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Digital resources for middle school mathematics.


Middle school mathematics have traditionally been a turning point for many students as they transition from the relative "safety" of elementary school elementary school: see school.  arithmetic to the more challenging demands of high school algebra algebra, branch of mathematics concerned with operations on sets of numbers or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as , pre-calculus and statistics. This paper reviews a number of digital educational resources, available as CD-ROMs or Internet websites and chosen to conform to Verb 1. conform to - satisfy a condition or restriction; "Does this paper meet the requirements for the degree?"
fit, meet

coordinate - be co-ordinated; "These activities coordinate well"
 Massachusetts Department of Education curriculum standards, which may help teachers, students and their parents meet these challenges and smooth the transition from middle school to high school mathematics.


American schools and math teachers at all levels continue to grapple with to enter into contest with, resolutely and courageously.

See also: Grapple
 the almost indefatigable disdain that many students feel toward the study of mathematics and science. While studies have shown that proficiency in mathematics is a key component for success in a computer-based economy (Olson, 2006), all but the most gifted American students continue to lag behind their European and Asian counterparts in mathematical ability. The prevailing attitude among many young people is that math is boring and for "nerds" only, a prejudice that is unfortunately reinforced in the popular media. To counteract this attitude many educators contend that math activities should be made fun or "cool" to encourage students to participate. However, this approach fails to recognize that students are often intimidated in·tim·i·date  
tr.v. in·tim·i·dat·ed, in·tim·i·dat·ing, in·tim·i·dates
1. To make timid; fill with fear.

2. To coerce or inhibit by or as if by threats.
 by the perceived difficulty of math and that many lack the self-confidence to stay motivated and master mathematical concepts (McCallister,, 2003). This anxiety toward math can be compounded at the middle school level as students transition from elementary arithmetic Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. Most people learn elementary arithmetic in elementary school.  to intermediate mathematics concepts like algebra, geometry and probability. To address math anxiety and build self-confidence, student performance must be improved through skill-building exercises that strengthen problem solving problem solving

Process involved in finding a solution to a problem. Many animals routinely solve problems of locomotion, food finding, and shelter through trial and error.
 abilities and through instruction that provides the feedback necessary to master more difficult concepts.

This paper reviews a number of digital educational resources, available as CD-ROMs or Internet websites, which can be used in the classroom, or independently at home, to improve student performance in math and build self-confidence in problem solving. The purpose of this paper is not to endorse any particular product, but to provide examples of resources that satisfy general criteria that the author has proposed to meet the challenges of middle school mathematics instruction. Reviews are based on the author's independent use of each resource, and in each case an Internet web address is indicated through which the reader may access a demonstration version of the resource for further investigation.

Challenges of Middle School Mathematics

A number of factors conspire con·spire  
v. con·spired, con·spir·ing, con·spires

1. To plan together secretly to commit an illegal or wrongful act or accomplish a legal purpose through illegal action.

 to make middle school mathematics instruction challenging for teachers, students and parents alike. Teachers at the middle school level are generally faced with students emerging from different elementary schools and with varying backgrounds and competency COMPETENCY, evidence. The legal fitness or ability of a witness to be heard on the trial of a cause. This term is also applied to written or other evidence which may be legally given on such trial, as, depositions, letters, account-books, and the like.
 in all subjects, including math. Students taking math at the middle school level are transitioning from the familiar skills base of elementary school arithmetic into the more challenging study of pre-algebra and geometry, whose concepts and "formalisms" (equations and data representations) can be intimidating in·tim·i·date  
tr.v. in·tim·i·dat·ed, in·tim·i·dat·ing, in·tim·i·dates
1. To make timid; fill with fear.

2. To coerce or inhibit by or as if by threats.
 and confusing at first even to the best students (Booth, 1988). Finally, for parents of middle school students this level of mathematics can often present the first real performance obstacle to children who had previously done well in school, and parents will naturally seek to provide both the right encouragement and the right resources to help their children succeed.

Mathematics Curriculum Standards

The digital resources reviewed in this paper were chosen to conform to the Massachusetts Department of Education curriculum framework for students in grade 7 (Mass DOE, 2004) which is representative of standards used around the country at the middle school level. These standards require that students demonstrate skills in (i) number sense and operations (e.g., fractions, percentages, estimation), (ii) patterns, relations and algebra (e.g., simple linear equations), (iii) geometry (e.g., congruent con·gru·ent  
1. Corresponding; congruous.

