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Topic: Mathematical enterprise 

Mathematics

 Standard 9.  Understands the general nature and uses of mathematics
  Level II (Grade 3-5)
   Benchmark 2.Understands that mathematical ideas and concepts can be represented concretely, graphically, and symbolically
    Knowledge/skill statements
     1.Understands that mathematical ideas can be represented concretely
     2.Understands that mathematical concepts can be represented graphically
     3.Understands that mathematical concepts can be represented symbolically
  Level III (Grade 6-8)
   Benchmark 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
    Knowledge/skill statements
     1.Understands that mathematicians often represent real things using abstract ideas
     2.Understands that mathematicians work with abstract symbols to learn about the concept those symbols represent
  Level IV (Grade 9-12)
   Benchmark 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
    Knowledge/skill statements
     1.Understands that mathematics began long ago to solve practical problems
     2.Understands that mathematics evolved from a focus on real world problems to relationships between abstractions and the real world
     3.Understands that mathematics evolved from a focus on abstractions and the real world to the relationships among abstractions
   Benchmark 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
    Knowledge/skill statements
     1.Understands that propositions are generated from a small set of rules
     2.Understands that one of highest values in mathematics is simplicity
     3.Understands that some mathematicians try to identify the smallest set of explanatory rules
   Benchmark 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
    Knowledge/skill statements
     1.Understands that pure mathematical theories sometimes have very practical applications
     2.Understands that theories in mathematics are influenced by practical issues
     3.Understands that real world problems sometimes generate new theories
   Benchmark 5.Understands that new mathematics continues to be invented even today, along with new connections between various components of mathematics
    Knowledge/skill statements
     1.Understands that new connections are made every day between areas of mathematics
     2.Understands that new mathematics continues to be invented every day
   Benchmark 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
    Knowledge/skill statements
     1.Understands that mathematics and science value order
     2.Understands that mathematics and science value imagination
     3.Understands that mathematics and science value honesty and openness
     4.Understands that mathematics and science operate under common principles
     5.Understands that mathematics and science value collegiality
   Benchmark 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
    Knowledge/skill statements
     1.Understands that mathematics rules should not contradict each other
     2.Understands that mathematicians operate by choosing rules and following them