The Test of Mathematical Abilities for Gifted Students(TOMAGS) is a standardized, norm-referenced testdesigned to assess mathematical talent in children 6through 12 years old. The TOMAGS requires stu-dents to use mathematical reasoning and problem-solving skills to understand how to communicatemathematically to solve problems. The test wasdesigned to identify students who have talent or gift-edness in mathematics and should not be used fordiagnostic purposes. The Primary level is for stu-dents 6 through 9 years old; the Intermediate level isfor students 9 through 12 years old. The TOMAGScan be group administered by teachers, counselors,psychologists, and other individuals. Scoring is easy,and guidelines are provided to assist the examiner ininterpreting the results. Reliability and validity arestrong and support its use as an identification instru-ment of giftedness in mathematics.

In this chapter, we discuss the National Councilof Teachers of Mathematics (NCTIVI) standards, char-acteristics of children who are gifted in mathematics,assessment of talent in mathematics, and a descrip-tion of the TOMAGS and its uses.

1Overview of the TOMAGS

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National Council of Teachersof Mathematics StandardsThe NCTM standards were most important to thedevelopment of the TOMAGS. In 1989 the NCTM pub-lished the Curriculum and Evaluation Standards forSchool Mathematics. The standards were an attemptto specify national, professional standards for schoolcurricula in mathematics. Thousands of members ofthe NCTM were involved in the drafting, reviewing,and refining of the standards during the 3 years ittook to develop and subsequently publish the docu-ment (Crosswhite, 1989).

The standards are statements that can be usedto judge the quality of mathematics curriculum orevaluation methods and are based on the followingfive goals (NCTM, 1989):

1. "Learning to value mathematics" (p. 5)2. "Becoming confident in one's own ability" (p. 6)

3. "Becoming a mathematical problem solver" (p. 6)

4. "Learning to communicate mathematically" (p. 6)

5. "Learning to reason mathematically" (p. 6)

These goals were developed with the intent that stu-dents become mathematically literate and developthe ability to explore, to conjecture, and to reasonmathematically. There are 54 standards divided intofour categories: K through 4, 5 through 8, 9 through12, and evaluation.

The 1 3 curriculum standards f o r Grades Kthrough 4 are (1) mathematics as problem solving,(2) mathematics as communication, (3) mathematicsas reasoning, (4) mathematical connections, (5) esti-mation, (6) number sense and numeration, (7) con-cepts of whole number operations, (8) whole numbercomputation, (9) geometry and spatial sense, (10)measurement, (11) statistics and probability, (12)fractions and decimals, and (13) patterns and rela-tionships. The 13 standards for Grades 5 through 8are (1) mathematics as problem solving, (2) math-ematics as communication, (3) mathematics as rea-soning, (4) mathematical connections, (5) numberand number relationships, (6) number systems andnumber theory, (7) computation and estimation, (8)patterns and function, (9) algebra, (10) statistics, (11)probability, (12) geometry, and (13) measurement.

The first three curriculum standards for bothGrades K through 4 and 5 through 8 address math-ematics as problem solving, mathematics as commu-nication, and mathematics as reasoning and arereflected i n the TOMAGS's construction. Problemsolving includes developing and applying strategiesand approaches to solve problems, and verifying andinterpreting results. Mathematics as communicationincludes reading, writing, and discussing ideas usingthe language of mathematics including signs, sym-bols, and terms of mathematics. Mathematical rea-soning includes making conjectures, gathering evi-dence, and building an argument to support logicalconclusions.

In addition to the curriculum standards, thereare 14 evaluation standards. The evaluation stan-dards stress that several aspects o f assessmentshould receive increased attention, whereas othersshould receive decreased attention. For example,"focusing on a broad range of mathematical tasksand taking a holistic view of mathematics" shouldreceive increased attention, whereas "focusing on alarge number of specific and isolated skills organizedby a content-behavior matrix" (NCTM, 1989, p. 191)should receive decreased attention. The TOMAGSdoes this very well by focusing on a broad array ofmathematical tasks with many items incorporating

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several mathematical concepts. This is discussed inmore detail i n the section on content validity inChapter 6.

The evaluation standards are further categorizedinto three sections: general assessment, studentassessment, and program evaluation. For the pur-poses of the TOMAGS, the first two sections, generalassessment and student assessment, are most criti-cal. General assessment consists of three standards:alignment, multiple sources of information, and appro-priate assessment methods and uses. Student assess-ment consists o f seven standards: mathematicalpower, problem solving, communication, reasoning,mathematical concepts, mathematical procedures,and mathematical disposition.

