This week the Thomas Fordham Institute released a report claiming “alignment glitches” between the Next Generation Science Standards and the Common Core State Standards for Mathematics. Fordham’s original report on the standards suffered from a number of issues, but many of those boiled down to differences in educational philosophies. Unfortunately, this report suffers from far more serious problems.

The Fordham report notes three supposed issues with the math in the NGSS:

- Misalignment: situations where the math required by the NGSS exceeds the math required by the Common Core for that grade level.
- Missed opportunities to include more math.
- Superficial connections between the science and math content.

I’m going to focus on the “misalignment” criticism because that is clearly the most serious of the three. Nearly all of the examples they give of this supposed shortcoming don’t hold water. Let’s go through them one by one.

**Supposed misalignment #1: **

4-PS3-1: Use evidence to construct an explanation relating the speed of an object to the energy of that object. [Assessment Boundary: Assessment does not include quantitative measures of changes in the speed of an object or on any precise or quantitative definition of energy.]

The Fordham report claims that students would need to use quadratic functions to meet this standard, which would be well beyond 4th grade math. It also claims that without math, students cannot construct an explanation.

This is nonsense. The relevant disciplinary core idea here is that “the faster a given object is moving, the more energy it possesses.” There are numerous non-quantitative ways a 4th grade student could explain how this is true. For example, cars traveling at faster speeds in an accident obviously suffer greater damage. A faster bowling ball will knock down pins more easily than a slower bowling ball. An experiment could easily be done by rolling or launching an object at varying speeds and observing increasingly greater effects for higher speeds.

Most importantly, the NGSS does not attempt to align this standard to Common Core math. So this can in no way be interpreted as an “alignment glitch.”

**Supposed misalignment #2: **

4-PS4-1: Develop a model of waves to describe patterns in terms of amplitude and wavelength and that waves can cause objects to move. [Clarification Statement: Examples of models could include diagrams, analogies, and physical models using wire to illustrate wavelength and amplitude of waves.] [Assessment Boundary: Assessment does not include interference effects, electromagnetic waves, non-periodic waves, or quantitative models of amplitude and wavelength.]

This one is just bizarre to me- the clarification statement specifically describes the kinds of models that are meant here, and yet the Fordham report goes off about trigonometric functions? The NGSS links this to the CCSSM math practice standard “Model with mathematics.” In this case, Appendix L extends this to identify a CCSS standard related to drawing points, lines, and line segments, all of which seems to me to be a perfect example of “modeling with mathematics” on a diagram to illustrate the different properties of a wave (including amplitude & wavelength).

**Supposed misalignment #3: **

MS-PS4-1: Use mathematical representations to describe a simple model for waves that includes how the amplitude of a wave is related to the energy in a wave. [Clarification Statement: Emphasis is on describing waves with both qualitative and quantitative thinking.] [Assessment Boundary: Assessment does not include electromagnetic waves and is limited to standard repeating waves.]

Again, the Fordham report claims that this requires “trigonometric and quadratic functions.” This is simply untrue. Here’s the relevant physical principle: the energy of a wave is proportional to the square of the wave’s amplitude. To understand this, students simply need to be able to do something like “write and evaluate numerical expressions involving whole number exponents” which coincidentally is a 6th grade math expectation in the CCSS. Although this mathematical connection is not explicitly made in the NGSS, the connections regarding ratios (for example, ratios related to number of crests/troughs within a specified length of the wave) and giving examples of non-linear functions all seem very relevant here.

**Supposed misalignment #4:**

MS-PS2-2: Plan an investigation to provide evidence that the change in an object’s motion depends on the sum of the forces on the object and the mass of the object. [Clarification Statement: Emphasis is on balanced (Newton’s First Law) and unbalanced forces in a system, qualitative comparisons of forces, mass and changes in motion (Newton’s Second Law), frame of reference, and specification of units.] [Assessment Boundary: Assessment is limited to forces and changes in motion in one-dimension in an inertial reference frame and to change in one variable at a time. Assessment does not include the use of trigonometry.]

The Fordham report claims that the standard requires the use of vectors, which is not grade appropriate. After much discussion in the report, their beef seems to be simply that the PE doesn’t read “Plan an investigation *of one-dimensional motion* to provide evidence that the change in an object’s motion depends on the sum of the forces on the object and the mass of the object.” But that is clearly described in the assessment boundary: the motion is in a single direction. And regardless, as the Fordham report says, meeting this PE for 1-dimensional motion is completely grade-appropriate, so this is not a case of any meaningful “misalignment” but nit-picking over formatting and wording.

