2 min read

Learning is Path Dependent

Prior to the summer of 2022, my son had no interest in rockets. Trains? Yes. Fire trucks? Yes. Planes? Hell yes. Planes with space shuttles on them? Sure. But rockets, space, astronauts? Nothing like this could hold his interest.

Then we went to the Houston Space Center. After this experience, he thinks of nothing but rockets. He builds rockets and launch pads. He holds “rocket launches” with these. He lectures his father on both real and made-up rocket history. He asks questions about rockets. He interprets everything with a vaguely phallic shape as a potential rocket.

What would have happened if we had gone to the submarine museum or seen a Cirque de Soleil show instead? There’s no guarantee that he would be into naval history or contortionism. But there was no guarantee he would be this into rockets after visiting the Space Center, either.

Something about the experience changed the trajectory of his interests — and, consequently, his learning. He enters school with certain conceptual ideas, through which he can make analogies. Rockets need fuel and engines; many use boosters; many use stages; they require heat shields to re-enter the atmosphere. He enters school with basic knowledge of historical facts: the space race was an aspect of the cold war between the Soviet Union and the United States; the United States was the first country to reach the moon; he can recite snippets of Kennedy’s famous “we go to the moon not because it is easy, but because it is hard” speech. Rockets, at least for the foreseeable future, will shape how he understands new concepts and new facts and new ideas.

Another child, in another circumstance, will enter school with a different background. They will have a different set of cognitive resources to leverage as they learn math and science and reading. For no other reason than that we happened to visit the Space Center one day.

Another example of the path dependence of learning comes from Ken Koedinger’s lab at Carnegie Mellon. As students were learning fractions, they had one group learn to add fractions before learning to multiply them. And another group learn to multiply fractions before learning to add them. Usually, teachers teach students to add fractions first. But adding fractions, even if it’s conceptually easy, is procedurally quite difficult: you have to find common denominators, convert the fractions using those denominators, remember to add only the numerators, and then (potentially) reduce the fraction to its simplest form. Multiplying fractions is comparatively simple. You just have to multiply the numerators and then multiply the denominators. Then simplify, if needed.

The students who learned to multiply fractions first performed better on both multiplying and adding fractions. Why? The traditional way (addition first, then multiplication) leads to interference. Students get confused about which procedures go with which idea. The procedure for multiplying “settles” in students’ minds more quickly — at least that’s the authors’ best interpretation — leading to students who learn multiplying first to perform better on both multiplying and adding fractions. This result is not atypical in the research on learning: same material, different sequence, different outcome.


Patel, R., Liu, R., & Koedinger, K. (2016). When to Block versus Interleave Practice? Evidence Against Teaching Fraction Addition before Fraction Multiplication. Cogsci 2069-74. [I'm missing some information from this citation, but hopefully there's enough to find it]