美国的初等数学教育

我正在教本科生Math220（Preliminary Calculus），发现很多学生的基础代数能力确实比较糟糕。今天信箱里来了一封群发信件，大家闲来无事不妨看看：）
Math in MD


David Hamilton wrote: At the end of my 220 exam a student asked could he
leave the answer to #3 unsimplified as

0.1 X 8

because "he was not very good with decimals" (calculators were not allowed).

Use of calculators was promoted as making long division/multiplication
obsolete but in fact the result is that many  students cannot do the
simplest numerical thinking.

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That students arrive on campus NOT fluent in arithmetic and overly
calculator dependent (big time) is the fault of the middle schools, high
schools and the MD State Dept. of Mis-Education.

That students arrive in Stat 100 and Math 220 NOT fluent in arithmetic and
overly calculator dependent (big time) is partially the fault of our Math
Dept.'s only partially functional placement exam.  Also median grade on
Precalculus exam given, by Denny Gulick a few years ago, to Math 140
students on Day 1 of class was an absurdly low  40%.

The demonizing of arithmetic and the over emphasis of over using
calculators by most colleges of education across our country and by The
National Council of Teachers of Mathematics (NCTM) has resulted in
Arithmetic being squashed down so much that students can't learn it.

Steven Leinwand was the co-chairman of the U. S. Dept. of Education's
Expert Panel (on textbooks) and the top mathematics adviser at
Connecticut's Department of Education and was he was a consultant to The
MD Math Commission (2001) [on K-12 math education in MD].  (No
mathematician from the flagship campus was a consultant.)
   In his article "It's Finally Time to Abandon Computational Algorithms",
he began:
"It's time to confront those nagging doubts about continuing to teach our
students computational algorithms for addition, subtraction,
multiplication, and division [like 23 x 37]. It's time to acknowledge that
teaching these skills to our students is not only unnecessary, but
counterproductive and downright dangerous! And it's time to proclaim that,
for many students, real mathematical power, on the one hand, and facility
with multi-digit pencil and paper computational algorithms, on the other
hand, may be mutually exclusive." ...
"Today, real people in real situations regularly put finger to button and
make critical decisions about which buttons to press, not where and how to
carry threes into hundreds columns."

1989 NCTM Standards: "This is not to suggest that valuable time should be
devoted to exercises like  (17/24) + (5/18)".

Recommendation #11  Rationale:  "… for many students the right mathematics
in HS is not the narrow pre-calculus curriculum … "  Instead teach "data
analysis, statistics, probability, discrete math, …"

Without "the narrow pre-calculus curriculum" as background, only
superficial courses in "data analysis, statistics, probability, discrete
math" and high school physics are possible.

The message delivered at a 1999 staff development session for Algebra I
teachers in Montgomery County, was summarized to me as "Do not worry about
the students understanding algebra -- Just be sure they can put anything
on their Hand calculators".


Silver Chips, the student newspaper of Blair High school, Montgomery 
county, MD 2/18/2003

State and county math standards hurt student performance
http://silverchips.mbhs.edu/inside.php?sid=2639

Changes in the Algebra I curriculum brought about by the new Maryland
High School Assessment Tests (HSAs) and a push by [Montgomery County
Public Schools] MCPS to have more middle school students take algebra
have caused many students to be seriously unprepared for higher level
math, according to numerous Blair teachers and administrators.
&
Shortly before the introduction of the HSA, MCPS mandated changes to the
Algebra I curriculum to align the course with the tested material. & "We
don't think the material is what they need to know to be successful,"
said Blair algebra lead teacher Maria Costello.

Changes in the curriculum are cited as a main cause for students'
deficiencies in basic algebra, which are manifesting themselves in
higher level math courses that require an understanding of concepts
taught in Algebra I. "Our Algebra II students are worse than ever. Our
Pre-Calculus students are worse than ever. It's falling apart as we go
up the ladder," said Costello.

