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FOILisBAD.nb
FOIL is a BAD Idea
David Fowler, University of Nebraska-Lincoln
Ÿ The Distributive Property
For real numbers r, s, and t, the operations of multiplication and addition are combined by the Distributive Property:
r Hs + tL = r s + r t
Hs + tL r = s r + t r
It's important for students to begin thinking about the combination of these two operations. The associative and commutative
properties deal with a single operation; for example, x + y = y + x.
Note: for this discussion, the "real numbers" don't need further definition. To most students, and some teachers, the real numbers
are "the numbers."
The Distributive Property, sometimes called the "Distributive Law," is one of the initial obstacles to students trying to learn
algebra. As pre-service teachers in their first practicum experience soon find out, their students, usually in eighth, ninth, or tenth
grades (in the latter case usually repeating algebra) have difficulties of the following kinds.
1. Arithmetic weakness. Given an example, such as
3 ´ (7 + 8) = 3 ´ 7 + 3 ´ 8,
the student is unable to correctly perform either the addition or multiplication operations. The example thus has no meaning for
them.
2. Notational confusion. The use of parentheses for grouping may not be clear to the student. (This will continue to be a problem,
when parentheses indicate an ordered pair of numbers, such as (a, b) or a function f(x), or a parenthetical sentence, such as this
one.)
The different conventions of indicating multiplication; for example, by the "times sign" ´ (which on a white board looks the
same as a letter x), or by a raised dot x·y (which may confuse ELL students), or by juxtaposition xy (which is standard for
mathematics teachers but avoided by many computer programmers) result in different ways of writing a distributed expression:
r ´ (s + t) = r ´ s + r ´ t
r · (s + t) = r · s + r · t
r(s + t) = rs + rt
To an experienced college student, these three expressions mean the same thing, but to a novice, they can cause considerable
confusion.
3. Lack of conceptual understanding. The student may not grasp the meaning of "distributive" as an adjective and the connection
to "distribute" as a verb. The teacher may tell the student that "r is distributed over the expression s + t, but this adds nothing to
the student's understanding.
4. Failure to apply working memory. The student may not be sufficiently motivated to keep track of the chain of computations
involved in comparing the left and right sides of a distributive example.
My UN-L colleague Meixia Ding and co-author Xiaobao Li have published an interesting analysis of the ways Chinese and
United States textbooks introduce elementary students to the distributive property:
Ding, Meixia; Li, Xiaobao (2010) Comparative Analysis of the Distributive Property in U.S. and Chinese Elementary Mathematics Textbooks.
Cognition and Instruction, v28 n2 p146-180.
FOILisBAD.nb
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Ÿ Distributing a binomial expression
For a particular school or school district, the generalization of the distributive property to include the product of a single element
and a series of terms, such as
r (s + t + u + v) = r s + r t + r u + r v
may or may not be included in the curriculum. However, the multiplication of two binomial expressions, such as
Hr + sL Hx + yL
is an essential part of any basic algebra course. Students should understand that the distributive property can be applied iteratively. Here the expression (x + y) is distributed from the right, and then r and s are distributed from the left:
Hr + sL Hx + yL = r Hx + yL+s Hx + yL = r x + r y + s x + s y
And of course, we can initially distribute the expression (r + s) from the left:
Hr + sL Hx + yL = Hr + sL x + Hr + sL y, etc.
Again, students may have difficulty in understanding, especially if they fail to understand the distributive property for the reasons
given above. Reason 4 may particularly apply, since the number of terms is larger, and the pattern of distribution may not be
clear.
Ÿ An unfortunate acronym: F-O-I-L
Faced with the difficulty of teaching students to compute the product of two binomial expressions, some teachers fail to point out
the underlying distributive property. Instead, they apply a pseudo-algorithmic procedure identified by the acronym "FOIL". The
procedure is applied in the following fashion:
Given the expression
Hr + sL Hx + yL
1. Multiply the first terms, where "first" means the left-hand variable within each parenthesis, in this case r and x, giving the
result
r x.
2. Add the product of the outer terms, where "outer" means the left-hand variable of the first pair of terms and the right-hand
variable of the second pair, and add that to the previous result, giving the new result
r x + r y.
