agreed or disagreed with the following
statement: “To succeed at freshman
mathematics at my college/university,
it is important to have knowledge of
and facility with basic arithmetic algorithms—for example, multiplication,
division, fractions, decimals, and
algebra (without having to rely on a
calculator).” I got 93 responses (we’re a
small community), and they all agreed
with the statement.
One ex-dean wrote, “That it is even
slightly in doubt is strong evidence of
very distorted curriculum decisions.” A
professor at the National University of
Singapore wrote, “Without such facility,
no one gets to enter the university.”
(This comment calls for a cultural
aside. Even though Johns Hopkins is
a selective university, we have no way
of knowing whether our entering students know arithmetic because both
the SAT and ACT tests allow the use of
calculators.) From Germany, “What a
question! The answer is, of course, yes!”
From Japan, “I thought it was a joke for
you to have asked our opinion about
such a self-evident truth.”
It’s Elementary
Should elementary schools stand on
their state standards and give students a
weak preparation in arithmetic over the
objections of college teachers? Wouldn’t
it make more sense to listen to the
mathematicians who, in time, will get
those students in their classrooms?
In a humble attempt to get people
to do research on the subject, I gave an
arithmetic test to my 200-plus Calculus
III students on the first day of class.
These were mostly freshmen who had
taken the AP calculus test and received
credit for a full year of calculus. There
were only 10 questions, and the results
are revealing. Of the students who
missed 3 or more questions, almost
50 percent either dropped the course
or did badly on the final exam. Of the
33 students who were clueless about
how to do the long division problem,
11 ended up on probation in later years.
The Common Core
standards put a
majority of state
standards to shame.
(There’s only a 1 in 100 chance of that
happening at random.)
Students don’t really add, subtract,
multiply, and divide a lot in advanced
mathematics courses, so why do we
mathematicians insist on this? If you’re
going to study mathematics in college,
you have to learn to study it in K– 12
—and that starts in elementary school.
Like it or not, arithmetic is the
foundation of mathematics. Studying
slides, flips, and turns year after year
(as many state standards stipulate) will
not prepare a student for college mathematics, nor will related problems appear
on the college placement test that students have to take. Students must study
arithmetic. The standard algorithms for
whole numbers are the only really big
theorems that students can be taught in
elementary school. It is deep, beautiful,
and powerful mathematics. Master these
algorithms with understanding, and
you’re ready to go.
Math standards have also failed to
recognize the importance of fractions.
Fractions are core to mathematics, yet
there isn’t enough about them in any of
the standards to criticize. Only 15 states
even mention common denominators.
The problem is that without a solid
foundation in fractions, students have
little hope of succeeding in college-level
mathematics.
Although it may seem that I’m
ignoring or skipping over high school
mathematics, it’s rather that I’m
focusing on arithmetic as the needed
foundation—and that foundation is
taught in elementary school. You see,
I’m not a “back-to-basics” person.
Rather, I’m a “build-the-foundation”
person.
The Common Core Standards:
How Do They Stack Up?
For the most part, state standards are
a thing of the past because most states
have adopted the Common Core State
Standards. So how will that change
things?
The Common Core standards put a
majority of state standards to shame.
For one thing, they set arithmetic as
a priority. Students have to memorize
some number facts and learn the
standard algorithms. For example, we
see that by the end of grade 2, students
must know from memory all the sums
of any two one-digit numbers. However,
those very same standards don’t stipulate committing the corresponding
subtractions to memory. Teachers will
have to take care of that on their own.
The same thing happens in grade 3 for
the multiplication tables—there’s no
division. What were the standards creators thinking?
In grade 3, the focus on addition and
subtraction continues. There we have,
“Fluently add and subtract within 1,000
using strategies and algorithms based
on place value, properties of operations, and/or the relationship between
addition and subtraction.” That phrase
“strategies and algorithms” is potentially a problem, though; it sounds like
an attempt to undermine the standard
algorithms.
Finally, in grade 4, we have the
proper capstone standard for adding
and subtracting whole numbers: “
Fluently add and subtract multidigit whole
numbers using the standard algorithm.”
However, doing this in 4th grade after
students have nailed down fluency with
adding and subtracting—using who-knows-what technique—is problematic
