The science of reading tells us that novices learn to read through explicit and systematic instruction in decoding of words. The same cognitive principles that guide the science of reading are those that undergird all knowledge acquisition for novices, in the case of this post, specifically math. This post aims to distill some of the research around how novices learn, what it takes to become an expert, and the implications for the classroom.
Students as “Experts”
According to D.T. Willingham, in order to think as an expert does, students would have to understand how the parts of [a] problem related to one another, and know which parts are important and which are not (77). Given that experts think abstractly after nearly 10 years of practicing, to expect students to be experts in anything by the time they leave K-12 education is a wholly unrealistic one.
Willingham also writes that, “Cognition early in training is fundamentally different from cognition later in training” (97). In teaching novices any subject, they do not have the same arrangement of knowledge as experts in their long term memory. Even if they have automated knowledge,, such as fact families, multiplication tables and the like, I’d wager that until the later stages of a bachelor’s degree, (perhaps even beyond that degree), we are constantly in the gray area of knowledge acquisition in moving from novice to expert.
This to say: The focus of school should be to give students a solid foundation of knowledge in the basics of the field so that should they choose to continue their education in college, they have enough knowledge in their long term memory to begin the acquisition of expert thinking.
We should move to a place of feeling satisfaction when students have a solid grasp of the surface structure of knowledge rather than disappointed if they “only” know how to solve problems fluidly.
How Much Conceptual Work Should We Do?
Per Kirchner et al, decades of research clearly demonstrate that for novices (comprising virtually all students) direct explicit instruction is more effective and more efficient than partial guidance. Partial guidance is what we would see with “real-world” or authentic problems, and conceptual tasks.
Without fluid and automated fact knowledge, formulas, or even something as basic as vocabulary terms, frustration is likely to ensue. Further, cognitive load theory tells us that more stuff in students’ long term memory (though practice, drilling, retrieval and quizzes) means more space in the working memory (where you think about stuff) to do conceptual problems. From this, it may seem that we should never give students opportunities to practice manipulating or creating knowledge– and surely, naysayers of cognitive science will claim this, but it is categorically false.
Tom Sherrington notes that he, “can’t imagine a truly great curriculum where students do not, at some point, have a range of hands-on experiences, learn to make choices, explore ideas independently to find patterns and rules or develop original ideas”. From a cognitive perspective, Willingham, agrees.
Willingham writes that while we shouldn’t expect these activities to have much of a cognitive impact having students attempt to think like experts may have some benefits, such as increased motivation and having student feel proud of their work. In his own words,
“nothing terrible is going to happen… but the likely outcome will be that they won’t do it very well” (109).
Takeaway: Assignments that emphasize conceptual knowledge have their place.
We should weigh the affordances of each method, ensure that conceptual work does not replace or diminish the importance of having strong foundational knowledge taught through direct and explicit instruction, and understand the lack of cognitive benefits of having novice learners attempt to mimic experts.
Cognitive Science >> Curriculum
The issue with the current math discussion is that there is an underlying de emphasis of procedural knowledge all in the name of making math relevant. Students are expected to have conceptual knowledge as outcomes, but are not given the fundamental knowledge that comes from fluid, and practiced procedural and surface knowledge. We focus our attention on crafting students to think like experts when cognitive science tells us that they aren’t ready for this yet.
This to say: Just implementing a curriculum will not save us. Unless administrators, and teachers alike begin to understand the underlying cognitive principles that guide curricula, a few years from now it’s quite possible that this will be seen as a fad, another ‘silver bullet’ wannabe. Teachers should have tools to make our work about teaching versus curriculum planning, but we also need the knowledge to adapt these curricula and spot inconsistencies with what research shows us.
Just as students need to build their knowledge, so must teachers.
More Technical: Paul A. Kirschner, John Sweller & Richard E. Clark (2006) Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching, Educational Psychologist, 41:2, 75-86, DOI: 10.1207/s15326985ep4102_1
Less Technical, by same authors: Putting Students on the Path to Learning: The Case for Fully Guided Instruction:
Willingham, Daniel T. Why Don’t Students like School?: a Cognitive Scientist Answers Questions about How the Mind Works and What It Means for the Classroom. Jossey-Bass, 2010.
Sherrington, Tom. “Mode A Mode B = Effective Teaching and a Rich Enacted Curriculum.” Teacherhead, 8 June 2018, teacherhead.com/2018/04/22/mode-a-mode-b-effective-teaching-and-a-rich-enacted-curriculum/