3. Add the product of the inner terms, where "inner" means the right-hand variable of the first pair of terms and the left-hand
variable of the second pair of terms, and add that to the previous result, giving the new result
r x + r y+ s x .
Add the product of the last terms, where "last" means the right-hand variable within each parenthesis, and add that to the previous result, giving the final result
r x + s x + r y + s y.
The combination of "first, outer, inner, last" results in the acronym. Notice that each letter in the pair of binomials has two labels;
for example, r is both "first" and "outer", while s is both "last" and "inner."
Ÿ Why do teachers promote this procedure?
1. In a few cases, this is the only procedure a teacher may know. These are cases where the teacher needs mathematical development.
2. Mathematics teachers are often desperate to make their subject "interesting," by using cute mnemonics, procedures with catchy
acronyms, or songs, rhymes, etc. to engage their students.
3. The teacher sincerely believes that the FOIL procedure helps students who would not otherwise be able to understand and
perform the calculation of multiplying two binomials.
This third reason can be generalized to include many behaviors in mathematics education at all levels. Doug Rohrer and Harold
Pashler, in a recent article in the journal Educational Researcher,
"Recent Research on Human Learning Challenges Conventional Instructional Strategies,"
raise the question: Why Are Inferior Strategies So Popular? One answer that possibly applies to this case, is that strategies
producing a lower rate of student error in the training stage are preferred by teachers and students alike, although these strategies
may be less effective in producing long-term learning. Students can quickly learn to identify the "first, outer, inner, last" letters in
a parentesized string of characters, and combine them in pairs. This gives students and teachers alike the sense that they have
mastered this objective and can move to the next topic.
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FOILisBAD.nb
This third reason can be generalized to include many behaviors in mathematics education at all levels. Doug Rohrer and Harold
Pashler, in a recent article in the journal Educational Researcher,
"Recent Research on Human Learning Challenges Conventional Instructional Strategies,"
raise the question: Why Are Inferior Strategies So Popular? One answer that possibly applies to this case, is that strategies
producing a lower rate of student error in the training stage are preferred by teachers and students alike, although these strategies
may be less effective in producing long-term learning. Students can quickly learn to identify the "first, outer, inner, last" letters in
a parentesized string of characters, and combine them in pairs. This gives students and teachers alike the sense that they have
mastered this objective and can move to the next topic.
Few teachers are in a position to test their teaching methods through any systematic methodology. When they feel that an
instructional approach is successful, their evidence is essentially short-term and not comparative.
Ÿ What's bad about FOIL?
The FOIL procedure disguises the underlying mathematics. The student does not see the distributive property, and gets the
impression that algebra is essentially a system for rearranging sequences of letters.
The FOIL procedure promotes the idea that mathematics is a collection of tricks, and that deeper understanding is not necessary.
The FOIL procedure may lead to confusion when the student is asked to square a binomial; for example, Hx + yL2 . The student
sees no "inner" term, and concludes that the answer is x2 + y2 .
The FOIL precedure prevents students from appreciating deeper patterns in mathematics, as shown in the next paragraph.
The FOIL procedure does not generalize easily. After calculating the square of a binomial expression, an important step
(although not always shown to students) is the expansion to higher powers, and the eventual connection to the "Pascal" array:
1
x+y
x2 + 2 x y + y2
x3 + 3 x2 y + 3 x y2 + y3
x4 + 4 x3 y + 6 x2 y2 + 4 x y3 + y4
x5 + 5 x4 y + 10 x3 y2 + 10 x2 y3 + 5 x y4 + y5
x6 + 6 x5 y + 15 x4 y2 + 20 x3 y3 + 15 x2 y4 + 6 x y5 + y6
Ÿ Should teachers mention FOIL?
Teachers will need to mention the FOIL procedure, because some textbooks already include it, and there is also a good chance
that some student will have heard about FOIL from some other source.
There is no harm in describing the FOIL procedure, provided the teacher first presents the correct concept of the distributive
property. The teacher should also be sure students understand the generalization of the distributive property, and the relationship
to Pascal's Triangle.
Students may then appreciate the teacher showing the FOIL procedure, as a sort of mathematical oddity.