I was an unremarkable math student for most of my K–12 years. That said, I have a few very distinct memories of my math journey.
In second grade, I remember being sent to the third-grade class for reading but being in the low group for math. Being in the low group meant a lot of worksheets. You know the ones, with rows and rows of the same type of problem. My third-grade memories are pretty much exclusively of multiplication flashcards. So. Many. Flashcards. Nothing else stands out from elementary school, and although I generally remember middle school as harder, I could do the math.
Then came high school. And with it, algebra. I have an almost visceral memory of that class. When the letters and numbers started hanging out together, nothing made sense. Previously I had been able to get through by following the procedures my teachers taught. But in algebra, my teacher’s words turned into the “Wah, wah, wah” of Charlie Brown’s teacher, and the string of letters and numbers on the board were just that: a jumble of symbols with no meaning. With enough repetition and a lot of anxiety, I was able to follow the steps to get the right answer enough times to pass. I didn’t have a clue what I was doing or why I was doing it and, honestly, both of those questions seemed irrelevant.
I know my story is familiar enough to the point of being trite, but it speaks to the way many of us were taught math. We were taught to “do” math, not understand it, and to practice a set of predetermined steps to get the right answer. I was lucky to eventually encounter a math methods teacher in college who helped me rewind my math experience by giving me the opportunity to explore, enjoy, and—who’d have thought?—understand math.
I bring up my experience not only because it is so common but also because it provides context for modern math standards. In creating the Common Core State Standards (CCSS) and other college-and-career-ready standards, education leaders recognized the need for students to move beyond just “doing” math, as my colleague Ted Coe explains. To become critical thinkers who can apply math to solve real and unique problems, students must be given the opportunity to actively engage with and understand the key concepts of mathematics.
To support students in engaging deeply in mathematics, the creators of college-and-career-ready standards identified three key shifts: focus, coherence, and rigor. Focus refers to the deliberate narrowing in on the most important mathematical topics to support deep understanding of these critical ideas, rather than providing a shallow experience with a wide array of topics. Coherence speaks to the deliberate structure built into the standards through a conscious progression of understanding that builds across grades and a conscious connection between topics within grades. The third shift, rigor, is arguably the shift that best addresses the issues with how I and so many other others were taught math. Rigor in math speaks to the need for standards and curricula to support students in not just being able to do math, but also being able to engage with and understand it.
To support students in engaging deeply in mathematics, the creators of college-and-career-ready standards identified three key shifts: focus, coherence, and rigor.
The developers of the CCSS were clear on what they meant by rigor. It does not mean making the content harder or introducing concepts and skills at an earlier age. Rather, it is about supporting a deep, rich understanding of key mathematical topics and concepts. They identify rigor in math as being composed of three aspects: conceptual understanding, procedural skills and fluency, and application.
To support deep understanding of mathematics, college-and-career-ready standards are designed to appropriately balance the development of all three aspects of rigor in math within and across grades. The intent is to transition away from the overemphasis on simply memorizing and repeating procedures and toward developing conceptual understanding as a critical foundation for understanding procedural skills and fluently applying them to solve meaningful problems.
Rigor is a way of describing the cognitive complexity, or the cognitive demand, required to complete a task. Rigor in math is often associated with difficulty, but difficulty is not the same as cognitive complexity. Consider the following two tasks: inflating 100 balloons and putting together a 100-piece puzzle. Inflating 100 balloons may be tedious, and it would likely take some time. Whether you are using a pump or blowing them up by mouth, it might get difficult after a while, but the task is not cognitively demanding. Completing a 100-piece jigsaw puzzle may not take long to complete, but the task requires a strategy and decision-making. Even though completing the puzzle may be easier for an adult than a six-year-old child, the kind of thinking required is the same for both adults and children.
If you have been teaching for a while, you are likely familiar with other measures of cognitive complexity. Bloom’s Taxonomy was developed in the 1950s and revised in the early 2000s, and it was the dominant framework for years. With the onset of college-and-career-ready standards, the need for a framework specifically for assessments surfaced. Standardized tests at the time were measuring how students think about mathematical content and procedures but did not measure the depth of understanding relative to explaining one’s thinking, justification of processes, and transfer to other contexts.
In 2002, Norman Webb developed the Depth of Knowledge (DOK) framework to categorize assessment tasks according to the level of cognitive complexity required by the content of the task. While the DOK framework is useful to ensure assessments have a range of cognitive demand, it does not provide a way to examine the balance of conceptual understanding, procedural skill, and application.
How rigor in math supports deep understanding
If you are like me, you were taught to divide fractions using some version of the “keep, change, flip” rule whereby to divide two fractions you keep the first one the same, change the division symbol to a multiplication sign, and flip the final fraction to its reciprocal. Does this approach work? Yes. Do you know why? Probably not. This is a common example of a procedure-focused, instrumental understanding of a concept.
In 1976, Richard Skemp introduced the terms “instrumental understanding” and “relational understanding” to mathematics education. He explained “instrumental understanding” as “rules without reasons” and relational understanding as “knowing both what to do and why.” Various research studies have since shown the positive impact of relational understanding.
Students who develop a mental web of interrelated concepts are better able to understand and recall concepts, are more successful at problem-solving, and demonstrate the ability to generate original ideas. Developing students’ relational understanding helps them tackle unique problems and build their confidence. When taught using a traditional approach—the teacher teaches a procedure or skill, and students practice the skill with a series of similar problems—students don’t learn to reason; they learn to apply a discrete skill to a narrow set of cookie-cutter problems. This type of instrumental understanding promotes what CCSS coauthor Phil Daro terms “answer-getting.”“Keep, change, flip” is a prime example of answer-getting.
To become critical thinkers who can apply math to solve real and unique problems, students must be given the opportunity to actively engage with and understand the key concepts of mathematics.
When fraction division is taught as a stand-alone procedure, most students can get the answer, but they may not be able to apply this meaningless procedure to more complex tasks. This is problematic given that understanding of fractions and division are a key predictor of success in algebra. This is where the thoughtful balance of conceptual understanding, procedural skill and fluency, and application come into play.
The college-and-career-ready math standards approach to fractions utilizes rigor to promote relational understanding. When students are introduced to fractions, they are not taught just to identify the numerator and the denominator, or to abstractly think of two-thirds as “two out of three.” Instead, they are taught to understand fractions as units created by partitioning and iterating shapes or number lines.
Students apply the strong sense of the unit fraction developed in third and fourth grade to division, working initially with whole numbers and unit fractions only. This purposeful limitation builds understanding that 6 ÷ 1/3 translates to “How many one-thirds are there in 6?”—a simple concept to model. This understanding is then applied to problems like 6 ÷ 2/3 and, ultimately, to dividing fractions by fractions.
Scott Adamson demonstrates how the purposeful progression of conceptual understanding ultimately reveals the “keep, change, flip” procedure to students in a way that allows them to internalize it and apply it more successfully in novel situations in a video on YouTube.
Where to start?
Knowing how rigor in math supports student understanding is only the first step. To effectively incorporate rigor into your classroom, you need to understand how it relates to the other shifts, how to determine the rigor expectations in your standards, and how to incorporate aspects of rigor purposefully and appropriately into your practice.
Next month, we will be releasing a 60-minute, self-paced aspects of rigor eLearning designed to provide K–8 classroom teachers, intervention specialists, coaches, learning specialists, and instructional leaders guidance on incorporating aspects of rigor into instruction. It will help you continue your journey and your students’ journey into a deeper understanding and enjoyment of mathematics.