Chapter Five
Computers and Calculators In Mathematics Education
I Was Just Thinking...
Take any threedigit number and write it down twice to make a sixdigit number (i.e., 123123). Use a calculator to show that the sixdigit number is divisible by seven without a remainder. Then show that the sixdigit number is divisible by eleven without remainder. Finally, show that the sixdigit number is divisible by thirteen without a remainder. Can you explain why this works?
NCTM Standards On Computers and Calculators
... some aspects of doing mathematics have changed in the last decade. The computer’s ability to process large sets of information has made quantification and the logical analysis of information possible in such areas as business, economics, linguistics, biology, medicine, and sociology. Change has been particularly great in the social and life sciences. In fact, quantitative techniques have permeated almost all intellectual disciplines. However, the fundamental mathematical ideas needed in these areas are not necessarily those studied in the traditional algebrageometryprecalculuscalculus sequence, a sequence designed with engineering and physical science applications in mind. Because mathematics is a foundation discipline for other disciplines and grows in direct proportion to its utility, we believe that the curriculum for all students must provide opportunities to develop an understanding of mathematical models, structures, and simulations applicable to many disciplines...Because technology is changing mathematics and its uses, we believe that–
• appropriate calculators should be available to students at all times;
• a computer should be available in every classroom for demonstration purposes;
• every student should have access to a computer for individual and group work;
• students should learn to use the computer as a tool for processing information and performing calculations to investigate and solve problems.
(NCTM Standards, pp. 78)
____
The major topics addressed in Chapter 5 include:
• Uses of calculators in elementary mathematics instruction.
• Criteria for selecting classroom calculators.
• Instructional applications of computers.
• Characteristics of good educational software.
• How calculators and computers can be used by students to solve problems.
ENIAC, the first commercial digital computer, was developed in 1945 at the University of Pennsylvania as a direct outcome of the second world war. During the next two decades, the applications of computers in teaching and learning were explored. Due to the expense of computers at that time, these studies had little effect on classroom practice.
The current revolution in education computing is a by product of the "space race" that made available low cost, miniaturized electronic components. The first microprocessor, a computer on a chip, was developed in 1971. Since then, significant advances in microcircuits and information storage has produced inexpensive desktop computers that rival the capabilities of the largest computer in the 1971.
Over onemillion personal computers are currently being used in elementary schools throughout the United States and most provide sets of calculators for student use. In 1990, the "typical" elementary school had about 20 personal computers for instructional use (Becker, 1990). Although approximately twothirds of the total computer time is devoted to basic skills instruction, students are increasingly using them as "tools" to analyze data and word process. Inexpensive, handheld programmable calculators are used in secondary classrooms to construct graphs and evaluate equations. By the start of the next century, we will see computers influencing our society as profoundly as the cultural revolution caused the invention of the Gutenberg printing press 400 years ago.
Instructional Uses of Calculators
The National Council of Teacher of Mathematics (NCTM) Standards (1989) recommended that programs at all levels take full advantage of calculators and computers in mathematics instruction. Based on a long series of research studies, NCTM issued a detailed Position Statement on Calculators in the Mathematics Classroom (1987). The report recommends that all students have an appropriate calculator for mathematics learning and that calculators be integrated into classwork, homework, and assessment. Ready access to calculators in school would reduce the amount of time spent on computation drills. This would allow increased emphasis on problem solving, mathematical reasoning, and applications.
Table 5.1 lists five important instructional uses for calculators and research studies which support each (NCTM, 1987).
Table 5.1
Instructional Uses of Calculators
Instructional Use 
Research Support 
Concentrate on the problemsolving process rather than on the calculations associated with problems. 
Bitter and Hatfield (1992) found that the integration of calculators into the middleschool curriculum improved problem solving performance, especially for girls. (Also see Bartalo, 1983; Comstock & Demana, 1987.) 
Gain access to mathematics beyond the students’ level of computational skills. 
Allinger (1985) found that lowachieving students participated more effectively in mathematics class with access to a calculator. (Also see Suydam, 1982 & VanDevender & Rice, 1984.) 
Explore, develop, and reinforce concepts including estimation, computation, approximation, and number properties. 
Dossey (1990) found that students can use "fast pencils" to more effectively develop mathematics concepts and procedures. (Also see Hope & Sherrill, 1987.) 
Experiment with mathematical ideas and discover patterns. 
Usiskin (1978) argues that the calculator can be an used effectively in exploratory activities and group investigations. (Also see Bell, 1978.) 
Perform tedious computations that arise when working with real data in problemsolving situations. 
Dick (1988) found that regular use of a calculator helps students judge the reasonableness of a solution. (Also see Suydam, 1982.) 
Several states have taken the position that calculators should be available for instruction and testing of all children in grades K12 (California State Department of Education, 1992; Carter & Leinwand, 1987). Calculators have been employed to improve students’ understanding of basic concepts, provide additional skill practice, and support real problem solving in the classroom.
Learning to calculate accurately is an important part of growing up in our complex society. Many events in everyday life involve working with various kinds of numbers. Counting, making change, comparison shopping, figuring sales tax, filing income taxes, doing carpentry, and working in the garden all require an ability to manipulate numbers sensibly. It is not only necessary to estimate approximate answers and calculate accurately when required, but it is also important to have a good understanding of the underlying concepts in order to know when to apply the appropriate procedure. Getting the right answer does little good if you solve the wrong problem!
When inexpensive calculators first became available in the mid1970s, there was concern about their potential negative effects on the development of children's computation skills and mathematics learning. Since the early 1980s, several hundred studies of classroom calculatoruse have uncovered virtually no negative effects on mathematics learning (Driscoll, 1981; Suydam, 1982; Hembree & Dessart, 1986). In fact, research results have shown the calculator to be a useful tool to support the teaching of mathematics concepts, procedures, and problem solving at all grade levels (Bitter & Hatfield, 1992; Wheatley & Shumway, 1979).
Twentyfirst century children will still need to learn the basic facts and computation algorithms. The understanding of number operations and facility in computing small numbers is fundamental to the efficient application of arithmetic to real world problems. However, solving problems involving fractions, decimals, or very large and small values can be facilitated by using a calculator. Students can focus more attention on understanding problems and determining appropriate solution procedures if a calculator is available to handle the routine calculations. Students may also be more willing to risk tentative solutions if they are not confronted with several minutes of tedious arithmetic in order to test their conjectures (for a sample activity, see the section "Solving Problems with Calculators").
From a Different Angle (Sidebar Dialogue)
ST: I read that students are allowed to use calculators on standardized exams. Do you think that is a good idea?
CT: Well, it seems to be. The NCTM Standards are recommending that math lessons focus on problem solving at all grade levels. This means that accurate, paperandpencil computation will be stressed less than in the past. Calculators are necessary if we want our kids to spend their time understanding problem situations and working towards solutions. On an exam, if you want to test problem solving ability, calculators level the computation playing field.
ST: But won’t using calculators too early make it more likely that students will not develop a good understanding of number operations – place value, when to use addition or division in a problem?
CT: Sure if they are introduced indiscriminately. We will need to increase our emphasis on estimation and mental arithmetic at the same time. We don’t want children entering 2 • 10 into a calculator to find the answer. One of the most important roles for calculators occurs after number concepts have been developed using manipulative activities. Calculators can then be used whenever accurate answers are needed for exercises that can not be easily accomplished mentally.
Calculator Applications. Many calculator activities and strategy games have been designed to give children practice discovering patterns, thinking logically, and checking results (Immerzeel and Ockenga, 1977; Bitter & Mikesell, 1990). For example, calculators can be effectively employed to
• Develop the concept of placevalue (e.g., enter the value 68 341 into a calculator, what number must be subtracted in order to leave a zero in the thousands place while all the other digits remain the same?).
• Carry out tedious calculations or those which exceed students’ current level of ability (e.g., a second grader might recognize the need to add the costs of ten items purchased at the store but need help with the computation.).
• Practice estimation skills (e.g., enter 142 ÷ 16, estimate the solution, then press = to check the accuracy.).
• Give instant feedback for basicfact drills (e.g., enter 3 + 4, think of the answer, and press = to check solution.) Note: Inexpensive preprogrammed calculators are available which embed this activity in a game format.
• Enhance insight into why algorithms work (e.g., to demonstrate that 24 • 39 = 936, first calculate the subproducts 4 • 39 and 20 • 39, then add the results 156 + 780 = 936).
• Introduce new concepts such a percent, negative numbers, and square root (e.g., entering the subtraction exercise 4  7 = displays the solution 3).
• Play numbersequence strategy games (e.g., for two players sharing a calculator, see who can get to 21 first by starting at 0 and alternately adding 1 or 2 to the displayed sum).
