Dept.
of Mathematics,
University of
Maryland, College
Park, MD 207424015
Telephone:
301 405 5120 FAX: 301
314 0827 email: jdancis@math.umd.edu
This
paper presents a method for teaching calculations with fractions in a
manner
that is easily understood, easily remembered, and which will reduce the
need for memorization and homework/drill, and
will reduce students making "dumb" mistakes.
A
1999 Department of Education statistical analysis by C. Adelman, showed
that success in algebra in high school, is the best predictor for a
college
student earning a college degree. A
major difficulty in learning algebra is learning to do "symbolic"
calculations
with fractions (like adding a/b + c/d).
In
turn, a major reason for this is ineffective training
with
"arithmetic" fractions (like adding 1/3 + 1/7).
Students
being ready, willing and able to comfortably and competently calculate
with fractions, in all sorts of courses from engineering to high school
chemistry to college geography and
nutrition, should be the norm.
This
cries out for dramatic solutions for improving instruction
in fractions. The purpose
of this article is to present a case for a nontraditional,
understandingbased
approach to teaching calculating with fractions, which will result in
students
remembering how to do them. This
should increase the success rate in Algebra and the math and nonmath
courses
which use Algebra.
"Dumb"
mistakes do not just happen; this article will show how the traditional
instructional methods setup students to both make "dumb" mistakes and
forget too much over the summer. This
article also presents an instructional method which helps students
avoid
making "dumb" mistakes. College
calculus students are also plagued by "dumb" mistakes with fractions.
I
advocate teaching calculations with fractions in the simplest way,
using
only the most basic formulas, having students memorize very few
equations
and involved procedures. What
I have to say is not profound, but significant and useful.
The
reaction of several mathematicians to this article has been: "Of
course,
this is the way to teach algebra. Isn't
this the way it is taught"? Unfortunately not. (It would have save me
the
effort of writing this article.)
Our
approach has students repeatedly practicing algebraic
reasoning as
they make repeated use of basic rules^{[2]}instead
of memorizing a massive
number
of formulas. The basic rules
are rules like "Equals plus equals are equals" and a^{3}
= a x a
x a.^{ }
In
addition, students should memorize some common arithmetic synonyms
involving
fractions and decimals like 2/4
= 1/2 = 50%. (Dr. Bernice
Kasner pointed this out to
me.)
Like
many college professors, Dr. Frances Gulick, (a lecturer in mathematics
at the Univ. of MD) was already using many of the ideas expressed in
this
article. After reading this
article, she modified her teaching style. Dr. Gulick noted: I
have tried to be very careful to state the basic principles being used
in solving algebra problems and I have generally insisted that my
precalculus
students follow the guidelines [of this article]."The result of
the continual emphasis on basic concepts has been that a larger number
of students have succeeded in simplifying complex fractions, adding
fractional
expressions and solving equations correctly. It
also means that students now know what to expect when I ask questions
such
as 'what do we have to do next?' " (Emphasis added)
Hiebert
and Carpenter^{[3]} noted: "In
order to learn [mathematics] skills so that they are remembered, can be
applied when they are needed and can be adjusted to solve new
problems,
they must be learned with understanding". (Emphasis
added.) Traditional textbook
mathematics instruction basically teaches skills and calculation
procedures
without teaching understanding, without teaching when and how to use
the
skills and without teaching how to think through simple problems.
My
views are consistent with the views of mathematicians of the 19th
century: "By
1893 the mathematicians, who served on the Math Conference of the
Committee
of Ten, could not contain their disdain for the oldfashioned
arithmetic
texts. They were unimpressed by claims that these books instilled
mental
discipline. Rather, these mathematicians
saw these books as perversely designed to obscure the power of
mathematics
... . Instead of emphasizing
general principles, the traditional books wallowed in a
multiplicity
of special techniques and terminology."^{[4]}
