Mathematics software for high school students. (evaluation) Ronni Geist; Harry Geist.
Having served as a college learning resources specialist and a high school mathematics teacher, when we were given the opportunity to review the packages for this article, we were eager to learn the "state of the art" of high school mathematics software.
As you will see, some of what we found was superior, some mediocre. We also learned that just as you shouldn't judge a book by its cover, you can't judge software by its packaging or its price.
One suggestion: where a variety of programs are combined into a series, look at what is included on each disk before purchasing the entire package. We found that the quality of programming may vary significantly from disk to disk; in some instances, a disk may emphasize subject matter which may not be relevant. Success With Math
CBS Software's Success with Math is a series of four mathematics tutorials designed to help students in the primary and secondary grades reinforce their math skills. For this articke, we reviewed the two packages in this series for students on the secondary level: Linear Equations for grades 7 to 11 and Quadratic Equations for grades 9 to 12. (The other packages in this series are Addition and Subtraction for grades 1 to 4 and Multiplication and Division for grades 2 to 8.) Each of the packages we reviewed came with a small, brief User's Manual which provided a concise explanation of how the program operates. Linear Equations
Linear Equations is a self-paced program designed to provide drill and practice in solving linear equations. Written by Don Ross, a former high school mathematics teacher, this package requires no intervention from teacher or parents.
Once the program is booted up, the user enters his name and is then greeted personally by the computer. (Although not indicated on the screen or in the documentation, the computer truncates the name after the first ten letters.)
Next, the general concept of the program is described: "I am going to help you practice solving equations in the form AX + B = C" and the user is asked if he would like to review the instructions. If the answer is "yes," the program displays a series of ten screens which includes a step-by-step example demonstrating how the program expects the equations to be solved.
The program presents the problems on a divided screen: the upper portion of the screen displays the equation to be solved, the middle part of the screen is the "work area," and the lower third of the screen lists the five rules to be used whe solving the equations. The values for each equation are randomly generated; however, an algorithm is used to ensure that each problem will have a whole number as its answer.
To solve the equations, a series of specific steps must be followed. First, one of the five stated rules must be chosen:
1. Add the same term to both sides
2. Substract the same term from both sides
3. Multiply both sides by the same term
4. Divide both sides by the same term
5. Simplify both sides
Since every question in this package is of the same format, the program mandates a specific sequence be observed when employing the rules. First, addition or subtraction; second, simplification. Next, multiplication or division; and finally, simplification again. Thus the program stresses a particular method of solution, and the sequence results in the equation being rewritten in the form X = N, N being the solution.
Although this rigid sequence will solve these types of equations while providing students with a set routine to follow, it does not allow the student who sees an alternate method the opportunity to pursue it.
One nice feature which should be noted is the ability to choose either addition or subtraction (e.g., you can either add -7 or subtract 7) or multiplication by the inverse of the coefficient of X or division by the coefficient of X (e.g., multiplication by 1/3 or division by 3), when solving a problem. Unfortunately, however, when multiplying by a negative fraction, the negative sign must procede the fraction. That is, -1/3 cannot be written as 1/-3, although mathematically they are equivalent and technically speaking, 1/-3 is the way the inverse of -3 would be defined.
In solving a problem, if the student makes a procedural error, the program indicates the "preferred" rule to be chosen and then asks him to choose a rule again. Should the student make a computational error, the computer also provides an explanation indicating what value should have been chosen and then offers another chance to enter the value.
If an error is made in simplification, the computer displays an explanation along with the correct answer and then requests that this answer then be entered by the student. On an error like this (which might in fact be a typographical error rather than a ture mathematical error) we would have like to have seen the computer give the user a second chance to enter the correct value before displaying it. Unfortunately, this program has no provisions for stopping a student from committing the same error over and over again, nor does it recommend that the student seek outside assistance (teacher, parent, tutor) after numerous error have occurred.
The program keeps track of both procedural errors--choosing a rule which the program considers to be incorrect-- and computational errors--performing calculations incorrectly. After a problem is completed, a tally of the number of errors for that problem is displayed along with a cute message (e.g., "May the force be with you.")
Once the user has completed the number of equations he has requested, the program terminates, and he is left in Applesoft. To attempt additional problems, he must return the program. This, of course, means that no cumulative statistics are maintained.
