Project 3
Part 1
Owen, E., & Sweller, J. (1985). What do students learn while solving mathematics problems?. Journal
of Educational Psychology, 77, 272–284. doi:10.1037/0022-0663.77.3.272
Novice-Expert Distinctions in Problem Solving
Experts work forward from the problem givens to the problem goal. Choose equations that contain
only one unknown, thus allowing immediate calculation of that unknown. This procedure repeated
until the desired unknown was obtained. <> Novices using means-ends analysis work backward from
the goal to the givens. Choose an equation containing the goal. If a value for the goal cannot be
calculated because the equation contain other unknowns, these become subgoals, and the process
repeated until a value for an unknown can be calculated. The process can then be reversed allowing
problem solution.
Experts employ schemata. Schema: a cognitive structure allowing a problem solver to
categorize a problem and then to indicate the most appropriate moves for problems of that class. A
forward-working strategy. Novices, not having the required schemata, must use means-ends analysis
to solve the problems.
Hypothesis: the use of means-ends analysis retards the assimilation of mathematical principles as a
consequence of retarding the acquisition of schemata.
Experiment 1
Students presented with goal-modified problems would normally calculate the value of more
unknowns in each problem than would students presented with conventional problems. Groups
matched with respect to time spent on problem-solving activity.
Purpose: test the effects of reduced goal specificity on the assimilation of mathematical
principles. Principles: the trigonometric ratios of sine, cosine, and tangent. The ability to use
mathematical principles develops over some time. Possible reason: mathematical principles are
integral parts of schemata. Reducing the use of means-ends analysis by reducing goal specificity
should result in more rapid schema acquisition and increased knowledge of mathematical principles.
Primary measure: errors in the use of mathematical principles. If knowledge of problem-
solving operators develops in conjunction with schemata, then training on goal-modified problems
rather than on conventional ones should reduce errors on problems subsequent to training.
Method
Subjects: 20 10th-grade high school students who had been exposed to some trigonometry when in
the 9th grade.
Procedure. Instruction period to familiarize students with the basic principles of trigonometry. ->
Pretest to observe solution strategies and to determine their knowledge of trigonometric ratios. ->
Schema acquisition period during which each group was presented with problems of differing goal
specificity. -> Posttest using the same measures as the pretest to observe any differential changes
occurring since the schema acquisition period.
Experimental design and problem types. Two groups: the goal group was presented with
conventional problems; the no-goal group was presented with goal-modified problems. Core
Experiment 1: collection of schema acquisition problems that differed for each group with respect to
goal specificity.
Goal group: subjects were required to find the length of AD or BD. No-goal group: subjects
were required to find all the unknown sides in each of the 16 diagrams presented to the goal group.
Results and Discussion
Solution strategy on two-triangle problems: backward or forward from the verbal protocols. On
Part 1
Owen, E., & Sweller, J. (1985). What do students learn while solving mathematics problems?. Journal
of Educational Psychology, 77, 272–284. doi:10.1037/0022-0663.77.3.272
Novice-Expert Distinctions in Problem Solving
Experts work forward from the problem givens to the problem goal. Choose equations that contain
only one unknown, thus allowing immediate calculation of that unknown. This procedure repeated
until the desired unknown was obtained. <> Novices using means-ends analysis work backward from
the goal to the givens. Choose an equation containing the goal. If a value for the goal cannot be
calculated because the equation contain other unknowns, these become subgoals, and the process
repeated until a value for an unknown can be calculated. The process can then be reversed allowing
problem solution.
Experts employ schemata. Schema: a cognitive structure allowing a problem solver to
categorize a problem and then to indicate the most appropriate moves for problems of that class. A
forward-working strategy. Novices, not having the required schemata, must use means-ends analysis
to solve the problems.
Hypothesis: the use of means-ends analysis retards the assimilation of mathematical principles as a
consequence of retarding the acquisition of schemata.
Experiment 1
Students presented with goal-modified problems would normally calculate the value of more
unknowns in each problem than would students presented with conventional problems. Groups
matched with respect to time spent on problem-solving activity.
Purpose: test the effects of reduced goal specificity on the assimilation of mathematical
principles. Principles: the trigonometric ratios of sine, cosine, and tangent. The ability to use
mathematical principles develops over some time. Possible reason: mathematical principles are
integral parts of schemata. Reducing the use of means-ends analysis by reducing goal specificity
should result in more rapid schema acquisition and increased knowledge of mathematical principles.
Primary measure: errors in the use of mathematical principles. If knowledge of problem-
solving operators develops in conjunction with schemata, then training on goal-modified problems
rather than on conventional ones should reduce errors on problems subsequent to training.
Method
Subjects: 20 10th-grade high school students who had been exposed to some trigonometry when in
the 9th grade.
Procedure. Instruction period to familiarize students with the basic principles of trigonometry. ->
Pretest to observe solution strategies and to determine their knowledge of trigonometric ratios. ->
Schema acquisition period during which each group was presented with problems of differing goal
specificity. -> Posttest using the same measures as the pretest to observe any differential changes
occurring since the schema acquisition period.
Experimental design and problem types. Two groups: the goal group was presented with
conventional problems; the no-goal group was presented with goal-modified problems. Core
Experiment 1: collection of schema acquisition problems that differed for each group with respect to
goal specificity.
Goal group: subjects were required to find the length of AD or BD. No-goal group: subjects
were required to find all the unknown sides in each of the 16 diagrams presented to the goal group.
Results and Discussion
Solution strategy on two-triangle problems: backward or forward from the verbal protocols. On
