剑桥学习科学手册英文版.pdf(2)

perceptual cues with new internal goals
and/or external actions. Examples of English
versions of production rules are shown in
Table 5.1.
Production rules characterize how both
advanced and beginning students think or
reason in a domain. Students may acquire
informal, heuristic, or incorrect patterns of
thinking that are different from the con-
cepts and rules that are normatively taught
or presented in textbooks. Learning sci-
ences researchers have identified “informal”
or “intuitive” forms of thinking that students
may learn implicitly and outside of school
(cf. Lave, Murtaugh, & de la Rocha, 1984;
Resnick, 1987). Production rules can rep-
resent such thinking patterns as illustrated
by production #1 in Table 5.1, which repre-
sents an informal alternative to the formal
approach of using algebraic equations like
“8x = 40” (cf., Koedinger & Nathan, 2004).
Production rules can also represent heuris-
tic methods for discovering approaches to
solutions (Polya, 1957). Production #2 in
Table 5.1 does not suggest any particular
operation per se, but characterizes how a
good problem solver may think through a
plan of action before selecting a particu-
lar operation or theorem to apply. Non-
traditional strategies can be represented in
production rules, as illustrated by #3 in
Table 5.1, which characterizes the use of a
graphical rather than symbolic strategy for
solving an equation.
The if-part of a production rule can
help identify when the knowledge students
acquire is not at the right level of general-
ity. For instance, production #4 in Table 5.1
is too specific – it shows students how to
combine like terms in an equation when
coefficients are present (e.g., 2x + 3x → 5x)
but not when a coefficient is missing (e.g.,
x – 0.2x). Alternatively, students sometimes
acquire productions that are too general.
Production #3 in Table 5.1 represents how
students may learn to combine numbers by
the operator between them (e.g., 2*3 + 4 =
x → 6 + 4 = x) without acquiring knowl-
edge that prevents order of operations errors
(e.g., x*3 + 4 = 10 → x*7 = 10).
cognitive tutors 65
Table 5.1. Example Production Rules
Production Rules in English Example of its Application
1. Correct production possibly acquired implicitly
IF the goal is to find the value of quantity Q and
Q divided by Num1 is Num2 THEN find Q by
multiplying Num1 and Num2.
To solve “You have some money that you
divide evenly among 8 people and each
gets 40” find the original amount of money
by multiplying 8 and 40.
2. Correct production that does heuristic planning
IF the goal is to prove two triangles congruent
and the triangles share a side THEN check for
other corresponding sides or angles that may
congruent.
Try to prove triangles ABC and DBC are
congruent by checking whether any of the
corresponding angles, like BCA and BCD,
or any of the corresponding sides, like AB
and DB, are congruent.
3. Correct production for a nontraditional strategy
IF the goal is to solve an equation in x THEN
graph the left and right sides of the equation and
find the intersection point(s).
Solve equation sin x = x 2 by graphing both
sin x and x 2 and finding where the lines
cross.
4. Correct but overly specific production
IF “ax + bx” appears in an expression and
c = a + b THEN replace it with “cx”
Works for “2x + 3x” but not for “x +3x”
5. Incorrect, overly general production
IF “Num1 + Num2” appears in an expression
THEN replace it with the sum
Leads to order of opera

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