Selecting a Calculator. Calculators with a range of features are available for less than five dollars each. Just like a pencil, every elementary student should have a calculator available during mathematics lessons. This is especially true for Grade 36. If calculators are not available at school, students can be encouraged to bring a calculator from home for their personal use in school and to take home for homework (make sure you have a secure place to store the calculators when they are not in use).
When selecting a calculator for primary aged children, consider the following criteria:
• Good quality keys and a large, easytoread, eight to tendigit display with a backspace key.
• Solarpowered with a backup battery.
• Pressing any operation key ( +, , •, ÷) when carrying out a chain of calculations should cause the pending operation to be executed (e.g., entering 2 + 3 + should display the value 5.).
• The availability of a memory register (memory + or ) which makes twostep computations easier because the result of an intermediate step can be stored and recalled later (e.g., the exercise, (3 + 4) • (5 + 6) = ? can be solved by pressing keys in the following sequence: 3 + 4 = (answer 7) Memory + 5 + 6 = (answer 11) • Memory Recall (MR) = (answer 77)).
• The availability of a constant mode, which means that the calculator will repeat an operation if the key is pressed two or more times (e.g., to show that 4 • 3 can be calculated by adding 4 + 4 + 4, press 4 + + + to display the answer 12.).
Additional desirable functions for Grade 46 students include:
• A square root key that causes the number in the display to be replaced with its square root (e.g., entering 25 and then pressing the square root key gives the answer 5.).
• A change sign key (+/) that allows the user to change the sign of any number displayed (e.g., entering the value 3 +/ will display 3.).
• A reciprocal key that causes the number in the display to be replaced with its reciprocal (e.g., entering the value 4 followed by the reciprocal key gives the answer 0.25 or .).
• Capable of calculating using standard fractions without conversion to decimal form.
• Capable of calculating integer quotients and displaying whole number remainders.
• Capable of using parentheses to specify the order of operations.
Calculator for Grades K3. Select a calculator for Grades K3 that has large keys and easy to read display. Several companies produce calculators with a protective slip cover that are packaged for schools in sets of thirty. Texas Instruments supplies the Educator™ Basic Overhead Calculator that makes it easy to demonstrate the matching TI108 student calculator for an entire class.
Calculator for Grades 46. The Texas Instruments Math Explorer™ is an example of a calculator specifically developed for use by Grade 48 students. Unique features enable the calculator to be used more flexibly for mathematics instruction. It is capable of calculating with fractions without conversion to decimals. Simplifying fractions can be accomplished by entering a common divisor for the numerator and denominator, or it will automatically reduce the fraction to lowest terms and show the user the common factor used in completing the operation. Another useful feature is the integer divide key. Division using this feature gives the result as a whole number quotient and a remainder, rather than the typical decimal solution. This facilitates division of time and other measures when solving problems (e.g., 16 feet ÷ 5 = 3 and 1 foot remaining, rather than 16 ÷ 5 = 3.2 as would be shown on a typical calculator). The Math Explorer™ also has a handy backspace key, a fixed decimal mode for working with money, a percent key, memory keys, four exponent functions to find square roots and powers of 10, parentheses to help with order of operations, and runs on solar cells (see Figure 5.1). The Educator™ Intermediate Overhead Calculator which matches the keyboard layout of the Math Explorer™ is also available for wholegroup demonstrations.
Figure 5.1
New art of Math Explorer™ calculator.
Solving Problems with Calculators
Calculators are useful problemsolving tools in the classroom. In the following example, children can use a calculator to complete routine calculations, freeing them to focus attention on the underlying structure of the problem.
Suppose you wish to construct a box from a 1 • 1 meter piece of cardboard. Boxes are formed by cutting identical squares from each corner and folding each side into position (see 5.2).
Figure 5.2
Insert art of square with corners cut out.
As you vary the size of the four identical squares cut from each corner, different boxes can be constructed. Which box will have the greatest volume? First, let’s explore some questions students may ask as they try to understand this problem.
1. Will the boxes constructed using the method described will have different volumes?
2. How many different boxes can be constructed?
3. Is there a limit to how large the volume can get?
4. What is the largest square that can be removed from each corner and still make a box? The smallest?
5. How do you compute the volume of a box?
6. How accurate should the answer be?
To answer these questions, students could draw sketches and compute the volumes of the resulting boxes using a calculator or they could actually construct boxes using paper or cardboard until they are satisfied that the boxes do represent a range of volumes and that it is impossible to make them all. One possibility is to simplify the problem by allowing only squares with edges that are whole centimeters (1 cm, 2 cm, 3 cm, and so on) or multiples of 10 centimeters (10 cm, 20 cm, 30 cm, and so on) to be cut from the four corners. The resulting volumes could be computed using a calculator and the results compiled as in Table 5.2.
Table 5.2
Box Volume
Original Edge (cm) 
Size of cut (cm) 
New Length (cm) 
New Width (cm) 
New Height (cm) 
Volume (l • w • h) 
100 
10 
80 
80 
10 
64 000 
100 
20 
60 
60 
20 
72 000 
100 
30 
_ 
_ 
_ 
_ 
100 
40 
_ 
_ 
_ 
_ 
100 
50 
_ 
_ 
_ 
_ 
Continue the solution in Table 5.2 and find the box with a maximum volume. Next, find the size of the square to the nearest centimeter that results in the largest volume for the box. Try to answer the six questions posed in the problem introduction. Use a calculator.
Computers and Learning
Mathematics teachers today find it increasingly important know about the uses and the limitations of personal computers as tools for learning. While it is not essential for most teachers to know how to write computer programs (a list of special words and symbols that direct the operation of a computer), knowing how to operate a computer, integrate its use into classroom lessons, and effectively evaluate instructional software are now considered essential skills for beginning teachers.
Personal Computer Components. Several versatile personal computers are marketed for personal and classroom use. Prices range from under $100 for relatively simple devices to several thousand dollars for remarkably versatile systems. Computer selection depends primarily on the intended applications. The most important criteria to consider when buying a computer is to ensure it runs the software you wish to use. Processor power, memory, and cost should be secondary considerations.
All personal computers have the following hardware features in common:
• Input device – keyboard, mouse, turboball, joy stick, graphics pad.
• Output device – monitor, projection display, printer.
• Microprocessor and memory – central processing unit (CPU), random access memory (RAM), and read only memory (ROM).
• Data storage device – floppy disks, hard disks, CDROMS, optical disks.
Invent a metaphor for each of the 4 categories of components (i.e., input device is like a pencil or telephone keypad). Write a short story about how you might use the four components of a computer to help you solve a problem like planning a camping trip.
Types of Educational Software
A convenient way to classify computer use is specify how initiative, or control over the operation of the program, is distributed between the user and the computer (Levin & Souviney, 1983). Initially, the most common use for computers in classrooms was to provide automated drillandpractice and as an environment for students to learn how to write programs.
Computer DrillandPractice Software. When using drillandpractice software, the user responds to questions or situations posed by the computer. The response, generally a value or multiple choice selection, is evaluated by the program and the computer initiates the next appropriate display. This type of application places most of the initiative for sequencing, pacing, and content within the program itself. The user responds to predetermined screen displays controlled by the computer program.
Addition Logician is an example of drillandpractice software. Part of the Minnesota Educational Computing Consortium (MECC) Mastering Math series, this program contains a set of four addition drills and one review program for grade 3 students. Randomly generated exercises are presented at an appropriate level of difficulty. After successfully completing 5 examples, the user is presented with an interesting game play against the computer. Students are given a session score at the end of each lesson and a record is kept for review by the teacher.
Computer Programming. Learning to program a computer places most of the initiative with the user. The program’s list of instructions must be detailed since a computer only does as it is directed, nothing more and nothing less. Programmers must list all the instructions to control the operation of the computer and the organization of the screen displays. More advanced programming languages give useful error messages when the programmer makes a mistake, but in general, the programmer must exhibit a high level of initiative and skill when writing a program.
Programming Languages. BASIC, a general purpose programming language developed over thirty years ago at Princeton University, is available on all popular personal computers. It is reasonably easy to use since most BASIC statements and commands are written in English. A more recent language, Logo, was developed at MIT as a graphicsoriented programming environment for children (Papert, 1981). Using Logo commands, young programmers direct an electronic turtle to draw figures on the screen. Many other languages such as Pascal, C, and modern scripting environments like Hypercard are available on personal computers for use by middle and high school students and professional programmers.
For experienced programmers, programming can also be a powerful problem solving tool. Inventing and adapting programs requires a clear understanding of the problem situation, logical thinking, and good organization skills – all useful components of effective problem solving. Examples of instructional applications of BASIC and Logo programming for grades 46 are presented later in this chapter.
