Intelligent Help at Every Step: The Impossible Mathematical Dream?

By: Professor John C. Miller, Director, Algebra Courseware Project
Department of Mathematics, The City College of CUNY, New York, NY 10031
Phone: 212-650-5148; e-mail:

Computer-based mathematics courseware can take very different forms, depending on the relative importance ascribed by its designers to various aspects and features. We will consider the following features, which we list here in alphabetical order:

We will examine how these features are implemented in currently available commercial basic algebra courseware, and also in xyAlgebra, a non-commercial courseware package produced by the Algebra Courseware Project at The City College of CUNY.

If the criterion is the effectiveness of the software in helping typical students learn algebra, then the order of importance of these features is:

  1. Step-by-step help
  2. Problem variety
  3. Course management
  4. Multi-media features.
  1. Step-by-step help is practical in no instructional medium other than computer based instruction, and is critically important for typical students who often cannot reliably emulate sample problems and extend the techniques therein to similar problems without help at various steps.
  2. A sufficient number and variety of problems is very important, as students build skills by progressing to incrementally harder problems and gradually assembling the pieces of knowledge which will eventually merge into overall mastery of larger domains of expertise.
  3. Course management features are important primarily to instructors, who are assisted in various of their administrative tasks by computer-based information on student performance and progress, although some aspects of course management do also support the instructor in providing better instructional support to students.
  4. Multi-media features, as implemented in current commercial courseware, are sometimes attractive, but usually peripheral, sometimes motivational, but only to selected students, and always time-consuming. On balance, because most college-age students are not particularly susceptible to inspiration via movie clips, and because these segments are so time-consuming, we regard multi-media features as currently implemented in commercial algebra courseware to be ineffective in helping students to learn algebra, and to be, on balance, a slightly negative factor for student progress.

Analysis of the current commercial products, Addison Wesley Longman's "Summit: Interactive Beginning Algebra" and Academic Systems Corporation's "Interactive Mathematics," suggests that the designers of these products regard multi-media features as the most important of the types of features being considered here, and that they consider problem variety and step-by-step help as the least important.

Commercial demonstrations of these products usually focus on their multi-media features, and secondarily on their course management advantages. They rarely volunteer to show actual problems as they would appear to a student.

Neither commercial program permits the student to enter step-by-step solutions to most problems. Typically, the student's response is limited to either a selection from a multiple choice list or the entry of a single expression in an "answer box." The single expression answer format has two profound limitations: (1) the programs have no way to determine the nature of the student's errors, and thus no way to offer individualize responses or assistance, and (2) the designers must make impossible decisions concerning how lenient to be in deciding exactly what answers to accept as correct. In Addison Wesley Longman's "Summit" the first linear equation in the practice module is of the form "3x - 8 = 11." If the student enters "x = 19/3" in the answer box, the program responds "Incorrect" because it expects the answer "19/3," without the variable, despite the arguable superiority of the "incorrect" answer. When Academic Systems' "Interactive Mathematics" poses the problem "Solve 3z + 7 = 8z - 3. z = [ ]," the answer box is too small to accept responses such as "2.0" or "-4/5" or "4/5," despite the likelihood that some students will seek to enter such answers. When "Interactive Mathematics" asks the student for the square of 7x + 2, it characterizes as "incorrect" the answer "(7x+2)(7x+2)," completely failing to distinguish between incomplete and incorrect answers. More important, both programs routinely send the student "off-line" to work out multiple step problems, and thus cannot offer any help to the student in finding errors or in completing a problem if the student becomes confused part way through a solution. The programs are both electronic versions of the old programmed workbook, in that they offer only a canned solution of each problem to students who need help, and they assume that students will be sufficiently competent and diligent to compare their solutions to the canned solutions and then to infer the reasons for their errors without additional program support.

The commercial programs are both limited in problem variety. In their "Practice" or "Apply" modules in the section on solving linear equations in one variable, each appears to offer about 8 problems requiring solution of such equations, possibly involving parentheses but not involving fractional, decimal or literal coefficients. "Summit" appears to vary the coefficients of these 8 problems randomly for students who try the problems a second time. "Interactive Mathematics" appears to generate the exact same 8 problems each time, and for all students. This is an insufficient range of problems, or problem forms, to permit a student to learn (1) how to add or subtract terms on each side, (2) the shortcut of transposing terms, (3) the technique of separating constant terms from terms containing the variable, (4) the technique of collecting terms, (5) the technique of dividing both sides by a non-zero constant to obtain the variable alone, and (6) how to deal with the complications which ensue when the equation also involves one or more parenthesized sub expressions on one or both sides.

By contrast, "xyAlgebra" by the Algebra Courseware Project at The City College of CUNY supports step-by-step solutions to virtually every problem, and changes its solution strategy for each problem depending on the steps entered by the student, so that it can assist the student in completing any problem by the student's preferred method. It verifies each step of each answer for the type of "equivalence" appropriate to that problem type, and immediately warns the student whenever a "non-equivalent" step is entered. The student clicks "finished" to indicate that no more steps are needed, and the program suggests appropriate additional steps if its solution algorithm finds that the solution so far is incomplete.

Where "Summit" and "Interactive Mathematics" appear to have 8 problems, or problem forms, on elementary types of linear equations, xyAlgebra has 27 such problem forms, each able to generate dozens or hundreds of numerically distinct problems. A similar disparity appears to exist in most major topics.

The verbal problems section in xyAlgebra also offers step-by-step help in (a) the selection of needed variables, (b) the construction of expressions for intermediate quantities needed to obtain suitable equations, (c) the assembly of those expressions into the requisite equations, and (d) the solution of those equations. The verbal problem solving algorithm changes its strategy based on the steps entered by the student, allowing it to assist the student to solve each problem by the method preferred by that student.

The course management module of xyAlgebra gives the instructor an overview of every student's current location in the course, total time and number of sessions worked, date of last session, and exam scores. It displays averages and medians. It allows the instructor to modify the course for individual students, and to pass messages to them. However, xyAlgebra offers no multi-media features except an optional audio reinforcement for correct responses, and thus can run on any IBM-compatible computer manufactured within the last decade.

To obtain xyAlgebra without charge, please contact the author.