2. Mathematics
a. Coinciding exactly when superimposed: congruent triangles.

 angles, Cartesian coordinates Cartesian coordinates (kärtē`zhən) [for René Descartes], system for representing the relative positions of points in a plane or in space. ), (iv) measurement (e.g. area/perimeter of polygons), and (v) data analysis, statistics, and probability (e.g., graphical representations of data, mean, median and mode). Within each subject area students are required to interpret the problem they are being asked to solve, and frequently to represent their solution graphically using computer-based applications like graphing calculators Graphing Calculator may refer to:
  • Graphing calculators, calculators that are able to display and/or analyze mathematical function graphs.
  • NuCalc, a computer software program able to perform many graphing calculator functions.

Digital Resource Selection Criteria

In reviewing the educational CD-ROMs and Internet websites below consideration was given to selecting mathematics resources that can be used to address the concerns of middle school teachers, students and parents described above. To ensure that resources effectively address these concerns the following criteria are proposed:

Resources must provide the means for students with weaker backgrounds in certain areas to catch up to their more experienced classmates Classmates can refer to either:
  •, a social networking website.
  • Classmates (film), a 2006 Malayalam blockbuster directed by Lal Jose, starring Prithviraj, Jayasurya, Indragith, Sunil, Jagathy, Kavya Madhavan, Balachandra Menon, ...
. By satisfying this criterion mathematics resources should "level the playing field" for students coming from weaker elementary school instruction, and make the job of teaching a "performance diverse" student population easier. Such resources should force weaker students who require additional "drill and practice" to improve basic math skills and become solid problem solvers, thereby improving their self-confidence in math.

Resources must allow students to transition more successfully from basic arithmetic to the challenges of pre-algebra. By satisfying this second criterion resources should take some of the anxiety out of pre-algebra study, and make students more confident in tackling subjects that at first might appear intimidating. These resources should also serve as applications that students can "grow into" in subsequent grades so that skills mastered at the entry level can be applied to studying more complex subjects later on.

Resources must demonstrate to students that skills acquired in mathematics can be applied in an engaging fashion to other areas including science and social studies. By satisfying this third criterion resources will show students that mastery of mathematical skills can be applied in a meaningful context, which is important for children at the middle school level who are just beginning to define the interests they will carry into their professional lives. This element should encourage students to look more broadly at the ways in which mathematics can be applied toward their own emerging interests.

Resources must be straightforward for students and teachers to use and offer educational opportunities not easily found in other classroom resources. These final considerations address the practical side of using digital resources in the classroom, or at home, by acknowledging that learning to use these applications often requires a considerable time investment on the part of the teacher and the student, as well as a monetary investment for school districts and parents. To make the time and money invested worthwhile, digital resources should offer a unique learning opportunity and not simply duplicate the kind of instruction already available through textbooks and pencil-and-paper exercises. Resources must also be easy to use so that students can devote their time to learning the concepts and not to figuring out how to use the software or website.

Exemplary Digital Mathematics Resources

Each digital resource reviewed below is first characterized by its purpose, target audience, method of delivery, the company that created the resource, the web address of its product description and any available online demonstration. These items are followed by a description of the mathematics concepts presented and how the software or website functions to instruct students about these concepts. Each resource is also evaluated with respect to the general criteria above and specifically by any unique opportunities that the resource might bring to mathematics instruction for students and teachers. Given the volatility of educational resources, especially on the Internet, only resources from companies or educational groups that are well-established or have exhibited long-term "staying power" are considered.

Destination Math Mastering Skills & Concepts IV." Advanced Mathematics Destination Math Mastering Skills & Concepts V: Pre-Algebra Drill-and-practice and problem-solving exercises for grade 6-8. Available as CD-ROMs or via a subscription website from Riverdeep Corporation. Demo versions 1. demo version - An early, barely-functional version of a program which can be used for demonstration purposes as long as the operator uses *exactly* the right commands and skirts its numerous bugs, deficiencies, and unimplemented portions.
 available at