Significant time is needed for elementary students to become proficient enough to make programming an efficient problem solving tool. Due to limited access to computers and an already busy teaching schedule, elementary teachers focus less on programming than other instructional applications of computers.
Software tools, or mixedinitiative software, fall near the middle of a computer use continuum as shown in Figure 5.3. Such software takes greater advantage of the interactive capabilities of the computer than computer drillandpractice applications yet requires significantly less instructional time to become proficient than programming. Tutorials, simulations, tools, and scripting environments are finding widespread applications in elementary classrooms.
Figure 5.3
ComputerUse Continuum
Low User Mixed High User
Initiative Initiative Initiative
Computer Tutorials Simulations Tools Scripting Programming
Drilland Environments Languages
Practice
Computer Tutorials. Unlike drillandpractice software, that provides drill for previously taught procedures, computer tutorials attempt to teach currently unknown concepts. Tutorials generally rely on multiple choice responses to computer displayed questions. Students are guided through the predetermined subject matter based on the accuracy of their responses. More sophisticated tutorials identify students who exhibit a particular error pattern and divert them to another part of the program to review the relevant concept or skill. Then, when an appropriate level of performance is attained, the students are returned to the point the instructional sequence was interrupted. Tutorials are also used to reinforce previously taught concepts and procedures.
Fraction Bars Computer Program is a set of seven disks providing practice solving problems involving fractions (Scott Resources). As a tutorial, three examples of each type of exercise are given at the beginning of each lesson. Randomly generated exercises are then presented, first using graphic displays and then using abstract examples. Word problems, games, and a quiz are also included on each disk (for Grades 46).
Computer Simulations. Activities which enable participants to experience key features of real world situations or games which otherwise would be to inconvenient, expensive, or dangerous (i.e., medical experiments) are called simulations. As an interactive medium, personal computers are particularly effective at supporting simulations.
Simulations developed especially for instruction are those which have embedded tasks to guide the user through a series of carefully sequenced activities. These programs present problems to solve, introduce motivating games, ask questions, offer suggestions, and provide various kinds of user support. The following example is used in many elementary classrooms.
The Market Place gives elementary students practice coordinating several factors involved with running a successful bicycle shop, lemonade stand, or other business (MECC). Students determine the cost of various raw materials and the selling price of finished products. The program supplies random weather conditions which may affect sales, and the user can check their profits over a period of several days. The goal is to identify the correct costs and prices to maximize profits (for Grades 46).
Computer Tools. A computer can be used as a tool to accomplish a wide range of tasks. Students and teachers can use a software tool to assist with:
• Writing
• Constructing graphs and charts
• Designing graphics and animation
• Composing music
• Managing information
• Telecommunicating
Word processors, graphic design programs, and music composers extend the features of traditional tools. Tools respond only to the userinitiated commands; they do not pose questions or offer suggestions. A writer can enter and easily rearrange the lines of a poem; an artist can design an animated cartoon and adjust elements in each frame; or an astronomer can identify a constellation as viewed from the South Pole or from the moon. Each of these functions can be activated by loading a program from a disk into the computer memory. Such tools do not provide specific content instruction but offer a supportive environment to carry out common tasks more easily.
Young writers may initially express ideas more freely since using a word processor (e.g., Bank Street Writer, The Writing Center, AppleWorks, or FrEdWriter) makes it easier to later revise and polish their work than is possible with traditional technology (Mehan, et al., 1986). Individuals lacking the motor skills to play a musical instrument or draw using penandpaper may produce creative works with the aid of music or graphic design software that provides greater control over the composition and revision process.
Routine telecommunication access to massive data bases, such as library catalogues, stock market statistics and airline schedules, is now possible through use of appropriate software and a modem, a device which connects a computer with a telephone line. Electronic mail systems have become inexpensive enough to compete with regular mail. At the elementary school level, computerpal letters and school newspapers are being routinely exchanged between computers over telephone lines by children in classrooms separated by thousands of miles (Levin, Riel, Boruta, & Rowe, 1985).
Data base management software allows the user to enter items of information about each member of a group For example, personal information about each class member could be entered (name, address, telephone number, birthday, place of birth, parent name(s), number of pets, and number of siblings). Students could then ask the data base questions about the class. For example, the data base could be asked to systematically search for and print a list of everybody with a birthday in October.
Spreadsheet software is like a two dimensional calculator. While a calculator works with only one set of values at a time, spreadsheets allows the user to enter numbers (and text) in separate locations on the screen (called cells), do calculations on whole rows and columns of values at one time, and show the results on the screen. Classroom applications for these versatile tools are being designed by teachers for use typically in Grades 46. For example, the formulas for the area (A = _r2) and circumference (C = 2_r) of a circle can be entered in the same spreadsheet, so that for any value r entered in the first column, the resulting area and circumference will be displayed in columns two and three. Spreadsheets enable students to generate several examples during mathematical investigations. This resulting tables can lead to a search for growth patterns and generalizations about the relationship between the formula (i.e., why are the area and circumference equal when the r = 2?).
r 
_r2 
2_r 
1 
3.14159 
6.28298 
2 
12.56637 
12.56637 
3 
28.27432 
18.84955 



Botanical Gardens is a software tool that simulates a genetics laboratory (WINGS for Learning/Sunburst). It allows students to systematically test variables which effect plant growth. Several fictitious plant seeds are available for testing in the greenhouse. Users can create their own plants for exploration in the genetics lab and read about the plants in a Library. Results are graphically displayed on the screen but can not be easily printed.
Scripting Environments. Simplified programming languages which are specifically designed to help students and teachers to develop their own educational software are called an scripting environments. These highlevel scripting languages require minimal programming experience on the part of the user.
Hypercard is a popular example of a scripting environment. The user is presented with a stacks of onscreen file cards. Each card can contain text windows, graphics, and buttons. Each button can be programmed so that when it is pressed, a new card appears in the front. By carefully planning what is on each card and programming buttons to jump the other cards in as desired, tutorials, drills, tools, and simple simulations can be created for instructional purposes. Thousands of hypercard stacks have been created by educators to accomplish specific classroom tasks.
Classroom Computer Use
Teachers have successfully used a single machine with a class of students by setting up a computer learning center within the classroom (MillerSouviney, 1985). Pairs of children can be scheduled throughout the day to work on software selected to complement ongoing instruction. Effective uses of computers include individualized drillandpractices, concept development, and problem solving experiences. Teachers should avoid using computer games as, for example, a reward for good behavior because it is an inefficient use of the machine. To insure equity of access for all children in the class, activities like programming that demand considerable computer time may need to be scheduled on a voluntary basis during lunch or after school.
A computer can also be used to demonstrate a concept during a lesson. By connecting the computer to an overhead projection display pad or large screen monitor, the teacher can display graphs or other computergenerated visuals. For example, one advantage of a computer demonstration over traditional overhead transparencies is that professionallooking graphs can be displayed in realtime using actual information provided by the class. In a single class period, students could construct, display, and print separate graphs that summarize student television viewing preferences, eating habits, and family membership. Subsequent discussion of these graphs is likely to be more meaningful because the class directly participated in their construction. When reviewing software for classroom use, it is a good idea to consider its possible realtime display applications as well as its instructional uses.
Some schools have established a central computer facility which can be shared by all classes. Teachers either bring their entire class at a scheduled time, or, if a computer resource teacher is available, periodically send smaller groups for instruction.
Whether computers are available in a laboratory or in individual classrooms, it is important for every student to have equal access to them. As computers become more widely available, the problems associated with equitable scheduling are likely to diminish. Facility with computers will become more and more a basic skill, and it will be increasingly important for teachers to develop effective techniques to integrate this technology into everyday classroom activities.
Solving Problems with Computers
Computer supported problem solving applications can be classified into three categories:
• Problem solving software
• Software tools
• Programming
Each application demands a different level of teacher expertise and computer use (Kantowski, 1983). While it is preferable that teachers have facility with all three types of applications, with appropriate software even teachers with limited programming ability can effectively use computers in their mathematics classrooms.
Problem Solving Software. A number of quality software products are available which support the development of mathematical thinking and problem solving. The following example presents the user with a series problems associated with a particular environment. Students are encouraged to use problem solving strategies (guessandtest, working backwards, systematic listing, constructing a table, and so on). Varying amounts of help are displayed on the monitor depending on the program’s difficulty level and the user’s experience. Problem solving software can be incorporated into a learning center containing one or more computers or can be the focus of a teacherdirected activity for the whole class. It is preferable to select software that aligns with the ongoing mathematics curriculum rather than adjusting the instructional sequence to accommodate a particular program.
