Should we teach children to memorize the multiplication tables?
A new article argues that mathematical reasoning and complex problem-solving benefits from a foundation of well-memorized math facts
Highlights
• In the mid-1990’s, many educational jurisdictions began to de-emphasize the requirement that students memorize math facts and procedures (e.g., multiplication tables, procedure for multi-digit addition).
• Since then, research has revealed that mathematical reasoning and problem-solving benefits from having an existing foundation of well-memorized math facts and procedures.
• Students who do not have math facts and procedures memorized are more likely to struggle with math problems because of working memory limits.
Introduction
For decades, educational experts have been divided over the value of memorized math facts. Should children learn math facts (e.g., 4 + 9 = 13 and 7 x 3 = 21) by rote? In other words, should students rehearse these facts until they are fully committed to memory?
A recently published paper, Designing mathematics standards in agreement with science, written by Hartman, Hart, Nelson and Kirschner (2023), brings us closer to resolving the debate. The paper offers a review of the topic and explains why research now suggests that a limited amount of memorization is highly desirable.
The article begins with some historical context. During the 1960s and 1970s, children were expected to memorize fundamental math facts and algorithms. However, a shift occurred in 1989 when the National Council of Teachers in Mathematics (NCTM) introduced new math teaching standards. These standards recommended the use of calculators at all grade levels and emphasized the development of reasoning skills. At the same time, NCTM suggested that teachers reduce the time allocated to the rote memorization of math facts and algorithms. The NCTM’s recommendations were influential and many U.S. states and Canadian provinces modified their curricula in response (Hartman, Hart, Nelson & Kirschner, 2023). Today, it’s not unusual to find middle and high school students who don’t know their multiplication tables.
Implicit in NTCM’s 1989 recommendations was a belief that mathematics instruction should refrain from memorization-related activities in favour of deep thinking, reasoning and problem-solving. On the surface, this seems reasonable. However, there is increasing evidence that their new standards were shortsighted. In fact, they may have made deep thinking and problem-solving harder for students to do.
Studies conducted since 1989 have consistently shown that students are more likely to succeed in mathematics when they develop automaticity1 in basic math facts. This finding has been demonstrated across different grade levels and student populations. Mastery of math facts has been linked to enhanced math learning and problem-solving performance in elementary schools (e.g., Cumming & Elkins, 1999, Lin & Kubina, 2005) and even predicts academic success at the college/university level (e.g., Powell et al., 2020, Hartman & Nelson, 2016). Research also indicates that low-achieving math students experience significant sustained improvement in standardized test scores after developing an automatic recall of math facts (Pegg, Graham & Bellert, 2005, Stickney, Sharp & Kenyon, 2012).
The counterintuitive relationship between memorized math facts and deep thinking
Calls for “more memorization” may seem counterintuitive. Isn't rote learning the antithesis of deep thinking? With calculators readily available, what is the point of having students memorize math facts?
The Hartman et al. (2023) paper provides a compelling explanation of how memorizing certain math facts facilitates deep thinking and advanced reasoning. Fundamentally, our working memories are extremely limited in size, while our long-term memory is vast. By having key math facts and procedures immediately accessible in long-term memory, students need not waste cognitive resources on low-level calculations. This allows them to better focus their intellectual energies on the problem at hand. Simple math facts, such as 3 x 7 = 21, can be retrieved effortlessly and instantly from long term memory.
In contrast, students who haven’t memorized math facts must temporarily interrupt their higher-level problem-solving processes to manually calculate (or type into a calculator) the product of 3 x 7. This is a diversion that pulls attention away from the actual problem, consumes space in working memory, and increases the risk of error (e.g., a student who counts to 22 while using a counting-up strategy to calculate 3 x 7).
Proficiency in any field, not just mathematics, is built upon a foundation of automatized knowledge. For example, a novice piano player will find it laborious to play even simple songs. Initially it takes concentration to strike the proper key with the correct finger at the right moment. However, with practice and repetition, piano playing gradually becomes more automatic. Different chords and note progressions are executed with minimal conscious thought, freeing the musician's working memory to infuse the music with emotion or improvise with different tempos and dynamics. This higher-level of performance is only possible because the foundational skills have been automatized.
