When Learning Math, Sometimes Less Is More

One paradox of learning math is that “less” can indeed be “more.” Many students took advanced classes like calculus in high school. However, if I ask which college class they placed into, it’s often at a lower level than calculus.

Classroom observations support these placements: some students aren’t sure if 30% and 0.3 are equivalent; some think that adding one-third plus one-fourth yields one-seventh. It’s puzzling that high schools had moved these students into advanced classes. Something went wrong, probably well before senior year, and schools should slow things down and try to remedy it. Aspiring science students should not arrive in college unprepared for calculus despite having taken calculus.

The preferred explanation – that these students must have taken a placement test on a bad day –  is easily dismissed. We use multiple factors to place students into math classes, including high school grades and multiple tests. It takes more than a family crisis, missed bus, or flu to bar a well-prepared student from calculus. Some students do panic during math tests, but anyone who advanced to calculus in high school should have succeeded on more than a few math tests along the way. Tests are not the culprits.

As puzzling as the placement scores is the number of high school students taking calculus. Calculus was once the ubiquitous starting point of college math, not a common capstone to high school. If K-12 education had improved in the meantime, and more students quickly mastered algebraic skills, then, by all means, teach them calculus. But what if more students are merely sent along with modest, baseline grades in algebra because parents and principals want more children to study more advanced material?

Students often struggle with the most fundamental concept in algebra: changing the letter we use for a variable doesn’t change the rules. Students might be comfortable solving an equation for a variable named “x” (a common name for variables in algebra), but they worry if I label it “E” or “v” (common labels in physics). The equivalence of these situations is among the most important lessons of algebra. As Juliet said, “That which we call a rose by any other name would smell as sweet.”  

Many college-bound students would benefit from more gradual high school math curricula. Let them spend twelfth grade thoroughly revisiting algebra, sharpening core skills, and applying those skills to a variety of problems. Athletes lift weights and run laps before stepping onto the playing field, and aspiring science students should drill on fundamentals before attempting advanced topics. Succeeding in calculus at the start of college is better than paying college tuition to learn algebra that should have been mastered in high school.

Gradual curricula might be much more beneficial for college-bound students – but what about school administrators? How do they advance their careers if fewer students take advanced math classes? Schools are evaluated on whether students reach milestones within the K-12 system, not on what happens after they leave. It looks good to have more students taking advanced classes.

Well, what if K-12 administrators’ careers and budgets depended on how students performed in college? I suggest this not in a vindictive spirit toward high schools, but from experience with receiving educational feedback. In many universities (mine included), the science and engineering departments regularly interact with the industries that hire our graduates, and industry input helps us improve our teaching.

High schools should similarly consider what happens when underprepared students enter college. Instead of pushing them through rapid math sequences without mastering basic skills, let students progress more gradually. Learning a few key things well is more important than learning many things superficially. Less really can be more.