Linear Equations provides good drill and practice in solving one specific type of linear equation. The program encourages students to follow a fixed order of operations and reinforces these procedures. The random generation of equations provides a variety of problems in this format and offers immediate guidance and feedback when an error is made.
It should be noted, however, that in the study of algebra, linear equations can take other forms (e.g., AX + B = CX + D or C = AX + B). Also, high school mathematics problems do not always have whole numbers as answers; fractions and decimals are a reality of life.
While this program accomplishes its stated goals effectively, its area of concentration is only a small part of the study of linear equations in today's mathematics curriculum. Quadratic Equations
The highest level package in the CBS Software Success with Math series is Quadratic Equations, a comprehensive, self-paced tutorial similar in structure to Linear Equations. This package, written by the same author, develops skills in solving quadratic equations in the form AX.sup.2 + BX + C = 0. As with the previous program, a series of instruction screens is available, if desired, when the program is first started. This program offers the user a choice between equations which are "Easier" and "More Difficult." The difference between these problems is that in the easier equations, the coefficient of X.sup.2 is always 1, whereas in the more difficult equations, the coefficient is either 2 or 3.
As in the previous program, a specific sequence of rules must be followed: first, whenever possible, the user must divide both sides of the equation by a common factor. Following this, the left side of the equation must be factored and then each of the factors set to zero. The two resulting linear equations are then solved for X, yielding the solutions to the original equation.
The procedure for solving quadratic equations is virtually identical to that used in Linear Equations. Unlike Linear Equations, however, if the user factors the equation improperly, the program offers a second chance. Should he fail again, the program branches to a detailed description of how to factor the left side of the equation. This is a very useful feature and, in our minds, makes this package superior to Linear Equations. It should be noted, however, that repeated computational or procedural errors are allowed to continue endlessly, without recommendations to seek outside help.
Overall, this package meets its stated objectives; however, again, there are other forms of quadratic equations which are not covered here, including non-factorable quadratic equations, quadratic equations with factors with a coefficient of X.sup.2 larger than 3, and quadratic equations which are not in the standard format. Algebra Series
Microcomputer Workshops produces a series of six computer programs on varying topics of high school algebra. This Algebra Series includes disks entitled Equations, Solving Quadratic Equations, Simultaneous Linear Equations, Factoring Algebraic Expressions, Binomial Multiplication, and Graphing Linear Functions. The first two programs in the series, Equation and Solving Quadratic Equations, are virtually identical to the package distributed by CBS Software, except for very minor differences in format. (These packages were written by the same author who wrote the CBS Software.)
Although not as slickly packaged as the CBS Software product, the documentation with the packages from Microcomputer Workshops is more explicit and includes objectives, a program description, and a demonstration problem, although we did find a couple of typographical errors in their photocopied pages.
All of the programs in this series follow the structure described in the previous section. Simultaneous Linear Equations
In the Simultaneous Linear Equations program, the student is presented with two equations, each in the form AX + BY = C and asked to solve for X and Y. All of the coefficients are integers and all of the solutions are integers which fall into the range of -5 to +5. To solve the equation, the user must first transform one or both of the equations so that the coefficients of one of the variables are equal or opposite. Then the equations must be added or subtracted to eliminate one of the variables. The resulting equation must be solved for the remaining variable and that value substituted in either of the original equations to determine the value of the other variable.
Once the user has solved the equations, the program performs checks in both equations to show that the values are correct. In addition, the two equations are displayed as line graphs showing that the solution is the point of intersection of the two lines. This is an excellent feature of the program. An error tally is maintained for the following types of errors: eliminating a variable, solving linear equations, and computational errors. Factoring Algebraic Expressions
The Factoring Algebraic Expressions program offers students practice in factoring algebraic linear and quadratic expressions. The user is given the choice of five different types of problems. Easy Quadratic Trinomials, Hard Quadratic Trinomials, Difference of Two Squares, Common Factor, or a Mixture of the above types.
This program offers the user the option of assistance if he is unsure of what to do. For example, in factoring a trinomial, by typing H for HELP, the user can test several factors without having the attempts count against him. The program also displays the product of the factors so that the user can compare his work with the original expression.
