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Research Base of

By J. Jeffrey Richardson, Ph.D.

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Table of Contents

Purpose and Overview .............................................................................. 1 Number Fact Skill Acquisition ................................................................... 3 Computers, Learning, and Instruction ........................................................ 9 Instructional and Management Features of Math FACTMASTER? Online ....................................................... 12 Field Evaluation of the Math FACTMASTER Program .......................... 16 Summary ................................................................................................. 21 References ............................................................................................... 22

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Purpose and Overview

ath FACTMASTER? Online is a computer-based, online environment designed to help students in grades 1–6 achieve—and their teachers to manage—the goal of number fact mastery.

Math FACTMASTER Online provides a learning environment for the student and a management system for the teacher.

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Math FACTMASTER Online provides a learning environment for the student and a management environment for the teacher. The learning environment is tailored to each student to help him or her master the basic number facts. The management system is designed for classroom teachers, school principals, and district superintendents to help them manage student mastery in order to meet established performance goals. Mastery of the basic arithmetic number facts is an important stage in mathematics development because it frees the student’s attention for focusing on higher-order concepts, such as multi-digit computation and problem solving. Mastery means immediate recall, fluency—automaticity—in responding to a given number fact problem with the correct answer. The ability to recall a fact from memory means that the student does not need to spend conscious attention in obtaining the answer.

Math FACTMASTER Online is based on a key research finding that people learn by linking new knowledge to what they already know.

Two key elements in the learning environment are essential to efficient learning and reliable determination of mastery. First, response time must be measured. And a cutoff must be set that is short enough to indicate mastery. Three seconds, given allowances for keyboarding in the answer, is considered a reasonable standard measure for mastery. It is too short to allow the student to count out or think through the correct answer and thus assures that the fact is recalled from memory. Second, the learning environment needs to help students learn and commit facts to memory. Math FACTMASTER Online is based on a key research finding of cognitive psychology—people learn by linking new knowledge to what they already know. In applying this finding, Math FACTMASTER Online tracks which facts a student has mastered. The software then carefully selects a new fact for the student to work on that is related to a fact the student knows by one of the common thinking strategies covered in standard classroom textbooks and discussed later in this document.

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Math FACTMASTER? Online also gives students practice in the higher-order task of problem solving by presenting story problems using mastered facts. Since the student’s attentional resources are not distracted by wrestling with producing the number fact, he or she can concentrate on problem solving. Math FACTMASTER Online management system makes it feasible to individualize number fact mastery for the whole class, throughout the entire school year, and from year to year. Math FACTMASTER Online provides simple, at-a-glance reports that detail the level of mastery and rate of progress for the student, class, or school. Math FACTMASTER Online provides educational accountability.

A controlled evaluation of the Math FACTMASTER program predicts achievement gains beyond normal growth of 66% for 1st & 2nd grades and 134% for 3rd grade.

A controlled evaluation of use of the Math FACTMASTER program predicts grade level achievement gains above and beyond normal expected growth of 66% for first and second graders and 134% for third graders.

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Number Fact Skill Acquisition

B

efore children are ready to master the basic number facts, they first must understand the underlying concepts of number and number operations (addition, subtraction, multiplication, and division). This understanding is rooted in everyday life before children begin school as they encounter innumerable concrete instances involving quantity, counting, equality, conservation of number, and reversibility. As the meaning of the whole numbers and their operations are further developed in the early grade school arithmetic lessons, students are ready to tackle mastery of the basic number combinations.

The Instructional Objective of Number Fact Mastery

Children will use their understanding to develop quick recall. —2006 NCTM Standard

National Standards for Number Fact Mastery The National Council of Teachers of Mathematics (NCTM, 2006a), in their document, Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, renews and restates the instructional objective that students master the basic number combinations. Specifically: ? Grade 2. “Children use their understanding of addition to develop quick recall of basic addition facts and related subtraction facts” (p. 14). ? Grade 4. “Students use understandings of multiplication to develop quick recall of the basic multiplication facts and related division facts” (p. 16). The emphasis on “quick recall” is not new to the NCTM or to the field of mathematics education. In fact, the president of NCTM recently wrote, “[l]earning the basics is certainly not ‘new marching orders’ from the NCTM, which has always considered the basic computation facts and related work with operations to be important” (NCTM, 2006b). In fact, prior NCTM standards clearly state: ? “In grades K–4, students can model, explain, and develop reasonable proficiency with basic facts and algorithms. Children should master the basic facts of arithmetic. Practice designed to improve speed and accuracy should be used, but . . . only after children have developed an efficient way to derive the answers to those facts” (NCTM, 1989, p. 38). ? “By fluency we mean that students are able to compute efficiently and accurately with single-digit numbers” (NCTM, 2000, p. 84).