ProblemSolving Strategies is a good example of computer supported problem solving activities (MECC). The disk contains four programs, Diagonals, Squares, Thinking With Ink, and Pooling Around. The first two programs are tutorials which show applications of guessandtest, systematic listing, and simplifying strategies. Diagonals asks students to find the number of diagonals in regular polygons. In Squares, the user must find the total number of squares in a fivebyfive grid. Thinking With Ink is a game in which students must minimize the cost of painting a map where adjacent countries can not be painted the same color. Pooling Around explores the number of times a pool ball hits the edge of the table and has the student look for patterns that predict in which pocket it will end. The software supports the problem solver by generating lists, tables, and graphic displays on request (see Figure 5.4). These programs take good advantage of the interactive graphic capabilities of the computer, redrawing the maps and pool tables while simultaneously updating lists or tables (for grades 46).
Figure 5.4
Insert art of Pooling Around screen.
Software Tools. Computer software tools, originally created to improve productivity in businesses, have been cleverly adapted for use in schools. The software tool emulates traditional technology (a typewriter, pen and art board, or accounting pad) yet extends the capability of each. Some systems allow interaction between tools. For example, software allows a column of values in a spreadsheet to be automatically displayed in graph form or an entire spreadsheet to appear in a wordprocessing document (e.g., ClarisWorks for the Macintosh).
MECC Graph and Exploring Measurement, Time, and Money are software tools that allow children to enter information, and the computer creates a nicely formatted graph which can be printed (MECC). Like the calculator, a graph tool gives students greater freedom to concentrate on the structure of problem situations.
For example, suppose a graphing tool like MECC Graph was available to help solve the box construction problem introduced in the previous section. The volume for each box could be quickly graphed. The graph in Figure 5.5 shows that the maximum volume occurs when the edges of the corner squares are about 20 centimeters in length. If we calculate in 1centimeter steps, the volume of the boxes that result from removing squares 15 through 25 centimeters on an edge will be more accurate (see Figure 5.6).
Figure 5.5
Insert art of Box Volume Graph 050 cm
Figure 5.6
Insert art of Box Volume Graph 1525 cm
The answer (17 cm) is more clearly apparent when displayed in graph form. Even greater accuracy can be accomplished by computing volumes of boxes resulting from removing squares with edges between 16 and 18 centimeters in 1 millimeter steps (i.e., 160 mm, 161 mm, and so on). While it would be too time consuming to have students draw many trial graphs displaying increasingly accurate representations of this problem, using graphing tool provides a handy alternative. Such graphs also serve as excellent aids when students are asked to communicate and justify their solutions as recommended in the NCTM Standards.
Primary level students can also use graphs to help them solve problems and explain their results. For example, suppose a class was asked to find the total number of pets owned by students in the room. Each student could fill out, in class or for homework, a pet checklist listing common household pets (cats, dogs, snakes, fish, etc.). Students could then enter their results on a prepared bulletin board graph (see Figure 5.7). The teacher, or a student, could then enter the information into a graphing tool, print the graph, and duplicate a copy for each student. The graph could then be used to answer a series of related questions.
Figure 5.7
Insert graph of number of pets.
The graph could then be used as a basis for a series of related problems:
• Who has the fewest pets?
• Who has the most pets?
• How many pets are there altogether?
• What is the difference between the most and fewest pets?
• How many students have pets, 1 pet, 2 pets, etc.?
New graphs could be made showing the number of each type of pet. Questions could be explored based on the popularity of each pet (see Figure 5.8).
Figure 5.8
Insert graph of pet types.
Programming. Programming is a third way that computers can be used in the teaching of problem solving. Without a program, a computer is like an airplane without a pilot. A computer’s potential to carry out tasks is activated only when it receives a detailed list of instructions it can understand. Programs written by students or professionals are stored on a disk for future use.
Many computer languages have been designed which simplify the task of writing programs. Two popular languages used at the elementary level are BASIC and Logo. The BASIC language is available for all popular computers. It is relatively easy to learn how to write BASIC programs which manipulate numbers and text. Logo is a graphicsoriented language which allows even very young children to draw geometric figures on the monitor and save them on disk.
It is important to actually try the following programs on a computer. Though you can learn how the logic of a program is developed by reading the description, if you are going to introduce programming to elementary students, you should experience the process of entering and executing a program yourself. Programs are developed to solve some problem. BASIC and Logo programming are introduced by presenting a simple program to solve a problem. We then revisit new versions of the problem and adjust the program to help with the solution.
BASIC. The following examples are written in Applesoft BASIC for the Apple //e, //c, or //gs computer. However, each program will work on other computers with minor changes. See the user's manual supplied with the computer for specific details.
For novices, it is often helpful to introduce programming with a completed program. Students are then asked to change specific values and observe the effect on the output displayed on the screen. In the following BASIC program, notice that each line begins with a number. These are called line numbers. The computer reads the line with the smallest number and executes the instruction which follows. It then reads the next line, executes that instruction, and so on.
BASIC Count Program
1 LET COUNT = 1
2 PRINT COUNT
3 LET COUNT = COUNT + 1
4 IF COUNT <25 THEN GOTO 2
5 END
This BASIC program will print the counting numbers 1 through 24. The first line instructs the Apple to set aside a location in memory named COUNT and to put the value 1 in it. The second line instructs the Apple to display the value in COUNT on the screen. The next line is the BASIC instruction for taking the current number in COUNT and replacing it with the current value increased by 1. The fourth line asks the computer to make a decision: If the value in COUNT is smaller than 25, then jump back to line 2, when the value in COUNT reaches 25, the computer goes on to line 5 and stops. By carefully keeping track of the values in COUNT, it is possible to predict what numbers the computer will display as output on the screen.
5.1 OnLine Activity: Turn on the Apple Computer. Hold down the CTRL (Control) key and press RESET. You should see a blinking cursor. Type NEW and press RETURN to clear the memory. Enter the BASIC Count Program into the computer. Type RUN and press RETURN. Make a record of the output. Next, retype line #1 as follows and press RETRURN (this will change only Line 1 and leave the other lines unchanged):
1 LET COUNT = 5
What numbers do you think will be displayed on the screen when you run this new program? Why? Type LIST and press RETURN. Notice line #1 has been replaced as above. Now type RUN and RETURN. Make a record of the displayed numbers. Change the value in line #1 to 10, 17, 20, and then 25. Run the program and record the results after each change. Change the value in line #1 back to 1. Try changing the value added to COUNT in line #3, to 2, 3, and then 5. Run the program each time and record the results. Change the value 25 in line #4 to 50, 100, and then 1000. Run the program each time and describe the results. Try to make the program count down instead of up.
Novices can explore simple BASIC programs and attempt to unravel the logic used by the programmer. The following examples introduce the BASIC statements FOR...NEXT and INPUT. Programs like these can be used to provide practice in logical thinking and pattern recognition. See the references at the end of the chapter for sources of additional programs.
BASIC Count #2
1 FOR COUNT = 1 TO 25 STEP 1
2 PRINT COUNT
3 NEXT COUNT
4 END
BASIC Count #3
1 INPUT X
2 FOR COUNT = 1 TO X STEP 1
3 PRINT COUNT
4 NEXT COUNT
5 END
The following BASIC program generates the table of volumes for the Box Construction problem presented earlier. Notice the inclusion of a REMember statement. These statements are not processed by the computer but serve as reminders to the programmer about the purpose of the program or particular lines of code. Also, note that line numbers are multiples of ten to allow room for additional lines if the program needs to be altered later. Note that EDGE = 100 is the length of edge of original cardboard square; HEIGHT = height of box; LENGTH = length of box; WIDTH = width of box; and VOLUME = volume of box.
Maximum Box Volumes
5 REM BOX VOLUME
10 LET EDGE = 100
20 PRINT "HEIGHT", "DEPTH+WIDTH", "VOLUME"
30 FOR HEIGHT = 1 TO EDGE/2 STEP 1
40 LET DEPTH = EDGE(2*HEIGHT)
50 LET WIDTH = DEPTH
60 LET VOLUME = DEPTH*WIDTH*HEIGHT
70 PRINT HEIGHT,DEPTH,VOLUME
80 NEXT HEIGHT
90 END
5.2 OnLine Activity: Enter the program to compute volumes for the box construction problem. Press CTRLS when you want stop the program to view the results on the screen. If a printer is available, type PR#1, press RETURN, and RUN the program to display the output on the screen once again. Compare the resulting table to your previous answer. Change the value of EDGE from 100 cm to 1000 mm. What is the EDGE length, to the nearest millimeter, of the identical squares removed from each corner that results in the maximum box volume? Remember that the results will be expressed in millimeters and cubic millimeters. Use CTRLS to stop the display when you want to view the values on the screen (it takes a long time to print out the results on paper). Can you change line 30 in the program to display only the values in a narrow range around the largest volume?
