2005 State Report Card New Jersey Clarity: 2.17 C Content: 1.17 D Reason: 0.50 F Negative Qualities: 0.75 F Weighted Score: 1.15 Final Grade: D 2000 Grade: C 1998 Grade: C
New Jersey did itself no favors by revising its already middling math standards in 2002; it has dropped a full letter grade in our review. These standards do have some positive features, though. They are generally straightforward and clear. The field properties of rational and real numbers are well developed. Counting principles are carefully developed. The geometry standards for grades 2-8 are well written. Memorization of the basic arithmetic facts is explicitly required of elementary school students, and they are expected to carry out some whole number computations by hand.
However, use of the standard algorithms of arithmetic is not required, and hand calculation is undermined by standards that require the use of calculators at all grade levels. The 2002 “Questions and Answers” document makes clear that state mathematics assessments allow even elementary school students access to calculators during state exams:
Question: “Under the mathematical processes standard, indicator 4.5F4 says that students will ‘use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions.’ For what grade levels is this a reasonable expectation? Some teachers claim that they do not let their students use calculators until grade five or six, thinking that this will force them to become proficient at pencil-and-paper computation.”
Answer: “Calculators can and should be used at all grade levels to enhance student understanding of mathematical concepts. The majority of questions on New Jersey’s new third- and fourth-grade assessments in mathematics will assume student access to at least a four-function calculator. Students taking any of the New Jersey Statewide assessments in mathematics should be prepared to use calculators by regularly using those calculators in their instructional programs. On the assessments, students should be permitted to use their own calculators, rather than the school’s calculators, if they so choose. . . .”
The same document explains that students will be examined partially on their understanding of manipulatives:
Several of the questions on the mathematics assessments will assume student familiarity with various commonly used manipulatives, including but not necessarily limited to the following: Base ten blocks, Cards, Coins, Geoboards, Graph paper, Multilink cubes, Number cubes, Pattern blocks, Pentominoes, Rulers, Spinners, and Tangrams.
These directives are not the excesses of an isolated document. The New Jersey Framework lists among its goals the incorporation of calculators into the early grades and the integration of manipulatives, normally reserved for the elementary grades, into high school, as indicated in this passage:
Young children find the use of concrete materials to model problem situations very natural. Indeed they find such modeling more natural than the formal work they do with number sentences and equations. Older students will realize that the adults around them use calculators and computers all the time to solve mathematical problems and will be prepared to do the same. Perhaps more challenging, though, is the task of getting the “reverse” to happen as well, so that technology is also used with young children, and the older students’ learning is enhanced through the use of concrete models. Such opportunities do exist, however, and new approaches and tools are being created all the time.
The Framework adds that “algebra tiles are used to represent variables and polynomials in operations involving literal expressions” for high school students.
This agenda is fundamentally anti-mathematical. Mastery of basic skills is essential to learning more advanced topics. Manipulatives can be effective pedagogical tools in the early grades, but ultimately the power of mathematics lies in its abstract nature. Promoting algebra tiles in place of the more powerful and abstract distributive property in the high school grades is an impediment to learning mathematics, not an aid.
Incomplete and Inappropriate Content
Moving to specific content, the treatment of algebra in high school is weak. There is no mention of solving two or more linear equations simultaneously by algebraic methods, of algebraic manipulations of rational functions, or of completing the square for quadratic polynomials. The treatment of trigonometry and conic sections is skimpy, and there is no mention of complex numbers.
Displacing such foundations, a strand of standards is devoted to “Discrete Mathematics—Vertex-Edge Graphs and Algorithms.” In second grade, this strand includes the standard, “Play simple two-person games (e.g., tictac- toe) and informally explore the idea of what the outcome should be.” It continues into the high school grades with a focus on graph theory.Also deviating from mainstream topics are standards for middle and high school students devoted to fractals and tessellations.
Further compromising middle and high school standards is a premature focus on topics more appropriately reserved for calculus courses. These include optimization problems, studying “slope of a line or curve,” continuity, and monotonicity of functions, all with a heavy reliance on graphing technology.