**Questions and Answers Related to the Revised
Core Curriculum Content Standards in Mathematics
Adopted by the New Jersey State Board of Education on
July 2, 2002 and Readopted on October 6, 2004**

Q: |
In some cases, Cumulative Progress Indicators are described using four numbers and a letter (e.g., 4.4.7B1), and in other cases they are described with only three numbers and a letter (e.g., 4.5E2). Why is this? |

A: | In most cases, it is routine to use four numbers and a letter representing the content strand to identify a particular Cumulative Progress Indicator (CPI). For example, in 4.4.7B1, the first 4 represents mathematics (if it were a science standard, the first number would have been a 5); the second 4 refers to standard 4 (data analysis, probability, and discrete mathematics); the 7 specifies the grade level; the B refers to strand B (probability); and the 1 specifies CPI 1 (interpret probabilities as ratios, percents, and decimals). The other example was from standard 4.5 (mathematical processes), the one standard for which the CPIs are not divided by grade. The expectations for this standard are intended to address all grade levels. Therefore, in indicator 4.5E2, the 4 again refers to mathematics; the 5 refers to standard 5 (mathematical processes); the E refers to strand E (representations); and the 2 specifies CPI 2 (select, apply, and translate among mathematical representations to solve problems). It is expected that students will demonstrate this process with respect to content appropriate for their own grade levels. |

Q: |
Why is there so much duplication between grades 3 and 4, between grades 5 and 6, and between grades 7 and 8? |

A: | In developing the standards, care was taken to avoid unnecessary repetition of CPIs across the grade levels. However, indicators were originally written only for grades 2, 4, 6, 8, and 12. Subsequently, indicators for grades 3, 5, and 7 were developed as an appendix to the standards document, and then still later were included in the document itself to facilitate the development of grade-level assessments as mandated by the federal No Child Left Behind legislation. Thus, grade-three indicators, for example, were not developed independently, but were extracted from the grade-four indicators. That is, the grade-three indicators duplicate those CPIs from grade four that are also developmentally appropriate for grade-3 students. Likewise, the indicators for grades 5 and 7 duplicate some of the CPIs from grades 6 and 8. If New Jersey were not developing assessments in mathematics for grades 3, 5, and 7, those grade-level indicators could have been omitted, and the remaining document containing indicators for only grades 2, 4, 6, 8, and 12 could have stood alone. |

Q: |
Can I find all of the things my students must know and be able to do in the CPIs for my grade level? |

A: | No, some of the things that they must know and be able to do are in the indicators for earlier grade levels. A fifth-grade teacher, for example, would need to look at the CPIs for grades two and four, along with five, to identify the knowledge and skills that fifth-grade students are expected to have achieved. CPIs are not generally repeated across grade levels (except for some CPIs in 3-4, 5-6, and 7-8, as noted in the previous question). |

Q: |
Dollars and cents notation does not appear in the CPIs until grade 4, in indicator 4.1.4B6. Only cents notation appears in the corresponding grade-three indicator 4.1.3B5. Is this a mistake? Shouldn’t students be expected to have familiarity with the dollar-sign ($) before grade four? |

A: | It would certainly be appropriate for a student to have contact with the dollar-sign prior to grade four. The dollars and cents notation under the numerical operations strand in 4.1.4B6 is related to "decimals through hundredths" under the number sense strand in 4.1.4A1. Dollars and cents notation was consciously omitted from the grade-three indicators, so that teachers would not expect all students to perform operations with decimals at that grade level. |

Q: |
"Congruence (same size and shape)" is included in the grade-two indicator 4.2.2A1. "Congruence" then appears as part of the grade-four indicator 4.2.4A3. How should these differences in wording be interpreted? |

A: | The intent of the standard is that all students should become familiar with both the concept of congruence (same size and shape), and the term, congruence. However, although the term will undoubtedly be used in early grades, the focus in grade two should be on the concept—recognizing or drawing figures that have the same size and shape—rather than on necessarily using the term, congruence. Similarly, the grade-two indicator 4.3.2D1 specifies that students will understand and use the concepts of commutativity and associativity without necessarily using the terms. It is not until grade four (4.3.4D1) that students should be expected to routinely use the formal terminology associated with these concepts. |

Q: |
Some common manipulatives seem to involve shapes with which students are not expected to have familiarity until higher grades. One such manipulative is pattern blocks. While triangles and hexagons are included in the grade-three indicator 4.2.3A2, two of the pattern block shapes—rhombi and trapezoids—are not introduced until grade five, in indicator 4.2.5A2. Does this mean that pattern blocks should not be used by students in grades 2, 3, and 4? Does it mean that such manipulatives will not be used on the statewide assessments at grades three and four? |

A: | 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:
Pattern blocks, for example, are an extremely valuable tool for exploring patterns as early as kindergarten. Students in the early grades are expected to have a basic familiarity with the various pattern-block shapes (including the triangle, the rhombus, the square, the trapezoid, and the hexagon). They will not be expected to demonstrate a theoretical understanding of the characteristics of the trapezoid and the rhombus until grade five, according to indicator 4.2.5A2. |

Q: |
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. |

A: | 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. To more specifically answer the question, while the types of patterns and the types of solutions will vary by grade level, it is expected that all students, at all grade levels, will use calculators to explore patterns and validate their solutions to grade-appropriate problems. At the same time, students should also be carrying out some arithmetic operations without calculators—using pencil and paper and mental math. In this way, when faced with problems, students will have developed the necessary skills to select the appropriate computational method for the situation, based on the context and the numbers (indicator 4.1.6B4). |