Conceptual Bottlenecks Math Example 1

Solution

1. 1
   Solving: 'Find $x$ such that $x+3=7$.' Here $x$ is an unknown with a unique value ($x=4$). The task is computation.
2. 2
   Proving: 'Prove that $x+x = 2x$ for all $x \in \mathbb{R}$.' Here $x$ is a variable ranging over all reals. The task is logical reasoning for infinitely many cases.
3. 3
   The bottleneck: students accustomed to finding specific values must shift to reasoning about all values at once — a fundamental change in how $x$ is used.
4. 4
   Sign of the bottleneck: students try to verify 'for all' statements by checking a few examples.