Even knowledgeable math educators have a hard time defining the most important topics in the Common Core Math Standards and its offspring. The “Standards” is a complex and demanding document that its authors try to organize, link, and sequence by writing the progressions, and its proponents amplify in trainings and workshops. This complexity makes it hard for all but a few teachers to order and focus lessons on the critical concepts. They thus follow their textbook’s scope and sequence believing their authors have that special knowledge. And they march their students from page to page of equal weight, homogenizing and leveling the math concepts. Some will use public collections of test questions to enhance student success rates which only serves to increase focus on those topics that are easy to multiple-choice test. But in the end our students come away with the feeling and understanding that mathematics is the study of lots of small separate ideas and procedures that have little if any coherence.
Fortunately, technology offers us an easy way to find out what mathematical ideas the authors of the Common Core think our kids need to learn. Load its pdf or your state’s framework and use find (Control F) to count the number of times a word or concept is mentioned. I did this for the word fraction(s) and the number I get is 200 or so. Fraction(s) are mentioned in the 93 pages of the Common Core 200 times. It is obviously “important.”
I also counted the number of times ratio(s) is mentioned — 12 times! The authors of the Common Core are telling us something. And we can ask are they right? Should our students spend 20 times as much time on fractions as they do on ratios? Are fractions 20x more important?
Ratios are central to our lives and essential to our quantitative reasoning. Broadly speaking they are the division of two quantities. They take a wide variety of forms. Percents are ratios, rates like miles-per-hour or interest are ratios, averages including batting average in baseball are ratios as are statistical means and standard deviations, unit conversions like miles to kilometers or Euros to Dollars are ratios, most economic indicators even price per share are ratios, as are most significant scientific and mathematical quantities like mass and pi ratios. It is hard to think of any useful math that today that does not involve ratios.
In our schools ratio and proportions (proportions are ratios too of course) are the main new topics for 6th and 7th graders. They have to learn to solve a raft of traditional problems around motion, work, mixture, interest, and conversion. These are generally taught in silos as separate forms and algorithms that students are supposed to understand and master. This is just the beginning, because slopes are ratios which makes linear equations and linear functions fundamentally the study of ratios as is the whole of trigonometry. Calculus, too, is the study of ratios, for what is the derivative but a ratio (dy/dx).
Most of our work and life that uses quantitative reasoning revolves around ratios and remarkably little involves fractions unless you follow recipes at a bake shop or are a carpenter in the U.S. Fractions provided shortcuts to calculation in the analog age, but now in our digital age with machines doing the calculating, they are obsolete. And since they are numbers derived from ratios (rational numbers) isn’t it about time that we reverse that 20:1 ratio and prepare our kids for their future and not our past.
If you were to define a mathematics curriculum for the 21st century what concepts would you focus on — fractions or ratios?