Drills are a crucial but frequently neglected part of early mathematical education. Often, even when students do write drills as part of their elementary school education, these drills are limited to memorizing the multiplication tables, or doing simple addition. Still, in my experience as a math teacher it has not been uncommon to find a student in junior high who is in the habit of using a calculator for simple calculations like 3x4.
Field’s Medalist Terence Tao describes mathematical education as consisting of three phases: the “pre-rigorous” phase, lasting until the undergraduate level, where students are trained to perform computations, the “rigorous” stage, in undergrad and graduate school, where students are exposed to formal and precise proofs, and the “post-rigorous” stage, in late grad school, where mathematicians make use of well-developed mathematical intuition to solve problems. Drills are a tool for the pre-rigorous stage, and are vital for students even if they do not intend to reach the rigorous stage. Students may not intend to study in STEM fields, but they should still be comfortable performing simple arithmetical operations, reducing fractions, or turning fractions into decimals.
Drills are crucial because they help students develop automaticity in the skill being targeted. This means that if, say, the multiplication of 7 x 8 comes up as a subproblem inside a larger calculation, students do not need to stop and think about how to solve it – they immediately know that the answer is 56. This frees up their working memory to store information relevant to the larger problem, rather than finding multiples of 8. The number of items that can be stored in working memory is limited (individuals of average intelligence can store and manipulate about 5-6 objects at once in working memory, which increases with IQ). Therefore, it is important that students not get bogged down with performing simple calculations from earlier stages of mathematical education. For example, a student learning about the distributive property will have a much easier time solving (3x + 8)(7x – 9) if they have their multiplication tables memorized.
It is certainly ineffective to teach mathematics in a “plug and chug” manner where students plug values into formulas to arrive at results without understanding the underlying structure of the math. However, there are certain mathematical facts that simply must be memorized, such as the multiplication tables. Before starting to memorize the multiplication tables, students should be able to multiply using repeated addition, however once they possess this ability they should also memorize the result, since it will recur frequently in many mathematical problems. Rote memorization should only be used when absolutely necessary, however it is just as much of a mistake to eliminate it as it is to overuse it.
Drills come in two types: those for skill building, and those for memorization. Multiplication table drills are an example of a memorization drill, while fraction addition drills are an example of a skill building drill. Memorization drills often precede skill building drills, as basic identities must be learned before they can be put together into a more complex sequence. For instance, the 3x1 multiplication drill requires students to already have mastered their multiplication tables.
Drills are designed to be written without requiring scrap paper on which to perform auxiliary calculations. Drill problems should be simple enough to be solved mentally, writing only the answers on the page. The student’s working memory should not be overwhelmed by the amount of calculations they need to perform. Drills should also target the kinds of calculations that are most likely to recur frequently in later mathematical work.
Drills should be done repeatedly until students can achieve perfect or near perfect scores. This is preferable to simply assigning a handout, because handouts may contain an excess of problems for stronger students and an insufficiency of problems for weaker students. Drills allow the level of practice to be targeted to each student, enabling a form of mastery learning. Once a student has achieved a near-perfect score two sessions in a row, they can be considered to have mastered the content of the drill. Because spaced repetition optimizes recall over time, once students have mastered a drill they should continue to practice it, but over longer intervals. For instance, if a student is practicing a drill every day and eventually achieves a perfect score, they could then practice again after 3 days, then after a week, then after two weeks, then after one month. If they continue to achieve perfect scores over the spacing schedule, they can both be confident that they have mastered the material, and that they will not forget the skill a few months later.