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Why Are These Standards Important? Learning is cumulative and hierarchical. Larger concepts are built out of smaller ones. Because human attention is limited, in order to learn higherlevel concepts and skills, the amount of attention devoted to processing the lower-level sub-skills needs to be limited. In this way, arithmetic and reading are very similar. In reading, the underlying skills of phonetic awareness, phonics, and fluency must be mastered before comprehension can be the focus. In arithmetic, number sense, counting, the properties of combining numbers, and mastery of the number facts must be achieved before higherorder algorithms, such as multi-digit arithmetic, can be the focus. In sum, mastery of the basic number combinations makes written and mental multidigit calculations possible.

Learning the Number Facts

Readiness Most children entering school can solve word problems involving addition, subtraction, multiplication, and division by counting and “acting out” the problems (Carpenter, Ansell, Franke, Fennema, & Weisbeck, 1993).This demonstrates that an inherent mathematical competence is already present even in young children. At this stage, children begin to discover important relationships that underlie the number combinations that help them to remember the basic facts (Baroody, 1989).This is reinforced by classroom activities of joining, separating, and comparing sets of objects and begins the progression to learning the basic facts (Cathcart, Pothier,Vance, & Bezuk, 2001). Normal Developmental Sequence Children do not move from knowing nothing about the basic number facts to having them memorized. In fact, the basic number facts do not need to be memorized mechanically. Substantial international research now indicates that children move through a progression of different procedures to generate the answers for the basic addition facts (Fuson, 1992a). “Children begin with a conceptual understanding of number and the meanings of the operations. They develop increasingly sophisticated representations of the operations such as counting-on or counting-up procedures as they gain greater fluency. They also lean heavily on reasoning to use known answers such as doubles to generate unknown answers”(Kilpatrick, Swafford, & Findell, 2001, p. 195). As they get older, they use these methods more efficiently (Baroody, 1999a, 1999b).

Substantial international research now indicates that children move through a progression of different procedures to generate the answers for the basic addition facts.

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All children develop some intermediate and temporary procedures for producing answers to the basic number facts. Recall eventually becomes the predominant method, but the research has not distinguished between answers produced by recall and those generated by fast, non-recall procedures (Baroody, 1999a, 1999b). The various stages of non-recall procedures are: count-all, count-on, reason the answer using various thinking strategies, recall the answer or produce it very efficiently through procedural knowledge. Counting becomes abbreviated and rapid and students begin to develop procedures that take advantage of properties of arithmetic to simplify computation. Focused practice given this type of understanding then leads to rapid recall of the facts. To summarize, there are three steps to fact mastery: Step 1. Understanding the meaning of the operations, Step 2. Using thinking strategies that relate unknown facts to known facts, and Step 3. Developing immediate recall through drill and practice. The Role of Thinking Strategies as a Key to Meaningful Memorization The employment of thinking strategies by students is a key developmental step toward mastery of the basic number facts. Again, thinking strategies provide structure to organize the facts so that recall is easier. The key to thinking strategies is recomposing numbers into other numbers thereby turning a new and novel number combination into one that is easily related to and one that is familiar and known. Thinking strategies are based on the basic properties of arithmetic, which form a rich set of associations among the number facts. Students spontaneously discover and use these relationships to efficiently produce the basic number facts. Using thinking strategies helps develop a deep understanding of mathematics. Learning about the web of associations among the number facts helps in producing the answer to each and eventually assists in automating mental processes to the point of automatic recall, the goal for the number fact learning objective.

The thinking strategies are not arbitrary “tricks,” but rather manifestations of the fundamental properties of the whole number system.

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COMMON THINKING STRATEGIES FOR ADDITION ? Commutativitity (4 + 3 = ?, think 3 + 4 = ?) ? Counting-on from one addend when the other addend is small (+ 1, + 2) ? Relating a fact to the easily learned “doubles” (by counting-on or taking away from one of the doubled addends, 5 + 4 is like (1 + 4) + 4 is like 4 + 4 + 1 more) ? “Tens-frame,” where the addends are rewritten so that one of the addends becomes 10 (5 + 8 = 3 + 2 + 8 = 3 + 10 = 13) ? “Fact families,” where the four facts involving any pair of numbers are associated together (4 + 2 = 6, 2 + 4 = 6, 6 – 2 = 4, and 6 – 4 = 2) Arthur J. Baroody (1989), in A Guide to Teaching Mathematics in the Primary Grades, makes a number of informative observations about the thinking strategies: ? Teaching, using thinking strategies, makes learning the basic number combinations more interesting, validates children’s informal mathematical thinking, and makes basic arithmetic and number-fact mastery less threatening and overwhelming for children. ? A thinking-strategies approach “illustrates in a personally meaningful way that arithmetic problems can be solved in various ways.” ? A thinking-strategies approach encourages children to look for relationships and reason about numbers. ? A thinking-strategies approach more accurately reflects the nature of mathematics than does a rote memorization approach. ? A meaningful approach to number-fact mastery is more likely to foster analytic-thinking and problem-solving skills than a drill approach. ? Thinking strategies are children’s informal way of exploiting relationships or patterns. For example, children use easier facts to figure out and then remember harder facts. ? Mastering the number combinations is essentially a building process. Children master many new combinations by “seeing” how they are connected (related) to previously learned combinations (Baroody, 1987).

