Rote Learning and Runaway Standards: Our Broken Approach to Math Education
As a middle- and high-school math teacher, there is perhaps nothing more demoralizing than the realization that some of my students do not understand the four elementary arithmetic operations: addition, subtraction, multiplication and division. I don’t mean that students can’t perform these operations, rather that they don’t understand them: they do not understand the meaning of these operations or when it is appropriate to use them.
Many of these students think they understand math: they have memorized algorithms that involve “carrying” or “borrowing”; they remember that the “butterfly method” has something to do with fractions; they have studied the “rules” for integer operations. For many of these students, math has never meant anything more than rote memorization; they cannot even conceive of understanding because they have never experienced it. Without an understanding of underlying concepts, these students cannot recognize opportunities to apply them as part of a problem-solving process. Rote learners become utterly lost in the world of abstract mathematics, real-world applications and varied, multi-step problems.
How does this happen? To answer this question, let’s look at an example: basic subtraction. For most American elementary students, subtraction means the ubiquitous subtraction algorithm: subtracting place values and borrowing from the next largest place value when the difference would be negative (of course, an elementary student would not describe the process in quite the same way). Rote learners have memorized this algorithm without understanding how or why it works because they are simply not ready to understand it in this way (that is, the standard is not developmentally appropriate). If prompted, as they often are by standardized tests, these students will diligently apply this algorithm. The problem arises when these students are not prompted, as in the case of a dreaded “word problem” or, unfortunately, the “real world”. Without a conceptual understanding of subtraction, students cannot solve a problem without step-by-step instructions. This begs the question: what benefit do these skills offer to individuals, employers or our society at large if no one knows when to use them?
As an alternative, let’s consider a more conceptual strategy for mental subtraction, one that could serve as a replacement for a memorized algorithm. Subtraction, as a concept, can be carried out in two ways: by taking away (the difference of eight and five is found by beginning with eight and taking away five) or by counting up (the difference of eight and five is found by counting up from five to eight). Using the “counting up” method, a student could mentally find the difference of 83 and 27: first, we count up from 27 to 30 and remember how many we counted (three); next, we count up from 30 to 93, again remembering how many we counted (53); lastly, we combine our counts (56). Thus, the difference of 83 and 27 is 56.
Importantly, elementary-aged students are intellectually equipped to understand this conceptual strategy. This means that practicing this strategy will not only improve fluency but also deepen their understanding of subtraction as a concept, particularly if it is paired with a visual model. When faced with a real-world problem situation that calls for subtraction, students who have learned subtraction in this way will know exactly what to do. Later in the course of their math education, concepts like integer subtraction or the subtraction property of equality (which can be applied by subtracting the same quantity from both sides of an equation to isolate a variable) will be more easily understood.
Of course, just as simple subtraction might be mishandled in a classroom, similar pitfalls exist in the teaching of integer subtraction. Rather than reaching a conceptual understanding by subtracting negative numbers on a number line or by using integer tiles, students might memorize a rule for subtracting negative numbers: when they see two consecutive negative signs, many students learn to simply change them into an addition sign: 8 - (-5) becomes 8 + 5. Rules like these make neat substitutes for conceptual understanding and are enticing for teachers looking to achieve skill-based standards quickly and easily. While these strategies may pay off in the form of short-term results on a standardized test, it is not until later in a student’s education that the long-term disadvantages manifest in the form of an inability to solve problems or understand abstract extensions of these fundamental concepts.
This widespread issue, then, of neglecting conceptual understanding to focus on rote memorization has been fueled, in part, by our obsession with standardized testing. Teachers and students face enormous pressure to succeed on standardized tests and it is this pressure that so often compels teachers to take a shortcut to achieve a short-term result. It is painfully ironic, then, with all of this attention on standardized test preparation, that our students continue to underperform on a global scale, as measured by a standardized test. Clearly, our approach to math education is broken.
Unfortunately, the harmful pressure imposed by standardized tests is only growing in intensity. Legislators and bureaucrats continue to “raise standards” rather than doing the hard work to improve our broken system. As a result, a growing number of students are “left behind” as classrooms move forward at a dizzying pace to check skills off of a growing list. Drastic reform is urgently needed to take the emphasis off of a student’s performance on skill-based assessments and direct it toward each student’s conceptual understanding.