Logo. In the late 1960s, the MIT Logo Group headed by Seymour Papert developed the Logo computer language as a Piagetinspired learning environment for children. A key feature of this environment is a technological turtle, which draws graphics on the monitor under student control. Students can write Logo programs called procedures to guide the turtle's path. These procedures can be saved on disk for later use.
Logo is widely used in elementary schools. It is often used as an introduction to programming for young children and as a tool to aid the development of problem solving skills. Logo is an efficient environment for children to practice the problem solving strategies of subdividing problems into manageable units, searching for patterns, and evaluating solutions for errors in logic called bugs. The following activity uses the MIT (Terrapin/Krell) versions of Logo run on an Apple //e, //c, or //gs computers. Minor revisions may be necessary when using other versions of Logo.
First, let’s teach the turtle draw a square. Put the Logo Language Disk in the drive and turn on the Apple. When the screen displays WELCOME TO LOGO, type DRAW and press RETURN (make sure the CAPS LOCK key is down). You will see a small, triangular turtle in the center of the screen and a ? followed by a blinking cursor at the bottom (see Figure 5.9). Enter the following sequence of commands and watch the turtle draw a square. Press RETURN after each line:
FD 40
RT 90
FD 40
RT 90
FD 40
RT 90
FD 40
RT 90
Figure 5.9
Insert art of Logo screen.
FD 40 means FORWARD 40 units (or screen dots) and RT 90 means RIGHT TURN 90°. Type DRAW and press RETURN to make the turtle center itself on the screen and clear the screen. Try to predict what the following sequence will draw. Note that the commands can be entered on the same line separated by spaces. Enter the sequence and test your prediction (to correct a mistake, press the ESC key to back up, then reenter character).
FD 40 RT 120 FD 40 RT 120 FD 40
Notice that both sequences of commands repeat a pair of actions over and over. Logo lets the user carry out repetitions by employing the REPEAT command. Enter DRAW and press RETURN. Enter the following sequence of characters, including spaces, exactly as shown:
TO SQUARE
REPEAT 3 [FD 40 RT 120]
END
Notice the turtle draws the same equilateral triangle as before.
5.3 OnLine Activity: Try to use the REPEAT command to draw a square just like the one drawn earlier.
Logo also allows the user to save one or more procedures in memory and, if desired, on disk for later use. To save a procedure temporarily in memory, first pick a oneword name (e.g., SQUARE), enter TO SQUARE and press RETURN. This tells Logo that you want it to remember the procedure named SQUARE. You are now in the Logo editor, a mode where you can enter and later change, or edit, procedures. Enter the following procedure for making a square.
TO TRIANGLE
REPEAT 4 [FD 40 RT 9]
END
Use the ESC key to erase incorrect entries. Press CTRLC (hold down the CONTROL key and press C). If you entered the commands correctly, Logo will ask you to PLEASE WAIT, then the screen will display SQUARE DEFINED. This means Logo has memorized your procedure named SQUARE so you can use it later (Logo only remembers until you turn off the computer unless you also SAVE the procedure on disk. See the Logo Manual supplied with the program for directions).
You can now draw the square by simply entering the procedure name SQUARE and pressing RETURN. Type SQUARE and press RETURN. Repeat the process several times. What happens if you do not type DRAW and press RETURN after each run? Why?
To save the procedure named TRIANGLE, type TO TRIANGLE, press RETURN, and enter the following statement:
REPEAT 3 [FD 40 RT 120]
Press CTRLC after the procedure is entered. When Logo displays TRIANGLE DEFINED, run TRIANGLE. Type DRAW and press RETURN. Then run SQUARE. Notice Logo can remember more than one procedure at a time. Run TRIANGLE again without typing DRAW (see Figure 5.10).
Figure 5.10
Insert art of Logo screen with triangle and square.
Now let us explore the three values specified in the procedures named TRIANGLE and SQUARE.
TRIANGLE Æ REPEAT 3 [FD 40 RT 120]
SQUARE Æ REPEAT 4 [FD 40 RT 90]
The value preceding the left bracket [ indicates the number of times the commands inside the brackets should be repeated. The value following FD indicates the number of units the turtle will move forward. The value following the RT indicates the number of degrees the turtle should rotate to the right. The turtle can also move backwards (BK) and make left turns (LT). To make a pentagon (5sided polygon) or other regular polygon with sides 40 units in length, we only need to change the amount the turtle turns at each vertex. Notice in Table 5.3 that the turtle must turn a total of 360 degrees in its complete trip around a polygon (e.g., triangle: 3 • 120 = 360; square: 4 • 90 = 360). Therefore, the turtle must turn 72 degrees at each vertex to draw a pentagon (see Figure 5.11).
Table 5.3
Regular Polygons
Turtle Turn
Name #Sides Length (degrees)
Triangle 3 40 120
Square 4 40 90
Pentagon 5 40 ?
Hexagon 6 40 ?
Heptagon 7 40 ?
Octagon 8 40 ?
Nonagon 9 40 ?
Decagon 10 40 ?
Figure 5.11
Insert art of pentagon turtle trip.
5.4 OnLine Activity: Complete Table 5.3. Write Logo procedures to draw each of the regular polygons listed. How many degrees turn makes Logo sketch of the Heptagon (7sided figure) unique? (Hint: Run HEPTAGON then enter HT, for HIDETURLE, and press RETURN.) Systematically increase the number of sides to 12, 15, 18 and 24. What happens to the shape as the number of sides gets large. Notice that for these polygons, it is necessary to reduce the side length to about 15 units to keep the sketch from wrapping to the opposite side of the screen. Record your observations and discuss the results.
A wide range of classroom tested activities are available for Logo. See the reference and software section at the end of this chapter for additional suggestions.
Selecting Quality Computer Software
Thousands of instructional programs are now available for computer aided instruction (CAI). Good CAI software, like good books, requires considerable creativity and skill to produce. Quality commercial products can be costly, however, there are some excellent free public domain educational programs available (e.g., FrEdWriter is an easytouse classroom word processor). Like other instructional materials, software should be carefully evaluated prior to classroom use.
To help teachers select quality software, several national and regional organizations evaluate software products and report the results in catalogues. Local professional organizations of computerusing educators generally offer practical, uptodate information on educational software, and may provide access to a lowcost softwareexchange network.
It is always good practice to preview software before it is purchased to insure that it is easy to operate and can be effectively integrated into ongoing lesson objectives. Since quality software is generally costly, many companies offer preview policies for educators. Some software publishers supply multiple copy labpacks or site licenses of popular products at a substantially reduced perdisk cost. Such arrangements are necessary to satisfy the copyright requirements for school software use. Make sure the publisher agrees in writing to replace any product at low cost if it is defective or becomes damaged. Arguments for rejecting any software which display unacceptable characteristics are as true today as when Ann Lathrop reported them in 1982. Reject any software that:
1. Gives an audible response to student errors – no student should be forced to advertise mistakes to the whole class.
2. Rewards failure – programs that make it more fun to fail than to succeed.
3. Has sound that can not be controlled – the teacher should be able to easily turn sound on and off.
4. Has technical problems – is the software written so that it will not crash if the user accidentally types the wrong key; incorrect responses should lead to software initiated help comments.
5. Has uncontrolled screen advance – advancing to the next page should be under user control, not automatically timed.
6. Gives inadequate onscreen instructions – all necessary instructions to run the program must be interactively displayed on the screen (in a continuously displayed instruction window if possible).
7. Has factual errors – information displayed must be accurate in content, spelling and grammar.
8. Contains insults, sarcasm and derogatory remarks – students’ character should not be compromised.
9. Has poor documentation – demand the same quality teacher’s guide as with a text book or other teaching aid.
10. Does not come with a backup copy – publishers should recognize the unique vulnerability of magnetic disks and offer lowcost replacement.
You may find it helpful to use a software evaluation form like the one shown in Figure 5.12 to help with the selections of classroom software.
Figure 5.12
Evaluation of Instructional Courseware Form
A. DESCRIPTIVE INFORMATION
Title Version
Publisher Computer (brand/model)
Appropriate subject area(s)
Appropriate grade levels: (circle) K 1 2 3 4 5 6 7 8 9 10 11 12 College Teacheruse
Type of program (check all that apply):
__ drill/practice __ tutorial __ simulation __ educational game __ teacher utility
__ demonstration __ testing __ problem solving __ word processing __ game
__ other:
Describe the program (content, main objective, how students interact with program):
Describe management system, if any, including scoring or performance reporting and
the number of students/classes permitted by program:
B. EVALUATIVE INFORMATION:
STUDENT RESPONSE
Grade levels in which used: Subject:
Behavior observed which indicates learning took place:
Other reactions of students:
Any problems experienced, special preparation of students required:
EVALUATION CHECKLIST
Not
Yes No Applic.