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The reason that students are much less fluent in Arithmetic is that MCPS
has ordered teachers to allocate much less time to arithmetic
calculations. This is consistent with the MD state math curriculum. So
Larry washington reports that his child's then Grade 3 class in Montgomery
county allocated only 6 weeks for all multiplication from 2x3 to 23x 37
-- not enough time to learn multiplication. And Scott Wolpert
reported that his child, then in Grade 5 in Montgomery county, is struggling
with long division because the child did not learn the multiplication
table in Grade 3.

The MCPS syllabus includes solving equations like 4x = 2x + 10, but does
not specify the method. Solving via graphing is suggested. There is not
enough time to teach this two ways, both the old-fashioned
mathematicians' way as well as via graphing. [Too many topics] To teach
it the old-fashioned mathematicians' way requires students to do some
Arithmetic, of which many are not fluent. So only solving via graphing
is taught.


Relatedly, from a Homeroom column in Washington Post
http://www.washingtonpost.com/ac2/wp-dyn/A28031-2003Dec24?language=printer:

I now call the [MD Algebra] test a "pretend algebra" exam and fear that
it will undermine mathematics instruction throughout the state.

Take, for example, question No. 3:

Mary graphed the system of equations below.

y = 3/2 x + 7/2

y = -2/3 x + 7/3

Which of these best describes the relationship between the two lines?

A. They have no point in common.

B. They have one point in common.

C. They have two points in common.

D. They have infinite points in common.

This is one of the questions I thought I could use to sway Dancis [that
there is real Algebra on MD Algebra exam]. It is real algebra -- not
horribly difficult, but that doesn't concern me.

To answer the question correctly, you have to understand that each
equation represents a line and be able to plot it on a graph. You have
to understand that two intersecting straight lines have only one point
in common. It represents, in other words, a fair amount of algebraic
knowledge.

Except for one thing, which I didn't realize until Dancis pointed it out
to me. If you have your trusty graphing calculator -- which all kids
taking the exam have -- all you have to do is punch in the two equations
and see what pops up on the screen. You don't have to know anything
except how to use the calculator. &

Dancis agreed with me that the above question is a real algebra problem.
But by allowing graphing calculators to be used on the test, he said,
"They've found out how to trivialize it. This is why I call it pretend
algebra." And, he argued, if instruction focuses not on the math
required to solve the problem but on calculator tricks, that could
seriously undermine math instruction in the state.

Calculators allow kids -- and adults -- to do a lot more math than they
would without them. But calculators should supplement, not supplant,
basic mathematical knowledge. And I would like to know that kids can
graph a simple line without a graphing calculator before they graduate
from high school.

The head of math instruction for the state, Donna Watts, disagreed. "The
technology is there. It's not going to go away," she said. "There is a
limited population who can do math symbolically, the way mathematicians
do. If this is an exam for all students, we want to make it comfortable
for however students learn."

She made the analogy to cooking. "All students should be able to cook to
survive. Does that mean that all students should be able to cook like a
chef, or be able to follow a simple recipe?"

I would agree with her that the simple recipe standard is appropriate as
a requirement for all students. But we shouldn't then say that a simple
recipe means: "Remove from container and place in microwave for three
minutes." That is what we call "dumbing down" the standards.

 ><>><><><><><><><><>
 From Beware the MD Algebra Exam, [Item] 3. "The MD Algebra Test avoids
Algebra conceptual understandings, and problem solving" at
http://www.math.umd.edu/~jnd/Halloween.html:


The MD Algebra Test avoids Algebra conceptual understandings, and
problem solving.

(An expansion of my comments presented to the MD State Board of
Education on Oct. 30, 2001)

We will demonstrate this as we exam a problem from the MD HSA Algebra
sample Test.

Nice Problem. A tube of tooth paste costs 90 cents to make, and sells
for $2.50. The company has "fixed costs" (machinery or rent or
whatever). of $3000. How many tubes of toothpaste does the company need
to sell to cover/balance-out the fixed costs?

The profit on the sale of each tube is $2.50 0.90 = $1.60. Hence, the
company will need to sell 3000/1.60 = 1875 tubes. (O.K. to use a
calculator for the division only.)

