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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.