Dept. of Mathematics, University of Maryland, College Park, MD 20742-4015
Telephone: 301 405 5120 FAX: 301 314 0827 e-mail: firstname.lastname@example.org
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 non-traditional, understanding-based 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 non-math courses which use Algebra.
"Dumb" mistakes do not just happen; this article will show how the traditional instructional methods set-up 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 rulesinstead of memorizing a massive number of formulas. The basic rules are rules like "Equals plus equals are equals" and a3 = 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 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 old-fashioned 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." (Emphasis added.) This describes Traditional algebra books today.
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 wrote that "Material that is meaningful [understanding-based] 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 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 [understanding-based] 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. Understanding-based 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 understanding-based 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 rote-trained 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, understanding-trained 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 skill-based 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:
Calculations with Fractions
Example 1. The understanding method for adding fractions is to obtain a common denominator by simply calculating :
a + c = a x d + bxc = ad+ bc.
b d b d b d bd
This approach is easy to learn and remember.
In contrast, there is an traditional six-step cookbook version taught in many Grade 6 Arithmetic textbooks that starts with a long-winded 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:
+ 2 + 3 _ 2
on a Math S.A.T. test .
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 error-prone 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 hand-calculations.
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.
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 "cross-multiplication" which has the numbers 4 and 2 climb up from the cellar and walk across the equal sign, as if it were a bridge.
"Cross-multiplication" 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 to a variety of classes.
Problem 3. Solve x = 3x - 1.
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 "cross-multiply" 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 Pre-Algebra class, but only half the students could solve the equation on the May 1987 Math SAT test.
Problem 4. Solve for x: 6 + 6 + 6 = x
25 25 25 75
is to observe that the three terms on the right are identical and
6 + 6 + 6 = ( 3) 6 = 18;
25 25 25 25 25
this simplifies the equation to: 18 = x .
A common denominator is obtained by multiplying by 1 = 3/3:
Thus x = 54.
In contrast, solving 18 = x
using cross multiplication, has students multiplying 75x18 = 1350 and then dividing 1350/25 = 54.
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.
understanding method is to multiply by 1=8/8, since
an easily calculated common multiple of the denominators.
14 14 8
__8__ = __ 8__ x = 14.
1 1 8 4
The logic is easily understand in contrast to the common mystical scheme:
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
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.
the understanding method, (for hand calculations) described
will result in students learning fractions with much less effort,
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 how to change centimeters to meters, he just told them to move the decimal two places -- rather than illustrating the concept. ... 'Forty-five minutes later, only three of them got it.' ".
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.Multiply by 236.5: 236.5 centimeters = 236.5/100 m = 2.365 m.
common rote learning method
for remembering how to multiply (a+b)(c+d) is
to use the mnemonic FOIL
the sum of the products of the First,
Inside and Last terms, which correctly
F O I L
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 (30-2)x(20-1):
| | | \
| 2x19 | 28x19 | 19 |
|________|_____________| \ __
| | | / 20
| 2 | 28 | 1 |
|________|_____________| _ /
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
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 understanding-based 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 (x2y4)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 a3 = aaa, we see that
(x2 y4)3 = (x2 y4)(x2 y4)(x2 y4).
This enabled the child to quickly do the calculation.
How common is this? One in four students did not realize that 23 22 = 25 on a Math S.A.T. test.
Students, with understanding-based training, who forget what to do with the exponents, can fall back on the definition a3 = aaa, and quickly calculate:
23 22 =(2x2x2) x (2x2) = 25
Skipping reasoning steps. I catalog * the use of "cross-multiplication", * 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 skipping-steps 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.
to understanding-based instruction for hand calculations with fractions
Our children deserve better!
Copyright © 2007 by Jerome Dancis
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