What is numerical reasoning?

How a number series works

Holzman, Pellegrino and Glaser (1983) described number series solving in four steps: detecting relations, discovering the period, describing the pattern and extrapolating (Holzman, Pellegrino & Glaser, 1983). Their work built on Simon and Kotovsky (1963), who first introduced the rule schemes used to analyze letter series problems (Loe et al., 2018) (Simon & Kotovsky, 1963).

Take the opening example, 2, 5, 11, 23. Detecting relations: the gaps are 3, 6 and 12. Discovering the period: there is no repeating cycle, but each gap doubles. Describing the pattern: each term is twice the one before, plus 1. Extrapolating: 23 times 2, plus 1, gives 47. The trap is at the first step: a solver who looks only at the gaps can miss that the terms themselves are being multiplied.

A 2018 study turned that analysis into an automatic item generator built on five rule types: succession, parallel sequences, clusters, non-progressive coefficient patterns and complex progressive patterns (Loe et al., 2018). The first two are the ones most people meet first: a steady step from one number to the next, or two series interleaved and taking turns. The later types ask you to notice how the steps themselves behave.

All five rule types significantly predicted item difficulty, and together they accounted for 69% of the variance in estimated item difficulty (Loe et al., 2018). How hard it is to spot a relationship also depends on the arithmetic operations involved (Loe et al., 2018). Performance on the generated items reflected a single underlying factor, and scores correlated r = .60 and r = .66 (two forms) with a 16-item public-domain ability test (Loe et al., 2018). The number series guide works through each rule type with examples.

How numerical reasoning is measured

Numbers show up in many kinds of ability test, often in word-problem form. The ASVAB, used primarily to determine military enlistment eligibility, computes AFQT scores from four subtests, two of them numerical: Arithmetic Reasoning and Mathematics Knowledge (Official ASVAB: scores). The ASVAB and AFQT page covers how those scores work.

Employment tests mix numbers into short timed sets. CCAT questions come in three kinds: verbal, math and logic, and spatial reasoning, and calculators are not allowed (Criteria: CCAT prep). Wonderlic cognitive questions must be answered without a calculator, dictionary or other resources (Wonderlic: sample questions). The PI Cognitive Assessment reports a Numeric subscore alongside Verbal and Abstract Reasoning (Predictive Index: Cognitive Assessment). Even nonverbal batteries include numbers: the UNIT2 has a Numerical Series subtest among its six (PRO-ED: UNIT2).

The IQ Octopus test has a Quantitative Reasoning section of 3 items out of 25. The items are number series rather than arithmetic drills, and the test reports an estimated IQ for the whole set, not a separate numerical score.

Can numerical reasoning be improved?

Practice helps less here than you might expect. A 2018 meta-analysis of retest effects found that tests with numerical content showed a smaller first retest gain (0.18 SD) than tests with verbal (0.33) or mixed content (0.41) (Scharfen, Peters & Holling, 2018). Knowing the format still removes surprises, but number tests seem to reward familiarity less than other formats.

One surprising route is spatial training. A 2022 meta-analysis of 29 studies (N = 3,765) found that spatial training improved math performance by an average Hedges's g of .28 (Hawes, Gilligan-Lee & Mix, 2022). The effect grew as participants' ages increased from 3 to 20 years, and it transferred more to math outcomes closely aligned with the training than to more distant ones (Hawes, Gilligan-Lee & Mix, 2022). The spatial intelligence page covers that research.

For broad gains, schooling has the best evidence. A meta-analysis of 142 effect sizes found that an additional year of education raises cognitive ability by approximately 1 to 5 IQ points, with effects on all broad categories of ability studied (Ritchie & Tucker-Drob, 2018). If number series partly reflect math knowledge, as the network analysis above suggests, ordinary math learning may help on them too, though none of these studies tested that directly.

Try it

Take the free number sequence test for ten timed series, or work through the number series puzzles by difficulty with worked answers.