Practice Conceptual Bottlenecks in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
Specific concepts or ideas whose misunderstanding blocks progress across a wide range of related mathematical topics.
Gateway concepts—get these and everything else becomes easier.
Example 1
easyMany students struggle with the transition from 'find x' to 'prove for all x'. Explain why this is a conceptual bottleneck and give a concrete example of each type of problem.
Example 2
mediumA common bottleneck is understanding why \lim_{x\to a}f(x) does not require f(a) to be defined. Illustrate with f(x) = \frac{x^2-1}{x-1}, a=1.
Example 3
easyStudents often confuse 0.999\ldots with something 'just below 1'. Show that 0.999\ldots = 1 exactly.
Example 4
mediumWhy is it a conceptual bottleneck to move from 'numbers' to 'functions as objects'? Give an example where treating a function as an object is essential.