The Common Core’s Standards for Mathematical Practice significantly progress efforts to articulate how students should demonstrate their mathematical proficiency. These standards “describe varieties of expertise that mathematics educators at all levels should seek to develop in their students” (Standards for Mathematical Practice, 2010) by giving expectations for the depth of students’ understanding, the skills students should have, and how students should express what they know.
Unfortunately, the Standards for Mathematical Practice (SMP) are written in a manner that makes them difficult for many K-12 mathematics educators to understand which may result in an inability to implement these standards as their authors envisioned.
Each of the eight SMP is a paragraph that begins with the words “Mathematically proficient students” followed by the specific expertise students should demonstrate. The standards contain 3 to 10 sentences each with individual sentences running as long as 69 words in length. Agencies such as the Center for Disease Control (CDC), the state of Florida, and the United States Navy use the Flesch Reading Ease formula to measure how easy an English language passage is to read. Scores from this formula range from 0 to 100 with scores between 0-29 considered to be very difficult to read. For reference, Time Magazine articles during the 2000s averaged a score of approximately 55, the nursery rhyme “Humpty Dumpty” scores a 79.7, and Florida deems an insurance policy “readable” if “the text achieves a minimum score of 45 on the Flesch reading ease test.” (2011 Florida Statutes, 2011) Using this formula, as calculated by Microsoft Word 2010, the Standards average a score of approximately 32. Scores are as low as 17.5 and 15.6 for Standards 2 and 5, respectively making them especially difficult to read.
Accordingly, it has been my experience that when the Standards are reviewed during professional development using only the officially published format, many K-12 mathematics educators immediately feel overwhelmed by the way the SMP are presented and either had significant trouble understanding what the standards meant or could not agree on their meaning. As the SMP currently stand, educators may just skim them over or absorb incomplete meanings.
What is needed is a way to balance the Standards’ valuable ideas with simpler formatting, and a possible solution may come from the SMP’s introduction. In the introduction it states that the first standards are derived from the National Council of Teachers of Mathematics (NCTM) Process Standards. An example of the NCTM standards’ formatting is listed below.
Instructional programs from prekindergarten through grade 12 should enable all students to—
- Build new mathematical knowledge through problem solving
- Solve problems that arise in mathematics and in other contexts
- Apply and adapt a variety of appropriate strategies to solve problems
- Monitor and reflect on the process of mathematical problem solving
These standards are written in a manner that is more palatable by a broad range of educators. Accordingly, some suggestions for making the SMP more accessible to a broad range of educators include:
- Adjust each standard’s wording so that is scores at least a 45 using the Flesch Reading Ease formula.
- Break up each standard into a bullet list.
- Begin each bullet with a verb.
- Write each bullet using short sentences.
- State “Mathematically proficient students…” as a preface for each bullet.
- Label each bullet.
The last suggestion is not used by the NCTM, but since the SMP contain significantly more ideas, labels would better allow educators to reference specific parts of standards (such as MP 3.1 for the first bullet in Math Practice Standard 3). Educators would need to be informed, however, that the SMP listed in this format, does not constitute a checklist and that they should be continuously incorporated.
Clearly there is a degree of subjectivity involved when reformatting each standard and it is not my desire to change the authors’ intent. My goal was to copy the text word for word when possible. To illustrate the difference that these suggestions would make, Standard 2: “Reason abstractly and quantitatively” would go from:
“Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.”
Mathematically proficient students…
- MP 2.1 – Make sense of quantities and their relationships in problem situations.
- MP 2.2 – Decontextualize a problem by abstracting a given situation, representing it symbolically, and manipulating the representing symbols as if they have a life of their own, without necessarily attending to their referents.
- MP 2.3 – Contextualize a problem by pausing as needed during the manipulation process in order to probe into the referents for the symbols involved.
- MP 2.4 – Create a coherent representation of the problem at hand.
- MP 2.5 – Consider the units involved.
- MP 2.6 – Attend to the meaning of quantities, not just how to compute them.
- MP 2.7 – Know and flexibly use different properties of operations and objects.
The SMP do make significant progress towards unifying expectations as to how students should demonstrate mathematical proficiency. The irony is that while they go to great lengths to explain these proficiencies, they are written verbosely, causing the message to no longer be clear. We need to rethink the language used to increase the Standards’ readability because if mathematics educators cannot understand the standards, they will not be implemented as their authors intended.
UPDATE (11/10/2018) – Here’s my attempt at making more readable practice standards.
Melissa Canham and Nicholas Johnson contributed to this article.
“2011 Florida Statutes.” The Florida Senate. Web. 07 June 2012. <http://www.flsenate.gov/laws/statutes/2011/627.4145>.
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“NCTM Process Standards.” Process Standards. Web. 06 June 2012. <http://www.nctm.org/standards/content.aspx?id=322>.
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“Standards for Mathematical Practice.” Common Core State Standards Initiative. Web. 06 June 2012. <http://www.corestandards.org/the-standards/mathematics/introduction/standards-for-mathematical-practice/>.