(Emphasis added.) This describes
Traditional algebra books today^{[5]}.
The
pedagogical basis for our proposal is partially summed up by the
following
eloquent quotes (which were written about learning computer
programming,
but their statements apply equally to learning mathematics and other
subjects).
Charles
Kreitzberg and Len Swanson^{[6]} wrote
that "Material that is meaningful [understandingbased] learned may
be reformulated and used by
the learner" [that is, to attack and solve problems that are different
from the ones taught/drilled]. "Numerous
studies have shown that meaningful learning material is remembered
far
longer, recalled with less difficulty, and utilized more effectively
than
is material learned in a rote manner". (Emphasis
added.)
Similarly,
Ben Shneiderman^{[7]} wrote
"This syntactic [memorized formula] knowledge must be acquired
through
'rote learning', must be rehearsed frequently, and is subject to
forgetting." In
contrast "This semantic [understandingbased] knowledge ... is
acquired
through 'meaningful' learning, is resistant to forgetting and is
[computer] language independent." (Emphasis
added.)
There
are two extreme instructional methods for teaching hand calculations. I
advocate the first. Traditional
textbooks use the second. There
is also the extreme Reform method of not teaching hand calculations
with
fractions and doing everything on hand calculators.
#1. Understandingbased
and meaningful learning or "semantic" learning methods have students
learning
the justifications for their calculations,  why it is correct to do
or
believe something. This type
of learning emphasizes concepts (ideas) and general principles, each
useful
in many situations.
#2. Rote
learning or "syntactic" learning methods have students acquiring
memorized
formula knowledge. This type
of learning emphasizes procedures ("Just
tell me what to do") and much
special terminology and techniques (each only useful for a single
situation).
Traditional
instruction usually does not provide understandingbased
explanations of mathematics which tell the why's and the wherefores.
When,
explanations are provided, it is done quietly, and then ignored by the
textbook forever after.
Rote
learning has
students spending large amounts of time mindlessly doing dull exercises
in a rote manner. It is called "Cookbook" instruction since students
are
given recipes of steps/calculations to do. It
results in students memorizing an excessive number of formulas which
are
easily confused or garbled or forgotten. (See
Example 1 below)
When
rotetrained students do not remember a formula, they are trapped, or
they
simply "create'' a formula (See Example 1 below), often an incorrect
one,
and then proceed to calculate (or rather miscalculate). Then
the incorrect or garbled formula will be used in the calculations for
several
problems which embeds the incorrect formula in the student's memory. This
results in students collecting all sorts of misconceptions about
mathematics
and making a wide range of mistakes. Remedying
these misconceptions is difficult.
Students
often think that they know how to do a problem and are unpleasantly
surprised
when their answers are marked wrong. In
contrast, understandingtrained students will know (not merely think)
that
their answers are correct because they understood each step in their
calculations.
Problems
3 and 4 and 6 below, epitomize the fact that skillbased instruction
leaves
many students stymied when confronted with a problem that is only
mildly
different from the ones they have been programmed to do.
This
understanding based method, with its repeated use of basic rules, is the
antitheses of
the 1970s
Back
to Basics movement and the 1980s traditional approach with their
over
emphasis on rote learning and their over emphasis on memorizing too
many
specialized formulas. This
understanding based method is not the "New Math" of the 1960's or the
"Reform
Math" of some 1990's classes. It
does describe a way to teach the understanding urged by the Principles
and Standards for School Mathematics (PSSM). The
PSSM was issued in 2000 by the National Council of Teachers of
Mathematics
(NCTM) (the professional society of school mathematics teachers).
Of
course, initial instruction in fractions should be largely pictorial. This
article assumes that the students have already been taught/learned the
pictorial representations for simple fraction calculations with single
digit numerators and denominators. They
