Toward Understanding and Remembering[1] How to do Hand Calculations with Fractions

by Jerome Dancis

Dept.  of  Mathematics,  University  of  Maryland, College Park,  MD  20742-4015

Telephone: 301 405 5120  FAX: 301 314 0827 e-mail:

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 rules[2]instead of memorizing a massive  number of formulas.  The basic rules are rules like "Equals plus equals are equals" and  a3 = a x aa.  

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 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."[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 [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[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 [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:

Understanding-based 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 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:  

To add  1/14 +  1/16,  one first finds the least common multiple of  14  and  16;  one way to do this is by writing out the multiple of  14  and 16, until one observes a match:
14. 24. 42, 56, 70, 84, 98, 112
16, 32, 48, 64, 80, 96, 112,               Here,  112  is the LCM.

This method is often quite impractical and unwieldy.  This method provides  6th graders with many unnecessary difficulties for adding fractions, which necessitates its reteaching in Grades 7 and 8 Arithmetic classes.

Warning from the very good Mathematics Framework for California Public Schools – [K-12] Ch 3 P. 154 []:
“The addition of fractions in terms of the least common multiple [LCM] of the denominators has struck fear in students for many generations and should never have been used for the definition of adding fractions. Finding the least common multiple [LCM] is a special skill that should be learned, but it is not how students should think of the addition of fractions.“

Finding the LCM and LCD is difficult for adding or subtracting fractions, whose denominators do not factor easily, for example, 1/169 + 1/289.

Short cut (occasionally).  To find a common multiple of  8, 4  and 16, one observes that 16 is a multiple of  8  and  4,  hence  16  is a common multiple of  8, 4  and 16 (and it is the least common multiple).  This is an example of the following rule:

Rule:  If a number is a multiple of two others, it is a (least) common multiple of the three.

This methods is rarely, if ever, used in Algebra.  Why frighten 6th graders with a method, they will not need in Algebra.

Simplest form.  Sometimes, it is useful to convert fractions to simplest form, sometimes it is not useful.  In college, answers are not required to be in simplest form.

 A book may state that final answers must be in simplest form.   This is NOT a useful requirement; it is pedantic.   It is NOT useful to “simplify”  88/100  =  22/25.   We can get a handle on  88/100;  it is almost  90%.  What is an easy way to get a handle on  22/25?  Similarly, we can get a handle on   1+ 88/100;  it is almost  190%  or  200%.  What is an easy way to get a handle on  1 + 22/25?  Only, find the least common multiple and only reduce fractions to lowest terms when useful.

Comparing fractions.  To compare (find the larger of) two fractions, say,  (1/5)  and  (2/9),  simply rewrite them with a common denominator.   For  5  and  9, a common multiple is  45,  then:
 (1/5)  =  (1/5) x (9/9)  =  9/45  while  (2/9)  =  (2/9)x(5/5)  =   10/45.  
Thus   2/9  >  1/5.

Some books teach students to use a number line to compare fractions.  This works well only for like fractions and a few other special cases.  But, without already knowing, which is the larger  (1/5)  or  (2/9),  there is no way to properly place  (1/5)  and  (2/9) together on a number line.   So, students, who try to follow the instruction to use a number line to compare say,  (1/5)  and  (2/9), will be frustrated; they have walked into a trap.

Later, when in Algebra class, the rote learning  
method has students memorizing the equation:

  +  c   =    ad + bc.  
b      d             bd 

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:

+ c  
=           ac      .
b    d             ad + bc  

    Most high school students remember how to add fractions correctly; but a sizable fraction do not.  One in five incorrectly added the fractions:

1  +  2   3    _    2  
5     10      15       20

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 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:

x  =   3 
4       2

Understanding  Method #1: "Isolate  x"  by multiplying both sides by  4.  

 4 x = 4 . 3   which simplifies to  x =  6.

    4         2  

 Method #2:  "Clear the denominators" also by multiplying both sides by  4.  

 Method #3: Obtain a common denominator by multiplying by 1 = 2/2: 

 x = 2  3  = 6.         Hence  x = 6.

 4    2  2     

 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[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 "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.

 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.  

 Method #2:  "Clear the denominators" by multiplying both sides by 4.

This yields  2x = 3x - 4, which is easily solved.

 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.[12]  

Problem 4
.  Solve for  x:      6     =   

                                            25   25   25        75

One understanding  method is to observe that the three terms on the right are identical and hence 
  +    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:

3  18    =  54  =  x
3  25        75     75

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          8  

__8__    =  __ 8__       =    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 well4:

14/8 ÷ 4  =  14 x 4  =  14  = 7         WRONG

                            8             2

Again, an understanding method is to multiply by 1=8/8:

14         8  

__8__    =  __ 8__        =     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.

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.

