Reasoning matrices: from Raven to the rules

Where the format comes from

The British psychologist John C. Raven published the Progressive Matrices in 1938. Each problem is a grid of figures, usually three by three, with the last cell empty. The figures change according to one or more rules, and the task is to find, among several options, the one that completes the grid. The problems get harder little by little, hence the “progressive” in the name.

Raven worked from the ideas of Charles Spearman, who in 1904 proposed a general factor common to intellectual tasks and later described the core of that factor as the ability to perceive relationships and infer what is missing. The matrices try to measure this with as little prior knowledge as possible: no words, no arithmetic and no facts to remember.

Why it became the classic format

Two reasons explain the place matrices hold in intelligence tests. They depend little on language and schooling, which makes it possible to give the same problem to people from different countries and backgrounds, even though no test is entirely free of culture. And performance on them correlates well with performance on many other reasoning tasks, which is why they often serve as a measure of fluid intelligence.

Culture, as it happens, shows up in matrices in a curious way: the Flynn effect, the rise in scores over the 20th century, was especially strong on precisely this kind of test.

On IQNorth, matrices are the most heavily weighted area: 10 of the 37 scored questions. The test's structure is inspired by ICAR, a public-domain battery that also uses matrices, and our figures are original.

The rules that appear

In 1990, Patricia Carpenter, Marcel Just and Peter Shell studied in detail how people solve Raven's matrices. They recorded eye movements, asked participants to think aloud, and wrote computer programs that solved the problems. They found that almost every item combines a few types of rule, applied along the rows or the columns:

• Constant: an attribute repeats unchanged along the row or column.
• Progression: an attribute increases or decreases from one cell to the next, such as the number of figures or their size.
• Addition or subtraction: the third figure results from combining or removing the lines of the first two.
• Distribution of three values: three shapes or fills appear once in each row, in different orders.
• Distribution of two values: a line appears in two cells of the row and is missing from the third.

An item's difficulty came mainly from the number of rules in play at once and from how hard it was to see which attributes went together. The program that mimicked the best participants needed something more: keeping track of goals and subgoals while testing one rule at a time. The authors linked differences between people to this goal management in working memory and to the ability to abstract relationships.

A worked example

The matrix below is a public exercise from our training, not a question from the test.

The shape changes from column to column (circle, triangle, pentagon) and stays the same within each column; the quantity changes from row to row (1, 2, 3). Answer: 3 filled pentagons.

In Carpenter's terms, these are two simple rules combined: a constant shape in each column and a progression of quantity across the rows. Harder items combine three or four rules, including lines that add up or cancel out, and then the work of holding everything at once weighs much more.

How our figures are made

IQNorth's matrices are not drawn by hand. Each one starts from a combination of rules written in code, and the program draws the figures from it. This makes it possible to control what changes from one question to the next: how many rules, of what type, and which cell is left empty.

Once generated, each figure is checked to make sure it has only one correct answer. That sounds obvious, but it is not trivial: when two rules interact, a wrong option can sometimes satisfy everything too. If that happens, the figure is discarded.

The wrong options are designed with care as well. A known problem with matrix tests is that sometimes you can find the answer without looking at the grid, just by comparing the options with one another. Our distractors are built so that this does not work: to reach the answer, you have to understand the grid's pattern.

Training uses a separate bank, made with the same generator and the same safeguards, and the program checks that no training matrix repeats one from the test.

See also: Matrices on the test.