General Design:
___ ___ ___ 1. Effective, appropriate use of the computer?
___ ___ ___ 2. Design based on appropriate instructional strategies?
___ ___ ___ 3. Follows sound instructional organization?
___ ___ ___ 4. Fits well into the curriculum?
___ ___ ___ 5. Free of programming errors, reliable in normal use?
___ ___ ___ 6. Publisher's objectives are stated and met?
General Design: __ EXCELLENT __ GOOD __ OK __ POOR __ NOT USEFUL
Content:
___ ___ ___ 7. User may choose from levels of difficulty?
___ ___ ___ 8. Branches to easier of harder material in response to student performance?
___ ___ ___ 9. Factually correct?
___ ___ ___ 10. Punctuation, spelling, and grammar are correct?
___ ___ ___ 11. Free of excessive violence or competition?
___ ___ ___ 12. Free of stereotypes  race, gender, ethnic, age, handicapped?
___ ___ ___ 13. Interest, difficulty, typing, and vocabulary levels are appropriate and are commensurate with student skills?
___ ___ ___ 14. Program data, speed, word lists, etc., can be adapted by instructor?
___ ___ ___ 15. Responses to student errors are helpful?
___ ___ ___ 16. Responses to student errors are nonjudgmental?
___ ___ ___ 17. Responses to student success are positive and appropriate?
Content: __ EXCELLENT __ GOOD __ OK __ POOR __ NOT USEFUL
Ease of Use:
___ ___ ___ 18. Program may be operated easily and independently by student?
___ ___ ___ 19. Expected user responses for program operation are consistent?
___ ___ ___ 20. Instructions within program are clear, complete, concise?
___ ___ ___ 21. Instructions can be skipped or called to screen as needed?
___ ___ ___ 22. Instructions on how to end program, start over, are given?
___ ___ ___ 23. Menu allows user to access specific parts of program?
___ ___ ___ 24. Answers may be corrected by user before continuing with program?
___ ___ ___ 25. Pace and sequence can be controlled by user?
___ ___ ___ 26. Screens are neat, attractive, wellspaced?
___ ___ ___ 27. Well designed graphics are used appropriately?
___ ___ ___ 28. Sound is appropriate, may be turned off?
___ ___ ___ 29. Classroom management, if any, is easy to use?
Ease of Use: __ EXCELLENT __ GOOD __ OK __ POOR __ NOT USEFUL
Motivating Devices (Check all which are used):
___ color ___ personalization ___ timing
___ scoring ___ random order ___ graphics for instruction
___ sound ___ game format
___ graphics for reward
___ other:
Motivational Devices: __ EXCELLENT __ GOOD __ OK __ POOR __ NOT USEFUL
Documentation (Check all which are provided in the package):
___ instruction manual ___ instructional objectives
___ instructions appear on the screen ___ tests
___ describes required hardware ___ workbooks
___ procedures for installation ___ student worksheets
___ teacher's guide ___ suggested activities
___ other:
Documentation: __ EXCELLENT __ GOOD __ OK __ POOR __ NOT USEFUL
EVALUATION COMMENTS
Describe any special strengths of the program:
Weaknesses/Concerns/Questions:
Compare with similar programs:
OVERALL OPINION
__ EXCELLENT __ GOOD __ OK __ POOR __ NOT USEFUL
Educational Software Preview Guide by the Teacher Education and Computer Center and California Library Media Consortium for Classroom Evaluation of Microcomputer Courseware, Copyright 1986 by California Department of Education. Reprinted by permission.
Summary
• The availability of inexpensive computing devices will play an increasing role in mathematics instruction into the next century.
• Handheld calculators and personal computers are being used in basic skill instruction, concept development, and as a problem solving tool.
• Traditional drillandpractice software and programming applications are being complemented with computersupported tutorials, simulations, software tools, and scripting environments like Hypercard. These newer instructional applications take better advantage of the interactional capabilities of the computer than drillandpractice software, yet require much less handson computer time to be effective than is required for programming instruction.
• Good software tutorials, simulations, and tools are as difficult to produce as other quality instructional materials and should therefore be carefully evaluated prior to purchase.
• Educationcomputing magazines, commercial directories, and computerusing educator groups are good sources for advice on the quality of commercial and public domain software products.
Course Activities
1. Interview a teacher, principal, and a parent about their views on using calculators and computers in elementary schools. Compile a list of positive and negative aspects of each. Should the school provide calculators for everyone? Should computeruse be considered a basic skill? Should calculators be available during quizzes and standardized tests? What is the most effective uses of technology in the classroom. Write a short essay summarizing your conclusions about the use of technology in mathematics instruction.
2. Visit a school or other facility which uses computers for instructional purposes. Describe the type of interactions observed. Are all the user inputs of the multiple choice variety? Does the student need to use paper and pencil to work problems? Are graphics used to assist with concept development or are they just decorative? Does initiative pass between the software and the user? Could the same task be as easily accomplished with workbooks or paper and pencil? Discuss the results of your observations with your peers.
3. Review a piece of educational software using an appropriate evaluation form (see Figure 5.12). Share the results with other members of your class.
4. Check to see if a list of recommended mathematics software is available for use in local schools. How was this list compiled? Were local teachers involved in the selection? Were standard evaluation criteria employed? What is the distribution of the software products according to the userinitiative scale discussed in this chapter? Are schools abiding by the copyright laws with regard to instructional software? Discuss your findings.
5. Review the calculator curriculum resource materials listed in the references. Using file cards, describe, along with its learning objective, one appropriate calculator activity for each grade level K6. Try out several of these activities with a peer or, if possible, with an appropriate group of children. Discuss the effectiveness of each activity. Exchange these activity cards with your peers and begin a curriculum resource file for mathematics teaching ideas.
6. Select an instructional objective which can be addressed using a calculator (e.g., greaterthan/lessthan relationships). Write a lesson plan using one of the formats presented in Chapter 2 (or other appropriate form). If possible, try out the lesson with a small group of children. Discuss the changes you would make if you taught the lesson again.
7. What effect do you think the availability of inexpensive calculating devices will have over the next decade on the content included in elementary mathematics curriculum? Will we stop teaching arithmetic? What will be the role of fractions and decimals? Will computers eliminate the need for textbooks? Teachers? Write an essay supporting your conclusions.
8. Using several centimetersquared paper rectangles (20 cm • 30 cm), cut a series of identical squares from each corner and fold the resulting forms into boxes. Using a handheld calculator or computer, construct a table showing the length of the edge of each square and volume of the resulting box. Graph the results. What is the size of the removed square, to the nearest centimeter, which creates the box with the greatest volume? Try to a develop general rule that relates the length and width of any rectangle to the size of the removed squares which maximize the volume of the resulting boxes. Work in groups to solve this classsize problem.
9. If you have access to a computer, enter a BASIC program which generates the volume of boxes resulting from cutting equal squares from the corners of a 1 m • 1 m cardboard square (adapt the examples given in this chapter). Change the values in the program to generate tables showing the resulting volumes to the nearest 10 cm, 1 cm, and 1 mm.
10. Write a Logo procedure that draws a square with a triangle roof. Describe two other Logo tasks appropriate for the primary level and two for grades 36 (see the reference section for Logo curriculum resources).
11. Read one of the Arithmetic Teacher articles listed in the reference section. Write a brief report summarizing the main ideas of the article and describe how the recommendations for instruction might apply to your own mathematics teaching.
Children’s Books
Grades K3
Anno, M., & Nozaki, A. Anno’s magic hat tricks. Philomel Books. An imaginative book that uses delightful graphics to introduce young readers to binary logic, a fundamental component of computer processing.
Grades 46
Anno, M. Socrates and the three little pigs. Philomel Books. Children use counting, addition and multiplication to learn how a computer program works.
Personal Computer Software
Software 
Publisher 
Description 
Type 
Math Strand 
Grade Level 
A Chance Look 
WINGS for Leaning/ Sunburst 
A probability simulation which shows graphs of experiments with coins, cubes, and spinners. 
Simulation/ Tool 
Probability and Statistics 
46 
Addition Logician 
MECC 
Four addition drills and one review program. 
Drill & Practice 
Whole Number Operations 
23 
AppleWorks 
Claris 
An easy to use integrated data base, spread sheet and word processing system which allows the transfer of files among the three tools. 
Tool 
All 
36 
ArithMagic 
QED 
Three problems, Diffy, Tripuz, and Magic Squares, presented in a game format that give students practice with computation and recognizing patterns. 
Simulation 
Problem Solving 
46 
Base Ten Blocks 
Learning Box 
A simulation of base ten blocks which models computation algorithms. 