are ready to learn fraction calculations for all fractions.
Up
to now we have been vague. What,
we mean will be describe by the following examples:
Understandingbased
instruction.
Calculations
with Fractions
Example
1. The
understanding method
for adding fractions is to obtain a common denominator by simply
calculating^{[8]}
:
a + c = a
x d + bxc = ad+
bc.
b d b d b d bd
This
approach^{[9]} is
easy to learn and remember.
In
contrast, there is an traditional sixstep cookbook version taught in
many
Grade 6 Arithmetic textbooks that starts with a longwinded method of
finding
the least common multiple of the denominators:
Later,
when in Algebra class, the rote learning
method
has students memorizing the equation:
and
then just plugging into it; no
thinking or understanding required. They
use this rote method to do 20 addition of fraction exercises for
homework. In
Algebra 2, the students need to be reminded of the formula and do
another
large set of problems. The
rote method needs endless rehearsing which is boring.
My
child learned this "rote" method correctly, but not
surprisingly,
a year later, she mildly garbled the
formula as:
Most
high school students remember how to add fractions correctly; but a
sizable
fraction do not. One in five
incorrectly
added the fractions:
on
a Math S.A.T. test^{[10]}
.
Of
course, it was mostly the better educated high school students who took
the SAT exams. Instead of
accepting the fact that one
in five college bound students could not add fractions, a less
errorprone
instructional method should be tried.
Then
Connecticut could once again require students to learn how to add 1/3 + 1/7
by
hand. It is not uncommon for students in college engineering calculus
classes
to still doing calculations with fractions incorrectly. This
reduces their chances for success.
This
and the other Math SAT I problems discussed in this paper were given
back
in the 1980s when students had to do calculations by hand.
Currently,
with students using hand calculators on the Math SAT I, no such
problems
will be given. But, students
still need to learn the math concepts used for such handcalculations.
Problem
2. Solve
the equation:
Understanding Method
#1: "Isolate x" by
multiplying both sides by 4.
4
x
= 4 . 3 which
simplifies
to x = 6.
4 2
Understanding
Method
#2: "Clear the denominators"
also by multiplying both sides by 4.
Understanding
Method
#3: Obtain a common denominator by multiplying by 1 = 2/2:
x
= 2 3
=
6.
Hence x
= 6.
4
2 2
4
Understanding
Method
#4: Use fraction synonyms:
x/4
= (3/2) = 3 (1/2) = 3(2/4) = 6/4;
or
simply: x/4 = (3/2) =
6/4
Thus: x/4
= 6/4 and x =6.
These
approaches are easy to teach, learn and remember, as all they use are already learned
basic rules. This plethora
of understanding methods
exemplifies the PSSM "goal" of using a variety of methods.
In
contrast, the common method uses "crossmultiplication"
which has the numbers 4 and 2 climb
up from the cellar and walk
across the equal sign, as if it were a bridge.
"Crossmultiplication"
is a procedure that some students will forget or garble over the summer. Worse,
it trains students to violate what I have named "Epstein's
Rule":
Epstein's
Rule. It is
not all right to (or to teach students to) move numbers around in an
equation.
"Subtracting
5 from both sides of an equation"
uses the basic Rule: Equals minus equals are equals; instead of "moving
the 5 to the other side" where it magically gets transformed to 5.
Violating
Epstein's Rule is an invitation and a common reason for creative
mistakes.
The
following example was circulated by Dr.
Jerome Epstein^{[11]} to
a variety of classes.
Problem
3. Solve x
= 3x  1.
2 4
Many
students (with standard instruction) learn how to do this problem, predictably
many do not. Dr. Epstein
observed that the most common creative error made was to
incorrectly
"crossmultiply" the two terms next to the equal sign while leaving the
"1" alone, thereby obtaining the incorrect
equation:
4x
= 6x  1.
Problem
3 was not solved by any of the (mostly Grade 10) students in
the
second year of an "integrated" algebra and geometry course. It
was solved by only one in three (middle class) Grade 10 students in
Lincoln
County, Ontario, Canada. (Reported by Epstein)
This
problem was solved correctly by only one in four students in a calculus
course for business majors at Hofstra University. (Hofstra
is a private university in
Hempstead, N.Y. with generally middle class suburban students.)
The