If the user becomes totally lost, typing S for SOLUTION initiates a routine by which the computer leads him step-by-step through the factoring process. Once the computer completes this explanation, the user is returned to the original problem where he is required to complete it on his own.
These features are extremely useful and present the student with a clear explanation of how to solve these problems without having to seek outside assistance or refer to a textbook. This helps make this program completely self-contained, and would have been a welcome feature to the other programs reviewed thus far. Binomial Multiplication
The first thing we noticed about the Binomial Multiplication package was that the lettering appears on the screen in both upper- and lowercase, instead of all capitals. This was a welcome relief to our eyes! Along with the standard set of instructions, this package includes a short lesson on how to perform Binomial Multiplication using the FOIL method. (FOIL is a mnemonic which stands for First, Outside, Inside, Last, and describes the standard way in which this skill is taught.)
The descriptive lesson effectively uses the high-resolution screen and employs arrows, circles, and rudimentary animation to illustrate how the FOIL method is performed and to explain why it works. This explanation is well-constructed; it is easy to follow and clearly written and could certainly serve as a student's first introduction to this topic.
The package also provides drill and practice exercises using the FOIL method. The user is led through the problems step-by-step. Should an error be made, the program allows for a second chance before offering correction and guidance in solving the problem. In this package, an error summary of Multiplying Terms and Combining Like Terms is maintained by the computer. Graphing Linear Functions
The final package in this series, Graphing Linear Functions, gives the student practice in solving equations and locating and plotting points which will appear on the graph of the equation. In this program, equations can appear either in the form Y = MX + B or in another form which must be solved for Y.
Once the equation is in the required format, it is necessary to locate three points which satisfy the equation. To do this, values for X must be chosen so the computer can calculate the corresponding value of Y. All values must be within the range of -10 to +10. Also, only whole numbers may be used; this is an unfortunate restriction, since it is possible to graph points with fractional coordinates.
After three acceptable points have been found, a 20 x 20 coordinate grid is drawn on the screen, and the user must plot each of the points. A flashing X-cursor, centered on point (0,0) appears and can be moved using the I, J, K, M keys. Pressing the RETURN key plots a point on the graph. If the cursor is not in the proper locatin when RETURN is pressed, the computer provides another chance to plot the point and offers assistance in the form of either Sound or Silent Clues. The Sound Clue is a beep whose pitch becomes higher as the cursor approaches the correct location; the Silent Clue is a screen display of the coordinates of the point where the cursor is located. Both of these are very innovative and useful features.
After all three points have been plotted, the X and Y axes are redrawn with tick marks instead of grid lines, and the three points and the line connecting them are drawn. In this program, the following types of errors are automatically tallied by the computer: Solving for a Variable, Selecting Points, Plotting Points, and Calculation Errors.
This package is very well designed, again using the hi-res graphics screen for lowercase video and smaller numbers when fractions appear in the equations. As in the other packages, a strick order for solving and graphing the equations is required; in this particular package, this structure works quite well. This one package teaches many skills: solving equations, selecting points, plotting points, and graphing the equation. Graphing Linear Equations is a very useful learning tool.
The complete Algebra Series of six disks is available only for the Apple; the first two disks in the series, Equations and Solving Quadratic Equations are also available for the IBM PC, TRS-80, Commodore 64/Pet, and Atari computers. Master Math
Master Math, a six-disk package from PMI Incorporated, is a mixed bag of mathematical topics. This series can be purchased as a complete set, or specific disks may be purchased individually. The documentation with this package claims that "Master Math contains very program you will ever need to get you, your pupils, or your son or daughter through a high school math course;" however, we found this package to fall short of its promise. Of the six disks in the package, only the first three offer instruction and tutorials on various math skills; the last three disks provide a series of test problems which encompass a wide range of math topics.
Although PMI is a Maine-based company, this series was apparently written by an Englishman. As a result, some of the terminology differs from that which is normally taught in American schools (e.g., the American trapezoid is referred to as a trapezium and the American parallelopiped is the British cuboid), and some of the written symbols are not those commonly used here in the States (e.g., set signs appear as parentheses instead of "curly brackets" or braces). These factors could be a source of confusion to students. Numbers, Logs, and Antilogs
The first disk in this series focuses on Numbers, Logs, and Antilogs. This disk begins with an extremely basic concept (counting the number of digits to the left and right of a decimal point in a number) and goes on to introduce the concept of logarithms and the use of a table to determine the logarithm of a number. This is followed by a discussion of antilogarithms and how to use the antilog table, and then the package teaches how to use logarithms in calculations.