Problem-solving in mathematics works in a similar fashion. As Wu (1999) explains, “The automaticity in putting a skill to use frees up mental energy to focus on the more rigorous demands of a complicated problem” (Wu, 1999, p. 2).
Summary
The paper Designing mathematics standards in agreement with science makes a compelling case that math standards need to be modernized to bring them into alignment with evidence-based research. Specifically, we need to better emphasize the need for students to be able to quickly recall core math facts. Understandably, some educators have expressed concern that young learners will find rote learning exercises difficult, frustrating or off-putting. Fortunately, there has been a great deal of research on how teachers can foster math automaticity through activities that are low-stakes and even fun. Caron’s (2007) article, Learning Multiplication: The Easy Way is one example of a friendly and non-threatening activity that can promote the memorization of single-digit multiplication facts.
Teacher education programs sometimes tell new math teachers to "teach for understanding, not rote learning." This needs be recognized as a false dichotomy. Hartman, Hart, Nelson and Kirschner (2023) make a compelling case that to fully prepare students for higher-level mathematics and complex problem-solving, we must teach for understanding AND ensure that students have mastered their math facts.
References
Caron, T. (2007). Learning Multiplication: The Easy Way. The Clearing House, 80(6), 278-282.
Cumming, J. J., & Elkins, J. (1999). Lack of automaticity in the basic addition facts as a characteristic of arithmetic learning problems and instructional needs. Mathematical Cognition, 5, 149–180. Available at: https://www.tandfonline.com/doi/abs/10.1080/135467999387289
Hartman, J. R., Hart, S., Nelson, E. A., & Kirschner, P. A. (2023). Designing mathematics standards in agreement with science. International Electronic Journal of Mathematics Education, 18(3), em0739. Available at: https://www.iejme.com/download/designing-mathematics-standards-in-agreement-with-science-13179.pdf
Hartman J. R. & Nelson E. A., (2016), Automaticity in computation and student success in introductory physical science courses [online], available at: https://arxiv.org/ftp/arxiv/papers/1608/1608.05006.pdf
Lin, F. & Kubina, R. (2005). A preliminary investigation of the relationship between fluency and application for multiplication. Journal of Behavioral Education, 14(2), 73-87.
Pegg, J., Graham, L., & Bellert, A. (2005). The effect of improved automaticity of basic number skills on persistently low-achieving pupils. In H.L. Chick & J.L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (PME), Vol. 4 (pp. 49–56). Melbourne: PME. Available at: https://files.eric.ed.gov/fulltext/ED496946.pdf
Powell C. B., Simpson J., Williamson V. M., Dubrovskiy A., Walker D. R., & Jang B. (2020), Impact of arithmetic automaticity on students’ success in second-semester general chemistry, Chem. Educ. Res. Pract., 21(4), 1028–1041. Available at: https://pubs.rsc.org/en/content/articlehtml/2020/rp/d0rp00006j
Wu, H. (1999). Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education. American Educator, 23(3), 1–7.
Automaticity means being able to immediately recall a fact or procedure without any significant mental effort. It usually requires repetition and practice to achieve automaticity.
Great post! As a middle school English teacher, I can tell you that many students do not know their basic math facts. Ask them to computer simple figures--basically anything on a classic multiplication table--and they can't. For years students have lectured *us* because we have analogue clocks. And they can't read them.
This can't be too far apart from knowledge building on the English side of things.
We quit teaching spelling for the same logic: spell checkers and "memorization is bad." I've kept my own database of misspellings for the past four school years, and the average person would be absolutely horrified. Even many future AP students would flunk elementary spelling from the 1990's.
But it's system-wide. Fad after fad have kicked out the ladder from future generations.
(I remember an article years back about a Nobel Prize winner arguing (a) revision of Math standards, and (b) calculators for elementary students. What nonsense! Then again, years of complaining "More teachers should blog" is why I finally started a Substack...)
If I were in a flippant mood, I would ask those who make light of memorization to trust their lives or those of loved ones to a surgeon who goes to Google at every step of a procedure.