What is the TOMA-3? The Test of Mathematical Abilities explained

The TOMA-3, or Test of Mathematical Abilities, Third Edition, is a "norm-referenced" test of mathematics achievement published by PRO-ED in 2013 for students aged 8 years 0 months through 18 years 11 months. Norm-referenced means a student's performance is compared with a national sample of same-age peers rather than a fixed passing mark. It is given by teachers, educational diagnosticians, school psychologists, and other trained examiners, to a group or one student at a time, and takes 60 to 90 minutes. The TOMA-3 is not an IQ test. It measures what a student has learned in mathematics, and it is typically used alongside a measure of general ability when a school team wants to know whether math performance lags behind what a student's reasoning would predict. This article covers what the test contains, how it is scored, what the norms look like, and how it fits next to an IQ test.

According to PRO-ED, the TOMA-3 is designed to "identify, describe, and quantify mathematical deficits in school age children." In practice, it answers two questions: is this student significantly behind peers in mathematical knowledge, and if so, how far behind? The publisher describes the second answer in three bands (below average, poor, or very poor), which gives an evaluation team a shared vocabulary for the size of a problem.

The test was written by Virginia Brown, Mary E. Cronin, and Diane P. Bryant. The third edition replaced two earlier subtests, updated the norms, and expanded the technical documentation. PRO-ED lists the TOMA-3 at its "Level A" qualification tier, which its qualifications policy defines as having no special purchase requirements. That does not mean anyone can interpret the results well. PRO-ED's policy makes the "user" responsible for administration, scoring, and interpretation, and in schools the TOMA-3 usually forms one piece of a larger evaluation led by a qualified professional.

Because the TOMA-3 can be given to a whole class, it also works as a screener. A school might test a grade level to find students who need a closer look, then follow up individually with a diagnostic instrument such as the KeyMath-3 or a broad achievement battery like the KTEA-3.

The TOMA-3 has four core subtests, which combine into the overall composite called the "Mathematical Ability Index," plus one supplemental subtest.

• Mathematical Symbols and Concepts: Multiple-choice questions about mathematical signs, symbols, words, and phrases. This subtest replaced the Vocabulary subtest from the second edition and checks whether the student understands the language of mathematics.

• Computation: A series of written problems that increase in difficulty.

• Mathematics in Everyday Life: Multiple-choice questions about how mathematics is used in daily situations. It replaced the General Information subtest from the TOMA-2.

• Word Problems: Increasingly difficult story problems that require the student to read, decide which operation applies, and carry it out. Word problems draw on reading comprehension as well as math, which is one reason examiners look at this subtest next to a student's reading scores.

• Attitude Toward Math (supplemental): Rather than solving problems, the student responds to statements about math instruction and about their own ability and achievement, marking one of four boxes that run from "Yes, definitely!" to "No, definitely!" This subtest does not enter the Mathematical Ability Index. A 2021 meta-analysis in Psychological Bulletin by Barroso and colleagues, pooling 747 effect sizes from studies published between 1992 and 2018, found a correlation of about negative 0.28 between math anxiety and math achievement, an association that begins in childhood. That is a modest relationship, so a poor attitude score is context, never a cause.

The publisher reports that the TOMA-3 yields a Mathematical Ability Index, subtest "scaled scores," "percentile ranks," age and grade equivalents, and "standard errors of measurement." A scaled score places a subtest result on a common scale so subtests can be compared. A percentile rank is the percentage of the norm sample the student outscored. The standard error of measurement estimates how much a score would bounce around on retesting, and a careful examiner uses it to build a confidence band rather than treating one score as exact. The PRO-ED product page does not state the metric of the index or the scaled scores; the examiner's manual defines both, and evaluators should read a TOMA-3 score against the metric printed there rather than assuming a scale.

Age and grade equivalents deserve a caution. They are easy to explain to parents but lack equal intervals and exaggerate small differences at the extremes, and most measurement specialists advise against using them for eligibility decisions. Standard scores and percentiles are the numbers to lean on.

The TOMA-3 norm sample consists of 1,456 students aged 8-0 through 18-11 living in 21 states, and PRO-ED describes it as demographically representative of the 2011 U.S. school-age population. The publisher reports that reliability coefficients (content, time, and scorer) are consistently high and were calculated by age and for subgroups including sex, several racial and ethnic groups, gifted students, students with attention-deficit/hyperactivity disorder, and students with a mathematical learning disability. On validity, PRO-ED cites content-description, criterion-prediction, and construct-identification evidence, and the third edition added sensitivity, specificity, and ROC/AUC analyses, the statistics that show how well a cutoff score separates students who have a math learning problem from those who do not. The publisher also states that the composite has neither floor nor ceiling effects.

Two limitations follow. A sample of 1,456 spread over eleven age years is roughly 130 students per year, adequate for a group-administered achievement test but thinner than the norm samples behind the major individually administered batteries. And the norms are anchored to 2011 figures, so the test is now more than a decade old. Both are reasons to pair it with other data rather than resting a decision on it alone.

