Math is “ruthlessly cumulative”: The truth about fluency and the promise of differentiated practice

MIT Professor Steven Pinker made a statement that has stuck with me. He observed that math is “ruthlessly cumulative” (1997). Phil Daro expanded on this by noting that, when studying history, you can surely learn a great deal about World War II without deeply knowing about World War I, but there’s absolutely no way to have any success in algebra without an unwaveringly firm grasp of number sense. Certainly, the reality of pre-requisite skills is manifest in all domains, but, as most math teachers will tell you, no domain is as “ruthlessly cumulative” as mathematics.

We often acknowledge this when focusing on basic arithmetic operations or “math facts.” Caron (2007) points out that without the mastery of math facts, “students are virtually denied anything but minimal growth in any serious use of mathematics or related subjects for the remainder of their school years.” It may be helpful, however, to expand our focus on and concern about fluency in mathematics well beyond basic facts (e.g., addition, multiplication, etc.) to far more advanced concepts.

As Pinker (1997) notes, “Calculus teachers lament that students find the subject difficult not because derivatives and integrals are abstruse concepts—they’re just rate and accumulation—but because you can’t do calculus unless algebraic operations are second nature, and most students enter the course without having learned the algebra properly and need to concentrate every drop of mental energy on that.”

It’s an eye-opening observation, especially in light of the critical role math achievement plays in allowing students to advance in the growing fields of science, technology, engineering, and math (STEM). Without a strong foundation, they will struggle to succeed in the higher-level courses needed at each incremental level. Ultimately, they won’t have the option to even enter these in-demand fields, let alone compete in them.

The Unglamorous Truth about Creativity

So how do we make our students mathematically competent? Pinker (1997) notes that “the way to get to mathematical competence is similar to the way to get to Carnegie Hall: practice.”  The problem, as Pinker (1997) also notes, is that “drill and practice, the routes to automaticity” are, by some, undervalued and put down as “‘mechanistic’ and seen as detrimental to understanding.”

But, ask yourself: Would a world-class violinist tell someone that the scales she practiced years before taking the stage at Carnegie Hall were of no benefit?

Lemov, Woolway, and Yezzi (2012) echo this sentiment, stating, “Many educators perceive drilling—which they characterize with the pejorative ‘drill and kill’—to be the opposite, the enemy of higher order thinking,” but, “as cognitive scientists have shown, it’s all but impossible to have higher order thinking without strongly established skills and lots of knowledge of facts.”

Seeking to foster creativity and focus on deeper understanding, we may have lost sight of this fact:

“Creativity, it turns out, is often practice in disguise, and to get more of it, it often helps to automate other things” (Lemov, Woolway, and Yezzi, 2012). “This synergy between the rote and the creative is more commonly accepted in many nations in Asia,” (Lemov, Woolway, and Yezzi, 2012) while “Americans have developed a fine dichotomy between the rote and the critical: one is good, the other is bad” (Rohlem and Le Tendre, 1998). Yet, the truth is that “creativity often comes about because the mind has been set free in new and heretofore encumbered situations” (Lemov, Woolway, and Yezzi, 2012).

How to Drill—The “No-Kill” Approach

So, what does this mean for us?

• We must begin to acknowledge that the concepts of fluency and automaticity extend well beyond basic math facts.
• We must also acknowledge that practice, which has been “generally seen as mundane and humdrum, poorly used and much maligned,” is actually worthy “of deep, sustained reflection and precise engineering” (Lemov, Woolway, and Yezzi, 2012).

With these ideas in place, the true challenge for educators is in the full application. How do teachers manage all of the extensive practice necessary on hundreds of skills for the tens to hundreds of students they serve? This is where technology shines, making the seemingly impossible doable.

Since its inception, Renaissance has been committed to the “best use of technology,” which we believe is in the collecting, storing, and reporting of information. Why believe in something so seemingly dull? It is the key to giving students more time to learn and enabling teachers to target instruction without increasing their paperwork. This gets us excited.

References

Caron, Thomas A. (2007). Learning multiplication the easy way. Clearing House, 80(6), 278–282.

Lemov, D., Woolway, E., & Yezzi, K. (2012). Practice perfect: 42 rules for getting better at getting better. San Francisco: Josey Bass.

Pinker, S. (1997). How the mind works. New York: Norton.

Rohlem, T. & Le Tendre, G. (1998). Teaching and learning in Japan. Cambridge: University Press.