Instruction organized around thinking strategies has been shown to be very effective in promoting mastery of basic facts.

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Because they are based on the properties of arithmetic, thinking strategies can be found for each of the four operations. Subtraction becomes related to addition through understanding of the part-part-whole relation and its reversibility. For addition and multiplication, students invent strategies in which they derive related products from products they know (Kilpatrick, Swafford, & Findell, 2001, p. 191). Rathmell (1978) described several thinking strategies that children use with learning multiplication and division basic facts such as: repeated addition, skip counting, splitting the product into known parts, one-more-set, twice-as-much, facts-of-five, and pattern strategies (5-times table ends in 0 or 5, 9-times table answer digits sum to 9, etc.). In programs incorporating the fact-families strategy, addition and subtraction are taught together. The addition facts for a family normally are learned first, then the subtraction (Cathcart, Pothier,Vance, & Bezuk, 2001, pp.153–154). In spite of their power, thinking strategies should not become another memorize-math-by-rote routine. “One danger is that children might learn strategies by rote, so that mindless memorization is replaced by equally mindless ‘strategies’ ” (Cobb, 1985, pp. 141).

Children’s invention of thinking strategies should be applauded. After all, it is a sign that a child is really thinking about mathematics.

“The invention of thinking strategies should be applauded. After all, it is a sign that a child is really thinking about mathematics”(Baroody, 1989, p. 243). In fact, finding and describing patterns are a hallmark of mathematical thought. One technique is to have students verbalize and share strategies they invent or prefer for a given problem. This takes advantage of the notable variation in the procedures children use to solve simple addition problems (Carpenter & Moser, 1984; Fuson, 1992a, 1992b). While students spontaneously generate thinking strategies, some procedures can be taught that students might not otherwise think to use. Explicit teaching on the use and selection of thinking strategies is necessary (Cathcart, Pothier,Vance, & Bezuk, 2001, p.149). It has been shown that students can learn new strategies and generalize them to new facts (Fuson & Secada, 1986; Leutzinger, 1979; Steinberg, 1985; Thornton, 1978). Exercises that explore the thinking strategies enable students to focus on the strategies, reasoning, and patterns rather than only on the answer (Cathcart, Pothier, Vance, & Bezuk, 2001, p. 160). Specifically, “Instruction that is organized around thinking strategies has been shown to be very effective in promoting mastery of basic facts” (p. 152).

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Drill on arithmetic facts does not necessarily lead to recall. Drill must be preceded by sound instruction.

The Final Step—Committing the Number Facts to Memory First, a strong caveat on this final stage of mastering the number facts: drill on arithmetic facts does not necessarily lead to recall (Brownell & Chazal, 1935). In spite of drill, children tend to maintain whatever procedures have satisfied their number needs. “Practice is important, but it is not effective if other understandings and skills are not already in place” (Cathcart, Pothier, Vance, & Bezuk, 2001, p. 148). Drill does not supply children with more mature ways of dealing with number combinations. Drill must be preceded by sound instruction (Brownell & Chazal, 1935). Students should not begin drill-and-practice on a given fact until they understand the fact and have developed a habit of using an efficient thinking strategy to produce the answer to the problem. However, even as students take advantage of meaningful relationships between the number facts to answer them correctly, developing fluency in producing answers requires practice. Drill should be targeted on those problems that are actively being learned (Brownell & Chazal, 1935). As more time is spent practicing a given problem, the problem stimulus and the strategy both become cues for retrieval of the correct answer from long-term memory. Drill activities should include an approximately equal mix of facts children have and have not yet learned.

Skill in working with the number facts grows as they solve more problems.

Memorization and Problem Solving Are Linked The National Research Council (Kilpatrick, Swafford, & Findell, 2001) supports the connection between memorization and problem solving. At the root of number-fact fluency lies the conceptual understanding and the ability to identify and accurately represent situations in which the combinations (whether adding, subtracting, multiplying, or dividing) are required (p. 190). One of the most meaningful contexts for young children to begin to develop proficiency with the number facts is working with word problems. Skill in working with the number facts grows as they solve more problems (p. 183). “Providing word problems as context for adding and discussing the advantages and disadvantages of different addition procedures are ways of facilitating students’ problem-solving ability and improving their understanding of the addition process” (p. 191). Studies have found that when instruction is focused on problem solving, children not only become better problem solvers but they also master more combinations than children whose instruction focused on drill and practice alone (p. 219).

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Computers, Learning, and Instruction

It is now possible not only to provide individualized, interactive, learnercentered instruction, but also to record, track, and report instructional progress.