forgetting and garbling of formulas continues even among the better
students
who advance into engineering calculus classes in college.
This
problem can be solved using three of the same understanding methods
as was used for Problem 2.
Understanding
Method
#1: "Isolate x" by
adding 1(x/2) to both sides. This
yields:
1= (3/4)x
 x/2 = [(3/4)  1/2] x
= [(3/4)
 2/4] x = x/4.
Then: 4/4
= 1 = x/4. Thus
x = 4.
Understanding
Method
#2: "Clear the denominators"
by multiplying both sides by 4.
This
yields 2x = 3x  4, which
is easily solved.
Understanding
Method
#3: "Obtain a common denominator"
again by multiplying both sides by 4.
The
next problem should be an easy one for a PreAlgebra class, but only
half the students could
solve the equation on the May 1987 Math SAT test.^{[12]}
Problem
4. Solve
for x: 6 + 6 +
6
= x
25 25 25 75
One
understanding method
is to observe that the three terms on the right are identical and
hence
6 + 6 + 6 = (
3) 6
= 18;
25 25 25
25 25
this
simplifies the equation to: 18 = x .
25 75
A common denominator is obtained by multiplying by 1 = 3/3:
Thus x
= 54.
In
contrast, solving 18 = x
25 75
using cross
multiplication, has students multiplying 75x18
= 1350 and then dividing
1350/25
= 54.
Liping
Ma reports that only 9 of her small sample of 23 US school teachers did
the following problem correctly:
Problem
5. (Dividing
a fraction by a fraction) Divide 14/8
by 1/2.
An
understanding method is to multiply by 1=8/8, since
8 is
an easily calculated common multiple of the denominators.
14
14
8
__8__ = __ 8__ x
= 14.
1
1
8
4
2
2
The
logic is easily understand in contrast to the common mystical
scheme:
"Ours
not to reason why/we just invert and multiply."
These
instructions violate Epstein's Rule.
When
dividing 14/8 by 4,
the
question is: How to "invert" 4?
Problem
6. (Dividing
a fraction by an integer) Divide 14/8
by 4.
Dr.
Frances Gulick has observed numbers of college students do this by
"inverting" the 4 incorrectly
as
well, 4:
14/8
÷ 4 = 14
x 4 = 14 =
7 WRONG
8
2
Again,
an understanding method is to multiply by 1=8/8:
14
14
8
__8__ = __ 8__ x
=
14 .
4
4
8 32
The
NCTM response, to the low level of students skill at using fractions,
had
been to prescribe decreased attention to fractions in algebra.^{[13]}
Following
the understanding method, (for hand calculations) described
herein,
will result in students learning fractions with much less effort,
thereby
allowing decreased
attention to fractions but with increased student fluency.
Calculations
with Decimals
Example
7.
Multiply 3.45 x 2.8
An
understanding method is:
3.45
x 2.8 = (345/100) x (28/10)
=
(345 x 28)/1000 = 9660/1000
= 9.660
In
contrast, the standard algorithm is:
Step
#1. Drop the decimals.
Step
#2. Multiply the integers:
345
x 28 = 9660
Step
#3. Count the decimal places
dropped: 2 + 1 = 3
Step
#4. Insert this number of
decimal places: 9.660
The
counting of the decimal places becomes a mystical rote procedure. Also,
it violates Epstein's Rule
by having the students moving decimals. Such
a specialized procedure is easily forgotten or garbled.
A
student in a Georgia high school Algebra class noted: "I know how to
change
centimeters to meters [I learned it in middle school],
just
remind me, do I move the decimal left or right?"
"When
Grant Scott, a biology teacher, had to teach (his chemistry students)
at
Howard High School^{[14]} how
to change centimeters to meters, he just told them to move the decimal
two places  rather than illustrating the concept. ... 'Fortyfive
minutes
later, only three of them got it.' ".^{[15]}
Problem
8: Change
236 centimeters to meters.
Since
100 centimeters make
a meter, just like 100 cents
make a dollar, not surprisingly 236 centimeters
make 2.36 meters, just like 236 cents make $2.36
and 236% makes 2.36.
Problem
9: Change
236.5 centimeters to meters.
Start:
100
centimeters
= 1 meter.
Divide
by 100:
1 centimeter =
1/100 meter.
Calculating Products
A
common rote learning method
for remembering how to multiply (a+b)(c+d) is
to use the mnemonic FOIL
for
the sum of the products of the First,
Outside,
Inside and Last terms, which correctly
yields:
Using
a mnemonic is mystical. This
mnemonic is no help with (a+b+c)(d
+e).
A
pictorial understanding method
for remembering how to multiply (a+b)(c+d)
employs the fact that a region's area is the sum of the areas of its
parts. Consider
this large rectangle divided into four rectangles with their areas
written
in the middle:
a
+ b
______/\____________
/
a
b \
____________________ _