Achievement and intelligence are related but distinct, and the TOMA-3 sits on the achievement side. Deary, Strand, Smith, and Fernandes followed more than 70,000 English schoolchildren from age 11 to 16 and found that a latent general intelligence factor at 11 correlated 0.81 with a latent factor of national exam achievement at 16. Of 25 subjects, mathematics was the most strongly tied to "g," the general factor IQ tests estimate, with about 59 percent of its variance explained. That is a strong relationship and still far from perfect, and the gap is what an achievement test exists to describe.

The gap shows up in two ways. The first is the specific learning disability pattern. In the Journal of Developmental and Behavioral Pediatrics, David Geary reports that roughly 7 percent of children and adolescents have a mathematical learning disability and another 10 percent show persistently low math achievement despite average abilities in other areas, and that these deficits cannot be attributed to intelligence. Butterworth, Varma, and Laurillard, writing in Science, describe the core problem in "dyscalculia" as a deficit in understanding sets and their numerosities. A student in this group can post an average IQ score and a poor TOMA-3 score, and that discrepancy is one signal a school team looks at. The formal steps are described in the RIOT IQ article on how dyscalculia is diagnosed; no single test, and certainly no online test, makes that determination.

The second is the pattern Dr. Thomas Coyle discusses in the video above. Coyle studies "ability tilt," the within-person difference between a student's math and verbal scores. Writing in the Journal of Intelligence, he reports that math tilt (math higher than verbal) predicts STEM majors, jobs, and grades, while verbal tilt predicts humanities outcomes, even though tilt is unrelated to general intelligence. Two students with the same IQ can have very different profiles. The TOMA-3 has no verbal counterpart, so it cannot produce a tilt score by itself, but a math achievement score set beside a verbal ability score does something similar.

So a TOMA-3 result should be read in context. A low Mathematical Ability Index next to a low IQ score tells a different story from the same index next to a high IQ score.

The TOMA-3 measures learned mathematics, so it is sensitive to what a student has been taught. A student who moved between curricula, missed school, or is learning English will often score low for reasons that have nothing to do with a learning disability. The Word Problems subtest depends on reading, and the Attitude Toward Math subtest is self-report, which means it can be answered to please the examiner.

For adults, the picture is different. The TOMA-3 stops at 18 years 11 months, and there is no adult edition. An adult who wants to understand their own cognitive profile needs an ability measure rather than a school achievement test. The RIOT IQ test, developed by RIOT IQ with psychometrician Dr. Russell T. Warne, is designed for adults 18 and older. It contains 15 subtests across six cognitive indices (verbal reasoning, fluid reasoning, spatial ability, working memory, processing speed, and reaction time), takes about 52 minutes, and reports scores on the mean-100, standard-deviation-15 scale used by professional IQ tests. Comparing the verbal index with the nonverbal indices gives an adult a rough sense of whether their own profile tilts toward or away from verbal ability. It is not a diagnostic instrument and does not replace an individually administered evaluation by a qualified clinician. For a normed, professionally developed estimate of general reasoning ability, the RIOT IQ test is available online.

TOMA-3 is the Test of Mathematical Abilities, Third Edition, published by PRO-ED in 2013 and written by Virginia Brown, Mary E. Cronin, and Diane P. Bryant. The previous edition was the TOMA-2.

The publisher lists 8 years 0 months through 18 years 11 months. There is no adult version.

PRO-ED reports 60 to 90 minutes. It can be given to a group or to one student.

No. It is a norm-referenced test of mathematics achievement, whereas an IQ test measures general reasoning ability. The two are strongly related but distinct, and evaluators often compare a TOMA-3 score with an IQ score to see whether math performance is in line with a student's overall ability.

A composite Mathematical Ability Index, subtest scaled scores, percentile ranks, age and grade equivalents, and standard errors of measurement. The Attitude Toward Math subtest does not enter the composite.

Not by itself. A low score is one piece of evidence that a qualified professional weighs together with an ability measure, classroom data, and developmental history. Diagnosis is a clinical decision, never a single test score.

1. Barroso, C., Ganley, C. M., McGraw, A. L., Geer, E. A., Hart, S. A., & Daucourt, M. C. (2021). A meta-analysis of the relation between math anxiety and math achievement. Psychological Bulletin, 147(2), 134–168. pmc.ncbi.nlm.nih.gov

2. Butterworth, B., Varma, S., & Laurillard, D. (2011). Dyscalculia: From brain to education. Science, 332(6033), 1049–1053. pubmed.ncbi.nlm.nih.gov

3. Coyle, T. R. (2018). Non-g factors predict educational and occupational criteria: More than g. Journal of Intelligence, 6(3), 43. pmc.ncbi.nlm.nih.gov

4. Deary, I. J., Strand, S., Smith, P., & Fernandes, C. (2007). Intelligence and educational achievement. Intelligence, 35(1), 13–21. eric.ed.gov

5. Geary, D. C. (2011). Consequences, characteristics, and causes of mathematical learning disabilities and persistent low achievement in mathematics. Journal of Developmental and Behavioral Pediatrics, 32(3), 250–263. pmc.ncbi.nlm.nih.gov

6. PRO-ED, Inc. (2013). TOMA-3: Test of Mathematical Abilities–Third Edition, Complete Kit [product page]. proedinc.com

7. PRO-ED, Inc. (n.d.). Qualifications policy.. proedinc.com

Hero photo: A student working through a paper test with a pencil. Photo by Alison Wood, via Wikimedia Commons, CC BY 3.0 (cropped).