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he merger of the PC with the Internet has made possible a reemergence of computer-assisted instruction: individualized learning involving students interacting with a computer program. Coupled together— instructional software and web-accessible records and reports—a new horizon in educational computing has been reached. It is now possible not only to provide individualized, interactive, learner-centered instruction, but also to record, track, and report instructional progress. Just as many people now have personal Internet-based access to their banking and shopping records, school administrators can now maintain detailed “accounts” of individual achievement and use this information to help every student, class, and school succeed in meeting established educational goals.

Instructional Design

Instructional design has to do with how the computer interacts with students to achieve stated learning outcomes. While computers began their classroom role following the stimulus-response paradigm of James Skinner and behaviorist psychology, over the past two decades, this approach has been replaced with the modern, cognitive science approach to learning and instruction. Learning is seen as an active, constructive process where new information is connected to prior knowledge. Above, in the section on number fact skill acquisition, we reviewed how this model of human cognition has impacted theories of learning and instruction. Instructional design for computer-based learning simply reflects this understanding in the details of how learning environments are constructed. Evolving beyond the “programmed learning” of the 1960s, past “page turning” software of the 70s, and the games of the 80s and 90s, the current design goal of modern instructional software is to act as an intelligent tutor. The field of Intelligent Tutoring Systems (ITS) is the most advanced form of computer-based instructional design (Burns & Capps, 1988). Math FACTMASTER? Online follows the design principles of ITS. An ITS software program contains three key modules: 1. The expert module, which captures the competence of expert performance. In Math FACTMASTER Online, expert knowledge is represented and reflects what we actually know about how people learn and retrieve the number facts. As described above, this is a progression from counting, to thinking strategies, to rapid recall.

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2. The student model, which captures the state of the learning both as a subset of expert performance and as common inexpert behaviors to be unlearned. In Math FACTMASTER? Online, the student’s expert performances and inexpert behaviors are captured in the mastery table that encodes what the student knows: which facts are mastered, which are not, which are untried, and which of these are currently the focus of instruction. 3. The tutor module, which incorporates the pedagogical skills expert teachers use in teaching a specific instructional objective. In Math FACTMASTER Online, this is captured by a number of design principles (further described later), including: fluency or automaticity as the mastery criterion, the use of thinking strategies as the way to introduce new facts, the provision of a rich variety of problem formats, and the continual review of mastered facts. In the next section, the instructional features of Math FACTMASTER Online are explained in detail. But first, a discussion of research findings about technology and instruction is presented.

Research Findings About Technology and Instruction

A recent authoritative review of learning from technology-assisted instruction (Fletcher, 2003) cites the following general lessons based on four decades of educational research on the use of computers in instruction: ? Technology can be used to teach. Studies that show “significant difference” in learning outcomes focus on the progress of students, using no other instructional input than the “time on task”—time students spend using the software. One study of mathematics instruction with 69 Native American and 297 hearing-impaired students showed gains in performance on a standardized test could be predicted almost exactly based on time on task—to the nearest tenth of a grade placement based on total time in the instruction. ? Technology can be used to increase instructional effectiveness. The control vs. experiment treatment paradigm often results in findings of “no significant difference” because this design has inherently poor statistical power in educational settings due to small sample and effect sizes. However, a meta-analysis of 233 studies showed an average computer treatment effect equivalent to moving a 50th percentile student to the 65th percentile. Intelligent tutoring systems have produced effect sizes equivalent to 50th to 80th percentile.

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Because instructional software can tailor instructional sequence, pace, difficulty, content, and style to the needs of the individual student, fewer students are left behind.

? Technology can be used to ensure that all students learn. Because instructional software can tailor instructional sequence, pace, difficulty, content, and style to the needs of the individual student, fewer students are left behind. This is especially important in today’s world of annual classroom and school performance assessments. ? Technology can be used to reduce time needed to reach instructional objectives. Time is saved when students do not have to spend time on material they already know. An analysis of time savings in military training suggests an average of about 30% reduction in time to reach instructional objectives. ? Students prefer technology-based instruction. Early studies have found that students prefer technology-based instruction (Greiner, 1991). This statement hardly requires validation in today’s technology-savvy generation.

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Instructional and Management Features of Math FACTMASTER? Online

Math FACTMASTER Online is a learning tool for the student and a management tool for the teacher.

Math FACTMASTER Online is a learning tool for the student and a management tool for the classroom teacher, school principal, and district superintendent. The interactive capability of computer software has enabled Math FACTMASTER Online to provide a number fact mastery environment tailored to the needs of individual students since the program was first developed in 1986. More recently, the networking of computers has made it possible to collect, aggregate, and report information about individual students, classrooms, and schools to provide a management environment to support achievement of educational goals by teachers, principals, and superintendents. This section reviews both the instructional and the management capabilities of Math FACTMASTER Online.