We found several problems with this disk: First, the examples given on the screen use a logarithm table that is not standard for American students--nor is it the same table included in the documentation! This is unfortunate and could serve as yet another source of confusion for students trying to learn these skills.
Second, the disk we tested did not appear to work properly. Although the documentation indicates that there are lessons on powers and roots on the disk, we were unable to access these lessons.
Finally, the major emphasis on this disk, calculating with logarithms, is a topic which was declined in importance in the modern day math class, with the advent of calculators and computers. While logarithms are still taught today, the emphasis is on theory, rather than calculation. Modern Algebra And Set Theory
The second disk focuses on Modern Algebra and Set Theory. On the first part of the disk, explanations of the various set functions (Union, Intersection, Complement, and so forth) are given in text, without graphic representation. In addition, several of the notations used to indicate these functions, such as the intersection of two sets (*), subset (*), and the empty set (*), do not appear in the screen explanations. Instead, the user is instructed to refer back to the documentation to determine what these symbols look like. This, we feel, is a poor practice.
Units on the second half of the disk use hi-res graphics to demonstrate these same functions visually. We found this to be a distinct improvement; however, it would have been far more effective to integrate these graphic representations with the textual descriptions.
On the first two disks, an attempt is made to motivate students by rewarding correct answers with a series of games which can be played only if a correct answer is given to a problem. While the concept of reward is a good one, the games supplied are extremely primitive and would not, we believe, be especially appealing to the level of student for whom this package is designed. Areas and Volumes
The best part of this series is the third disk, which deals with Areas and Volumes. Using the hi-res graphics screen, through the use of extremely well-designed geometric figures, the four lessons on this disk take the student through the formulae for determining the area and volume of simple geometric shapes (square, rectangle, triangle) to more complex shapes (parallelogram, trapezoid, and circle) and then to arcs, spheres, cylinders, prisms, parallelopipeds, pyramids, and cones.
Each lesson on the disk has an accompanying review test, and the computer keeps track of the user's performance. Another nice feature on this disk is that problems are not programmed to come out evenly; as areas and volumes are figured, the user needs a pencil and paper for the computations, and he must perform various calculations to derive the correct answers.
Disks four through six in this series consist of examination problems on a variety of topics. Disk four includes such topics as: factoring algebraic expressions (incorrectly referred to as equations in the documentation), simple and compound interest, statistics, trigonometry, differentation, perentages, number bases, exponents, and profit and loss calculations.
Topics on disk five include: simplification of algebraic expressions, ratio and proportions, properties of polygons, graphing linear and quadratic equations, statistics, solving algebraic equations, vectors, exponents, geometry, matrices, number bases, and solving simultaneous equations.
Disk six includes test problems on: geometry of a straight line and triangle, lowest common multiple, mapping functions, calculations with fractions, congruent triangles, algebraic functions, and curency conversions. The documentation for all of these disks includes brief descriptions of each of these topics and, for some topics, supplies useful diagrams which do not appear on the screen.
The test programs on these three disks all follow the same basic format. The user is asked a question on the given topic. If an incorrect answer is entered onthe first try, a hint is provided as assistance. If an incorrect answer is entered again on the second try, a brief explanation of the method for solving the problem is given along with the correct answer. Scores are displayed after each group of five questions is answered; however, no cumulative tally is maintained.
These three disks are useful for drill and practice, but they assume that the user already has a firm grasp on the subject matter being tested. CIRCLE 418 ON READER SERVICE CARD Algebra Arcade
In our opinion, the best package of those we reviewed is Algebra Arcade, a new program from Wadsworth Electronic Publishing Company. This program was created by four mathematics professors under a National Science Foundation grant to develop computer programs for use in the classroom and illustrates how the computer can be creatively programmed to help make learning less tedious and more fun. The package combines the mathematical skills of graphing equations in algebra with the excitement of an arcade game.
Algebra Arcade is designed to teach students to plot equations on a coordinate system. The user can choose which type of equation "family" to practice: lines, quadratic equations, third degree equations, or equations, involving the sine, cosine, tangent, exponent, logarithm, integer, absolute value, and arctangent functions. The user builds the equation he wants, and the computer graphs the equation on the display screen.
