Focus, Coherence, and Rigor
These three words are the keywords (buzzwords/jargon, if you prefer) of the Common Core State Standards. The standards were designed with these three concepts in mind.
The Common Core standards were designed to focus on the skills that are truly integral to learning and understanding math. The writers actually cut out some skills that were determined to be less important.
One study that was performed recently involved determining the number of skills students are taught in their math classes all over the world, and testing students from all over the world with the tests we use here in the States. The researchers found that the US has one of the highest (if not the highest) numbers of skills taught. However, they lag behind other countries on the results of these tests. Note: you’d think that students in other countries would struggle with skills they had not been explicitly taught. Not so.
The writers of the Common Core believe that this disparity is caused by the “mile wide, inch deep” curriculum we currently have here in the US. We try to teach everything and don’t have enough time to teach anything well. (Take AIMS prep as an example – we spend months teaching students everything they could possibly expect to see on the AIMS tests, but lots of them don’t understand it well enough to actually perform on the test.)
Focus in the Common Core standards means attending to fewer topics in greater depth at any given grade level, giving teachers and students time to complete that grade’s learning. The focus in the standards means that we’re giving ourselves the time to work through the sophistication of the skills we are introducing. Once students have a complete understanding of a skill, they should be able to apply the skill in new ways, so we don’t have to explicitly teach those new ways.
Example: least common denominator. Students have been learning to add and subtract fractions using LCD for … I don’t know … a very long time. But does it really matter if a student uses the least common denominator? Isn’t it more important that she choose some common denominator as long as she can accurately find equivalent fractions? In fact, when she gets to algebra about 5 years down the road, she will have to add a/b + c/d, and will need to understand that the common denominator here is bd.
So the LCD has been removed from the CCSS. It’s not there. Common denominator is, of course (how else do you add fractions), but if you’re adding fourths and sixths, it doesn’t matter if you use twelfths (the LCD) or twenty-fourths (another CD). Even least common multiple in only mentioned once, briefly.
Right now, we teach math in a very disjointed manner. “Math” is basically a (very large) group of unrelated skills that students have to memorize. There’s no relation between the idea of splitting a cookie in half to share with your sister, the fraction 5/6, and the use of fractions in algebra. All of these are related concepts, but a select few of us can actually recognize them as related concepts. I probably couldn’t even have given this example until I started teaching math at a tutoring center 5 years ago.
Coherence in the CCSS refers to the idea that math tells a story, and we want students to see that story. We want students to understand how these concepts are interrelated within and across grade levels. Math is not just some random collection of things that we want students to memorize. It’s a process that gives order to our world, or better yet, shows us how to interpret the order that already exists in our world.
Now, there are two ways to approach coherence when discussing education standards. First, we can teach the math concepts in their logical order (what seems logical to people already fluent in math, that is.) Alternatively, we can teach concepts in the order that accounts for the natural development of the child. The writers of the Common Core attempted to balance these two options.
There are three main extremes that people will go to when discussing how to teach math – conceptual understanding (students need to understand the concepts of what they are doing), procedural fluency (students need to be able to compute accurately and quickly), and meaningful applications of mathematics (students need to see how math relates to real life and solve real-life problems using math). The “rigor” in the Common Core partly refers to the blending of all three of these goals.
It also refers to the challenge of the standards – they should be progressively more difficult to challenge students and keep them learning and moving forward on the continuum of understanding.
Yes, Common Core does require students to be fluent in their basic addition, subtraction, multiplication, and division facts. Yes, I am in favor of this. Yes, I know kids have calculators on their phones, but I don’t want them growing up and not understanding how “6 of one” and “1/2 dozen of the other” mean the same thing. Or not understanding that 5% off each item is the same thing as 5% off all the items.
Next post: How Common Core is attempting to change the way we teach math.