Instructional Features

Allows for Access Through Any Internet-Connected Computer The major innovation of Math FACTMASTER Online is that a student can access Math FACTMASTER Online from any Internet-connected computer that has the run-time environment installed. This makes it possible for the student to access the program from any computer—whether he or she is in a classroom, a computer lab, or a library. The run-time environment may be downloaded from the web so students also may use the program at home. In all cases, no matter where Math FACTMASTER Online is accessed, the student’s personal data is retrieved over the Internet and used to control the presentation of practice sessions. Once a session is complete, the results of the session automatically upload over the Internet and are securely stored for future use. Complies with National Standards Math FACTMASTER Online assists both students and teachers in achieving the new 2006 National Council of Teachers of Mathematics standards for number fact acquisition: 1. students use understanding to arrive at solutions to the basic number fact solutions, and 2. students practice until they are proficient at rapidly recalling the answers from memory. The pedagogically-sound and research-based instructional features employed by Math FACTMASTER Online are designed to meet these standards.

In all cases, no matter where Math FACTMASTER Online is accessed, the student’s personal data is retrieved over the Internet and used to control the presentation of practice sessions.

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Judges Mastery by Response Time The criterion for mastery of the number facts is automatic recall. Simply getting the correct answer—by counting, reasoning, or even recall—is not sufficient. The goal is retrieval of answers fluently, automatically, and without conscious effort. The reason for this goal is that it frees the finite human capacity to focus attention for application to higher-level skills, such as multi-digit arithmetic or problem solving. Adults employ a nearly error-free fact retrieval process and do not reconstruct or calculate the number facts (Aschraft & Bataglia, 1978).

Math FACTMASTER Online judges mastery based on response time.

Math FACTMASTER? Online judges mastery based on response time (3 seconds is considered a good benchmark of automaticity). The response time cutoff may be set faster or slower at the discretion of the teacher to accommodate individual differences. Identifies Fluent Facts In order to manage number fact acquisition, one of the tasks Math FACTMASTER Online accomplishes is to identify which facts are mastered or not. When a fact is first presented, the program notes whether it is mastered or not. If it is mastered, it becomes part of the pool of mastered facts. If it is not yet mastered, it becomes a part of the pool of “placed” facts. These placed facts are matched to mastered facts by simple thinking strategies to select new facts for the student to work on. Builds on Student’s Prior Knowledge—The Thinking Strategies The basic tenant of modern cognitive psychology is that learning is a process of relating new knowledge to old, of building relationships to what we know (Glass, Holyoak, & Santa, 1979). Because of this, what determines how fast we learn something is not the number of times we practice it but how the practice helps us organize the new knowledge with what we already know (Kintsch, 1977). Rote memorization fails to recognize this basic characteristic of human memory. This principle, when applied to learning the number facts is usually called a “thinking strategies approach.” As described previously, numerous studies have shown that teaching the thinking strategies approach facilitates the learning and retention of the basic number facts, particularly the so-called harder facts. “A strategies approach helps students organize the facts in a meaningful network so that they are more easily remembered and accessed” (Isaacs & Carroll, 1999). It has been shown that children can generalize a thinking strategy rule to new problems after being taught to use the rule.

A strategies approach helps students organize the facts in a meaningful network so that they are more easily remembered and accessed.

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A set of thinking strategies is employed by Math FACTMASTER? Online in selecting new facts. These strategies are typically included in arithmetic basals. A very common strategy is “commuted pairs” (3 + 5 = 5 + 3). “Counting on” and the “tens-frame” are other common textbook thinking strategies for addition. A common strategy for multiplication is “doubles” (6 × 7 = 3 × 7 doubled). For subtraction and division, Math FACTMASTER Online uses the familiar “fact family” strategy. Provides Continuous Review During the learning process, continued practice is needed to maintain fluency of mastered facts. Therefore, Math FACTMASTER Online includes mastered facts in every daily practice session. Fluency is maintained by always selecting the slowest of the set of mastered facts for students to practice. Varies Problem Format Varying the problem stimulus helps people learn. Having variety and richness in how problems are presented gives the mind more cues to use in retrieving facts from memory (Martin, E., 1972). During an interactive computer session, Math FACTMASTER Online presents the number facts in four contexts: standard fact sentence, open sentence, true/false, and story problems. Story problems are an especially important learning format because they involve the higher cognitive skill of applying knowledge. Finally, Math FACTMASTER Online provides daily, off-line practice “homework” worksheets tailored to the student’s stage of learning.

Teachers and administrators may access the Teacher Options and Reports of Math FACTMASTER Online from any Internet-connected computer.

Management Features

Reports and Teacher Options Are Accessible to Teachers and Administrators from Any Internet-Connected Computer With Math FACTMASTER Online, the ability to access reports, enroll students, and manage student settings is now independent of location. Teachers and administrators may access the Teacher Options and Reports of Math FACTMASTER Online from any Internet-connected computer— whether he or she is in the classroom, a school office, or at home. No runtime environment is needed to manage teacher options and to view reports. Settings Meet Individual Needs Teachers can adjust the Math FACTMASTER Online learning environment to meet individual needs. The two most important settings are number of problems per session and the mastery cutoff. Using these settings, a teacher can adjust Math FACTMASTER Online to be a successful learning experience for all learning abilities.

