Conceptualizing mathematical proficiencies
Students can exhibit a wide range of competencies in mathematics that
go beyond the traditional “procedural vs. conceptual” debates. What do
we expect quality student work and progress to look like? How can we
understand what our students are actually learning? What are we really
assessing in mathematics?
Chapter 4 of the book, Adding It Up: Helping Children Learn Mathematics,
offers five interconnected ways to think about student progress:
 conceptual understanding —comprehension of mathematical
concepts, operations and relations
 procedural fluency —skill in carrying out procedures flexibly,
accurately, efficiently and appropriately
 strategic competence —ability to formulate, represent and
solve mathematical problems
 adaptive reasoning —capacity for logical thought, reflection,
explanation and justification
 productive disposition —habitual inclination to see mathematics
as sensible, useful and worthwhile, coupled with a belief in
diligence and one's own efficacy.
The full chapter includes more detailed descriptions of the strands
and how they are interrelated. It can be accessed online at The National Academies Press.
Once you feel comfortable with these proficiencies, try analyzing some
of the tasks you use in your class with these in mind.