Course IV is a comprehensive mathematics program that teaches basic skills, math reasoning, conceptual understanding and problem solving for the following math topics: fractions, decimals, percents, integers and operation order. Course V is a comprehensive mathematics program that teaches the fundamentals of algebra, geometry, radicals and exponents, ratio and proportion, statistics and probability. Each set of topic exercises begins with a concept review section followed by three levels of increasingly challenging word problems that are set in an imaginary place Noun 1. imaginary place - a place that exists only in imagination; a place said to exist in fictional or religious writings
fictitious place, mythical place
 but have real-life applicability. The review section may be skipped and students can choose the desired level of difficulty for the problem sets. Students solve the problems by choosing between multiple choice items or by inputting short answers. The multiple choice format serves as excellent training for standardized tests A standardized test is a test administered and scored in a standard manner. The tests are designed in such a way that the "questions, conditions for administering, scoring procedures, and interpretations are consistent" [1]  and often includes wrong answers that result from common mistakes. Excellent explanatory and interactive feedback is available for each problem that allows students to solve the problem as the software reveals the solution. The software is easy to use and navigate, and the animated graphics See animation.  are attractive and sufficiently "cool" for middle school students and do not overwhelm o·ver·whelm  
tr.v. o·ver·whelmed, o·ver·whelm·ing, o·ver·whelms
1. To surge over and submerge; engulf: waves overwhelming the rocky shoreline.

 or detract from detract from
verb 1. lessen, reduce, diminish, lower, take away from, derogate, devaluate << OPPOSITE enhance

verb 2.
 the instruction. Both packages serve as excellent examples of how drill-and-practice software tools can elevate el·e·vate  
tr.v. ele·vat·ed, ele·vat·ing, ele·vates
1. To move (something) to a higher place or position from a lower one; lift.

2. To increase the amplitude, intensity, or volume of.

 weaker students into the realm of problem solving, as well as strengthen the math skills of students at all proficiency levels. The corrective interactive feedback offered for each problem provides a detailed level of explanation that could probably not be replicated by the most patient of teachers.

Website with free and subscription access offering mathematics homework assistance and skills practice for students in grades 6 through college. Maintained by the HotMath Corporation.

This website offers math homework assistance by providing solutions to odd-numbered problems in many of the leading mathematics textbooks used at the middle school through college levels. Recognized publishers include McGraw-Hill, Prentice-Hall, and Holt, Rinehart and Winston, and with a subscription students can access each problem page in a participating textbook. In addition the "Hotmath Team" offers an extensive series of "free-access" problems that cover expressions, operations, number theory, fractions, pre-algebra, geometry, statistics and probability. Solutions to all problems are explained first by offering students hints and then revealing steps in the solution in a hint-then-solution step fashion. The solution graphics are spare but sufficient and include only text, equations and simple line graphs In graph theory, the line graph L(G) of an undirected graph G is a graph such that
  • each vertex of L(G) represents an edge of G; and
  • any two vertices of L(G
. In addition to homework solutions and explanations, the site also offers exercises designed specifically for graphing calculators, and can provide students with access to online math tutors. Math textbooks often include answers to problems "in the back of the book" as a way for students to check their work independently, however, the answers themselves often do not offer any insight into the solution process. provides students with a unique service by expanding answers to math problems into full-blown step-by-step explanations, giving students the homework help they need and reinforcing their understanding of key concepts and solution strategies. The website also offers more motivated students a unique "look ahead" to advanced mathematics topics taught at the high school and college level like advanced algebra, pre-calculus and calculus calculus, branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit—the notion of tending toward, or approaching, an ultimate value. .

Free website providing links to homeschooling home·school or home-school  
v. home·schooled, home·school·ing, home·schools
To instruct (a pupil, for example) in an educational program outside of established schools, especially in the home.
 math resources for students in K-12. Created and maintained by former math teacher and homeschooler Maria Miller.

This website is advertised as a comprehensive math resource site for homeschooling parents and teachers. It offers links to a number of other math resource websites as well as downloadable math worksheets, interactive tutorials, games and curriculum guides. Because the website is geared toward homeschoolers it can provide students and parents (who may wish to augment their children's classroom instruction) with access to math instruction tools from a variety sources that have been found to be effective for home-schooled students who function essentially as "independent learners".

Math Arena Advanced

Pre-algebra games for grades 7 through 10 available on CD-ROM CD-ROM: see compact disc.
 in full compact disc read-only memory

Type of computer storage medium that is read optically (e.g., by a laser).
 from Sunburst Technology. Online demonstration available at

Math Arena Advanced offers twenty kinds of games at two levels (easy and hard) that allow students to practice a variety of pre-algebra concepts. In a first set of games called "time trials" students race against the clock to solve visual math problems of data graphing, linear equations, line graphs, algebra and manipulation of 2-D and 3-D shapes. A second set of games called "buzzins" allow students to compete against each other to solve probability, order of operations In arithmetic and algebra, when a number or expression is both preceded and followed by a binary operation, a rule is required for which operation should be applied first. From the earliest use of mathematical notation, multiplication took precedence over addition, whichever side of a , data graphing, and translating word problems into equations. The graphics are humorous and attractive and are accompanied by snazzy snaz·zy  
adj. snaz·zi·er, snaz·zi·est Slang
Fashionable or flashy.