The three standards in the first section, generalassessment, are particularly pertinent to the devel-opment o f the TOMAGS. Standard one, alignment,refers to the degree to which assessments matchthe curriculum's goals, objectives, and mathemati-cal content. Standard two, multiple sources of infor-mation and uses, directs assessors to present tasksthat demand different kinds of mathematical think-ing. Standard three, appropriate assessment meth-ods, requires examiners t o select instruments o nthe basis of their use and the developmental leveland maturity of the student. The general assessmentsection further states that when comparing gen-eral mathematical capability of a student with that ofother students or a national norm, the examiner willwant to use highly reliable tests designed for maxi-mum discrimination. The TOMAGS meets all of thesecriteria.

Characteristics of Students WhoAre Gifted in MathematicsIn developing the TOMAGS, we also considered thecharacteristics of gifted mathematics students. Thesestudents have many of the following abilities:

• t o recognize and spontaneously formulate prob-lems, questions, a n d problem-solving s teps(Greenes, 1981; Scruggs, Mastropieri, Monson &Jorgensen, 1985; O'Conner & Hermelin, 1979;Sternberg & Powell, 1983; Scruggs & Mastropieri,1984);

• t o distinguish between relevant and irrelevant infor-mation i n novel problem-solving tasks (Marr &Sternberg, 1986);

• t o see mathematical patterns and relationships(Cruilcshank & Sheffield, 1992; Miller, 1990);

• t o have more creative strategies for solving prob-lems (Devall, 1983; Miller, 1990; Shore, 1986; Dover &Shore, 1991);

• t o t h i n k abstractly a n d reason analytically(Cmilcshank & Sheffield, 1992; Marr & Sternberg,1986; Miller, 1990);

• t o be more flexible in handling and organizing data(Cruikshank & Sheffield, 1992; Devall, 1983; Greenes,1981; Miller, 1990; Shore, 1986; Dover & Shore, 1991);

• t o offer original interpretations (Greenes, 1981);

• t o transfer ideas generalized from one mathematicalsituation to another (Cruilcshank & Sheffield, 1992;Greenes, 1981; Miller, 1990);

• t o be intensely curious about numeric information(Cmikshank & Sheffield, 1992; Miller, 1990)

• t o quickly learn and understand mathematical ideas(Dover &Shore, 1991; Miller, 1990);

• t o reflect and take a longer amount of time whensolving complex problems or those with severalsolutions (Davidson & Sternberg, 1984; Miechen-baum, 1980; Sternberg, 1982; Wong, 1982; Woodrum,1975); and

• t o persist i n finding the solution t o problems(Ashley, 1973; House, 1987).

While each mathematically talented student maynot have all of these abilities, researchers agree thatthese students do not necessarily have the abilityto compute. Miller's (1990) definition emphasizesthis point by stating that mathematical talent "refersto an unusually high ability to understand mathe-matical ideas and to reason mathematically, ratherthan just a high ability to do arithmetic computa-tions." (). 2)

Grades and traditional achievement tests may notidentify gifted students with mathematical abilities.Mathematics classes frequently focus on computa-tional accuracy rather than problem solving, so chil-dren who achieve high grades in school may not bemathematically talented. Similarly, standardizedachievement tests concentrate on low-level tasks andalso may not identify gifted students (Romberg &Wilson, 1992). A need exists to assess problem-solvingand reasoning abilities of students who are gifted inmathematics. The TOMAGS was developed to meetthis need.

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Assessment of Giftednessin MathematicsAssessment of student knowledge of mathematics iscarried out for many purposes. The TOMAGS wasdeveloped to identify giftedness in mathematics. Themost critical feature of this type of assessment is thedegree to which i t discriminates among studentswho have varying degrees of talent in mathematics.

Over the last two decades there has been a sub-stantive increase in identifying children with math-ematical talent. The roots of this movement can betraced back to 1971, when Julian C. Stanley startedthe Study of Mathematically Precocious Youth (SMPY)at Johns Hopkins University (Stanley, 1991). Stanleyused above-level testing because most age-appropri-ate tests were not difficult enough for this population(see the next section for a more thorough discussionof this concept). He used as his original criterion forentry into the program a score of 500 to 800 on theScholastic Abilities Test-Mathematics (SAT-M) for stu-dents who were younger than 13 years old. In 1979 aunit was developed at Johns Hopkins, independentfrom SMPY, to conduct talent searches and academicprograms. The creation of this unit enabled SMPY toraise its lower score from 500 to 700 on the SAT-Mand concentrate its efforts on students scoring from700 to 800.

