Drill and practice is an outcome of associative, stimulus-response theories of learning based on Ivan Pavlov's experiments with dogs taught to salivate when they heard a bell tone. Such theories stress passive, receptive learning in which knowledge consists of a depository of information that is to be memorized. Teaching is informing learners of what they are to learn and having them practice it until they can automatically, without thinking, give the desired response to a given stimulus. For the number facts, figuring them out, as with or Tap & Tally™, is considered a “crutch” that allows kids to avoid the “hard work” of memorization.

Virtually all teachers use drill and practice to some extent in teaching the number facts. The usual tools for doing so are flash cards, speed drills (as with ), and worksheets. When used in conjunction with mental math and Tap & Tally™, drill and practice is beneficial in teaching the number facts. A major shortcoming of it, though, is that it leads to guessing when the correct response to a number fact doesn't “jump” into a learner's head. Thus if guessed incorrectly, the wrong answer is reinforced.

The main problem with drill and practice in teaching the number facts, however, is the common belief that it is effective—that once a fact is "drilled" into a student's head, it will remain there forever ready to be called up in an instant for the remainder of the student's life. Ask any elementary school teacher or middle school teacher who teaches math if they are satisfied with how well the kids they got at the beginning of the year knew their “facts,” in particular, the times table. We have asked that of hundreds of such teachers and have never been answered afirmatively. Logically, that leads to one of two conclusions: (1) Either every teacher before them did a terrible job of teaching the number facts, or (2) many of the kids the teachers got forgot some of the facts when they were not using them during the break between grades. We side with the latter conclusion. How about you? Brain research and “laws of forgetting” assure us that even things learned to a machine-like level tend to defy automatic recall if not practiced on a regular basis. The wisdom in the timeless refrain, “use it or lose it,” cannot be denied!

Mathematics is about thinking. It is about representing patterns symbolically, modeling quantitative events, making connections between assumptions and conjectures, and the like. Beyond a few definitions (like that for a fraction), the meaning of certain symbols (like = for the equal sign), and knowing the counting words (like one, two, three, ... ), much of arithmetic doesn't need to be remembered. Most of it, even some of the formulas, like LxW=A for the area of a rectangle, can be figured out when needed. Thus driving children to mindlessly know a number fact without thinking about it conflicts with the very spirit of mathematics.