[Origin unknown.]

 background music. Like its popular "younger sibling sibling /sib·ling/ (sib´ling) any of two or more offspring of the same parents; a brother or sister.

" Math Arena for grades 4 through 7, the Advanced package provides students at all performance levels with engaging drill-and-practice activities that require them to hone their math skills and visualize mathematical concepts in unique ways.

Shodor Project Interactivate

Free website offering interactive Java-based mathematics simulations for grades 6 through 8. Created and maintained by the Shodor Education Foundation.

This website is a great online resource for both teachers and students that offers lesson plans and Java-based activities designed to support number and operations, geometry and measurement, functions and algebra, and probability and data analysis concepts. The wide range of activities provided in each subject area progress from simple interactive simulations that introduce fundamental concepts to more challenging simulations and games that build on earlier concepts to teach complex topics and develop higher problem-solving skills. In all cases activities use standard equations to represent mathematics problems and simple animation and graphics to illustrate both simple and advanced concepts. Many activities are also correlated to specific lessons found in established middle school mathematics textbooks. Project Interactivate activities emphasize higher problem solving over skill building and challenge students at all levels of mathematical ability with unique interactive visualizations Interactive visualization is a branch of graphic visualization in computer science that studies how humans interact with computers to create graphic illustrations of information and how this process can be made more efficient.  of math concepts that cannot be replicated with classroom chalkboard discussions.

Graph Master

Data graphing toolkit for grades 4 through 8 available on CD-ROM from Tom Snyder Tom Snyder (May 12, 1936 - July 29, 2007) was an American television personality, news anchor, and radio personality best known for his late night talk shows The Tomorrow Show, on the NBC television network in the late 1970s and '80s, and The Late Late Show  Productions. Online tour and trial version available at

Graph Master is a versatile and easy-to-use toolkit for graphing, analyzing and comparing data with nine different graph types, including linear x-y plots, histograms, pie charts A graphical representation of information in which each unit of data is represented as a pie-shaped piece of a circle. See business graphics. , etc. In addition to a number of data sets included with the package, a number of ready-made data CDs are also available with social studies, nature and geography data applications. A related resource website also provides links to a number of other sources of data for analysis (see DASL See WebDAV.

DASL - Datapoint's Advanced System Language.

A cross between C and Pascal by Gene Hughes with custom features for Datapoint hardware (no stack). It is used internally by Datapoint.
 below). Graph master can serve as a practical math tool with which students can investigate, compare and interpret data in various graphical representations, and it also allows students to export their data graphs for use in multi-media presentations. The toolkit serves as a good "prequel pre·quel  
A literary, dramatic, or cinematic work whose narrative takes place before that of a preexisting work or a sequel.

[pre- + (se)quel.]
" to Microsoft Excel (tool) Microsoft Excel - A spreadsheet program from Microsoft, part of their Microsoft Office suite of productivity tools for Microsoft Windows and Macintosh. Excel is probably the most widely used spreadsheet in the world.

Latest version: Excel 97, as of 1997-01-14.
 and its plotting features, and the built-in data library demonstrates that math can be applied to the study of a wide range of subjects.

DASL: The Data and Story Library

Free website lib.stat stat
With no delay.


STAT Stat! Clinical medicine adverb Fast, quickly, immediately, schnell, vite Lab medicine noun sponsored by the Carnegie Mellon University Carnegie Mellon University, at Pittsburgh, Pa.; est. 1967 through the merger of the Carnegie Institute of Technology (founded 1900, opened 1905) and the Mellon Institute of Industrial Research (founded 1913).  Department of Statistics that offers access to real data sets for statistical analysis for grades 6 and higher.

The DASL data and story "library" was created as part of a larger effort to enhance the teaching of statistics using computers, and is one of the many data links suggested on the Graph Master resource webpage (above). The library's purpose is to provide data from a wide variety of topics so that statistics teachers can find real world examples that students will find interesting. Each dataset includes a "story" to explain why the information was collected and the original intent of the statistical analysis. This website offers data for all statistical skill levels and identifies data sets by appropriateness of analysis method, and gives students access to data that would be difficult, impossible or very time consuming for them to collect themselves. Some of the datasets are very unique: for example one is an 1854 "census" of the number of lunatics residing in each county in Massachusetts, and another dataset examines the relationship between human behavior and superstition superstition, an irrational belief or practice resulting from ignorance or fear of the unknown. The validity of superstitions is based on belief in the power of magic and witchcraft and in such invisible forces as spirits and demons.  on Friday the 13th Friday the 13th

regarded as unlucky day. [Western Folklore: Misc.]