Today many programs conduct talent searchesin mathematics. For example, a similar program isthe Elementary Student Talent Search conducted atCarnegie Mellon University (Lupkowsld-Shoplik &Kuhnel, 1995). These talent searches have one thingin common: All use above-age testing when identi-fying mathematical talent. A s mentioned earlier,above-level testing is used because grade-level testslack diff icult i tems. Whereas above-level testingappears to have validity with many children, i t hasbeen used primarily with children in Grade 4 o rabove. When assessing younger children, NCTMstresses the use o f developmentally appropriateinstruments, which makes above-level testing poten-tially problematic.

In recent years, educators have begun assessingfor mathematical talent using alternative methods inaddition to above-level testing. Sheffield (1994) dis-cussed several important methods that are essentialin the assessment of talent in mathematics. Althoughmany of these, such as student interviews, are notsuitable for inclusion in a standardized test, onemethod discussed is using open-ended questions.The TOMAGS uses an open-ended question formatwith developmentally appropriate problems.

Needs in Identifying Gifted ChildrenStanley (1976) pointed out that tests generally usedto identify gifted children were inappropriate becausethey failed t o have enough "ceiling." Grade- andage-appropriate tests are often too easy for gifted chil-dren. Testing the child's limits can be achieved only ifthe test is difficult enough to determine the extent ofthe child's knowledge. For example, a second-gradechild capable of sixth-grade math work is not going toshow his or her true abilities in mathematics by takinga second-grade achievement test. In addition, two stu-dents who earn the top score on the test may, in fact,have very different abilities in mathematics if the testis too easy. In this case, the test does not accuratelymeasure what students are able to do (i.e., it does notdiscriminate among students with high ability). Inaddition, if a test does not have enough difficult items,a student missing only one item may receive a per-centile rank that eliminates him or her from entry intothe gifted mathematics program when, in fact, the stu-dent has high ability in mathematics. Unfortunately,most tests used to identify students as gifted are notdeveloped for use with gifted children, but for usewith heterogeneous groups of children.

The TOMAGS was constructed specifically forchildren gifted in mathematics. The two levels of thetest—Primary and Intermediate—have many difficultitems that allow examiners to test the limits of giftedchildren.

We conclude these three introductory sections onbasic concepts by referring our readers to the contentvalidity section in Chapter 6. In that section, qualita-tive and quantitative evidence is presented that sup-ports the alignment of the NCTM standards with theabilities that are measured by the TOMAGS and theformats in which the abilities are tested.

Description of the TOMAGSThe TOMAGS was developed to address some of theconcerns presented in the previous sections andto identify children who demonstrate giftedness inmathematics. The TOMAGS consists of open-endedquestions presented i n a problem-solving format.The TOMAGS Primary consists of 39 problems and

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the TOMAGS Intermediate consists of 47 problems.Campbell and Bamberger (1990) defined mathemati-cal problem solving as envisioned in the NCTM stan-dards as "students actively involved in constructingmathematics .. . [and] apply[ing] new mathematicalknowledge" ( ) . 15). The TOMAGS was designed toestimate a child's capacity t o use mathematicalknowledge in novel situations and, in some cases, toconstruct new strategies to solve a problem.

The TOMAGS requires the child to work through aseries of problems that are aligned to the NCTM stan-dards. The test examines a sample of the child's abilityto use flexibility in mathematical reasoning and trans-fer already learned mathematical knowledge to newsituations. The test problems use, i n some cases,knowledge taught formally in school and, in othercases, knowledge not taught formally in school.

Uses of the TOMAGSOne of the major purposes for administering theTOMAGS is to obtain information that is helpful inidentifying children for gifted classes that emphasizemathematics. It is not intended for identifying chil-dren for classes emphasizing talent in leadership,visual o r performing arts, and/or academic areasother than mathematics.

Second, the TOMAGS may be used as a screeninginstrument with the entire pool of students beingconsidered for the gifted mathematics program or asa second-level screening instrument with only thenominated students. The TOMAGS provides normsfor both gifted and normal groups. "Borderline" chil-dren may be more accurately identified using thegifted norms.

Third, the TOMAGS may be used clinically toexamine a child's relative strengths and weaknessesin the individual constructs incorporated into thetest. In this way, a child's potential is not hiddenthrough the quantification of multiple test scores.

Fourth, the TOMAGS has value as a researchtool, especially for researchers who need standard-ized instruments to study talent in mathematics. Itsresults can be used to evaluate the effectiveness of agifted mathematics program or to test various theo-ries of mathematical talent.