Algebra skills make a difference in college — but arithmetic skills make a bigger and broader difference in life.
Look, algebra skills are important. Not only does a mastery of algebra give a student among the most powerful tools ever developed to model and predict quantitative phenomena, it also develops key logical and analytical skills. But algebra is a major obstacle to success in college-level math courses (and hence success in college in general), and it may be the biggest “dropped stitch” in students’ mathematical development. Peter Bahr ( Res High Educ (2012) 53:661–693 ), in a study of retention within college remedial math sequences, writes that
beginning algebra is a ‘‘low point’’ in the math sequence in terms of the likelihood of success on the first attempt among students who progress up to this course from a lower level.
Indeed, a study by the Community College Research Council shows that scoring in the highest quartile versus the lowest quartile on the Accuplacer algebra placement exam has an effect size in predicting students’ college credit attainment comparable to that of reading and writing placement exams (a 10-credit advantage for algebra vs. 10.6 for reading and 11 for writing). But all three of these tests were less predictive than a fourth placement exam: arithmetic, which conferred a 12.7-credit advantage to its top-quartile scores. The authors write:
the predictive power of these [algebra, reading, writing] placement tests on college credits earned was very low; the best-predicting test (ACCUPLACER Arithmetic) explained 6 percent of the variation in college credits.
So while efforts to improve students’ algebra skills will likely improve their chances at college success, it may be that improving their arithmetic skills would have an even bigger effect — and given algebra’s rarity in the public sphere and arithmetic’s pervasiveness, strong arithmetic skills are likely to provide the most benefit to the most people. To me, the single most crucial of these arithmetic skills is the arithmetic of ratios, proportions, and percentages.
As I wrote in the article Disenfractioned, one of the most important number skills for daily life, is the ability to attend to fractions conceptually. That is, to conceive of a number that does not represent an absolute quantity but rather a comparison between two quantities. This is one of the largest cognitive shifts in arithmetic, and is probably second only to the shift into algebra in the extent to which students’ success in high school and college mathematics depends upon it. But a lot of effort and time is devoted to calculating with, rather than conceptualizing, fractions. I put it this way:
Rather than treating fractions as a necessary annoyance of arithmetic on our way to algebra, geometry, and calculus […] slow down and explore how fractions and fractional reasoning help us make sense of real-world problems. Rather than reaching up toward higher levels of abstraction, reach out toward more diverse contexts of application. Contemplate before you calculate.
It’s not just math instructors who are stymied by students’ weaknesses with fractions and percentages. Others are starting to take notice. My colleagues who teach upper-level college accounting and management courses have told me that, more than algebra and calculus skills, at the top of their wish list for their students is stronger understanding of basic fractions and percents.
Outside of higher education, the trend is no less evident. New England meteorologist Dave Epstein tweeted his frustration this week:
So where do we begin rebuilding these skills? As I’ll suggest in the upcoming workshop “Seven Habits of Highly Numerate People (And So Can You!),” the first step is to catch yourself already using this reasoning in everyday life. How do you find a 20% tip? Most people aren’t fastidious enough to find an exact answer with a calculator, so typically they estimate — another undervalued numeracy skill — by rounding the bill to the nearest $10, casting aside the zero, and doubling what’s left. While it’s possible that many people do this without knowing “why it works,” at the least it is one slice of life where successful dealings with percents are more common than not. (Waitstaff can feel free to disagree with me on that last point.)
The more I learn about the upcoming Common Core State Standards for math, the more encouraged I am that the elementary school curriculum in these topics is moving in a positive direction. And that direction, according to the standards, is “early,” dwelling on fractions beginning with the third grade, including this standard (3.NF. a.3d) that I particularly appreciate (emphasis mine):
Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
This standard is a direct assault on what may be the biggest “bad habit” that students pick up in making the shift from numbers as quantity to numbers as comparison: denominator neglect. Proper fractions represent relationships of parts to wholes; likewise with percentages between 0% and 100%. But too often, the role of the whole is lost, and this is especially true of percentages everywhere they are reported, from popular media to academic journals to federal budget bills. One of the most useful habits to build context-aware numeracy skills is to pause at each percentage you encounter and ask yourself: “Percent of what?” Forcing the lazy side of your brain to hold two numbers, both part and whole, in your memory is the most fundamental prerequisite to a better understanding of fractions and percents.
The front lines of the battle for numeracy, then, begins in elementary school — but continues into and throughout adult life. Innumeracy preys on weak arithmetic skills. But fortunately, these skills may be the easiest to rebuild, and everyday life provides plenty of opportunities to practice them.