Simulation 
Whole Number Operations 
36 
Base Ten On Basic Arithmetic 
MECC 
Multiplication and placevalue practice 
Tutorial 
Whole Number Operations 
36 
Bank Street Laboratory 
Holt, Reinhart and Winston 
Tools for measuring and graphing variations in sound, light and temperature. 
Tools 
Problem Solving 
36 
Botanical Gardens 
WINGS for Learning/ Sunburst 
Allows the user to test variables effecting the growth of four fictitious plants and create four plants of their own to explore. 
Simulation 
Problem Solving 
56 
Bounce! 
WINGS for Learning/ Sunburst 
Students improve their pattern recognition skills by seeing, hearing, physically demonstrating patterns. 
Game/Drill & Practice 
Patterns 
4.6 
Challenge Math 
WINGS for Learning/ Sunburst 
Provides practice using basic whole number and decimal operations in an environment motivated by a space intruders, a dinosaurlike creature, and a mysterious mansion. 
Drill & Practice 
Whole Number and Decimal Operations 
46 
Calliope 
Innovision 
An idea processor for children and adults which uses light bulbs to store and reorganize thoughts and graphics. 
Tool 
All 
56 
Delta Drawing 
Spinnaker Software 
An easy to use turtle graphics environment. 
Programming 
Problem Solving 
K3 
Ecosystems 
Holt, Reinhart and Winston 
An ecological simulation. 
Simulation 
Problem Solving 
46 
Exploring Measurement, Time, and Money 
IBM 
A graphing tool for young children. 
Tool 
All 
26 
Fay: That Math Woman 
Didatech 
A set of lessons which provide computation practice using a number line model, 
Drill & Practice 
Whole Number Operation 

Fraction Bars Computer Program 
Scott Resources 
Set of seven disks which provide practice solving problems involving fractions. 
Tutorial/ Drill & Practice 
Fractions and Decimals 
56 
FrEd Prompts 
Steele Publishing 
A set of public domain writing environments which work with FrEdWriter. 
Tutorial 
All 
36 
FrEdWriter 
Steele Publishing 
A public domain word processor for classroom with online help and a simple onscreen prompting feature. 
Tool 
All 
36 
Geometric preSupposer –Points and Lines 
WINGS for Learning/ Sunburst 
Versatile tool which helps students with geometric constructions and measurements of figures and supports the testing of conjectures. 
Tool 
Geometry 
56 
Geometry Problems for Logo Discoveries 
Creative Pub. 
Problem solving activities employing turtle geometry. 
Tutorial 
Geometry 
46 
Gertrude's Puzzles 
Learning Company 
Practice with classification in a flexible problem solving environment 
Simulation/ Tool 
Patterns and Relations 
36 
Gertrude’s Secrets 
Learning Company 
Practice classifying objects in an easytouse microworld environment. 
Simulation/ Tool 
Patterns and Relations 
K3 
Get to the Point 
WINGS for Learning/Sunburst 
Students practice ordering, estimating, and computing decimal numbers. 
Drill & Practice 
Decimals 
46 
Graphing Equations 
WINGS for Learning/ Sunburst 
Exploration of linear and other equation graphs, includes Green Globs game. 
Tool 
Functions and Relations 
56 
Immigrant 
Harvard University 
AppleWorks simulation of 1800s life of immigrant families in the United States. 
Simulation 
Math Problem Solving 
56 
Interpreting Graphs 
WINGS for Learning/ Sunburst 
Provides experience relating graphs to real world events. 
Tutorial 
Math Communication 
56 
King's Rule 
WINGS for Learning/ Sunburst 
The user tries to discover the rule relating sets of numbers at varying levels of difficulty. 
Tutorial 
Functions and Relations 
46 
Logo 
Terrapin/Krell 
A programming language for children which uses turtle graphics. 
Tool 
Geometry 
26 
Logo Discoveries 
Creative Pub. 
Problem solving activities employing Logo programming and Logo graphics. 
Tutorial 
Geometry 
46 
The Marketplace 
MECC 
Several simulations, including Lemonade, which explore the economic features associated with commerce. 
Simulation 
Functions and Relations 
36 
Math Ideas With Base Ten Blocks 
Cuisenaire 
Computation practice using graphics of baseten blocks. 
Simulation/ Drill & Practice 
Whole Number Operations 
36 
Maya Math 
WINGS for Learning/ Sunburst 
A simulation of the Mayan number system and calendar. 
Simulation 
Number Sense 
36 
MECC Graph 
MECC 
A graphing tool which automatically draws a bar graph with scales and titles for a set of values entered by the user. 
Tool 
Math Communication 
26 
Partial Fractions 
WINGS for Learning/Sunburst 
Students guess answers for fraction multiplication and division exercises to develop an understanding of the relative size of answers. 
Drill & Practice 
Fractions 
46 
Pattern Poser 
WINGS for Learning/Sunburst 
Students create, explore, and extend patterns involving addition, subtraction, multiples, and factors. 
Simulation/ Drill & Practice 
Patterns 
46 
Place Value Place 
InterLearn 
Two games which use a multicolumn graphic calculator to explore place value concepts. 
Simulation 
Number 
26 
Problem Solving Strategies 
MECC 
Set of four problem solving situations employing the solution strategies of making table, looking for patterns, and guessandtest. 
Simulations 
Math Problem Solving 
56 
Right Turn: Strategies for Problem Solving 
WINGS for Learning/ Sunburst 
A problem solving experience involving making transformations on a 3 • 3 colored grid and predicting the results of flips and turns. 
Simulation 
Geometry 
56 
Robot Odyssey 
Learning Company 
Problem solving in a simulated world using robots designed by the user. 
Simulation 
Math Problem Solving 
56 
Rocky’s Boots 
Learning Company 
Exploration of electric circuits in a flexible, graphics oriented environment. 
Simulation 
Math Problem Solving 
46 
SimAnt 
Maxis 
A problemsolving simulation of backyard ant ecology. 
Simulation 
Problem Solving 
46 
Snooper Troops 
Spinnaker Software 
A series of engaging detective cases which require use of reasoning skills. 
Simulation 
Math Reasoning 
46 
Strategies in Problem Solving: Dinosaurs and Squids 
Scott, Foresman 
Presents four types of nonroutine problems appropriate for intermediate and middle school students and a tutorial on how to employ solution strategies. 
Simulation 
Math Problem Solving 
46 
Survival Math 
WINGS for Learning/ Sunburst 
A set of simulated activities requiring the use of arithmetic to solve real world problems. 
Simulation/ Drill & Practice 
Number Operations 
46 
Shark Estimation Games 
InterLearn 
Four estimation activities simulating a hunt for a hidden shark involving integer and decimal values on the number line and coordinate plane. 
Simulation/ Drill & Practice 
Estimation 
36 
Stickybear Math 
Weekly Reader 
A series of addition and subtraction drills with excellent graphics. 
Drill & Practice 
Whole Number Operations 
K3 
Taking Chances. 
WINGS for Learning/Sunburst. 
An introduction to probability involving machines that make colored marbles. 
Simulation 
Probability 
23 
Treasure Mountain 
Learning Company 
One of the Super Solver series of gametype activities which develop math, science, reading and reasoning skills. 
Game/Drill & Practice 
Math Reasoning 
36 
Winker’s World of Numbers 
WINGS for Learning/ Sunburst 
Students practice using number patterns to predict missing values. 
Game/Drill & Practice 
Patterns 
46 
Winker’s World of Patterns 
WINGS for Learning/ Sunburst 
Students help Winker the worm solve number, color, and word patterns as he inches about a puzzle grid. 
Game/Drill & Practice 
Patterns 
K3 
What Do You Do With A Broken Calculator 
WINGS for Learning/ Sunburst 
Solving arithmetic problems with a simulated calculator in which certain keys are inconveniently disabled. 
Simulation 
Whole Number Operations 
46 
Where in the World is Carmen Sandiego 
Broderbund 
An exciting detective activity that uses geographic and historical clues to find a missing person. 
Simulation 
Math Reasoning 
46 
The Writing Center 
Learning Company 
An easy to use wordprocessing and desktop publishing program including clip art and a thesaurus. 
Tool 
Problem Solving 
16 
World GeoGraph 
MECC 
A world geography data base with an easy graphing interface. 
Tool 
Problem Solving 
56 
References and Readings
Arithmetic Teacher Articles
Baker, D., Edwards, R., & Marshall, C. (1990). Teaching about exponents with calculators. Arithmetic Teacher, 38(1), 3840.
Bartalo, D. (1983). Calculators and problemsolving instruction: They are made for each other. Arithmetic Teacher, 30(5), 1821.
Battista, M., & Clements, D. (1990). Constructing geometric concepts in Logo. Arithmetic Teacher, 38(3), 1517.
Billstein, R., & Lott, J. (1986). The turtle deserves a star. Arithmetic Teacher, 33(7), 1416.