What distinguishes this program from those previously described is the way in which the topic is presented. After being shown a set of coordinates, a group of creatures called "algebroids" scurry onto the screen, scattering to various locations on the grid.
It is the user's task to come up with an equations which, when plotted, will pass through as many algebroids as possible. After the graph is plotted, a "whirlwind" follows the path of the graph, "knocking off" any algebroids through which it passes. Points are earned for each algebroid hit: 200 for the first, 400 for the second, and so on. However, the graph plotted must not hit the "shifty-eyed ghost" who also appears on the grid; if it does, the ghost turns into the dreaded Graph Gobbler who munches the equation, leaving the player scoreless.
The game can be played alone, or one-on-one. If, when playing against an opponent, the Graph Gobbler appears, the offending player is sent to "The Committee" which decides, randomly, if he should be penalized one turn, three turns, of if he can get off without losing a turn.
Each player (in both the one- or two-player games) starts off with ten turns; an extra turn is added by scoring 10,000 points, clearing all tne algebroids, or by finding the invisible Graph Gobbler. (Oh yes, we forgot to mention that when the tenth algebroid is eliminated, the Graph Gobbler "has a fit and disappears.")
At that time, the player has up to three tries to find the invisible Graph Gobbler by graphing equations. If one graph finds the hiding place, the player scores an additional 1000 points and earns an extra turn. But, if the player misses after all three attempts, he is sent to The Committee. When both players have used up all their turns, the one with the highest score is named the Algebra Arcade champion.
The program has several features which make it more enjoyable and challenging. It can be played either under keyboard control or by using a joystick; this is the first questions asked after the program is loaded. The program includes a Features Menu which can be used to change the characteristics of the game. Specifically, this menu includes the following options:
* Saving the current game on disk to continue it a later time.
* Changing the coordinates of the grid: They are preset at -5 to the left and bottom and +5 to the right and top. They can be reset to any values, seven characters or less in length, which may include a decimal point or [pi].
* Selecting certain equation families. The families include: the linear equation Y = MX + B, three different representations of a quadratic equation, a third degree equation, and an equation containing a sine function. An additional option allows you to build your own equation, using allowable symbols and functions which appear in a table on the display screen.
* Playing the game with or without the Graph Gobbler.
* Testing the length of a graph, to help determine the use of long, oscillating graphs which could make the game quite easy (although finding such graphs is no easy feat). This length test can be turned on or off.
* Adding an internal timer to encourage quick thinking: if this option is chosen, the regular score is multipled by a factor, dependent upon how quickly the equation was entered. The timer can be set to several speeds, or not used. If it is used, the algebroids change positions on the screen between turns. In an untimed game, algebroids not eliminated return to the same position at which they appeared during the previous turn.
If desired, these features can be changed as the game progresses. The options of coordinates and types of equations allow for a wide range of difficulty. High school freshmen can use this program to learn to plot straight lines, while seniors and even college students could use it to study the characteristics of higher degree, more complex equations. Thus, the package can be used by a student over a period of years, as his knowledge of mathematical concepts increases.
An additional option of the program is the use of a Practice Field on which equations can be tested before they are actually plotted on the playing field. In this way, a player who is just learning these skills can specify several different equations to be plotted on the practice field. By choosing values wisely, the player can gain valuable knowledge as to how various equations appear on the grid, so that when he chooses an equation to be plotted on the playing field, the number of points scored can be maximized.
The package includes a 26-page User's Guide that is professionally written and profusely illustrated. Included in the manual are: a quick description of the program, containing enough brief instructions to start a new user at play immediately; a detailed set of instructions, clearly describing all of the various options available to the user (including descriptions of how to enter equations, allowable symbols, and how the game is scored); suggestions on graphing equations and sample equations to try on the practice field; playing tips; and illustrations of some "interesting equations." Finally, there is a brief list of references and a bibliography for further study.
This package demostrates what good educational software can and should be: creative, enjoyable, and a valuable learning experience. We recommend Algebra Arcade highly. CIRCLE 419 ON READER SERVICE CARD
Products: Success with Math (computer program)
Algebra Series (computer program)
Master Math (computer program)
Algebra Arcade (computer program)