Word problems with fractions.  In word problems, one must be clear as to what the "base" is, that is, the fraction is being taken of precisely which number.  Sometimes, this is tricky, especially when the base is not specified explicitly.  Teachers should be trained to realize that the next problem is ambiguous, since there are two natural bases for the fraction, leading to two reasonable interpretations.

Problem 6-B. (Ambiguous)  I have two identical pizza pies, each cut into five pieces of equal size.  I eat one piece from each pie. What fraction have I eaten?

The two reasonable interpretations are:
(a) What fractional amount [of a pie] have I eaten?
(b) What fraction of the two pies have I eaten?

Yes, the wordings, of the two parts, are almost identical; training in precise reading is needed to understand the difference.

Answers.  (a)  Each piece is 1/5 of a pie.  I ate 1/5 of one pie, and 1/5 of the other pie; so the amount I ate was  1/5 + 1/5  =  2/5  of a pie.
(b)  I have eaten two of the ten identical pieces, so I have eaten  2/10  = 1/5 of the two pies.

Having a semantic discussion over an ambiguous, imprecise or unclear sentence (like "What fraction have I eaten?") is rarely useful; it is often counterproductive.  One should simply note the ambiguity and then explicitly state the "base" (here one pie or two) for the fraction.

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. ... 'Forty-five 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.

Multiply by 236.5:        236.5 centimeters  =  236.5/100 m   =  2.365 m.

2365%  makes  2.365

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:

Example 10.                                                              F     O     I      L
                                                             (a+b)(c+d) = ac + ad + bc + bd. 

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  |

|________|_____________|  _ / 

   2                   28

\______   _____________/

            \/  30

Thus:  30
x 20  =  600  =  Total area  =  Sum of areas of the 4 rectangles  =  38 + 28 x19 + 2 + 28.  
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  3/4  x  1/4.  Here is how this diagram can be used (in lower grades) to  calculate it  easily, and with understanding,

Example 11. (Multiplying "mixed numbers")
            5 3/4 x 4 1/4.

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  

3/4 x 4 1/4  = Total area  =  Sum of areas of the 4 rectangles  =  5x44x3/4 +5x1/43/4 x1/4.  
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[16].

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.  

Changing to understanding-based instruction for hand calculations with fractions is important!

Our children deserve better!

Copyright  © 2007 by Jerome Dancis

[1] Here the word "remembering" simply means  the common, positive English usage meaning of the word, not the pejorative "rote-remembering".
[2]In mathematics jargon, the basic rules are the axioms for an ordered field and the defining equations. This is also called "argueing from first principals".
[3]  Making Sense: Teaching and Learning Mathematics with Understanding (Heinemann 1977), Page 6.
[4]   Quoted from a 1998 presentation by David L. Roberts which  is essentially a summary of his doctoral dissertation. His email address is  
[5]  Paul Davis, "Teaching mathematics and Training seals", SIAM News, (March 1987) page 7.
[6] "A cognitive model for structuring an introductory programming curriculum in the proceedings of the National Computer Conference, 1974, pages 307-311.
[7]"Teaching programming: A spiral approach to syntax and semantics" in Computing and Education, Vol. 1 (1977) pp.193-197.
[8]  The basic rules used are:  b/b = 1, 1xa = a = ax1,  and the Distributive Rule, alias factoring  (x + y)z  =  xz + yz.
[9]   Beginning students might  include two additional steps which use the definition  a/b  =  a x 1/b. 
a + c  =  a d + b c  =  ad x + bc x 1  =  (ad + bc) x 1  =  ac + bd.

b    d      b d    b d              bd           bd                       bd            bd

[10]10 SATs, Third Edition (Ques. # 4, Part 6 of Test 1 given on May 2, 1987) data on Page 67.
[11] "What is the real level of our students or What do diagnostic tests really measure?",  Problem #21 unpublished, for a copy email Jerome Epstein <>.
[12]10 SATs, Third Edition (Ques. #  24, Part 6 of Test 1) data on Page 67.
[13]   According to the NCTM's Curriculum and Evaluation Standards for School Mathematics (the Standards) (1989).  On the web at
[14]   in upscale Howard county, Maryland
[15] from Linda Perlstein's  February 15, 1999, Washington Post,  front page article "Right Teacher, Wrong Class".
[16]10 SATs, Fourth Edition (Ques. # 4, Part 5 of Test 1 given on May 7, 1988) data on Page 61.