National and State Subject Matter Content Standards

¶ … National and State Subject Matter Content Standards for Math

According to the California standards for high school students, the geometry curriculum contains six critical components: "to establish criteria for congruence of triangles based on rigid motions; establish criteria for similarity of triangles based on dilations and proportional reasoning; informally develop explanations of circumference, area, and volume formulas; apply the Pythagorean Theorem to the coordinate plan; prove basic geometric theorems; and extend work with probability" (Common Core Standards, California Department of Education: 69). The elucidated standards are often quite specific in terms of how students are asked to apply basic concepts such as measuring angles; understanding the different properties of parallel lines; and manipulating various polygons. Not only must the students prove theorems but they must also be able to construct such shapes using a variety of methods in a hands-on fashion (Common Core Standards, 2013, California Department of Education: 70).

Deeper understanding as well as superficial understanding must be demonstrated; the core requirements are conceptual as well as highly specific. This can be seen in regards to the definition of congruency: students must "use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent" (Common Core Standards, California Department of Education: 70).

In the federal Common Core standards for education there is an equal stress upon being able to produce geometric theorems as well as applying the student's understanding to the operations of shapes. In many areas, common core standards are replicated verbatim on both websites. Regarding the need to model with geometry, even the same example is used both in the California and the federal standards, the first of which is to "use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder)" (Common Core Standards, Official Website CCSS.Math.Content.HSG-MG.A.1; California Department of Education 75). But on the California website, California-specific standards are noted in bold typeface and emphasize more the additional, specific hands-on applications students must demonstrate to show comprehension of core concepts, versus merely being able to restate ideas and perform basic operations. One such California-specific standard is: "verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve real-world and mathematical problems" (Common Core Standards, California Department of Education: 74).

Regarding Algebra I, the standards for the subject in California are defined as pertaining to three critical areas: "deepen and extend understanding of linear and exponential relationships; contrast linear and exponential relationships with each other and engage in methods for analyzing, solving, and using quadratic functions; extend the laws of exponents to square and cube roots; and apply linear models to data that exhibit a linear trend" (Common Core Standards, California Department of Education:60). In contrast, the federal Common Core standards begin with a more theoretical discussion of algebraic equations, what they contain, and algebraic distinction between equations and inequalities.