Nebraska could adopt next month new learning standards for math that state officials hope will lead students in kindergarten through high school to a deeper understanding of mathematics and cut back on the number of topics, especially in younger grades, the Omaha World-Herald reports.
A draft version of the proposed standards, arranged vertically here and horizontally here, has kindergartners counting to 100, while the state’s current standards have them counting only to 20. The standards start introducing students to linear algebra in seventh and eighth grades, while the former standards put off most of that until high school.
Teachers could spend more time on each standard, said Matt Larson, a math curriculum specialist for the Lincoln Public Schools and one of about a hundred educators who contributed to the development of the new standards. As students acquire more than a superficial understanding, he said, they’ll be less likely to “hit a wall” when they encounter more challenging math problems.
As with the Common Core standards, upon which Nebraska’s draft standards are partially based, we have few issues with the math standards in grades kindergarten through eighth grade. There’s a good emphasis on fluency, although Nebraska’s draft standards rely too much, we think, on the algorithms students use to do arithmetic rather than teaching a multi-pronged approach to problem solving. “The processes highlight the applied nature of math within the workforce and clarify the expectations held for the use of mathematics in and outside of the classroom,” the standards document states, apparently ignoring the mathematics underlying the arithmetic processes used in the workforce.
If seventh graders can “write a two-step equation to represent real-world problems involving rational numbers in any form” (MA 7.2.3.b), more power to them. This gets into abstract thinking, and many seventh graders still think in concrete terms. The Common Core, however, assumes seventh graders can process abstract thought as well (7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities).
When it comes to high school, though, there are a few issues. For starters, the standards stop at 11th grade, leaving 12th grade to “advanced topics,” and then, adding only the Laws of Sines and Cosines in geometry. As stated, these have limited application in the workforce and in college majors, even in mathematics. “Apply the Law of Sines and the Law of Cosines to find unknown measures in triangles,” Nebraska’s proposed standards say. This has next to nothing to do with “going deeper” and understanding the underlying mathematics that Nebraska officials have boasted about.
The Common Core goes deeper into the value of learning the Laws of Sines and Cosines. Plus, it gives those laws context:
The Pythagorean Theorem is generalized to non-right triangles by the Law of Cosines. Together, the Laws of Sines and Cosines embody the triangle congruence criteria for the cases where three pieces of information suffice to completely solve a triangle. Furthermore, these laws yield two possible solutions in the ambiguous case, illustrating that Side-Side-Angle is not a congruence criterion. Analytic geometry connects algebra and geometry, resulting in powerful methods of analysis and problem solving. Just as the number line associates numbers with locations in one dimension, a pair of perpendicular axes associates pairs of numbers with locations in two dimensions. This correspondence between numerical coordinates and geometric points allows methods from algebra to be applied to geometry and vice versa.
In other words, the Common Core takes into account the contribution of the Laws of Sines and Cosines toward a deeper understanding of mathematics. Nebraska’s standards just leave it short of a full explanation, even for advanced students. (Note that the Laws of Sines and Cosines in the Common Core are also designated as advanced topics.)
That’s just one example, really, but the standards leave so much open to interpretation by individual school districts that, in some places, especially in high school, the state might as well have no standards at all and just leave the teaching of high school math to local school districts.
That may be what’s intended, as Nebraska has always been a state heavily focused on local control. Standards, by definition, however, should describe what students should know, including the advanced topics that are intended for students taking that fourth year of math, as many colleges require.
Finally, while we like the designation of “advanced topics,” it seems more appropriate to list these topics in specific courses, as high schools teach math today. There’s algebra I, algebra II, geometry, trigonometry, and so forth. Again, if the idea is that local school districts will slice and dice the standards up in a way that suits them, the whole point of developing state standards for assessment seems off. The state administers the math test to 11th graders, by which time all content except the advanced topics should be known and understood.