Fact Retrieval vs. Problem Solving in the Brain
In general, 2nd-3rd graders will switching a counting strategy to math facts retrieval when doing arithmetic, but these researchers found "considerable variation in the mix of strategies used to solve these problems...Retrieval fluency related to grade level in this sample."
The study is also a good reminder that fact retrieval is not the same as problem solving...something that has tremendous implications for how we should design education.
If you haven't read this 'oldie-but-goodie' paper, check out Transforming Physics Education. Researchers found that "students receiving traditional instruction, master, on average, less than 30% of the concepts they did not already know at the start of class." Nearly all teachers overestimate the ability of their students to answer questions correctly after attending a lecture.
"The definitive conclusion is that no matter how "good" the teacher, typical students in a traditionally taught course are learning by rote, memorizing facts and recipes for problem solving; they are not gaining a true understanding."
The study's conclusions? "To move a student toward expert competence, the instructor must focus on the development of the student's mental organizational structure by addressing the 'why' and not just the 'what' of the subject...(introducing) ideas in terms of real-world situations or devices with which the students are familiar; recasting homework and exam problems into a form which the answer is of some obvious utility rather than an abstract number, and making reasoning, sense-making, and refelcting explicit parts of in-class activities, homework, and exams..."
Algebra Problem Solving paper: docs.lib.purdue.edu/jps/vol3/iss1/2/
Newman et al., Journal of Problem Solving