NRC’s Five Strands of Mathematical Proficiency

As part of my rising AP Calculus students’ summer work, they are to review six slides I created to illustrate the five strands of mathematical proficiency defined by the National Research Council (NRC).  For the purposes of this post, the slide corresponding to a particular strand is integrated into the task card text below.  Students received them as separate documents.

My primary purpose for this assignment is to expand high school students’ awareness of mathematics beyond that of mimicking the procedures demonstrated by their teacher for cookie cutter type “problems.”  While many may have touched upon aspects of these strands in earlier mathematics courses, I plan to make these practices explicit before the start of the school year.

A secondary purpose for this assignment is to improve students mathematical literacy.  While many secondary mathematics teachers incorporate reflections into their pedagogy, few require students to read or write formally about mathematics, or broach how to communicate and critically think in mathematics.  This benefits students who excel with literary modes of expression.

A tertiary purpose is to offer students a diverse set of work from which I may learn more about them as students of mathematics.  It also fosters an appreciation of mathematic’s complexity and simplicity, a duality of existence that confounds most students of mathematics, until the magical moment where they stumble upon a profound discovery that ignites a life-long pursuit of mathematics or science.

Questions for my readers:

Has anyone required their students to consider the NRC’s five strands of mathematical proficiency?  If so, in what ways?  And if so, how did it go?

What are your reactions to this assignment?

The text from the task card for this assignment with slides interwoven follows.


NRC’s Five Strands of Mathematical Proficiency


“Mathematics is one of humanity’s great achievements.  By enhancing the capabilities of the human mind, mathematics has facilitated the development of science, technology, engineering, business, and government.  Mathematics is also an intellectual achievement of great sophistication and beauty that epitomizes the power of deductive reasoning.  For people to participate fully in society, they must know basic mathematics.  Citizens who cannot reason mathematically are cut off from whole realms of human endeavor.  Innumeracy deprives them not only of opportunity but also of competence in everyday tasks.”

National Research Council. (2001). Adding it up: Helping children learn mathematics. J Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.


In 2001, the National Research Council (“NRC”) of the National Academies of Science published a report titled Adding It Up: Helping Children Learn Mathematics.  The report addressed the latest educational, psychological, and neurological research about teaching for mathematics proficiency, focusing on the interactions between teachers and students around educational materials and how teachers develop proficiency in teaching mathematics.  Their work significantly influenced subsequent research, as well as the pedagogical and curricular aspects of teaching of mathematics from kindergarten through community college.

In the report, the NRC chose the term mathematical proficiency to capture what it means for anyone to learn mathematics successfully.  Mathematical proficiency consists of the following five components, or strands. [1]

Conceptual Understanding:  The comprehension of mathematical concepts, operations, and relations.

Procedural Fluency:  The skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.

Strategic Competence:  The ability to formulate, to represent, and to solve mathematical problems.

Adaptive Reasoning:  The capacity for logical thought, reflection, explanation, and justification.

Productive Disposition:  The habitual inclination to see mathematics as sensible, useful, worthwhile, coupled with a belief in diligence and one’s own efficacy.

The most important observation the committee makes and stresses is that the five strands are interwoven and interdependent in the development of proficiency in mathematics (NRC, 2001, p. 116).


Review the slide set titled NRC Math Proficiency Standards located on the summer work page of the course website as well as the background information above.  Reflect upon each of the NRC-defined strands of mathematical proficiency. In no less than 250 words, submit a well-written, typed, reflection addressing the following prompt, preferably in a tabular format. [2]  Use one-inch margins, 12-point Times New Roman font, and double line spacing. [3]  Papers may be double-sided; consider using a landscape layout if using a table.  Make sure you include your name, period, date, and page number on each page.

Consider whether viewing mathematical proficiency as comprising these different strands might help you in your math, especially with calculus, and/or the AP Exam, as applicable.  Think about your strengths and weaknesses in each of these areas, and what you might do to strengthen your overall mathematical proficiency.

Writing Prompt:

What were your experiences in your math classes (past & present) along each of the strands?  Were your experiences positive?  Why or why not?  Could you benefit from keeping these different dimensions of mathematical proficiency in mind as you continue with your education? What parts could be challenging?  How could we address your challenges in the class?

Due Date:

This assignment is due on the first day of school.  Late work will not be accepted.

I will read your submission with great interest.  It will help me get to know you better.  I hope this provides you with an opportunity to get to know yourself a little better, especially as it relates to differing dimensions of mathematical proficiency.


The following rubric will guide my rating of your assignment.  If you feel I missed critical elements of your writing, please feel free to contact me afterwards.

[1] You can read more about these strands, and download the entire report for free, at the following site.

[2] Clearly delineate your earlier experience and current experience for each strand.

[3] Double-spacing not required in a table.  Font size can be 10-point in tables.

About Dave aka Mr. Math Teacher

Independent consultant and junior college adjunct instructor. Former secondary math teacher who taught math intervention, algebra 1, geometry, accelerated algebra 2, precalculus, honors precalculus, AP Calculus AB, and AP Statistics. Prior to teaching, I spent 25 years in high tech in engineering, marketing, sales and business development roles in the satellite communications, GPS, semiconductor, and wireless industries. I am awed by the potential in our nation's youth and I hope to instill in them the passion to improve our world at local, state, national, and global levels.
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7 Responses to NRC’s Five Strands of Mathematical Proficiency

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