**Supposed misalignment #5:**

MS-ESS2-6: Develop and use a model to describe how unequal heating and rotation of the Earth cause patterns of atmospheric and oceanic circulation that determine regional climates. [Clarification Statement: Emphasis is on how patterns vary by latitude, altitude, and geographic land distribution. Emphasis of atmospheric circulation is on the sunlight-driven latitudinal banding, the Coriolis effect, and resulting prevailing winds; emphasis of ocean circulation is on the transfer of heat by the global ocean convection cycle, which is constrained by the Coriolis effect and the outlines of continents. Examples of models can be diagrams, maps and globes, or digital representations.] [Assessment Boundary: Assessment does not include the dynamics of the Coriolis effect.]

This one is perhaps the most strange of all to me. The Fordham report seems to be stuck in the viewpoint that the “models” being referred to here are the kinds of climate models that professional scientists use, despite the explicit statement “Examples of models can be diagrams, maps and globes, or digital representations.” All these are obvious tools that are used to teach and understand this concept at the middle-school level. Most bizarrely, there is no attempt in either the NGSS or Appendix L to tie this to CCSS math at all! Look at the CCSS connections for this standard– MS-ESS2-6 appears only in the ELA connections. The misalignment here seems to be purely in the imaginations of the Fordham authors.

**Supposed misalignment #6:**

HS-ESS1-4: Use mathematical or computational representations to predict the motion of orbiting objects in the solar system.[Clarification Statement: Emphasis is on Newtonian gravitational laws governing orbital motions, which apply to human-made satellites as well as planets and moons.] [Assessment Boundary: Mathematical representations for the gravitational attraction of bodies and Kepler’s Laws of orbital motions should not deal with more than two bodies, nor involve calculus.]

Apparently this “draws upon rather serious college-level mathematics” according to Fordham. Or, maybe it means Newton’s law of universal gravitation and Kepler’s laws. Because after all, those are both mentioned in the additional information that goes along with the PE, and those are both standard high school science expectations. Fordham’s claim that there is “not much left in the solar system” if you limit to two bodies at a time discredits the fundamental importance and validity of these laws and I have to assume is intended more for laughs than as a serious critique.

**Supposed misalignment #7:**

HS-ESS2-6: Develop a quantitative model to describe the cycling of carbon among the hydrosphere, atmosphere, geosphere, and biosphere. [Clarification Statement: Emphasis is on modeling biogeochemical cycles that include the cycling of carbon through the ocean, atmosphere, soil, and biosphere (including humans), providing the foundation for living organisms.]

The Fordham authors seem to be hung up on the idea that a “quantitative model” used by grade school students must be the same as the type of model used by professional scientists, as their argument is that this kind of model is inappropriate for high school students. But there are many other kinds of models. Here’s an example of an end product of an assessment of this PE as I’d envision it, and as I’d guess any high school Earth Science teacher would envision it as well. This is a quantitative model that would require no mathematics a high school student couldn’t easily handle. A simple spreadsheet model would be another potential example. Look at the CCSS mathematics connections for HS-ESS2-6 as well- they are all in reference to choosing appropriate units, measurement accuracy, and similar concepts that seem highly relevant to creating this kind of representation.

**Supposed misalignment #8:**

HS-ESS3-3: Create a computational simulation to illustrate the relationships among management of natural resources, the sustainability of human populations, and biodiversity. [Clarification Statement: Examples of factors that affect the management of natural resources include costs of resource extraction and waste management, per-capita consumption, and the development of new technologies. Examples of factors that affect human sustainability include agricultural efficiency, levels of conservation, and urban planning.] [Assessment Boundary: Assessment for computational simulations is limited to using provided multi-parameter programs or constructing simplified spreadsheet calculations.]

The Fordham report makes similar criticisms about “nothing in the CCSSM” allowing a student to do this. The CCSS connections cited in the NGSS are related to math practices, which certainly seems reasonable. Here’s a great little simulation that I’d use with students for this. Spreadsheet calculations seem equally appropriate. I just don’t see what’s so “mis-aligned” about expecting students to apply basic ideas of addition, subtraction, division, ratios, equations etc. to understand ideas like per-capita consumption.

**The lone true misalignment: **

5-ESS2-2: Describe and graph the amounts and percentages of water and fresh water in various reservoirs to provide evidence about the distribution of water on Earth. [Assessment Boundary: Assessment is limited to oceans, lakes, rivers, glaciers, ground water, and polar ice caps, and does not include the atmosphere.]

The Fordham report rightly notes that percentages are not taught in the math CCSS until grade 6, so this is off by a year. Although you could theoretically graph a percentage without knowing how to calculate one, this is a definite misalignment, although hardly one that goes “well beyond” the math expected for the grade level. Perhaps this could be taught at the end of 5th grade in preparation for 6th grade?

**Conclusion**

There is simply no evidence for the kind of serious misalignment that the Fordham report claims. I have to assume that the report has shown the most clearly egregious examples the authors found to make its points, but the only real misalignment noted is a minor one. I don’t doubt that there are imperfections in the NGSS/CCSSM connections, and I wish that these connections had been included in the publicly released drafts of the standards, so that the science education community could have helped to troubleshoot these. But if there are serious alignment issues, the Fordham report has not identified them.

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