The recently released Appendix C, intended to clarify key choices made by writers of the Next Generation Science Standards (NGSS), addresses College and Career Readiness. It is lengthy and rich in self-praise and in repetition of claims made earlier in the evolution of the NGSS and their initiating Framework. The handling of one subhead—addressing mathematics and the alignment of these standards with the Common Core (CC) standards for mathematics—is a miniature of the Appendix as a whole: pious declarations of purpose for not-quite-compliant products. Our discussion below will serve as a critique of the whole.

The slighting of mathematics in the actual NGSS standards does increasing mischief as grade level rises.
*Photo by LianaAn* |

This section of the Appendix is entitled, “The Importance of Mathematics for College Readiness in Science.” The historic association of mathematics with science and science education is lauded and given robust support. An understanding is implicit throughout: that content-relevant mathematics *is* emphasized in the NGSS and aligned with the Common Core math standards. This section of the Appendix makes the observation that “calling for application of mathematics in a performance generally raises the level of rigor.” True. Hence we may ask, “Do the NGSS, in fact*,* call effectively for science-relevant math, and do those calls align, grade-wise, with CC-Math?”

Examination of mathematics in the NGSS was a key element of our reviews. Unfortunately, we found inconsistency between the strong NGSS (and Appendix C) assertions and what was actually found by the mathematicians, among others, of our reviewing group. Moreover, the NGSS producers, perhaps sensing disquiet or inadequacy, issued an Appendix L that is meant to demonstrate the adequacy of NGSS math and of its alignment with CC-Math.

Now Appendix L does indeed offer, finally, some nice examples of math integrated with science content. Yet as our distinguished mathematical (and K–12 math education) colleague Prof. W. Stephen Wilson has shown, it cannot cure the maladies of the NGSS in this realm:

###### Although Appendix L gives many worthwhile examples of grade-level mathematics applied to support and clarify the NGSS standards, it is, in essence, an after-the-fact corrective effort rather than the well-integrated mathematics that the NGSS should have produced...Much of the NGSS document was not written with mathematics in mind. When mathematics was included, it generally lacked useful specificity. Appendix L cannot and does not satisfactorily compensate.

The slighting of mathematics in the actual NGSS standards does increasing mischief as grade level rises, especially in the physical sciences. An especially sore point is high school physics, which simply cannot be taught or learned non-mathematically. There are several pious pronouncements in Appendix C, such as the following:

###### In particular, the best science education seems to be one based on integrating rigorous content with the practices that scientists and engineers routinely use in their work—including application of mathematics.

Such lip service is paralleled in the high school standards themselves:

###### Analyze data to support the claim that Newton’s second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration.

But, aside from the fact that this is *all *that the NGSS high school standards have to say about Newton’s laws—the foundation for understanding pretty much everything else in physics—the mathematical background is absent. Also, the “analyze-data” mandate (which data, whose data, what claim?)—the omnipresent “practices” part of this standard—is clearly more important, or so it reads, than Newton’s Law; but what is it supposed to mean in…practice?

It is charming to say, “… Students learn science more effectively when they actively engage in the practices of science.” But the practice of physics (and, to a lesser extent, chemistry) at the high school level is impossible without extensive attention both to ideas that can only be satisfactorily expressed mathematically and to gritty calculations based on those ideas. Such expressions and calculations are of course done in all high school physics courses, and they are valuable preparation—readiness—for any college science discipline. Yet high school physics is effectively absent from the NGSS.

Associated with the dearth of both mathematics and physics in the NGSS is the shortage of rigor. Appendix C is properly emphatic on rigor: “Both the Framework* *and the NGSS reflect current thinking about the need for greater depth and rigor in K–12 science schooling.” NGSS tries, it seems, to make space for depth and rigor by eliminating entire fields of physics and engineering, including thermodynamics, optics, electronics, and just about everything that has been learned since 1950. That might be forgiven if the remaining fields really were tackled in depth. But in fact the standards often forego depth and rigor in favor of hand waving.

A particularly egregious example is the treatment of energy, a concept mentioned throughout the NGSS and of the greatest importance in not only physics but in nearly all fields of science. Yet energy is never rigorously defined, although that can be neatly done at the high school level with just a bit of algebra. Many more examples, of not depth but of its opposite, can be found in the detailed chemistry, life-sciences, and earth-space-sciences sections of our final NGSS review. And that sampling is by no means exhaustive.

A drawn-out defense of the “practices” that now verbally dominate every standard, with attached bibliography supposed to cite empirical evidence that this “next-generation” pedagogy boosts student achievement, fails to convince those of us who have plenty of empirical support for recognizing the following:

1. The “practices” strategy is not “next generation.” It has been promoted and implemented nationally as “inquiry learning” or the like since the early 1990s.

2. It has had no notable effect on the (mediocre) performance of American students in national and international science assessments.

3. Students will not learn best if they practice science exactly as do real scientists. A firm conclusion from cognitive psychology contradicts that claim. Beginners don’t and can’t think (or “practice”) as do experts (scientists). The practices of experts exploit prior experience and extensive build-up in memory of scaffolding: facts, procedures, solutions to standard problems in the field, vocabularies—of *knowledge*, in short, superficial as well as deep. Such background must be acquired in an orderly way over time and form the grist for the mill of practice. There is as yet no demonstrably effective pedagogical short cut. You can’t practice on vacancy; you can only practice on stuff.

*Paul R. Gross is an emeritus professor of life sciences at the University of Virginia and a director and trustee emeritus at the Marine Biological Laboratory in Woods Hole, MA.*

*Lawrence S. Lerner is a professor emeritus at the College of Natural Sciences and Mathematics at California State University–Long Beach.*