Mona Morningstar

ANALYSIS OF RESPONSE DATA

This chapter presents the analysis of the group response data collected during 1966–67 and 1967–68 at the Stanford–Brentwood GAI Laboratory. The purpose of the analyses is to determine how the structural features of the problems affected student performance. The data include the proportion of students correct on the first response for each problem, the mean latency to a correct first response for each problem, and the number of students who responded to each problem. For the 1966–67 data, two series of linear regression analyses are reported; one uses a monotonic function of mean proportion correct on first response as the dependent variable and the other uses mean latency. A descriptive analysis of the mean proportion correct on groups of items from both the 1966–67 and the 1967–68 curriculums is also reported. This detailed descriptive analysis was made to obtain a more qualitative understanding of the factors affecting performance of children on the items under consideration. The regression analyses were used as guidelines to identify the relevant factors for descriptive purposes.

CURRICULUM AND STUDENTS

This chapter presents a description of the curriculum and the students participating in the Stanford mathematics programs in computer-assisted instruction during 1966–68. The curriculum material for each of Grades 1 through 6 was arranged sequentially in blocks to coincide approximately with the development of mathematical concepts introduced in several text series. There were 23 concept blocks in Grade 1 and 23 concept blocks for each of Grades 2 through 6. Each concept block included a pretest, five days of drill, a posttest, and sets of review drills and review posttests. Addition was first introduced in Block 103, Grade 1, with problems in horizontal format in both canonical and noncanonical form. Addition in a vertical format first appeared in Block 104. Subtraction problems in horizontal and vertical format with both canonical and noncanonical form were first presented in Block 105, Grade 1. Fractions were introduced for the first time in Block 215 in the second grade. The students were shown a matrix of letters and either were asked how many parts the matrix contained or how many letters were in a given part of the matrix. The first block of multiplication in Grade 2 (Block 220) and Grade 3 (Block 312) contained canonical and noncanonical problems in a horizontal format with a maximum product of 9. The second block of multiplication in Grade 3 (Block 3l8) and the first block in Grade 4 (Block 407) were similar with horizontal format and a maximum product of 8l.

AUTOMATON MODELS OF STUDENT PERFORMANCE

This chapter discusses several theoretical models of the performance of students engaged in solving arithmetic problems and the relationships these models bear to data obtained from the recorded responses of California school children who used the IMSSS blocks mathematics curriculum program during the 1967–68. The models are drawn from a general class of mathematical structures in the theory of automata and thus are termed automaton models. Using the work of Suppes as the starting point for the formulation, one shall assess the predictive abilities and the ranges of application of the various automaton models by comparing them to the recorded data from six concept blocks: two from addition, two from subtraction, and two from multiplication. The purpose of formulating and testing such models was to determine whether their structures and predictions could be used to improve the responsiveness of CAI systems to the difficulties each student individually experiences. The automaton formulation has been chosen because concise models of student performance from an information-processing standpoint had to be constructed.

ANALYSIS OF INDIVIDUAL STUDENT DATA

This chapter discusses several models of individual performance in the first-grade mathematics curriculum. Regression techniques were used to examine two basic types of models: a temporal model in which the prediction of an individuals performance for a given block of lessons was based on his performance for the immediately preceding blocks, and a conceptual model in which the prediction was based on performance in previous blocks of lessons of the same concept. Two performance measures were used: the proportion of problems the student answered correctly on the first response and the students average response latency to the first response. The chapter presents the studies of models of the sort that are of interest for several reasons. The models studied in this chapter concentrate on data from individual students. Second, there has been a tendency in the recent educational literature to claim that social and economic variables are more important in predicting student behavior than academic variables. Third, by using models of the type studied, one can ask questions of the kind that have become fashionable in discussions of cognitive style—questions about whether student variation is greater than curriculum variation.

CURRICULUM AND OPERATION OF THE LABORATORY

This chapter describes the curriculum and the operation of the laboratory made for the tutorial program in elementary mathematics that was developed and tested at Brentwood Elementary School in East Palo Alto in 1966. The lessons in the mathematics program covered the ordinary range of arithmetic topics, as well as a few topics less commonly found in first and second grades. The content and scope of the curriculum were drawn largely from Suppes with the addition of some topics, such as oral story problems, which cannot by their nature be adapted to a textbook format. As the programmed lessons were tutorial, many of the lessons were explanatory, relying on oral explanations synchronized with changing visual displays. The lessons were short, and explanations were simple and direct. Generally, the problems within one lesson were all of the same type; the first few were accompanied by explanatory audio messages, leaving the remainder as practice problems. Both explanatory and practice problems contained provisional audio messages that were heard only by the students who responded incorrectly or who failed to respond within a reasonable time.

Chapter 3 – REGRESSION MODELS AND RESPONSE DATA FOR 1966–67

In analyzing the response data for Stanford mathematics programs in computer-assisted instruction in 1966–67, the factors are identified that contribute to the difficulty of arithmetic problems, understands how students learn elementary arithmetic and why they have learning difficulties. This chapter presents not just the final results of the analyses done, but also the reasoning processes and the failures encountered in obtaining the final results. The complexity of the learning process increases the necessity for detailed exposition if the findings are to contribute to an adequate theoretical discourse on mathematics learning. The methods of analysis and results, especially the use of linear regression models, extend the earlier work of Suppes, Hyman, and Jerman and Suppes, Jerman, and Brian. Typical factors that were examined were the magnitude of the largest number in the problem, the number of times the sum of a column exceeded nine and the form of the equation in which the problem was presented.