Transformations of mathematical and stimulus functions.

Ninness et al. (2006) worked with eight adults who had no math trouble.

They used a computer program that showed a formula on top and three graphs below.

The learner clicked the graph that matched the formula.

After a few blocks the adults could match brand-new formulas to graphs without extra teaching.

The team then changed the rules or the points earned and watched the picks flip.

Seven of eight adults learned the matches in one short session.

When the rule said "pick the steeper line" or the points favored one form, their choices flipped right away.

The math meaning had jumped into the equivalence class, so a quick word or prize could steer it elsewhere.

Preston (1994) showed the same fast emergence with plain shapes, so the math result is not a one-off.

Barthelemy et al. (1989) first proved that once items are linked, the way we respond to one can spread to the rest; Ninness et al. (2006) now show this spread works for math meaning too.

Pérez-González et al. (2003) found that a new cue can re-route an equivalence relation; the 2006 paper adds that a one-sentence rule or tiny point change can do the same trick.

Fields et al. (2021) doubled success by moving the response window, while Ninness et al. (2006) doubled the power of the class by moving the reward window — two sides of the same coin.

You can use brief equivalence drills to give learners new math insight without long lectures.

After the links are set, a short verbal rule or token system can swing choices toward the form you want targeted that day.

Try it when you need students to see graphs and equations as the same thing, then nudge them toward the version that best fits the current task.