 \
 ac  bc

c 
___________________
\
__


 /
c+d
 ad  bd

d 
________________ _ _
/
Thus: (a+b)(c+d)
= Total area = Sum of areas
of the 4 rectangles = ac
+ bc + ad + bd.
This
diagram easily handles the case: (a+b+c)(d+e):
_________________________
_


  \
 ae  be
 ce 
e \
_______________________
  e+d
 ad  bd  cd
 d /
_______________________
_ /
a
b
c
\__________ _____________/
\/
a + b + c
Thus: (a+b+c) (e+d)
= Total area = Sum of areas
of the 6 rectangles = ae
+ be + ce + ad + bd +cd.
Of
course drawing a diagram takes time. Drawing
20 diagrams for 20 homework exercises of the form (a+b)(c+d) would take
forever. But
doing 20 homework problems is not
necessary. Four
should be enough to commit the diagram to memory. The
quiz could consist of one problem instead of three.
Also,
if a student forgets the general rule, he/she can always fall back on
drawing
a diagram.
This
diagram can also be used (in lower grades) to quickly calculate, with
understanding, 28x19
as (302)x(201):
______________________
_


 \
 2x19  28x19 
19 
_____________________
\
__


 /
20
 2 
28

1 
_____________________
_
/
2
28
\______ _____________/
\/ 30
Thus: 30x
20 = 600 = Total area =
Sum
of areas of the 4 rectangles =
38
+ 28 x19
+ 2 + 28.
Hence: 28 x 19 =
600
 38  2  28 = 532.
Multiplying
"mixed numbers" has caused such difficulty that the
NCTM 1989 standards stated: "This is not to suggest that valuable time
should be devoted to exercises like 5 3/4
x 4 1/4. Here
is how this diagram can be used (in lower grades) to
calculate
it easily, and with understanding,
First
one rewrites it as: (5
+ 3/4)
x (4 + 1/4).
_______________________
_ NOT
TO SCALE


 \
 5x4  4x3/4 
4 \
______________________
\
__


 / 4
+ 1/4

5x1/4  3/4 x1/4
 1/4
/
_____________________ 
_ /
5
3/4
\______ _____________/
\/
5 + 3/4
Thus: 5 3/4
x 4 1/4
= Total area = Sum of areas
of the 4 rectangles = 5x4
+ 4x3/4
+5x1/4
+ 3/4 x1/4.
Hence: 5 3/4
x 4 1/4 = 20
+ 3 + (1+ 1/4)
+ 3/16
= 24 + 4/16 + 3/16
= 24 7/16.
Students,
with understandingbased training, who forget a rule for exponents, can
fall back on the definition; the next example demonstrates this.
Example
12.
My traditionally trained child asked me how to do (x^{2}y^{4})^{3}.
I
asked "what does 'cubing' mean?"
He
responded: "Just tell me what to do with the exponents; do I add them,
multiply them, subtract or divide them"?
I
persisted: Since a^{3} =
aaa, we see that
(x^{2}
y^{4})^{3}
= (x^{2}
y^{4})(x^{2}
y^{4})(x^{2}
y^{4}).
This
enabled the child to quickly do the calculation.
How
common is this? One in four
students did not realize that 2^{3} 2^{2} =
2^{5}
on
a Math S.A.T. test^{[16]}.
Students,
with understandingbased training, who forget what to do with the
exponents,
can fall back on the definition a^{3} =
aaa, and quickly calculate:
2^{3} 2^{2}
=(2x2x2)
x (2x2) = 2^{5}
Skipping
reasoning steps.
I
catalog * the use of "crossmultiplication", *
the formula for the sum of two fractions, * the "invert and multiply"
scheme,
FOIL as well as other violations of Epstein's Rule, as "skipping the
reasoning steps". Their
use makes calculations mystical
and sets students up to make "dumb" errors. The
common training in skipping the reasoning steps, also results
in
many a college calculus student incorrectly
inventing/practicing more advanced versions of skippingsteps and
creative
algebra  to their detrement.
It
is absurd that so many students have so much difficulty learning to
calculate
with fractions and decimals; this
demands better textbooks and effective methods of instruction.
As
one of my students wrote: "[The college matrix algebra course] was all
taught and organized in such a way that I know I'll be able to remember
it all for years to come, unlike most other, less sensible classes"
(emphasis added).
Calculating
with fractions can and should be taught and organized in a manner which
enables students to remember it all for years to come.
Learning
to calculate accurately with fractions (via this approach) should
significantly
reduce the prevalence of mathematics anxiety and mathematics phobias. Also,
the reduced emphasis, on memorizing special formulas/procedures, should
significantly reduce the number of students who think that they are
inadequate
as learners of mathematics because they are unable to memorize all the
formulas.
Changing
to understandingbased instruction for hand calculations with fractions
is important!
Our children deserve better!
Copyright © 2007 by Jerome Dancis