Bobis, J.. (1991). Using a calculator to develop number sense. Arithmetic Teacher, 38(5), 4245.
Bradley, C. (1992). The four directions Indian beadwork design with Logo. Arithmetic Teacher, 39(9), 4649.
Bright, G. (1988). Estimating numbers and measurements. Arithmetic Teacher, 36(1), 4849.
Brown, S. (1990). Integrating manipulatives and computers in problemsolving experiences. Arithmetic Teacher, 38(2), 810.
Calculators Focus Issue. (1987). Arithmetic Teacher, 34(6).
Campbell, P. (1988). Microcomputers in the primary mathematics classroom. Arithmetic Teacher, 35(6), 2230.
Comstock, M., & Demana, F. (1987). The calculator is a problemsolving concept developer. Arithmetic Teacher, 34(6), 4851.
Corbitt, M. (1985). The impact of computing technology on school mathematics: Report on an NCTM conference. Arithmetic Teacher, 32(8), 1418.
Dick, T. (1988). The continuing calculator controversy. Arithmetic Teacher, 35(8), 3741.
Dubitsky, B. (1988). Making division meaningful with a spreadsheet. Arithmetic Teacher, 36(3), 1821.
Edwards, N., Bitter, G., & Hatfield, M. (1990). Data base and spreadsheet templates with public domain software. Arithmetic Teacher, 37(8), 5255.
Harvey, J. (1991). Using calculators in mathematics changes testing. Arithmetic Teacher, 38(7), 5254.
Hoeffner, K., Kendell, M., Stellenwerf, P., Thames, P., & Williams, P. (1990). Problem solving with a spreadsheet. Arithmetic Teacher, 38(3), 5256.
Kantowski, M. (1983). The microcomputer and problem solving. Arithmetic Teacher, 30(6), 2021, 5859.
National Council of Teachers of Mathematics. (1987). A position on calculators in the mathematics classroom. Arithmetic Teacher, 34(6), 61.
Newton, J. (1988). From patternblock play to Logo programming. Arithmetic Teacher, 35(9), 69.
Parker, J., & Widmer, C. (1992). Statistics and graphing. Arithmetic Teacher, 39(8), 4852.
(1992). Computation and estimation. Arithmetic Teacher, 40(1), 4851.
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Schielack, J. (1990). A graphing tool for the primary grades. Arithmetic Teacher, 38(2), 4043.
Sgroi, R. (1992) Systematizing trial and error using spreadsheets. Arithmetic Teacher, 39(7), 812.
Starkey, M. (1989). Calculating first graders. Arithmetic Teacher, 37(2), 67.
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Thompson, C, & Van de Walle, J. (1985). Patterns and geometry with Logo. Arithmetic Teacher, 32(7), 613.
Thompson, V. (1992). How to win people and influence friends: Calculators in the primary grades. Arithmetic Teacher, 39(5), 1517.
Wiebe, J. (1989). Calculator memory and multistep problems. Arithmetic Teacher, 37(1), 4849.
Wiebe, J. (1990). Turtle tips. Arithmetic Teacher, 37(9), 2830.
Wiebe, J. (1990). Data base programs in the mathematics classroom. Arithmetic Teacher, 37(5), 3840.
Wilson, K. (1991) Calculators and word problems in the primary grades. Arithmetic Teacher, 38(9), 1214.
Additional References and Readings
Allinger, G. (1985). Percent, calculators, and general mathematics. School Science and Mathematics. 85(7), 567573.
Becker, H. (1990). How computers are used in the United States schools: Basic data from the 1989 I.E.A. computers in education survey. Baltimore, MD: Johns Hopkins University, Center for the Social Organization of Schools.
Bell, M. (1978). Calculators in secondary mathematics. Mathematics Teacher, 71(5), 405410.
Billstein, R., Libeskind, L., & Lott, J. (1985). MIT Logo for the Apple (Terrapin/Krell). Menlo Park, CA: Benjamin/Cummings Publishing Co.
Bitter, G. (1982). The road to computer literacy, Part III: Objectives and activities for grades 4.6. Electronic Learning, 2(3), 4448, 9091.
Bitter, G., & Hatfield, M. (1992). Integration of the Math Explorer™ calculator into the mathematics curriculum. Monograph #1. Technology Based Learning and Research, Arizona State University.
Bitter, G., & Mikesell, J. (1990). Using the Math Explorer™ calculator: A sourcebook for teachers. Palo Alto, CA: Dale Seymour Publications.
California Department of Education. (1992). Mathematics framework for California public schools. Sacramento, CA: California Department of Education.
Campbell, M., & Fenwick, J. (1985). Exploring with Logo. Newton, MA: Allyn & Bacon.
Drescoll, M. (1981). Research within reach: Elementary school mathematics. Reston, VA: NCTM.
Dossey, J. (1990). Transforming mathematics education. Educational Leadership, 47(4), 2224.
Educational software preview guide. (1986). Educational Software Evaluation Consortium. Teacher Education and Computer Center, Region 15, San Diego County Office of Education, San Diego, CA.
Graves, D. (1978). Balance the basics: Let them write. New York: Ford Foundation.
Hansen, V. (Ed.). (1984). Computers in mathematics education: 1984 Yearbook. Reston, VA: National Council of Teachers of Mathematics.
Hembree, R., & Dessart, D. (1986). Effects of handheld calculators in precollege mathematics education: A metaanalysis. Journal for Research in Mathematics Education, 17, 8399.
Hope, J., & Sherrill, J. (1987). Characteristics of unskilled and skilled mental calculators. Journal for Research in Mathematics Education, 18(2), 98111.
Immerzeel, I., & Ockenga, E. (1977). Calculator activities for the classroom. Palo Alto, CA: Creative Publications.
Lathrop, A. (1982). The terrible ten in educational computing. Educational Computer, 2(5), 34.
Levin, J., Riel, M., Boruta, M., & Rowe, R. (1985). Muktuk meets Jacuzzi: Computer networks and elementary schools. In S. Freedman (Ed.),The acquisition of written language: Revision and response (pp. 160171). New York: Ablex Publishing Company.
Levin, J., & Souviney, R. (1983). Computers: A time for tools. Newsletter of the Laboratory of Comparative Human Cognition, 5(3), July.
Leuhrmann, A. (1982). Part V. Computer literacy, what it is; why it’s important. Electronic Learning, 1(5), 20,22.
Mehan, H., MillerSouviney, B., Riel, M., Souviney, R., Whooley, K., & Liner, B. (1986). The write help. Glenview, IL: Scott, Foresman and Company.
Mercer, C., & Mercer, A. (1985). Teaching students with learning problems. Columbus, OH: Charles E. Merrill Publishing Co.
Miller, S., & Thorkildsen, R. (1983). Getting started with Logo. Allen, TX: Developmental Learning Materials.
MillerSouviney, B. (1985). Computer supported tools for expository writing: One computer – twentyeight kids. Unpublished master’s thesis, University of California, San Diego.
Moore, M. (1984). Logo discoveries. Palo Alto, CA: Creative Publications.
Moore, M. (1984). Geometry problems for Logo discoveries. Palo Alto, CA: Creative Publications.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York: Basic Books.
Riel, M. (1982). Computer problem solving strategies and social skills of linguistically impaired and normal children. Dissertation Abstracts International, #8216102. (University Microfilms No. 821b, 102).
Suppes, R., & Morningstar, M. (1972). Computerassisted instruction at Stanford, 19661968: Data, models, and evaluation of arithmetic programs. New York: Academic Press.
Suydam, M. (1979). The use of calculators in precollege education: The state of the art. Columbus, OH: Calculator Information Center.
Usiskin, Z. (1978). Are calculators a crutch? Mathematics Teacher, 71(5), 412413.
VanDevender, E, & Rice, D. (1984). Improving instruction in elementary mathematics with calculators. School Science and Mathematics, 84(8), 633643.
Wheatley, G., & Shumway, R. (1979). Impact of calculators in elementary school mathematics: Final report. Washington D.C.: National Science Foundation.
Witte, C. (1984). Simple computer programs: My first programs. Torrance, CA: Schaffer.
PART II
Mathematics Strands
Part II addresses the mathematics strands and teaching practices recommended for Grades K6. The following topics are included:
• Cognitive abilities related to each mathematics topics.
• Mathematics concepts and skills with examples of appropriate teaching methodology.
• Classroom activities appropriate for use with elementary students.
• Problem solving experiences appropriate for use in elementary classrooms.
• Calculator and computer activities.
• Alternative assessment strategies.
• Ways to adapt instruction for children with special learning needs.
• Sample lesson plan.
• Course activities.
• Related children’s books, software, Arithmetic Teacher references, and additional readings.