Teachers can adjust the Math FACTMASTER Online learning environment to meet individual needs.

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Math FACTMASTER Online provides management reports to help teachers, principals, and superintendents manage progress in the number facts.

Individual Reports Since Math FACTMASTER? Online works by maintaining a record of what the student knows (the “student model”), this information is available to the teacher and student. These reports include the Mastery Table and the Response Time Table. There is a Mastery Table report and Response Time Table report for each one of the four operations (addition, subtraction, multiplication, and division). Classroom Reports Teachers are faced with managing the progress of all students in the classroom. To support this, Math FACTMASTER Online provides a report showing how each student is doing. This report lists every student with percent mastery for each of the four operations and recent progress. The report can be sorted, so that the teacher can instantly retrieve, for example, the names of students who have not progressed sufficiently in one of the four operations. Once identified, the teacher can then focus help for this group. School and District Reports Principals can track and manage progress in number fact mastery for all classrooms in their school through the All School report. This report shows the average percent mastery of each of the four operations and recent progress for each classroom group. A similar report would summarize school performance at the district level.

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Field Evaluation of the Math FACTMASTER? Program

During the spring semester of 2004, two Boulder, Colorado schools in the Boulder Valley School District participated in an experimental vs. control evaluation of the Math FACTMASTER program. These two schools (Foothills Elementary and Flatirons Elementary) were very similar, both having two or three rounds of self-contained, teacher-led classrooms. They were similar in socioeconomic makeup and standardized test performance in math. A total of 323 students at Foothills and 243 students at Flatirons participated in the study. Participation was roughly equal for boys and girls, and across grade levels first through fifth. A criterion-referenced, penciland-paper, timed test was administered to both groups at the beginning and end of the study. The first and second grade test lasted 4 minutes and had 90 addition and subtraction problems of progressing difficulty. The third, fourth, and fifth-grade test lasted 8 minutes and had 180 problems covering all four operations of progressing difficulty. For the experimental treatment, students at Flatirons used the Math FACTMASTER program, a software program from CURRICULUM ASSOCIATES?, Inc., for supplemental, individualized drill and practice in the basic number facts. For the control treatment, teachers at Foothills followed their normal classroom curriculum practices for the number facts.

Before the Experiment, How Did Grade Levels Compare?

The results1 below show how grade levels compared (i.e., all first graders vs. all second graders, etc.) at their respective schools in their performance on the pre-test (scores indicate total correct). Table 1: Pre-Test Grade Level Scores—Total Correct

Foothills (Standard Mean Deviation) 14.9 (1.3) 29.5 (1.8) 78.1 (4.1) 106.4 (3.5) 115.7 (4.3) Flatirons (Standard Mean Deviation) 9.9 (1.9) 24.5 (1.9) 59.1 (5.2) 87.8 (6.1) 125.6 (4.2)

Grade First Second Third Fourth Fifth

Max. Score 90 90 180 180 180

Note the increase in average score from one grade level to the next. The change from one grade to the next is one measure of student progress. A gender effect was found at both schools for first and second grade—boys on average scored 6.2 points more than girls. The effect was not found for third grade and beyond.

1

Statistical analyses conducted by the Statistics Consulting Service, University of Colorado at Denver.

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How Often Was the Math FACTMASTER? Program Used?

Table 2 shows how many software sessions were completed by grade level. A session was comprised of 10 timed number fact sentences tailored to the student’s unique number fact mastery history. Second and fourth grades used the program the most often. The software supports either {addition/ subtraction} or {multiplication/division}, or both. First, second, and third graders used mostly {addition/subtraction}. Fourth-graders used both versions. Fifth graders used mostly {multiplication/division}. Table 2: Software Use at Flatirons—Number of Sessions

Grade First Second Third Fourth Fifth Min. 3 3 3 6 0 25%-tile 7 22.75 14.5 23 28 Median 11 54 23 44 37 Mean 15.9 57.3 23.5 47.4 40.6 75%-tile 24 85.5 30.5 72 46.5 Max. 47 135 57 109 111

How Did the Two Schools Compare in Pre/Post Test Gains?

Pre/post test gains were greater at Flatirons (experimental) than at Foothills (control), except for fifth grade, as shown in Table 3. All differences were statistically significant. Since the software was only used at Flatirons, other between-school differences may have accounted for this outcome. However, as noted previously, the schools were very similar. Table 3: Between-School Differences in Pre/Post Test Gains

Grade First Second Third Fourth Fifth Foothills (Control) 5.67 1.73 -1.97 -1.85 18.14 Flatirons (Experiment) 9.59 8.41 11.10 19.12 11.93 Diff. 3.92 6.68 13.07 20.97 -6.21 df t-value (pooled) p-value 2.22 111 .0282 4.05 110 .0001 3.48 97 .0008 6.33 107 <0.0001 -1.88 106 .0632

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The effect of daily use of the Math FACTMASTER? program on annual achievement was estimated: 1st & 2nd grade–66% above normal grade-level growth, 3rd grade–134% above normal grade-level growth.

