The Standards for Mathematical Practice (SMP) are often cited as evidence that the current math content standards suggest a significant shift in mathematics education. This shift is frequently framed as a more holistic or progressive view of math education because the SMP capture the more enduring capacities and dispositions students should develop in math class. In other words, they purportedly paint a fuller picture about what it means to do math, or put differently, what it is that mathematicians do.
Yet, reading the SMP in their entirety it is difficult not to conjure up the stereotypical image of the lone mathematician wrangling with abstractions, obsessed with precision and structure, and quick to criticism. It is the mathematics of an elite few who can persevere. It is not a portrait of mathematics that is necessarily new, appealing, or even accurate.
This is not to say that the practices in the SMP are not important in mathematics – just that they are not the totality of mathematics. They are incomplete. And not only in the sense that they are not an exhaustive list of mathematical practices (though certainly that is true too), but also in the sense that each practice is itself incomplete. It only paints with half the color needed to capture the full vibrancy of mathematics. For each SMP, a complementing practice can be identified – a practice that is opposite (or nearly opposite), though not necessarily oppositional, and equal mathematical.