This Nice Problem was not on the sample MD Algebra test; -- well not
until all the conceptual understandings, and problem solving had been
removed and after it had been rewritten in a long-winded and pretentious
manner. ( I suggest that you read the first paragraph of the problem,
then jump to the Pedagogical Analysis, below.):

Problem #32 on the sample MD Algebra test. (on the web at
http://www.mdk12.org/mspp/high_school/look_like/algebra/v32.html):
"The income (y) for a particular toothpaste company is modeled by the
equation y = 2.5 x dollars, where y is the income for selling x tubes of
toothpaste. The cost of producing toothpaste is y = 0.9 x + 3000
dollars, where y is the cost of producing x tubes.

* How many tubes of toothpaste must be sold for the income to equal the
production cost? Use mathematics to justify your answer. (If you solve
the problem graphically, use the grid provided in the Answer Book to add
to your written response.)
(Suggested graphing window 0 < x < 3000, 0 < y < 5000.)

* What is the income and production cost at the point when they are equal?

* The company makes a profit when their income is greater than their
production cost. What is the least number of toothpaste tubes the
company can sell to make a profit? Use mathematics to justify your answer."

A Pedagogical Analysis of Problem #32:

A crucial part of problem solving is "setting-up" the equations for a
"word problem". Also know as "modeling and interpreting real-world
situations". This problem does not test this skill because the equations
are provided. In sharp contrast, read the mis-claimed stated-expectation
for this problem, on the state's website.:

Expectation 1.2: "The student will model and interpret real-world
situations, using the language of mathematics and appropriate technology."
(Click, on "view Core Learning Goal, Expectation and Indicator this item
tested" on right side at
http://www.mdk12.org/mspp/high_school/look_like/algebra/v32.html )

In fact, I counted only one of the 49 problems on the sample MD Algebra
test, which actually required the student to set up the equations.

Solving simple equations both by hand and with a graphing calculator, is
an important part of real Algebra. Here the equation 2.5x = 0.9x + 3000
needs to be solved. But the students do not need to do the simple
calculations; they are encouraged to use their graphing calculators
(which provide graphs of the functions). In fact, I counted only two
problems on the sample MD Algebra test, which required students to solve
equations, none, which required students to solve equations without a
graphing calculators.

Here, the thinking part was reduced to choosing the correct "window" to
view on the graphing calculator. Even that was deemed too hard as
suggested "window" ranges are supplied.

Economists use q for quantity and c for cost. Never the cryptic x for
quantity and y for cost as in this problem. A needed skill, in setting
up a problem, is to choose names of variables that assist in
understanding the problem and the equations. But then graphing c = 0.9q
+ 3000 on a graphing calculator requires some conceptual understanding
unlike y = 0.9x + 3000 which does not.

Another solution, which received the highest possible score when graded,
is presented at
http://www.mdk12.org/mspp/high_school/look_like/algebra/anchors/a32_score_3.html.
Here the student typed the two given equations into the calculator and
had the calculator list their table of values. The student then "scolled
through the table until [the numbers for both Y's] were the same."
Precious little [Grade 6] conceptual understanding and problem solving
involved.

This avoidance of conceptual understandings, and problem solving is in
sharp contrast to the Maryland State Dept. of Education statement:
"In all mathematics content standards, the emphasis is on achieving a
balance among memorization of facts, proficiency with paper and pencil
skills, appropriate use of technology, conceptual understandings, and
problem solving". On the web at
http://www.mdk12.org/mspp/standards/math/introduction.html.

A big No-No in real Algebra is never using the same variable to mean two
different things in the same problem. This problem violates this rule,
having y representing both "income" and "cost". This type of ambiguity
often confuses students. This suggests a problem writer, with little
understanding of the very basic algebraic concept of "variables" (the
x's and y's) in algebra. Of course, problem writers, who actually
understand Algebra would require more pay for each problem. This would
reduce the profits of the profit-making, test-writing company.

The following was added to the webpage for this problem between June
2001 and March, 2002 (I informed them of this and all the errors listed
above on Oct. 30, 2001):

"The variable y is used to represent both the income for selling x tubes
of toothpaste and the production cost for x tubes of toothpaste. This is
an error in the use of a variable."