Who Learned More?

Imagine two children learning about a new concept. One is guided to think through the process step by step while actively engaged with the concept; the other is given an equation to apply.

Take the concept of slope as an example:

Child #1 is asked to plot a position vs. time graph step by step, predicting where each new point would go. They painstakingly count the blocks of time and space: as they move one step forward in time, they move one step up in position, then two steps up, then three, then four—noticing a pattern emerging.

Their thought process: "First, as I moved forward one step in time, I had to move 1 step up in position, then two, then three…..This isn’t a straight line—it’s growing! The function must describe how this change happens. The line gets steeper… why? Oh! Because the position is changing faster over time. What does that mean? Maybe something is speeding up! I wonder what happens if the object is slowing down. I want to find out!"

Child #2 is simply given the equation Δy/Δx. They find two numbers, subtract, divide, and get an answer.

Their thought process: "I have to find y₂, then y₁. Now I have to find x₂, then x₁and subtract them. Now divide y₂ - y₁ by x₂ - x₁. This doesn’t make any sense. Why am I doing this? Math Sucks! Physics Sucks!"

Who truly learned?

This distinction is especially important at a young age when children are most receptive to learning and retaining information. The way they first experience a concept shapes their understanding for years to come. If we engage their natural curiosity and let them explore patterns, they develop an intuitive grasp of ideas—one that stays with them far beyond the classroom.

This pattern appears everywhere in education. Are we teaching students to manipulate equations, or are we helping them develop intuition? Are they discovering patterns for themselves or just following a set of steps?

Computation is important, but real learning happens when students engage deeply—predicting, questioning, and making sense of what they see. When we shift our focus from rote procedures to meaningful exploration, we transform passive learners into curious thinkers.

There’s a fundamental difference between teaching students to think versus teaching them to compute. And at a young age, that difference can shape the way they learn for life.