# List of Benchmarks for Mathematics |

**Standard 9.** | Understands the general nature and uses of mathematics |

| ** Level Pre-K (Grade Pre-K)** |

| 1. | Not appropriate at this level |

| ** Level I (Grade K-2)** |

| 1. | Not appropriate for this level |

| ** Level II (Grade 3-5)** |

| 1. | Understands that numbers and the operations performed on them can be used to describe things in the real world and predict what might occur |

| 2. | Understands that mathematical ideas and concepts can be represented concretely, graphically, and symbolically |

| ** Level III (Grade 6-8)** |

| 1. | Understands that mathematics has been helpful in practical ways for many centuries |

| 2. | Understands that mathematicians often represent real things using abstract ideas like numbers or lines; they then work with these abstractions to learn about the things they represent |

| ** Level IV (Grade 9-12)** |

| 1. | Understands that mathematics is the study of any pattern or relationship, but natural science is the study of those patterns that are relevant to the observable world |

| 2. | Understands that mathematics began long ago to help solve practical problems; however, it soon focused on abstractions drawn from the world and then on abstract relationships among those abstractions |

| 3. | Understands that in mathematics, as in other sciences, simplicity is one of the highest values; some mathematicians try to identify the smallest set of rules from which many other propositions can be logically derived |

| 4. | Understands that theories in mathematics are greatly influenced by practical issues; real-world problems sometimes result in new mathematical theories and pure mathematical theories sometimes have highly practical applications |

| 5. | Understands that new mathematics continues to be invented even today, along with new connections between various components of mathematics |

| 6. | Understands that science and mathematics operate under common principles: belief in order, ideals of honesty and openness, the importance of review by colleagues, and the importance of imagination |

| 7. | Understands that mathematics provides a precise system to describe objects, events, and relationships and to construct logical arguments |

| 8. | Understands that the development of computers has opened many new doors to mathematics just as other advances in technology can open up new areas to mathematics |

| 9. | Understands that mathematics often stimulates innovations in science and technology |

| 10. | Understands that mathematicians commonly operate by choosing an interesting set of rules and then playing according to those rules; the only limit to those rules is that they should not contradict each other |