In a recent online opinion column in Education Week (1/13/2020), math education professor Jo Boaler writes that her paper “Fluency Without Fear … builds a case … on the importance of moving away from speed and memorization toward number sense and conceptual thinking.”
But in fact, what scientists say about memorization and speed is the opposite of what Boaler claims.
The key surprising discovery of cognitive science in recent research: When solving problems, the human brain has remarkable strengths but also stringent limitations. Scientists have verified that during the steps of solving a problem, “working memory” (where your brain solves problems) has an essentially unlimited ability to apply facts and procedures that can be quickly recalled from long-term memory.
However, at each problem step, working memory can generally hold only seven or fewer elements of not-well-memorized data and relationships, each for 30 seconds or less (Clark, Sweller, and Kirschner, American Educator, Spring 2012).
This makes memorization a key part of math success. When solving a multi-step problem, if needed relationships must be looked up, calculated on a calculator, or even mentally calculated, storing the answer takes up space that is limited in working memory. This tends to overload working memory, problem data tends to drop out, and confusion tends to result.
Cognitive scientist Susan Gathercole warns “Poor working memory skills are relatively commonplace in childhood” and advises to “avoid working memory overload in structured learning activities.” She explains:
“The capacity of working memory is limited, and the imposition of either excess storage or processing demands in the course of an on-going cognitive activity will lead to catastrophic loss of information from this temporary memory system.” (Working Memory in the Classroom, 2008)
How can the brain work around working memory limits?
In 2008, a U.S. Presidential Commission reported on ways to improve math education. Five leaders in the study of how the brain solves math problems summarized the findings of science. They wrote that to get around the “bottleneck” of working memory’s limitations when dealing with not-well-memorized information, the “central” strategy is
“the achievement of automaticity, that is, the fast, implicit, and automatic retrieval of a fact or a procedure from long-term memory.” (Geary et al., National Mathematics Advisory Panel (NMAP). Report of the Task Force on Learning Processes to the , page 4-5)
What is needed, say the experts, is fast recall of memorized facts and procedures.
How do students achieve fast recall? By exerting effort to memorize. In their task force report, the cognitive scientists noted:
“Verbatim recall of math knowledge is an essential feature of math education, and it requires a great deal of time, effort, and practice.” (p. 4-xii)
Cognitive scientist Daniel Willingham explains:
“Does speed matter? It does. When working a complex problem you not only want to pull simple math facts from memory, you want to do so quickly, so that the other work can proceed apace. Indeed, adults with stronger higher-level math achievement retrieve math facts faster (Hecht, 1999).” (Science and Education blog, 7/2/2017)
Boaler claims “conceptual thinking” can take the place of “speed and memorization.” But scientists tell us that in the brain, to construct conceptual frameworks, small elements of new knowledge first must be stored in the neurons of long-term memory – “memorized” by effort at recall. Neurons gradually wire together if their knowledge is repeatedly recalled at the same time at steps during problem solving. (Hebb, 1949).
But empty neurons do not connect. Willingham writes:
“A teacher cannot pour concepts directly into students’ heads. Rather, new concepts must build on something students already know.” (American Educator, Winter 2009-10)
Boaler claims students can learn math without the hard work involved in memorizing fundamentals. Perhaps she is “influential” because that’s what we would all prefer to hear. No one wants Dickensian schools.
But scientists say what she is advocating simply does not work. Unless teachers ask students to exert the effort needed to achieve quick recall of fundamental facts and procedures, according to science, students will be severely handicapped in their ability to solve math problems above grades K-3, when problems involve more steps.
Education Week has been publishing a series of articles on “The Science of Reading.” It’s been incredibly informative for educators seeking to improve student learning. I do not, however, recall any article which covered the consensus of science on how students can and cannot learn math. Is my recollection mistaken?
In math, you have published the views of those who deny science. Do such views merit publication? Would you publish a column by those who argue against vaccination before students enroll in school? Why not?
As professionals, we as educators have a moral and legal obligation to do our best to follow scientific best practices. If we listen to science on the importance of memorization and speed in recall, student prospects in careers that require math will rise. Poor and minority children will especially benefit.
Please continue to help educators understand what science has discovered about learning.