See : Luck, Bad
. Much of the data and related "stories" are taken from social studies and historical contexts, and all data sets offer the opportunity to pursue advanced statistical analysis.


The digital resources reviewed in this paper were selected because they satisfy general criteria proposed to provide teachers and students with effective tools for mastering key middle school mathematics concepts. In particular, all resources support standard mathematics curricular objectives in ways that cannot generally be achieved via traditional teacher-at-the-chalkboard instruction or with pencil-and-paper homework assignments. The drill-and-practice and math gaming resources reviewed promote skill-building with the objective of turning all students into good math problem solvers. The online math homework support resource provides practical intervention for students who use a number of popular middle school textbooks. The homeschooling website provides links to math resources geared specifically toward independent learners. The interactive online resource uses unique graphical representations to illustrate and reinforce fundamental algebraic 1. (language) ALGEBRAIC - An early system on MIT's Whirlwind.

[CACM 2(5):16 (May 1959)].
2. (theory) algebraic - In domain theory, a complete partial order is algebraic if every element is the least upper bound of some chain of compact elements.
 and geometric concepts. The data graphing toolkit and data library website provide true-to-life applications of math skills that demonstrate the utility of mathematics for all fields of study. With these tools in hand teachers and parents can effectively address the challenges of middle school mathematics and help former elementary school students develop the proficiency to succeed later at high school level mathematics.


Booth, Lesley R. (1988). "Children's Difficulties in Beginning Algebra". In Arthur F. Coxford & Albert P. Shuttle (Eds.), The Ideas of Algebra, K-12, 1988 Yearbook (pp. 20-32). Reston, VA: National Council of Teachers of Mathematics The National Council of Teachers of Mathematics (NCTM) was founded in 1920. It has grown to be the world's largest organization concerned with mathematics education, having close to 100,000 members across the USA and Canada, and internationally. .

Massachusetts Department of Education (Mass DOE). (2004). "Supplement to the Massachusetts Mathematics Curriculum Framework Grades 3, 5 and 7 Grade Level Standards", May 2004. Retrieved November 2006 from

McCallister, Deborah A., Adrian Mealer, Peggy S. Moyer, Shirley A. McDonald, John B. Peoples. (2003). "Chattanooga Math Trail: Community Mathematics Modules Volume 1", The University of Tennessee at Chattanooga UTC was founded in 1886 as then-private Chattanooga University (later known as Grant College). In 1907, the university changed its name to the University of Chattanooga. In 1969, the university merged with Chattanooga City College to form the modern UTC campus as part of the University , July 2003. Retrieved November 2006 from ERIC at www.eric, 22/c1/d7.pdf.

Olson, Lynn. (2006). "Economic Trends Fuel Push to Retool re·tool  
v. re·tooled, re·tool·ing, re·tools
1. To fit out (a factory, for example) with a new set of machinery and tools for making a different product.

 Schooling: Leaders Want Tighter Links to Workplace, College Skills". Education Week, March 22, 2006, vol. 25, no. 28.

Shelley E. Scruggs, Boston University School of Education History
Established in 1918, the Boston University School of Education has a long tradition of commitment to the community:
  • In 1977, the Boston University School of Education and a selection of Boston-area school districts, social service agencies, and overseas

Shelley E. Scruggs, Ph.D., holds a doctorate in electrical engineering electrical engineering: see engineering.
electrical engineering

Branch of engineering concerned with the practical applications of electricity in all its forms, including those of electronics.
 and is a student in the Graduate Certificate in Instructional Technology There are two types of instructional technology: those with a systems approach, and those focusing on sensory technologies.

The definition of instructional technology prepared by the Association for Educational Communications and Technology (AECT) Definitions and Terminology
 program at Boston University Boston University, at Boston, Mass.; coeducational; founded 1839, chartered 1869, first baccalaureate granted 1871. It is composed of 16 schools and colleges. .
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Author:Scruggs, Shelley E.
Publication:Academic Exchange Quarterly
Date:Jun 22, 2007
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