How Did Amount of Software Use Impact Pre/Post Test Gains?

First and Second Grade. Each completed software session resulted in a 0.054-point improvement in total correct score above and beyond any improvement expected due to other factors (p<.0001). Extrapolating usage to one session per day for a 180-day school year, we would expect 10 points (95% confidence interval is 5.84 to 13.60) additional improvement, a 66% improvement above and beyond normal first-to-second grade growth shown in Table 1. (It should be noted that this extrapolation is beyond the scope of the data, see Table 2.) Third Grade. Each completed software session resulted in a 0.373-point improvement; with an extrapolated annual improvement of 67 points, equal to the grade-level growth at Flatirons from third to fifth grade (refer to Table 1, previous page). Fourth and Fifth Grade. Neither grade showed a significant effect due to software usage for either total correct or total attempted. A number of fourth and fifth graders scored near the perfect-score level of 180, so it is possible that for some fifth-grade students, the improvement in total correct was hindered by an insufficient number of questions on the exam or too much time to complete it. These results are summarized in Table 4 below. The expected annual increase for first, second, and third-grade growth is shown in Figure 2. Table 4: Pre/post Test Gains with Math FACTMASTER Extrapolated to a Full School Year (Daily Use for 180 Days)

Math Math FACTMASTER FACTMASTER Program Average Program Annual Gain Per Session Gain 0.054 9.59 0.373 67.00 -1.85 n.s. Annual Math FACTMASTER Program Effect 66% 134% n.s.

Grade 1st & 2nd 3rd 4th & 5th

Normal Annual Gain 14.6 34.6 28.7

n.s. = not significant.

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Figure 2. Additional Annual Progress Toward Mastery

Additional Annual Progress toward Mastery Due to Math FACTMASTER

(Extrapolated to Daily Usage)

120 Pre/Post Test Gain 100 80 60 40 20 0 MFM Normal 1st Grade MFM Normal 2nd Grade Normal 3rd Grade MFM

MFM = Math FACTMASTER Program

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Faculty Views Regarding Number Fact Acquisition

Twenty-three faculty at the two elementary schools responded to a series of 10 questions regarding their views of number fact acquisition. The results are presented below in summary form. Teachers were asked to respond to 8 of the questions on a scale from strongly agree to strongly disagree (scored as from 1 to 5 respectively). The results are shown in Table 5. Teachers also provided written comments, including: Drill is not appropriate developmentally for first graders, especially considering their need to be exposed to many math concepts, hands-on investigation, and problemsolving activities. Memorization of facts does not lead to meaningful application of these facts. Weekly time allocated to number fact mastery is often distributed throughout an integrated curriculum. Long-term retention was cited as an issue of concern. Table 5: Faculty Views Concerning Number Fact Acquisition

1 = Absolutely 2 = Pretty Much 3 = Somewhat 4 = Not Really 5 = Absolutely Not Average Response

Survey Questions

(N = 23 Teachers)

Is number fact acquisition an element of the curriculum for your grade level during some or all of the school year? Is the level of number fact acquisition and mastery uniform across all students in the class? Do some students still need work in this area? Do you know which students are better/worse in mastery? Do you know specifically what each student’s problem areas are? Do you individualize instruction to focus on each student’s needs? Could you confidently discuss with parents a student’s status in number fact mastery? Would a computer-based learning system that identifies mastery level of students and provides drill and practice individualized to each student be of use to you and your class?

X X X X X X X X

1.8 4.3 1.7 1.7 2.0 1.9 1.8 1.6

0

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Summary

The most important feature of Math FACTMASTER? Online is that it helps students and teachers achieve the national standards for number fact mastery. This has been the purpose and objective of the Math FACTMASTER program since it was first published in 1986. The instructional design of Math FACTMASTER Online is supported by 20 years of research on number fact acquisition. The two most important findings are that 1. understanding plays a vital and integral role in the learning and mastery of number facts, and 2. the ultimate learning goal for number fact mastery is automatic fluency in number fact retrieval from memory. With the introduction of Math FACTMASTER Online, a dramatic advance has been achieved in the usability of the program and the accountability it provides. Usability is greatly enhanced through the ability of students to access the program from any Internet-computer. Accountability in learning outcomes is supported through provision of a powerful set of Internet-accessible classroom management tools that enable teachers and administrators to observe—at a glance—progress toward number fact mastery for a student, a group of students with similar needs, a classroom, or an entire school, or school district. A controlled experiment was conducted in 2004 involving two elementary schools and 566 students. The results provided evidence that the Math FACTMASTER? program improved student learning when compared to the traditional classroom treatment. The data also showed that each session with the Math FACTMASTER program contributed to growth. When usage was extrapolated to daily usage over the full school year, substantial gains above and beyond normal expected growth were found. Math FACTMASTER Online is an effective technology-based instructional tool that helps its users on all levels meet established learning goals.

