusing,
since it involves a different application of completing the square -
we're suddenly not out to solve f(x)=0, we want to write f(x) =
(x+4)^2 - 8. This will be confusing when it comes up, and artificially
new to them. Of course, completing the square will be brushed under
the rug with the "h = -b/2a" trick, which will be a little bit
crippling for students who may need to complete the square in later
Calculus courses where completing the square is neccesary to do
partial fractions in any kind of generality.
- Example, p. 51: It deserves to be mentioned that integer polynomials
with rational roots, like 8x^2 -10x -3, can be factored by grouping if
the middle term is split into two parts. The task of factoring ax^2 +
bx + c, it can be shown with a few computations, amounts to splitting
up bx = mx + nx, with the condition that mn = ac. Blitzer would have
you write out pairs of binomials and guessing at the coefficients
until something works. Factoring by inspection quite often has its
merits, but the systematic approach has a certain mathematical
structure to it that's worth studying. Blitzer shuns it.
- Example, section 2.5: In this section, the focus of study is
transformations of functions. That is, given a function f and its
graph, we wish to find a graph of g, where g(x) = af(bx+c) + d.
Blitzer here makes the truly puzzling decision to always have b=1 or
b=-1, skipping over any discussion of how replacing x by 2x "speeds
up" the function. Blitzer tells 3/4 of a story and leaves this
whodunnit mystery of the symmetries between precomposition and
postcomposition unsolved for all time. Sometimes the b term can be in
some way factored out and grouped with a, but the students will
presumably have to reinvent the wheel in later courses where they
study the graph of, say, sin(3x).
Also puzzling about 2.5 is that 2.6 is where function composition is
introduced. This is stange, since these transformations are
compositions.
- Example, section 1.8: This section is on quadratic and rational
inequalities. The idea is to rewrite a given inequality as "p(x)/q(x)
> 0" or "p(x)/q(x) < 0", factoring, and using the factorizations of p
and q to conclude where the inequality is satisfied. The method
proposed? Find the zeroes of p and q and test points in between. This
can be awkward and tedious if the zeroes are, say, 4/17 and 8/33. The
alternate method, which favors examining the structure of the problem
over punching numbers into a calculator, is to make a quick +/- table
of the individual factors to determine the behaviour of the whole. The
alternate method is not even mentioned. Where as the +/- table method
is essentially a rigorous proof of the solution, the method of picking
points in between has a "trust us, kid" tone to it.
- Example, section 2.4: The discussion of even and odd functions makes
no mention whatsoever that x^n is and even function when n is even,
and an odd function when n is odd. Also completely missing is an
exploration of what happens when you add together two odd functions or
two even functions, which together yields a method that allows an
instant classification of any polynomial as even, odd, or neither.
* The difference quotient is given a massive hand wave - we're told
that "they play an important role in understanding the rate at which
functions change", and then we have to compute a few from time to
time. A sequence of exercises studying the difference quotient, the
slopes that they define, and what happens for small h, all in
anticipation of the concepts of "limit" and "derivative" would be
nice. No such luck. No investigation ensues ever of what a difference
quotient means or how it relates to linear equations.
* At the end of each chapter, there are review exercises. The harder
problems are consistently left out.
* Little math biographies are thrown in for fun - "Gauss was really smart! He figured stuff out! But you don't have to be smart. Here, have a cook book."
Anyway, this book contributes to the ongoing self-fulfilling prophesy
that our students will, for the most part, be intolerably dense, need
lots of hand-holding, and feel entitled to their feelings of scorn
whenever anyone asks them to think for themselves. Use this text,
write a problem on a test that even trivially varies from the format
of the exercises, and watch what happens!