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References

Aschraft, M. H., & Bataglia, J. (1978). Cognitive arithmetic: Evidence for retrieval and decision processes in mental addition. Journal of Experimental Psychology: Human Learning and Memory, 4, pp. 527–528. Baroody, A. J. (1987). Children’s mathematical thinking: A developmental framework for preschool, primary, and special education teachers. New York: Teachers College Press. Baroody, A. J. (1989). A guide to teaching mathematics in the primary grades. Boston: Allyn and Bacon, p. 235. Baroody, A. J. (1999a). Children’s relational knowledge of addition and subtraction. Cognition and Instruction, 17, pp. 137–175. Baroody, A. J. (1999b). The roles of estimation and the commutativity principle in the development of thirdgraders’ mental multiplication. Journal of Experimental Child Psychology, 74, pp. 157–193. Brownell, W. A., & Chazal, C. B. (1935). The effects of premature drill in third-grade arithmetic. Journal of Educational Research, 29, pp. 17–28. Burns, H., & Capps, C. G. (1988). Foundations of intelligent tutoring systems: An introduction. In M. C. Polson & J. J. Richardson (Eds.), Foundations of intelligent tutoring systems. Hillsdale, NJ: Lawrence Erlbaum. Carpenter, T. P., Ansell, E., Franke, M. I., Fennema, E., & Weisbeck, L. (1993). Models of problem solving: A study of kindergarten children’s problem-solving processes. Journal for Research in Mathematics Education, 24, pp. 328–441. Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15, pp. 179–202. Cathcart, W. G., Pothier,Y. M.,Vance, J. H., & Bezuk, N. S. (2001). Learning mathematics in elementary and middle schools, (2nd ed., p. 149). Upper Saddle River, NH: Merrill Prentice Hall. Cobb, P. (1985). Critique: A reaction to three early number papers. Journal for Research in Mathematics Education, 16, pp. 141–145. Fletcher, D. (2003). Evidence for learning from technologyassisted instruction. In F. F. O’Neil & R. S. Perez (Eds.), Technology applications in education, a learning view. Mahwah, NJ: Lawrence Erlbaum. Fuson, K. C. (1992a). Research on whole number addition and subtraction. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning, (pp. 243–275). New York: Macmillan. Fuson, K. C. (1992b). Research on learning and teaching addition and subtraction of whole numbers. In G. Leinhardt, R.T. Putnam, & R. A. Hattrup (Eds.), The analysis of arithmetic for mathematics teaching, (pp. 53–187). Hillsdale, NJ: Lawrence Erlbaum. Fuson, K. C., & Secada, W. G. (1986). Teaching children to add by counting-on with one-handed finger patterns. Cognition and Instruction, 3, pp. 229–260. Glass, A. L., Holyoak, K. J., & Santa, J. L. (1979). Cognition. Reading, MA: Addison-Wesley. Greiner, J. M. (1991). Interactive multimedia instruction: What do the numbers show? In Proceedings of the ninth annual conference on interactive instruction delivery, (pp. 100–104). Warrenton,VA: Society for Applied Learning Technology. Isaacs, A. C., & Carroll, W. M. (1999). Strategies for basicfacts instruction. Teaching Children Mathematics, 5, p. 9. Kilpatrick, J., Swafford, J., & B. Findell (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, D.C.: National Academy Press. Kintsch, W. (1977). Memory and cognition. New York: John Wiley & Sons. Leutzinger, L. P. (1979). The effects of counting-on on the acquisition of addition facts in grade one (Doctoral dissertation, University of Iowa, 1979). Dissertation Abstracts International, 40(07), 3765A. Martin, E. (1972). Stimulus encoding in learning and transfer. In A.W. Melton & E. Martin (Eds.), Coding processes in human memory, Washington, D.C.: Winston. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston,VA: NCTM.

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———. (2000). Principals and standards for school mathematics. Reston,VA: NCTM. ———. (2006a). Curriculum Focal Points for Prekindergarten through Grade 8 mathematics: A quest for coherence. Reston, Virginia: NCTM. ———. (2006b). Published Letter to the Editor, Wall Street Journal. Accessed 12/5/06 from http://www.nctm.org/ focalpoints/wsj_letter.asp. Rathmell, E. C. (1978). Using thinking procedures to learn basic facts. In M. Suydam (Ed.), Developing computational skills:1978 Yearbook of the National Council of Teachers of Mathematics. (pp. 13–38). Reston,VA: NCTM. Steinberg, R. (1985). Instruction on derived facts strategies in addition and subtraction. Journal for Research in Mathematics Education, 16, pp. 337–355. Thornton, C. A. (1978). Emphasizing thinking strategies in basic fact instruction. Journal for Research in Mathematics Education, 9, pp. 214–227.

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